Chapter 6 Etreme Values: Curve Sketching and Optimization Problems Specific Epectations Describe the ke features of a given graph of a function, including intervals of increase and decrease, critical points, points of inflection, and intervals of concavit. Determine, from the equation of a polnomial or a rational function, the ke features of the graph of the function, using the techniques of differential calculus, and sketch the graph b hand. Identif the nature of the rate of change of a given function, and the rate of change of the rate of change, as the relate to the ke features of the graph of that function. Determine, from the equation of a rational function, the intercepts and the positions of the vertical and the horizontal or oblique asmptotes to the graph of the function. Solve optimization problems involving polnomial and rational functions. Section 6., 6., 6., 6.6 6., 6., 6., 6.6 6., 6., 6.7, 6.8 6., 6.5, 6.6 6.7, 6.8 MODELLING M AT H Practical applications of calculus are often concerned with optimization. It is not merel the solution to a problem that is required, but the best solution. We want, not merel to clean up the oil spill, but to clean it up in the best possible wa, in the minimum time, with the minimum damage to the environment, and with most efficient use of resources. In studies of nature, we are often interested in etremes. When we stud the tides, we want to know which are the highest in the world, where are the tidal ranges the smallest, and for an particular location, when are the tides highest and lowest. Phsicists, who stud the fundamental particles of matter in giant particle accelerators, need to understand how the forces between particles increase or decrease as the distance between the particles changes. All of these applications of calculus involve finding maimum and minimum values of functions, which are known as etreme values. Graphs provide a powerful wa of visualizing the overall behaviour of a function, including the etreme values. In this chapter, we develop methods for determining the graph of a function, and methods for solving optimization problems. As ou will see, similar tools are involved in both.
Review of Prerequisite Skills. Solving radical equations Solve and check. a) b) c) d) e) f) g) h) i) j) k) l) = 5 = + = = = 8 + + = + = + 7 5 = 9 + + 6 + = = + +. Using technolog to solve equations (Section.5) Solve for using technolog. a) 6 9 b) 8 c) d) 6 e) 6 6 f) 5 5 g).. 5. h)..5. i). j). 5... Function notation Evaluate the following. a) If f (), find f (). b) If f () 5, find f (). c) If A() ( ), find A(5). d) If P ( )= +, find P().. Function notation Graph the points on f () defined b a) f () b) f () c) f () 5 d) f () 5. Function notation Answer true or false, and justif our response. a) If f (), then f () f (). b) If g(), then g() g(). c) If f( )= +, then f () f () d) If f( )=, then f () f (). 6. Interval notation (Section.) Translate into interval notation. a) 5 b) c) d) 7. Interval notation (Section.) Rewrite the interval notation as an inequalit. a) (.,.99) b) (, ] c) [, ] d) (, ) 8. Polnomial inequalities (Section.6) Solve for. a) ( ) b) ( ) c) ( )( ) d) 9. Word problems A rectangular picture, cm b cm, is to be framed with a mat such that the width of the mat is equal on all sides of the picture and the area of the mat is equal to the area of the picture. Find the width of the mat.. Word problems A paper tra is made from a sheet of metal that is cm wide and 5 cm long. The ends are folded up a height of h centimetres to contain the paper. If the capacit of the tra is 6 cm, determine the height of the tra. 5 cm cm h cm. Domain State an restrictions on the domain of the variable for each function. a) f () b) = c) t ( )= + MHR Chapter 6
d) = e) = ( )( + ) 5 f) h ( )= 5 g) f( )= h) = 6. Intercepts (Chapter ) Find the intercepts of each curve. a) b) 8 c) g() 7 d) s(t) t t 5 e) V(r) r 7r 6 f) 5 g) 6 6 h) f () 7. Intercepts (Chapter ) Find the intercepts of the curve and use them, together with the methods of Chapter, to sketch the curve.. Smmetr (Section.) State whether each function is even, odd, or neither. a) b) c) d) 5. Smmetr (Section.) Determine whether each function is even, odd, or neither. a) f () 6 b) 5 c) f () d) e) = f ) + f( )= g ( )= + t g) st ()= h) t + t t + 5 6. Division of polnomials (Section.) Find the quotient and the remainder for each division. + a) b) + + 5 c) d) + 5 Review of Prerequisite Skills MHR
6. Increasing and Decreasing Functions One of the most practical applications of calculus is to determine the best, or optimal, solution to a problem. This often involves determining the largest value or the smallest value of a function over a certain interval. For eample, a beverage compan designs containers to minimize the cost of producing them chooses industrial machiner and processes to minimize production time and cost selects routes and schedules to minimize transportation time and cost uses marketing strategies to maimize sales makes business decisions to maimize profits We eamine such eamples in detail in this chapter. To help us understand how to determine the maimum and minimum values of functions in the following sections, we first eplore how to determine where functions increase and where the decrease. Imagine that the graph of a function represents a hill, and ou are riding a mountain bike on the hill from left to right. If ou are riding uphill, the function is increasing. If ou are riding downhill, the function is decreasing. A function is increasing if, as we move from left to right along its graph, the -coordinates increase in value. A function is decreasing if, as we move from left to right along its graph, the -coordinates decrease in value. This definition can be stated formall as follows: A function f is increasing on an interval (a, b) if f ( ) f ( ) whenever in the interval (a, b). A function f is decreasing on an interval (a, b) if f ( ) f ( ) whenever in the interval (a, b). = f( ) } } f ( ) f ( ) = f( ) } } f ( ) f ( ) a b a Increasing on (a, b) Decreasing on (a, b) b MHR Chapter 6
The function in the figure is increasing from F to G, that is, for (., ). It is also increasing from H to K, for (,.). It is decreasing from G to H, for (, ). The definitions of increasing and decreasing functions epress precisel what is meant b those terms. However, using these definitions is not a practical wa to determine where a function is increasing or decreasing, since we would have to check ever pair of points. In the figure to the right, notice that, where the function is increasing, the tangents to the curve all seem to have positive slopes, and where the function is decreasing, the tangents all seem to have negative slopes. This suggests that perhaps the derivative of a function can be used to determine where it is increasing and where it is decreasing. You will eplore this in the following investigation. 6 K = f( ) G H F 6 Investigate & Inquire: Increasing or Decreasing and the Derivative. Sketch the graph of f () over the interval [5, 5]. If ou use a graphing calculator, select the Window variables for the -ais appropriatel.. Determine a formula for the derivative f ().. Using the same aes or window as in step, graph f.. Use the graph of f to estimate the intervals on which f increases and the intervals on which f decreases. 5. Use the graph of f to make a statement about the values of f on the intervals that ou found in step. 6. Repeat steps to 5 for three other functions of our choice. 7. Do ou think there is an connection between the values of the derivative of each function in steps and 6, and whether the function increases or decreases? If so, clearl describe the connection. Eplain wh ou think the connection is valid. Do ou think the connection applies to all functions? Eplain. The ideas considered in step 7 of the investigation lead to the following test: Test for intervals of increase/decrease: Suppose that f is differentiable on the interval (a, b). If f () for all (a, b), then f is increasing on (a, b). If f () for all (a, b), then f is decreasing on (a, b). 6. Increasing and Decreasing Functions MHR
Eample Intervals of Increase and Decrease Without graphing the function, find the intervals on which the function f () 9 is increasing and the intervals on which it is decreasing. Solution First we find the derivative of f (). f () 9 To find the intervals of increase and decrease, we need to determine the intervals for which f () and the intervals for which f (). We begin b determining the values of for which f (). 9 ( 6) ( 8)( 8) 8 or 8 8 or 8 The numbers 8 and 8 are boundar If the derivative did change sign in the interval, points that separate the domain into three it would have to cross the -ais first. Therefore, intervals: (, 8), (8, 8), and (8, ). Since it would have another zero. Since there are no other the derivative is continuous, it does not change zeros in the interval, there can be no change of sign. sign within an of these intervals. Thus, determining the sign of the derivative at a single test value within an interval indicates the sign of the derivative throughout that interval. An interval chart can be used to organize the information. Intervals Test values Signs of f( ) Nature of graph f( ) = ( 8)( + 8) 8 8 (, 8) ( 8, 8) (8 ) + decreasing increasing decreasing Thus, f is decreasing for (, 8) and for (8, ), and increasing for (8, 8). The graph of f () illustrates our findings. 8 (8, ) 8 8 f ( ) = + 9 8 ( 8, ) MHR Chapter 6
Eample Rate of Growth of a Cell Culture A cell culture eperiencing changing environmental conditions has a rate of growth modelled b the function r (t) t t, t (, 6), with r (t) measured in cells per hour. a) When is the rate of growth of the cell culture increasing? b) Does the rate of growth ever decrease? Eplain. Solution a) We begin b finding the derivative of r (t). Differentiating, r(t) 6t t. To find the intervals of increase, we first solve r( t)= t( t)( + t) r(t). 6t t t(6 t Intervals ) t( t)( t) Test values t,, or Signs of r( t) The numbers,, and separate the domain into Nature of graph four intervals: (, ), (, ), (, ), and (, ). As in Eample, we need onl test a single value in each interval to determine the sign of the derivative throughout the interval. Since the domain is restricted to t (, 6), the cell growth rate is increasing for t (, ). This is illustrated b the graph of r(t). b) The cell growth rate decreases for t (, 6). If the cells are in a controlled environment, it ma be that the nutrients necessar for growth are rapidl depleted after h. The cells ma also have a limited lifespan of or hours. (, ) (, ) (, ) (, ) + + increasing decreasing increasing decreasing 5 5 5 rt ( )= t t (, 56) 5 6 t 5 It was noted in Eample that a continuous function can change sign onl if it crosses the -ais at a zero. It is possible for a discontinuous function to change sign at a point of discontinuit, where it does not have a zero. For eample, the function = changes sign at its discontinuit at ; it is positive to the right of the discontinuit and negative to the left of the discontinuit. Therefore, in an analsis of the signs of a function, we should make note of the function s zeros and its points of discontinuit, as shown in Eample. = 6. Increasing and Decreasing Functions MHR 5
Eample Containing an Oil Spill MODELLING M AT H Oil spills in the ocean are contained b various sizes of rectangular booms. The formula b w relates the total length of boom required for a particular rectangular w design of area m and width w metres. For which intervals of the width, w, is the length of the boom decreasing? Solution b = + w w b = + w Boundaries for the increasing and decreasing intervals are found when b. For a rational function, points where b is undefined also define boundaries for these intervals. If b, w = w ± 58 If w, b is undefined. Note that in this eample, the function b(w) is not defined when w and the domain is onl meaningful when w. Therefore, the onl intervals that need to be considered are (, 58) and (58, ). Intervals Test values Signs of b( w) = + 58 (, 58) (58, ) + Nature of graph decreasing increasing The length of the boom is decreasing as the width changes from to 58 m. The graph supports this conclusion. b 5 b = w +w 6 8 w 6 MHR Chapter 6
Ke Concepts A function f is increasing on an interval (a, b) if f ( ) f ( ) whenever in the interval (a, b). A function f is decreasing on an interval (a, b) if f ( ) f ( ) whenever in the interval (a, b). Suppose that f is differentiable on the interval (a, b). If f () for all (a, b), then f is increasing on (a, b). If f () for all (a, b), then f is decreasing on (a, b). Communicate Your Understanding. Is it possible for a function to have neither intervals of increase nor intervals of decrease? If so, give an eample. If not, eplain wh.. Can an increasing function ever have an undefined slope? Eplain.. If a function describes the height of a projectile fired from the ground, what intervals of increase and decrease would ou epect to find?. a) In the eamples, wh were the given test values chosen instead of some other test values? b) Would different test values give the same results? Eplain. A B Practise. Given each derivative f (), which intervals need to be tested to determine where the function f () is increasing or decreasing? a) f () b) f () ( ) c) f () ( )( ) d) e) f () f () ( )( ) f) f () ( )( ) g) f () 9 h) f () 6 i) j). f( ) ( )( ) f ( ) Without graphing, determine the intervals of increase and decrease for each function. a) b) ( ) c) ( ) d) e) 6 f) g) 5 5 5 h) 8 i) 6 9 j) k) 6 = l) = + 5 + m) n) = = Appl, Solve, Communicate. Application Temperature fluctuations can often be modelled b functions. Suppose the temperature at a certain location t hours after noon on a certain da is T degrees Celsius, where T = t t + 8t + for t [, 5]. During which time intervals is the temperature falling?. Inquir/Problem Solving a) Show that the function f () is alwas increasing even though f () is not positive at ever point. Give an eample of another function with the same propert. b) Do the functions in part a) contradict the test for intervals of increase/decrease? Eplain. 6. Increasing and Decreasing Functions MHR 7
5. Communication The absorption of an initial dose, D, of a medication into the bloodstream can be modelled b the function Dt ()= D, + t where D is the amount of medication present t hours after ingestion. a) Find the intervals when the amount of medication is decreasing. b) Is the amount of medication ever increasing? Eplain. c) According to the model, does the medication ever completel leave the bloodstream? d) Does the function accuratel model the situation for all time? Eplain. 6. Inquir/Problem Solving The height, h, in metres, t seconds after a stone is hurled upward from the roof of a building m high, is given b h.9t 8t. a) During which interval is the stone rising? b) When will the stone hit the ground? c) For how man seconds is the stone falling? 7. Application Given f () and g(), determine the intervals of increase and decrease of h(), where a) h() f () g() b) h() g(f ()) c) h() f ()g() d) f( ) h ( ) = g ( ) 8. Application A manufacturing compan finds that the profit for a production level of motors per hour is p, (, ]. For what range of production level is the profit a) increasing? b) decreasing? C 9. Application, Communication The altitude, in metres, after t seconds, of an airplane at an airshow is given b the equation h t 5t, t [, 5]. a) Determine the intervals in which the airplane is descending and ascending. b) Describe the motion of the plane.. Application An electrical generator produces power, in watts, according to the formula R P =, where R is the resistance, in ( R + 5. ) ohms, of the circuit. During which intervals is the power a) increasing? b) decreasing?. Determine the range of values of k such that f () k is decreasing on the interval (, ).. Find an eample of a function that has a positive derivative throughout its domain, and et the function is not alwas increasing. Eplain wh our eample does not contradict the test for intervals of increase/decrease.. Suppose that f decreases for (m, n) and increases for (n, p). Under what conditions can ou conclude that f decreases for (m, n] and increases for [n, p)?. a) Determine the intervals of increase and decrease for the function 5 5 f( ) = ( + ) 5 b) Prove that ( ) 5 + > + for (, ). 8 MHR Chapter 6
Maimum and Minimum Values 6. Some of the most practical applications of calculus involve the determination of optimal solutions to problems. This involves determining etreme values, that is, maimum or minimum values of a function. Manufacturers want to maimize profits, contractors want to minimize costs, and phsicians would like to select the minimum dosage of a drug that will cure a disease. We need to develop a variet of tools for solving optimization problems, depending on whether we are given a verbal description, data, a graph, or an equation. Investigate & Inquire: Optimal Shape for a Can What is the most economical shape for a beverage can of fied volume? Suppose the required volume for a clindrical can is 55 cm. We need to find the height, h, and radius, r, in centimetres, that minimize the amount of metal used to make the can. Consider the possible dimensions.. Let r cm. What is the value of h, given V 55 cm and r cm? Sketch the can.. What is the surface area of the can? Label the sketch with this information.. Repeat steps and for r cm, cm, cm, 5 cm, and 6 cm.. Which radius in steps to results in the minimum surface area for the can? 5. How can ou refine the search for the radius that minimizes the surface area? 6. Do ou have the best possible value for the radius? How confident are ou that our answer is optimal? Volume of a clinder = πr h Surface area of a clinder = πr +πrh 7. What assumptions have been made regarding the construction of the can? What other factors might be necessar to consider when designing a can? 8. Describe, and suggest reasons for, an patterns that ou found in our search for the optimum radius. 6. Maimum and Minimum Values MHR 9
In Eample of Section 6.7, we will use calculus techniques to obtain the solution to the problem of designing the most economical can. As a first step, we develop some tools related to functions and their graphs. Consider the graph of the function f in the following figure. Note that the domain of f is [, 8]. Suppose the graph of f represents two hills with a valle between them. The point (6, 5) on the second hill is higher than an nearb point (sa for [5.9, 6.]). Thus, we call 5 a local maimum value of the function f. Similarl, 7 is a local maimum value of f, since the point (, 7) is higher than nearb points on the first hill. However, since 7 is greater than an other value of f in its domain, 7 is also called the absolute maimum value of f. Similarl, the value is called a local minimum value of f, since the point (, ) is lower than nearb points in the valle. Consider the point (8, ). Since is less than an other value of f in its domain, is called the absolute minimum value of f. These concepts are summarized in the following formal definitions. 8 6 = f( ) 6 8 A function f has a local maimum (or relative maimum) value f (c) at c if f (c) f () when is close to c (on both sides of c). A function f has an absolute maimum (or global maimum) value f (c) at c if f (c) f () for all in the domain of f. A function f has a local minimum (or relative minimum) value f (c) at c if f (c) f () when is close to c (on both sides of c). A function f has an absolute minimum (or global minimum) value f (c) at c if f (c) f () for all in the domain of f. Look again at the point (8, ) in the graph of f above. According to the definition of local minimum, we must consider values of f on both sides of 8. Since the graph is onl defined to the left of 8, (8, ) cannot be a local minimum. Thus, on a closed interval, neither local minimum values nor local maimum values can occur at the endpoints. However, absolute minimum and maimum values can occur at the endpoints. Local maimum and local minimum values of a function are often called local etreme values, or local etrema (sometimes called turning points). Similarl, absolute maimum and absolute minimum values of a function are often called absolute etreme values, or absolute etrema. Also, absolute maimum and minimum values are often called simpl maimum and minimum values. Etrema is the plural of etremum. MHR Chapter 6
Eample Determining Local Etrema From Graphs Determine the local and absolute etrema for each function. a) f () for [, ] b) for [, ] = f( ) 5 8 6 = 5 6 Solution a) Reading from the graph, there are local maimum values of f () 9 and f () 7. The absolute maimum value is 9. The local minimum value is f (6) 5. We can see from the graph that the absolute minimum value occurs at the rightmost endpoint, where. Thus, the absolute minimum value is. b) The local maimum value is f (). We can see from the graph that the absolute maimum value occurs at the rightmost endpoint. The function value at this endpoint is f () (). Thus, the absolute maimum value is. The local minimum values are f () and f (). Thus, the absolute minimum value is. Critical Points and Local Etrema How can we determine the maimum and minimum values of a function without graphing? You will investigate this problem net. Investigate & Inquire: Determining Local Etrema In Section 6., ou learned how to determine when a function is increasing and when it is decreasing. In this investigation, ou will eplore how to use this information to determine the local maimum and local minimum values of a function.. Determine the intervals of increase and decrease for each function. a) f () b) f () 7 c) f () d) f () 6 6. Maimum and Minimum Values MHR
. For each function in step, determine the value of f at the endpoints of the intervals ou determined. What do ou notice?. Graph each function in step and use the graph to determine the local maimum and local minimum values.. Eplain how to use the intervals of increase and decrease in step and our results from step to determine the local maimum and minimum values without graphing. 5. Note that the functions in step are all differentiable. Conjecture a test for finding the local maimum and local minimum values of a differentiable function using intervals of increase and decrease and the values of the derivative of the function at the endpoints of the intervals. Notice that, in this investigation, the value of f at the endpoints of the intervals of increase and decrease for each function is. For other functions, it could happen that the graph has a cusp (sharp corner) between intervals of increase and decrease. For eample, consider the function f (). We know that f is not differentiable at, that is, f () does not eist. But if we look at the graph, we can see that the function f () decreases for and increases for. We call an -value for which the derivative is equal to or does not eist a critical number. A number c in the domain of f is a critical number of f if either f (c) or f (c) does not eist. The point (c, f (c)) is called a critical point. It appears from the preceding discussion that local maimum and local minimum values occur at critical points of functions. This is true in general, and is called Fermat s theorem. Fermat s theorem: If f has a local maimum value or local minimum value at c, then either f (c) or f (c) does not eist. Be careful: just because f (c) or f (c) does not eist is no guarantee that f has a local maimum or minimum value at c. For instance, consider the graphs of f () and g ( )=. Window variables: [.7,.7], [.,.] or use the ZDecimal instruction. MHR Chapter 6
6 f ( ) = g ( )= 6 6 6 Notice that f has a horizontal tangent at the origin (that is, f () ), and g has a vertical tangent at the origin (that is, g() does not eist). Yet neither f nor g has a local maimum or local minimum value at. What Fermat s theorem sas is that if f has a local maimum or a local minimum value at c, then f (c) is either or does not eist; but the converse is not necessaril true, as the two previous graphs show. If we are looking for local maimum or local minimum values, Fermat s theorem tells us where to look: at the critical numbers. Once we have determined the critical numbers, how do we decide whether the function value at a critical number is a local maimum, a local minimum, or neither? To visualize this, think about the following graphs as representing hills and valles. local maimum value f ( ) > f ( ) < = f( ) = f( ) f ( )< f ( )> local minimum value In the graph on the left, as ou walk from left to right, first ou walk uphill, then ou reach the peak, and then ou walk downhill. That is, the function is increasing, then the local maimum is reached, and then the function is decreasing. In the graph on the right, it is the other wa around: first the function is decreasing, then the local minimum is reached, and then the function is increasing. 6. Maimum and Minimum Values MHR
Since f is increasing when f () and f is decreasing when f (), we obtain the first derivative test. The first derivative test: Let c be a critical number of a continuous function f. If f () changes from positive to negative at c, then f has a local maimum at c. If f () changes from negative to positive at c, then f has a local minimum at c. If f () does not change sign at c, then f has neither a local maimum nor a local minimum at c. Eample Determining Local Etrema Determine the local maimum and local minimum values of the function f (). Solution Since f is differentiable, the critical numbers are the -values for which f (). f () Setting the derivative equal to and solving for, we obtain or Thus, the critical numbers are and. We test f () at -values on the left and right of the critical numbers using an interval chart. Intervals Test values Signs of f( ) B the first derivative test, f has a local maimum at. Its value is f (). The first derivative test also indicates that f has a local minimum at. Its value is f (). The graph of f illustrates our findings. Nature of graph f ( )=( )( + ) (, ) (, ) (, ) + + increasing local maimum value at (, ) f ( ) > decreasing f ( ) < f ( ) < increasing f ( ) = + f ( ) > local minimum value at (, ) MHR Chapter 6
Consider the function f () with domain [, ]. Notice that the absolute maimum value occurs at the right endpoint of the domain, and the absolute minimum value occurs at the left endpoint of the domain. Now consider the function g() with domain [, ]. For this function, the absolute minimum value of occurs at the critical number, and the absolute maimum value of 5 occurs at the right endpoint of the domain. Therefore, if we are looking for the absolute etrema of a function for which the domain is a closed interval, we should test the values of the function at the critical numbers and at the endpoints of the interval. 5 g ( ) = + f ( ) = Eample Determining Absolute Etrema Find the absolute maimum and absolute minimum values of the function f () for [,.5]. Solution Using Fermat s theorem, we first determine the critical numbers of f. Since f is differentiable for all values of in its domain, we need onl determine the -values for which f (). f () 8 8 If f (), then 8 8 8( )( ) 8 or or The value is not in the domain [,.5] so we do not consider it. The values of f at the other two critical numbers are f () () () and f () () () The values of f at the endpoints of the interval are f () () () and f (.5) (.5) (.5).5 Considering the values,,, and.5, the absolute maimum is, which occurs at (an endpoint), and the absolute minimum is, which occurs at. Note that is also a local minimum value. The graph illustrates our conclusions. Absolute maimum value f ( ) = + Absolute minimum value Local maimum value 6. Maimum and Minimum Values MHR 5
Eample Acidit Caused b Eating Cand The ph in a person s mouth t minutes after ph is a measure of acidit: the lower eating cand can be modelled b the equation the number, the stronger the acid. P. t.t.t 6. for t [, ]. Determine the time when the ph is lowest (most acidic) after eating cand. Solution The derivative is P. 6t.8t.. To determine the critical numbers, we set the derivative equal to and solve for t:. 6t.8t.. 8. 8 (. 6)(. ) t (. 6). or 5. 5 We do not consider t 5.5 because it is outside the interval [, ]. Thus, the onl critical number is t.. P =.6 t +.8 t.. (,.) (., ) + decreasing increasing Intervals Test values Signs of P Nature of graph We use an interval chart to organize the information. From the chart, and the first derivative test, the ph has a local minimum at t. and its value is P(.) 5.. The ph values at the endpoints of the interval are P() 6. P() 6. Since both values are greater than 5., the absolute minimum ph of 5. occurs after. min. A sketch of the ph function illustrates our calculations. P 8 6 (., 5.) 5 5 5 t Ke Concepts A critical number of a function is an -value for which the derivative of the function is or does not eist. Local maimum and minimum values occur at critical numbers, but not all critical numbers give rise to local maimum or minimum values. Fermat s theorem: If f has a local maimum value or local minimum value at c, then either f (c) or f (c) does not eist. To find the absolute maimum and absolute minimum values of a continuous function with domain [a, b], a) Identif the critical numbers and endpoints of the interval. b) Find the values of f at the critical numbers of f in [a, b]. 6 MHR Chapter 6
c) Find the values of f at the endpoints of the interval. d) The greatest of the values from steps b) and c) is the absolute maimum value and the least of these values is the absolute minimum value. Use the first derivative test to determine whether a function has a local maimum value, a local minimum value, or neither, at a critical number. Let c be a critical number of a function f. If f () changes from positive to negative at c, then f has a local maimum at c. If f () changes from negative to positive at c, then f has a local minimum at c. If f () does not change sign at c, then f has neither a local maimum nor a local minimum at c. Communicate Your Understanding. Eplain the difference between an absolute maimum and a local maimum.. Is it possible for a function f to have a critical number at which f has neither a local maimum value nor a local minimum value? If it is possible, sketch a graph of a function that satisfies this condition; if it is not possible, eplain wh.. Name the two tpes of critical numbers and how the differ.. If a function is differentiable on an interval, and the interval contains no critical points, wh does the slope of the graph at one test point in the interval reveal the increasing/decreasing nature of the function over the entire interval? A Practise. Find the absolute maimum and absolute minimum values of each function. a) b) 6 8 9 6 5 6 6 8 B. Without graphing, find the absolute maimum and minimum values of each function. a) f () for [, ] b) f () for [, ] c) f () 9 for [, ] d) f () for [, ] e) f () 7 for [, ] f) f () for [, ] g) f () ( ) for [, ] 6. Maimum and Minimum Values MHR 7
h) f () for [, ] i) f () 7 for [, ] j) f () 5 5 for [, ] k) f () for [, ] l) f () for [, ]. Determine the critical numbers of each function and appl the first derivative test to determine whether the critical numbers correspond to local maimum or local minimum values. Determine the local maimum or local minimum values. a) b) ( ) c) ( ) d) 6 e) = f ) 8 g) = h) 6 i) 5 5 j) k) = ( ) l) =. Application Determine the local etrema, and sketch each function. a) b) c) d) e) f) 7 5 5 68 g) h) 6 Appl, Solve, Communicate 5. The volume of a shed in the shape of a rectangular prism with height metres is given b V 6 for [, ]. Determine the height of the shed that has absolute maimum volume. 6. The height, h(t), in metres above the ground, of a flare as a function of time, t, in seconds after the flare is fired, is given b the function h(t).9t t for t. a) Determine the height of the flare when it is fired. b) Determine the maimum height of the flare. c) When does the flare hit the ground, to the nearest second? 7. Application Weather forecasts in the winter refer to wind-chill factor. Wind chill is the additional cooling that occurs when skin is eposed to wind. One of the equations used to model wind chill is W.7v 8.v 58v 9 for v [, 5], where v is the wind speed in metres per second. The units for W are megajoules per square metre per hour. Determine the speed of the wind that gives a minimum wind-chill factor. Web Connection To learn more about modelling wind chill, go to www.mcgrawhill.ca/links/caf and follow the link. 8. Application The speed of air escaping through our trachea when ou cough is a function of the force due to chest contraction and the size of the trachea. It can be modelled b the equation S F(.9r r ), where F is the constant force of our chest contraction, r is the radius of our trachea, in centimetres, and S is the speed of the airflow, in centimetres per second. a) Determine the maimum speed of airflow that can be attained. b) Sketch the graph of S versus r (use F ). 9. Inquir/Problem Solving Three students, Lawrie, Gavin, and Rebekah, were asked to analse the function f( )= +. Lawrie said, There is a critical point at, and I have tested neighbouring points on either side. I tested and and found a negative slope to the left and a positive slope to the right, hence there is a local minimum at. Gavin countered, But I have found, using the first derivative test, that the adjacent critical point at is also a local minimum. How can that be? Adjacent critical points cannot both be minimum values. a) Rebekah sorted it out for them. What was her argument? b) Eplain an errors in Lawrie s and Gavin s eplanations. 8 MHR Chapter 6
MODELLING M AT H. A marine biologist in Burncoat Head, Nova Scotia, collected the following tidal data at hourl intervals from : until 9:. Time 5 6 7 Tide Height...8.5 8. 9 6. 8.9. Time Tide Height.8. 9. 5 6.5 6.9 7.8 8.8 9.9 a) Use regression to determine a quartic function to model the data. b) Find the times, between : a.m. and 7: p.m., when the height of the tide is increasing, and when it is decreasing. At what time is the height of the tide neither increasing nor decreasing? c) When, during the da, is the height of the tide a maimum? a minimum?. Application The predicted population, in millions, of a small countr has been modelled using the equation P, where is 5 the number of ears after, and [, ]. a) Determine the intervals of increase and decrease in population. C b) Determine when the population reaches a minimum. What is the minimum population? c) Sketch a graph of the population over the ear period.. Find the absolute maimum and absolute minimum values of the function defined b for [, ] f( ) = + + for (, ) for [, ]. Determine the absolute maimum and absolute minimum values of the function g ( )= 9 on the interval [ 55, ].. Determine the absolute maimum of = ( r + ) for [, r], where r is a constant. 5. Inquir/Problem Solving a) If a function is not restricted to a closed interval, can it have an absolute maimum or an absolute minimum? If es, give an eample of each. If no, eplain wh. b) Must a differentiable function on a closed interval have an absolute maimum? If no, give an eample. If es, eplain wh. 6. Communication a) If a differentiable function f () has f () and f (), will it have a local maimum? Eplain. b) Would our conclusion in part a) be the same if f were not differentiable for all values of? Eplain. Historical Bite: Fermat s Long-Unproven Theorem Fermat s theorem (page ) is not the same as the famous Fermat s last theorem, although the same mathematician, Pierre de Fermat (6 665), was responsible for both results. Fermat s last theorem states that the equation n + n = z n has integer solutions,, and z onl for n = or, solutions in the case n = being Pthagorean triples. Fermat wrote down no proof of this theorem, simpl stating in his notes that I have discovered a beautiful proof of this result, which unfortunatel this margin is too small to contain. Toda, mathematicians believe that Fermat was mistaken in believing he had a proof, because theor developed much later revealed complications that he would not have known about. It was not until 99 that Andrew Wiles (95 ) made headlines with a complete and highl comple proof of Fermat s last theorem. 6. Maimum and Minimum Values MHR 9
6. Concavit and the Second Derivative Test Imagine driving along a curved road from R to T as shown. As the car moves from R to S, the road curves to the left, and ou feel the seat of the car eert an additional force on ou to the left. This force tends to decrease as ou approach S, and reaches zero at S. The road then curves to the right and the additional force from the seat points to the right. In this section, we eplore curves that behave in this fashion, and points such as S. S T S T R R If we think of the road as the graph of a function, then the curve from R to S is described as concave upward (curving so as to enclose the space above it). From S to T the curve is concave downward, and from T on it is concave upward. To distinguish between these two tpes of curves, eamine the graphs of the functions f and g below. Each graph connects point A to point B, but the two graphs bend in different was. If we draw tangents to the curves at various points, the graph of f lies above its tangents and is called concave upward, while the graph of g lies below its tangents and is called concave downward. = f( ) B A = g( ) A B a Concave upward on ( a, b) b a b Concave downward on ( a, b) The graph of f is said to be concave upward on an interval (a, b) if it lies above all of its tangents on (a, b). The graph of f is said to be concave downward on (a, b) if it lies below all of its tangents on (a, b). MHR Chapter 6
For instance, the function shown below is concave upward on the intervals (, ), (, 5), and (5, 6.5), and concave downward on (, ) and (6.5, ). P Q R S 5 6 7 A point P on a curve is said to be a point of inflection if the curve changes from concave upward to concave downward or from concave downward to concave upward (that is, it changes concavit) at P. For instance, the graph just considered has three points of inflection, namel, P, Q, and S. Point R is a cusp but is not a point of inflection, since the concavit does not change at R. Also, notice that if a curve has a tangent at a point of inflection, as at points P, Q, and S in the figure, the tangent crosses the curve there. In the following investigation, ou will eplore the connection between concavit and tangents. Investigate & Inquire: Concavit and Inflection Points. Graph each function. Estimate the intervals on which the function is concave upward, the intervals on which it is concave downward, and the point(s) of inflection. a) f () b) f () 5 c) f () 5 6 d) f () 5 5. Calculate f () and f () for each function in step. Graph the function and its derivatives on the same set of aes.. For each graph in step, determine the sign of f () on each interval on which the graph is concave upward. Repeat this for each interval on which the graph is concave downward.. Make a conjecture about the sign of f () when the graph is concave upward. Repeat when the graph is concave downward. 5. Draw a vertical line through the point of inflection of each graph of f in step. Let the -coordinate of the point of inflection be a. 6. How does the first derivative behave near an inflection point (a, f (a))? Conjecture a test for inflection points using onl the first derivative. Test our conjecture. 7. How does the second derivative behave near an inflection point (a, f (a))? Conjecture a test for inflection points using onl the second derivative. Test our conjecture. 8. What is the value of the second derivative at each inflection point ou found? 6. Concavit and the Second Derivative Test MHR
Note from the investigation that the sign of the second derivative affects the direction of concavit. If the second derivative is positive, the graph is concave upward. If the second derivative is negative, the graph is concave downward. This can be understood b thinking of the second derivative as the rate of change of the slope of a graph. Thus, if f (), the slope of the graph of f is increasing at. Eamine the following graph of a function that is concave upward on (a, b). We can see that the slopes of the tangents to the function are increasing as increases. The slopes on the = f( ) left are negative, increasing through negative values up to, and then increasing through positive values. A similar argument can be made for a function that is concave downward on an interval. Test for concavit: If f () for all (a, b), then the graph of f is concave upward on (a, b). If f () for all (a, b), then the graph of f is concave downward on (a, b). 5 f ( ) f ( ) f ( ) f ( ) f ( 5 ) It follows from the test for concavit and the definition of a point of inflection, that there is a point of inflection at an point where the second derivative changes sign. Note that the second derivative must be zero or not eist at a point of inflection. This compares closel with Fermat s theorem (page ), which relates critical numbers to the first derivative. As with Fermat s theorem, the condition f or does not eist does not guarantee that a point of inflection eists, as Eample shows. Eample A Function With Zero Second Derivative but no Point of Inflection Show that the function f () satisfies f () but has no point of inflection. Solution Since f (), we have f () and f (), so f (). But for both and, so the concavit does not change and there is no point of inflection. 6 8 f( ) = Eample Concavit and Points of Inflection for a Cubic Function a) Determine where the curve f () 5 is concave upward and where it is concave downward. b) Find the points of inflection. c) Use this information to sketch the curve. Draw the tangent at the point of inflection. MHR Chapter 6
Solution a) If f () 5 then f () 6 5 and f () 6 6 6( ) We use an interval chart to determine when f () is negative and when it is positive. The curve is concave downward (since f () ) for (, ) and concave upward (since f () ) for (, ). b) The curve changes from concave downward to concave upward when, so the point (, 5) is a point of inflection. c) Net, we determine the critical numbers of f (). Since f is continuous, we need onl consider -values for which f (). 6 5 = ± ( 6) ( 6) ( )( 5) () 6± = 6 Intervals Test values Signs of f( ) Nature of graph concave downward f ( ) = 6( ) (, ) (, ) + concave upward Since there is no real value of for which f (), the function has no critical points. Hence, the function is either alwas increasing or alwas decreasing. We test a particular point to determine which. Since f () 5, the function is alwas increasing. This information, together with parts a) and b) and the -intercept, f (), allows us to sketch the curve. We also need to determine the equation of the tangent at the point of inflection. The slope of the tangent is m f() () 6 () 5 The equation of the tangent is m( ) 5 ( ) 8 6 6 f( )= +5 + (, 5) = + Note that the information regarding concavit is ver helpful in sketching the curve. 6. Concavit and the Second Derivative Test MHR
Eample Concavit and Points of Inflection for a Rational Function Discuss the curve = with respect to concavit and points of inflection. + Solution Let f( )=. + Then f( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) f ( ) ( ) ( ) ( ) We determine when f is negative and when it is positive in order to determine the concavit. To do this, we first determine when f (). ( ) = ( + ) We can multipl both sides of the equation b the denominator, which is alwas positive. ( ) =± Intervals Test values Signs of f( ) Nature of graph ( concave upward f( ) = ( + ) (, ( ( ( ) + + concave downward ( concave upward f ( )= + (.58,.75) (.58,.75) ( We consider the intervals,, and.,,, The curve is concave upward for and, and concave,,.5.5.5.5 downward for. The points of inflection are, and,., The graph verifies these calculations. MHR Chapter 6
The Second Derivative Test The following investigation eplores the possibilit of using the second derivative of a function to determine if a local etremum is a local maimum or a local minimum. Investigate & Inquire: Distinguishing Local Maima from Local Minima. Eamine the graphs of the functions f, f, and f from step in the investigation on page. Draw vertical lines through the local minimum and maimum points of each graph of f.. How do the derivatives f and f behave at these local etrema?. Can the second derivative be used to distinguish local maimum points from local minimum points? If so, conjecture a general rule.. Test our rule on two additional functions. Note, from the investigation, that the second derivative can be used to determine whether a local etremum is a maimum or a minimum value of a function f. We assume that f () eists and is continuous throughout the domain of f. The figure below shows the graph of a function f with f (c) and f (c). Since f (c), the graph of f is concave upward near c, and therefore lies above its tangents at (c, f (c)). But since f (c), this tangent is horizontal. Therefore, f has a local minimum at c. = f( ) (, c f()) c c f( c) =, f( c) > Similarl, as shown in the figure on the right, if f (c) and f (c), then the graph of f is concave downward near c, and therefore lies below its horizontal tangent at (c, f (c)). Thus, f has a local maimum at c. = f( ) (, c f()) c c f( c) =, f( c) < 6. Concavit and the Second Derivative Test MHR 5
Second derivative test: If f (c) and f (c), then f has a local minimum at c. If f (c) and f (c), then f has a local maimum at c. Eample Local Maimum and Minimum Values, Concavit, and Points of Inflection Find the local maimum and minimum values of the function =. Use these, together with concavit and points of inflection, to sketch the curve. Solution Let f( )=. Then f () 6 ( 6) f () ( ) To find the critical numbers, we set f () and obtain and 6. Then, to use the second derivative test, we evaluate f at these numbers: f () and f (6) 6 Since f (6) and f (6), f (6) 8 is a local minimum. Since f (), the second derivative test gives no information about the critical number. But, since the first derivative does not change sign at (it is negative on both sides of ), the first derivative test tells us that f has no maimum or minimum at. Since f () ( ), f () at and at. We determine the concavit using an interval chart as shown above to the right. Intervals Test values Signs of f( ) Nature of graph concave upward f ( )= ( ) (, ) (, ) (, ) + + 6 6 concave downward concave upward point of inflection (, ) ) f ( )= ) 6 8 point of inflection (, 6) Thus, f is concave upward on (, ) and (, ) and concave downward on (, ). The points of inflection are (, ) and (, 6). Now, we graph the function using this information. 8 local minimum (6, 8) 6 MHR Chapter 6
Eample shows that the second derivative test gives no information when f (c). It also fails when f (c) does not eist. In such cases, we must use the first derivative test. Ke Concepts The graph of f is called concave upward on an interval (a, b) if it lies above all of its tangents on (a, b). It is called concave downward on (a, b) if it lies below all of its tangents on (a, b). A point P on a curve is called a point of inflection if the curve changes concavit at P. Test for concavit If f () for all on (a, b), then the graph of f is concave upward on (a, b). If f () for all on (a, b), then the graph of f is concave downward on (a, b). At a point of inflection, f or f does not eist. However, this condition on f ma also be true at other points. Therefore, inflection points cannot be located simpl b setting f (). A point of inflection for f occurs when the sign of f changes at that point. Second derivative test If f (c) and f (c), then f has a local minimum at c. If f (c) and f (c), then f has a local maimum at c. The second derivative test does not appl when f (c) and when f (c) does not eist. Communicate Your Understanding. Referring onl to the characteristics of a graph, describe the terms a) concave upward b) concave downward. Draw two different curves, each with a point of inflection, so that one of the curves is differentiable at its point of inflection and the other is not. Mark each point of inflection and eplain wh it is a point of inflection.. Eplain each statement. a) If f is concave upward on an interval, then the slopes of the tangents are increasing from left to right on that interval. b) If f is concave downward on an interval, then the slopes of the tangents are decreasing from left to right on that interval.. Chris claims that, to find inflection points, all ou have to do is find where the second derivative is or undefined. Discuss the validit of this statement. 5. If f (c) and f (c), what can ou conclude about the point on the graph of f where c? 6. Does the second derivative test appl to the function f( )= at? If es, appl it. If no, eplain how to determine whether f has a maimum value, a minimum value, or neither at. 6. Concavit and the Second Derivative Test MHR 7
A Practise. a) State the intervals on which f is concave upward and the intervals on which it is concave downward. b) State the coordinates of the points of inflection. = f( ) d) State the -coordinates of an points of inflection of f. Eplain wh each is a point of inflection. = f ( ) 6 8 8 8. The figure is a sketch of f for a function f. State the intervals on which the graph of f is concave upward and the intervals on which it is concave downward. Determine the -coordinates of an inflection points of the graph of f. = f( ) 6 8. The following are graphs of the first and second derivatives of a function f (). For each graph, i) identif all the critical numbers of f and eplain if the relate to a maimum, a minimum, or ou cannot decide, using onl the second derivative test ii) for the numbers for which the second derivative test fails, use the first derivative test to decide the nature of the related point a) = f ( ) 6 = f ( ). The graph of the first derivative f of a function f is shown. a) On which intervals is f increasing? decreasing? b) On which intervals is f concave upward? concave downward? c) State the -coordinates of an local maimum or minimum points of the graph of f. 6 6 8 MHR Chapter 6
b) c).8.6 8 6 = f ( ) 6 8.. = f ( ) 6 8 = f ( )...6.8 = f ( ) 5. Find the intervals on which each curve is concave upward and the intervals on which it is concave downward. State an points of inflection. a) f () 5 b) 6 5 c) g() 6 d) e) f () 6.5 f) 6 g) h() 6 6 h) 5 5 i) f( )= j) k) = g ( )= + 6. Use the second derivative test to find the local maimum and minimum values of each function, wherever possible. B a) 5 8 b) f () c) g() d) f () 9 e) f) f () 5 5 5 g) h() ( 8 ) h) 7. Find the local maimum and minimum values of each function. a) g() 6 b) = ( ) c) h ( )= d) + 8. Use the first and second derivative tests to determine the maimum points, minimum points, and points of inflection for each function. Graph each function. a) f () 6 b) f () 6 c) 6 d) f () 8 5 e) 6 f) g) h) Appl, Solve, Communicate 9. a) Determine where f is positive, where it is negative, and where it is zero. Repeat for f and f. b) Is there an interval where f, f, and f? If so, state the interval. 8 6 5 = f( ) h ( ) 6. Concavit and the Second Derivative Test MHR 9
. A manufacturer keeps accurate records of the cost, C(), of producing items in its factor. The graph shows the cost function. = C ( ) a) Eplain wh C(). b) Eplain the significance of the point of inflection. c) Use the graph of C to sketch the graph of the marginal cost function, C ().. Application A tpical predator-pre relationship for foes and rabbits is shown in the graph.. Communication A software compan estimates that it will sell N units of a new product after spending dollars on TV ads during football games. The relationship can be modelled b N(), [, 5], where is measured in thousands of dollars. a) Find the minimum and maimum number of sales. b) Does more advertising alwas lead to more sales? Eplain.. Inquir/Problem Solving Water is poured into a cone-shaped vase, as shown, at a constant rate. a) Sketch a rough graph of H(t), the height of the water in the vase as a function of time. b) Sketch rough graphs of H(t) and H(t). c) Eplain the shape of all three graphs in terms of the shape of the vase. Eplain the concavit in the graph of H(t). Number of rabbits 8 (, 88) Ht () 6 Number of foes 8 6 Time (das). For the function f( ) = ( + ), determine a) the intervals of increase and decrease b) the local maimum and minimum values c) the intervals of concavit d) the points of inflection a) Estimate the intervals on which the graph of the pre population is concave upward; concave downward. b) Estimate the coordinates of the points of inflection of the pre graph. c) Describe the predator-pre relationship in terms of maimum and minimum values, intervals of increase and decrease, and concavit. 5. Inquir/Problem Solving If possible, sketch the graph of a continuous function with domain R satisfing each set of characteristics. If it is not possible, eplain wh not. a) The first and second derivatives are alwas positive. b) The function is alwas positive and the first and second derivatives are alwas negative. MHR Chapter 6
C c) The first derivative is alwas positive and the second derivative is alwas negative. d) The first derivative is alwas negative and the second derivative is alwas positive. e) The first derivative is alwas positive and the second derivative alternates between positive and negative. f) The function is alwas negative and the first and second derivatives are alwas negative. 6. Application Use a graphing calculator or graphing software. Give results to three decimal places. i) Draw the graph of f. ii) Find all etrema for f. iii) Find all points of inflection for f. a) f () 6 5 5 8 9 b) ( + ) f( ) = ( ) ( ) 7. Is it possible to use the graph of the second derivative of a function to determine all the -coordinates of the maimum and minimum points without an knowledge of the first derivative? Eplain. 8. For what values of the constants c and d is (, ) a point of inflection of the cubic curve c d? 9. Graph several members of the famil of polnomials defined b f () c. a) For which values of c do the curves have maimum points? b) Prove that the minimum and maimum points of each of these curves lie on the parabola defined b.. Sketch the graph of a continuous function that satisfies all these conditions. f () f (), f () f() f () f () f () for (, ) and for (, ) f () for (, ) and for (, ) f () for ( ), f () for (, ) lim f( ), lim f( ). Sketch the graph of a continuous function that satisfies all of the following conditions. f () for (, ), f () for (, ) f () for (, ), f () for (, ) lim f( ) f is an odd function. Communication The function f () ( )( ) was graphed with technolog. a) Use calculus techniques to determine an local etrema and points of inflection for this curve. b) Do the window settings chosen show all the relevant features of the graph? Eplain. c) If our answer in part b) was no, give appropriate window settings for graphing this function using technolog. 6. Concavit and the Second Derivative Test MHR
6. Vertical Asmptotes MODELLING M AT H Rational functions are used to model man situations involving economics and medicine. Man phenomena in nature, such as the intensit of light as it travels through space, the force of gravit, and electrostatic force, follow an inverse square law, that is, a function of k the form f( )=. Investigate & Inquire: Force Between Two Charged Particles Consider two electricall charged particles, A and B. A is fied in its position at the origin and B is free to move along the -ais. The magnitude of the electrostatic force F between these two particles is inversel proportional to the square of the distance between them. This force, measured in newtons, is given b F ( )=, where is the position of B, in metres. What happens to the magnitude of the electrostatic force as B comes close to A on either side of A? In the following steps, ou will eamine the force F ( )= for close to.. Construct a table of values for F for values of close to, on both sides of. You can do this b hand, or using a spreadsheet, a graphing calculator, or graphing software.. What happens to the electrostatic force, F, as the distance between the two small particles decreases from m to almost m for positive values of? for negative values of?. Graph F for the interval [, ]. a) What happens to F() as approaches from the left and from the right? b) Does lim F ( ) eist? Is F continuous at? Eplain our reasoning. c) What is the magnitude of the electrostatic force when?. If ou were to draw the line (the -ais), would the graph of F intersect this line? Does drawing this line show anthing else about the graph of F? From the investigation, note that the closer we take to, the larger becomes. In fact, it appears that, b taking close enough to, we can make F() as large as we like. Since the values of F() do not approach a particular number as approaches, we sa that lim does not eist (as a numerical value) and write lim. We sa that the line is a vertical asmptote of the graph of F ( )=. MHR Chapter 6
Notice in the investigation that F is not defined at, and so F is not continuous at. This tpe of discontinuit is called an infinite discontinuit. (Refer to Chapter.) In general, if f() is defined on either side of the number a, we write lim f () if the a values of f () can be made arbitraril large (as large as we like) b taking sufficientl close to a, but not equal to a. This tpe of limit is called an infinite limit. The epression lim f ( ) is usuall read as the limit of f () as approaches a is a infinit or f () increases without bound as approaches a. This does not mean that is a number or that the limit eists. It simpl epresses the particular wa in which the limit does not eist, because the magnitude of g() groups arbitraril large as approaches a. = f( ) = g( ) a a These figures illustrate two kinds of infinite limits, lim f ( ) and lim g ( ). a a The latter means that the values of g decrease without bound as approaches a or that the limit of g() as approaches a is negative infinit. Once again, the limit does not eist, because the magnitude of g() grows arbitraril large as approaches a. One-sided infinite limits can also be defined, as shown in the figures below. The notation for each limit is shown with its diagram. In each case shown in the figures, the limit does not eist. = f( ) = f( ) = f( ) a a = f( ) a a lim f( ) a lim f( ) a lim f( ) a lim f( ) a Remember that a means that approaches a from the left, and a + means approaches a from the right. 6. Vertical Asmptotes MHR
The line a is called a vertical asmptote of the curve f () if at least one of the following is true: lim f( ) lim f( ) lim f( ) a a a lim f( ) lim f( ) lim f( ) a a a Eample Fundamental Limits Evaluate lim and lim, and state whether the eist. + Solution Consider the graph of =. Window variables: [.7,.7], [.,.] or use the ZDecimal instruction. We determine values of for -values close to, on either side of, selecting Ask mode on the TABLE SETUP screen of a graphing calculator. As approaches from the right, becomes increasingl large. Therefore, lim. If is close to, but negative, then is a negative number with an arbitraril large magnitude. Thus, lim. Therefore, lim and lim do not eist. + These limits can also be seen from the graph of =. The line (the -ais) is a vertical asmptote. Eample An Infinite Limit Find lim. 5 ( 5) Solution We select Ask mode on a graphing calculator to create a table of values to get an idea of the limit. We look at values of close to 5, on either side of 5. Let =. ( 5) We can conjecture from the table that lim. ( ) 5 5 MHR Chapter 6
Consider onl the term. If is ver close to 5, on either side of 5, then ( 5) is ( 5) ver small, but positive. This means that is ver large and positive. Therefore, ( 5) ( ) 5 is arbitraril large in magnitude, but negative. Therefore, our conjecture that the limit does not eist is correct: lim 5 ( 5) To find the vertical asmptotes of a rational function, we first determine whether the numerator and the denominator have an common factors. If the do, we simplif the function, noting an restrictions. We then find the values of where the denominator is zero, and compute the limits of the function from the right and the left at those values of. Eample Determining Vertical Asmptotes Find the vertical asmptotes of each function and sketch its graph near the asmptotes. + a) f( )= b) + + Solution a) First, we factor the denominator. + f( ) = + + + = ( + )( + ) The numerator and the denominator have a common factor of ( ). Thus, we simplif the function as follows: f( ) =, + g ( )= + 5 Now, we can determine the limit as approaches from both sides. lim f( ) = lim, + = + = Thus, f has a removable discontinuit at, and therefore is not an asmptote. However, is an asmptote. For, f (), and for, f (). f( ) = + + + 6 6. Vertical Asmptotes MHR 5
Thus, lim f( ) and lim f (). Now we can sketch the part of the + graph that lies near the asmptotes. Note the gap in the graph at. This is the removable discontinuit. b) First, we factor the denominator: g ( ) = + 5 = ( + 5)( ) Since the denominator is when 5 or, and the numerator and denominator have no common factors, the lines 5 and are vertical asmptotes. To help Intervals Test values Signs of g ( ) with the graph, we also consider when the numerator is zero (at ), since the graph might change sign. We use an interval chart to determine the behaviour of the graph on either side of the asmptotes and near. Thus, the -values of the function approach negative infinit as approaches 5 from the left and as approaches from the left. The -values of the function approach positive infinit as approaches 5 from the right and as approaches from the right. Thus, we have lim g ( ), lim g ( ), 5 lim g ( ), and lim g ( ). 5 Also, the -values change from positive to negative at. Now we can sketch the parts of the graph that lie near the asmptotes and near. g ( )= ( + 5)( ) 5 (, 5) ( 5, ) (, ) (, ) U + U + 6 6 We will be able to complete such graphs with the methods discussed in the following sections. g ( ) = + 5 Ke Concepts In general, if f() is defined on both sides of the number a, we write lim f ( ) if a the values of f () can be made arbitraril large (as large as we like) b taking sufficientl close to a. We write lim f () if the values of f are negative and their magnitudes increase a without bound as approaches a. 6 MHR Chapter 6
To indicate one-sided limits involving vertical asmptotes, we write = f( ) = f( ) = f( ) = f( ) a a a a lim f( ) lim f( ) lim f( ) lim f( ) a a a a to indicate the behaviour of the function on either side of the asmptote. Communicate Your Understanding. Eplain what is meant b a vertical asmptote.. What is the meaning of an infinite discontinuit? Use a graph in our eplanation.. a) Eplain how ou would find the vertical asmptotes of each function. i) + f( )= ii) g ( )= 5 b) Eplain how ou would sketch the graph of the function near each asmptote.. How can ou determine the behaviour of the graph of a function near its vertical asmptotes? 5. Eplain each epression in our own words. a) lim f( ) b) 6. Write a brief paragraph to assess the following reasoning. Suppose that lim f( ) and lim f( ). Since the left- and right-hand limits are equal, lim f( ) eists and lim f( ). lim f( ) A Practise. The graph of f is given. a) State the equations of the vertical asmptotes. b) State the following. 8 8 = f( ) i) lim f( ) ii) iii) 5 lim f( ) iv) v) lim f( ) vi) vii) lim f( ) viii) lim f( ) 9 lim f( ) 5 + lim f( ) + lim f( ) + 9 + 6. Vertical Asmptotes MHR 7
c) Find the following limits, if possible. If it is not possible, eplain wh. i) lim f( ) ii) lim f( ) 5 iii) lim f( ) iv) lim f( ). Give an equation for a function having a) vertical asmptote b) vertical asmptotes and c) vertical asmptotes 5 and 5, and f () for all other values of B. Find each limit. a) lim b) lim 5 ( ) 6 ( ) 5 + 6 c) lim 5 d) lim 7 ( ) ( ) + 7 e) + f ) 5 lim ( ) ( ) +. Find each limit, if possible. If it is not possible, eplain wh. a) lim b) lim c) lim d) lim 6 6 e) lim f ) lim + g) lim h) lim + i) lim + 5. Find the vertical asmptotes of each function and sketch the graph near the asmptotes. a) f( )= b) = + c) 5 + k ( ) = d) g ( ) = ( ) ( + ) e) = f ) = g) + 8 h ( )= h) = + 6+ 8 ( + ) i) + = j) q ( )= 9 6. Communication Sketch a graph satisfing each set of conditions. a) f (), f (), lim f( ), lim f( ) 9 b) f (), f () 6, c) g(), lim g ( ), Appl, Solve, Communicate 7. Application A consultant has issued an environmental report on the cost of cleaning up a propert that was previousl the site of a chemical factor. Costs can increase dramaticall depending on the percent of pollutants that needs to be removed. Her report gives the cost, C, in dollars, of removing p% of the pollutants from the site as 5 Cp ( ). p a) What is the cost of removal for half of the pollutants? 9% of the pollutants? b) Determine lim Cp ( ). c) Would it be affordable to remove all of the pollutants? Eplain. d) In light of our response to part c), is the model realistic for p in the entire domain [, ]? 8. Application Determine lim and + lim a) b evaluating f( )= for values of that approach from the left and from the right b) b using a graphing calculator or graphing software 9. Communication According to Newton s law of gravitation, the force of gravit between an two masses (m and m ) is given b the equation Gmm F =, where G is a positive constant, and d d is the distance between the centres of the masses as shown. m p lim f( ) d lim g ( ) m 8 MHR Chapter 6
Gmm a) Find lim. d d b) Eplain wh the limit found in part a) is not reasonable for normal objects of non-zero size. c) Black holes are postulated to be ver massive objects of ver small size. Eplain wh the limit found in a) would appl in this case, if we assume Newton s equation holds for such etreme conditions.. How small must be so that? 5. Find lim.. Inquir/Problem Solving According to Einstein s special theor of relativit, the mass, m, of a particle with velocit v relative to an observer is m m = v c where m is the mass of the particle when it is at rest and c. 8 m/s, the speed of light. C a) What happens to the mass as v c? b) It is thought that a particle that is travelling with a speed less than the speed of light can approach the speed of light but not reach or surpass it. Eplain wh this is so according to the relativistic equation for mass. p ( ). Inquir/Problem Solving Let f( ) =, ( a) n where n is a positive integer and p() is a polnomial function. Assume that a is not a factor of p(). a) Under what conditions is lim f( ) a and under what conditions is lim f( )? a b) Repeat part a) for lim f( ). a +. For an real number a, a a. a) What is the meaning of the smbol? Does it have a numerical value? b) Use the two functions f( ) = and ( ) 6 9 g ( ) =, and the properties of limits to ( ) show that the smbol does not represent a real number. Historical Bite: Who Invented Limits? The roots of calculus can be traced to the calculations of areas and volumes b earl Greek scholars such as Archimedes (87 B.C.) and Eudous (8 55 B.C.). The used the method of ehaustion as a precursor of our modern notion of limit. Although the limit idea was implicit in their work, the never actuall formulated the concept. Sir Isaac Newton (6 77) was the first to recognize the importance of limits that approach nearer than b an difference. It was left to the French mathematician Augustin Cauch (789 857), man ears later, to make these ideas more precise. 6. Vertical Asmptotes MHR 9
6.5 Horizontal and Oblique Asmptotes In Section 6., ou investigated the behaviour of the inverse square law for the electrostatic force between two charges, A and B, as B approached the fied charge A. Now ou will investigate the force as B moves farther and farther awa from A, on either side of A. Investigate & Inquire: Force Between Two Charged Particles. Consider two charged particles, A, fied at the origin, and B, with position on the -ais. Construct a table of values for the force F ( ) between A and B for positive and negative values of with large absolute value. You can do this b hand, or using a graphing calculator, graphing software, or a spreadsheet.. What happens to the value of the force as charge B moves farther and farther awa from charge A to the right? to the left?. Use a graphing calculator to graph F for large positive values of.. Repeat step for negative values of with large absolute values. 5. Do ou think the limit of the force eists as approaches infinit? Eplain our reasoning. 6. Repeat step 5 for the limit as approaches negative infinit. 7. Repeat steps to 6 for the function g ( ). 8. Make a conjecture about what conditions must be true for the limit of a general function h(), as approaches positive or negative infinit, to eist. MODELLING M AT H For the function F ( ), note that, as and, the values of F() get closer and closer to. On a graph of F(), the points approach the -ais. We can make the points get as close to the -ais as we like b taking positive or negative values for with arbitraril large absolute values. Similarl, the values of g() get closer and closer to as and. On the graph, the points approach the line. When the points approach a horizontal line in this wa, the line is called a horizontal asmptote. A number L is a limit at infinit for a function f if the values of f () can be made as close to L as we like b taking positive or negative values of with sufficientl large absolute values. We write this relationship either as lim f( ) L or lim f( ) L, whichever is appropriate. Again, the smbol is not a number. The epression lim f( ) L is read as the limit of f () as approaches infinit is L or the limit of f () as increases without bound is L. The epression lim f( ) L is read as the limit of f () as approaches negative infinit is L or the limit of f() as decreases without bound is L. 5 MHR Chapter 6
The line L is called a horizontal asmptote of the curve f () if either lim f( ) L or lim f( ) L, or both. For instance, note that in the investigation, the line is a horizontal asmptote of the function F ( )= and the line is a horizontal asmptote of the + function g ( )=. + The diagrams show several of the man was for a curve to approach a horizontal asmptote. = f( ) = L = f( ) = L = L = f( ) lim f ( ) = L lim f ( ) = L lim f ( ) = L = f( ) = L = L = f( ) = f( ) lim f ( ) = L lim f ( ) = L, lim f ( ) = lim f = L ( ) = lim f ( ) = L Note that it is possible for the graph of a function to cross a horizontal asmptote, as can be seen in some of the figures. The line L is a horizontal asmptote if the function approaches arbitraril closel to L as approaches either positive or negative infinit (or both). The graph of the function could cross the asmptote once or man times without affecting this condition. Note that the second-last function has two horizontal asmptotes. Eample Basic Limits at Infinit Find lim and lim. Find the horizontal asmptote of. Solution When is large, is small. We can demonstrate this b calculating the reciprocal of successivel larger numbers.... 6.5 Horizontal and Oblique Asmptotes MHR 5
In fact, b taking large enough values of, we can make as close to as we like. Therefore, lim. If we choose negative values for with large absolute values, is a negative number with small absolute value. Again, can be made as close to as we like b choosing negative values for with sufficientl large absolute values. Therefore, lim. Thus, the line (the -ais) is a horizontal asmptote of the curve =. Using a similar argument leads to the following important rules for calculating limits. If r is a positive integer, then lim and lim. r r Eample Long-Term Behaviour of Average Cost A photocoping store charges a flat rate of $ plus $.5/cop. a) Write a function f () to represent the average cost per cop. b) Determine lim f( ). c) What is the significance of this limit for the customer? Solution a) The cost of copies is.5. +. 5 The average cost per cop is f( ) =. b). 5. 5 lim lim lim lim 5. lim 5. () 5. 5. c) The more copies the customer makes, the closer the average rate per cop comes to $.5. The flat rate of $ becomes less and less significant as more copies are made. Window variables: [, ], [,.5] In the following eample, we compute a more complicated limit. Eample Limit of a Rational Function Evaluate lim. 5 5 MHR Chapter 6
Solution Keep in mind that the smbol is not a number. We cannot substitute this smbol into the epression and evaluate it. However, we can evaluate limits of the form lim. Therefore, r to evaluate the limit at infinit of a rational function, we first divide both the numerator and the denominator b the highest power of that occurs in the denominator, so that all limits will be in this form. (We can assume that we are not dividing b zero because we are interested onl in values of that have large absolute values.) In this case, the highest power of is, so we proceed as follows. lim lim 5 5 lim 5 5 6 The graph of = and its horizontal asmptote are shown in the figure. 5 + Note that the graph crosses the horizontal asmptote at a small value of. 6 = = 5 + Infinite Limits at Infinit Some rational functions approach horizontal asmptotes at infinit, but others do not. The following investigation will eplore the behaviour of rational functions further. Investigate & Inquire: Limits at Infinit of Rational Functions. For each function, i) draw a graph of the function ii) construct tables of values for for negative and positive -values with large absolute value iii) determine lim f( ) and lim f( ) a) + 5 + 5 + + = b) = c) = d) = 5 + +. Compare the degree of the numerator to the degree of the denominator for each function in step. What general conclusions can ou draw from our results? 6.5 Horizontal and Oblique Asmptotes MHR 5
. Repeat steps and for the following functions. a) b) c) = + + = d) = = + + + + 5. Repeat steps and for the following functions. a) 5 b) c) = + + + + 7 = = d) = + + + When evaluating limits at infinit for rational functions, it is often helpful to use a shortcut to determine if the limit is or a particular non-zero number such as, or if there is no horizontal asmptote at all. To do this, we compare the degree of the numerator to the degree of the denominator, as in the preceding investigation. For eample, note that in step of the investigation, the degree of the numerator is less than the degree of the denominator for each function. For this tpe of function, the limits at infinit are zero. For eample, lim and lim. Window variables: [9., 9.], [.,.] This makes sense because the absolute value of the denominator will increase at a far greater rate than that of the numerator for values of with arbitraril large absolute values, and so the limit is. Then, note that in step of the investigation, the degree of the numerator is equal to the degree of the denominator for each function. In each case, the limit at infinit is equal to the leading coefficient of the numerator divided b the leading coefficient of the denominator. For eample, in part b), lim. Note how quickl the graph approaches the asmptote in this eample. Window variables: [9., 9.], [, ] The terms that determine the degrees of the numerator and denominator, and, will dominate the behaviour of the function as the absolute value of becomes arbitraril large; all other terms will be small in comparison. Note that in step of the investigation, the degree of the numerator is greater than the degree of the denominator. There is no horizontal asmptote in this case. The numerator grows faster than the denominator and so the absolute value of the function increases 5 without bound. For eample, in part d), = has no horizontal asmptote. + 5 MHR Chapter 6
This is because the numerator will be dominated b 5 as the absolute value of gets arbitraril large, and the denominator will behave like, whose value increases much more slowl than 5 5 5. We write lim and lim to epress this. Infinite limits at infinit: The notation lim f( ) means that the values of f () increase without bound as increases without bound. There are similar meanings for the following epressions. lim f( ) lim f( ) lim f( ) Window variables: [.7,.7], [.,.] or use the ZDecimal instruction. = f( ) = f( ) = f( ) = f( ) lim f ( ) = lim f ( ) = All non-constant polnomial functions behave in this manner. For eample, we can see from the graphs of and that lim lim lim lim lim f ( ) = lim f ( ) = Window variables: [.7,.7], [, ] Window variables: [9., 9.], [75, 75] In the previous eamples, we studied vertical and horizontal asmptotes. However, there are man curves that approach asmptotes that are not horizontal or vertical as the absolute 6.5 Horizontal and Oblique Asmptotes MHR 55
value of becomes large. In step, part a) of the investigation, note that lim and lim. It also appeared that the graph was straightening out as the absolute value of became ver large. In fact, it appeared that the curve was getting closer and closer to the line. Such a line is called a linear oblique (or slant) asmptote. Window variables: [9., 9.], [9., 6.] The line m b is a linear oblique asmptote for a curve f () if the vertical distance between the curve and the line approaches as the absolute value of gets large for either positive or negative values of. We write this as lim[ f( ) ( mb)] or lim [ f( ) ( mb)] For rational functions, linear oblique asmptotes occur when the degree of the numerator is eactl one more than the degree of the denominator. The equation of the linear oblique asmptote can be found b dividing the numerator b the denominator, as in Eample. Eample Determining Linear Oblique Asmptotes Find an equation of the linear oblique asmptote for each curve. + a) = b) Solution a) Note that the degree of the numerator is more than the degree of the denominator. We can divide to obtain + = = + 7 = If the absolute value of is ver large, the value of approaches and becomes insignificant in comparison to the other two terms. Thus, the curve approaches the line. This oblique asmptote (in blue) can be seen on the graph. Note that we would also epect to find a vertical asmptote at. = + 8 8 8 = 8 56 MHR Chapter 6
b) We begin with long division. + + ) 7 + 6 8 Thus, 7 = 8 = + As in part a), we conclude that the oblique asmptote is. The graph shows the curve and the oblique asmptote added in blue. Again, we would epect to see a vertical asmptote at, resulting from setting the denominator equal to. 8 6 8 8 6 = 7 8 = + Eample 5 Determining the Asmptotes to a Curve Find all asmptotes to the curve =, and use this information to sketch the curve. Solution First, we ensure that the numerator and the denominator have no common factors. = ( + )( ) = Since there are no common factors, we set the denominator equal to to obtain the vertical asmptote. The degree of the numerator is more than the degree of the denominator, so the graph of the function has a linear oblique asmptote. Long division gives ) Thus, = = 6.5 Horizontal and Oblique Asmptotes MHR 57
Therefore, the equation of the linear oblique asmptote is. To help us sketch the curve, we find the intercepts. The -intercept is f (). The -intercepts occur when the numerator equals. ( )( ) or Using the above information, we sketch the curve, with the asmptotes shown as dotted lines. 8 = 6 8 6 = = 6 8 6 8 Ke Concepts Limits at infinit a) The line L is called a horizontal asmptote of the curve f () if lim f( ) L, lim f( ) L, or both. b) If r is a positive integer, then lim and lim. r r c) Infinite limits at infinit can be epressed as lim f( ) lim f( ) lim f( ) lim f( ) The line m b is a linear oblique asmptote for the curve f () if the vertical distance between the curve and the line approaches as the absolute value of gets large. A summar of the possible end behaviour of a rational function: a) if degree of numerator degree of denominator, the graph has a horizontal asmptote b) if degree of numerator degree of denominator, the graph has a horizontal asmptote other than c) if degree of numerator degree of denominator, the graph has a linear oblique asmptote d) if the degree of the numerator eceeds the degree of the denominator b more than, the graph will have neither a horizontal asmptote nor a linear oblique asmptote 58 MHR Chapter 6
Communicate Your Understanding. a) What does lim f( ) L mean? b) Eplain the difference between lim f( ) L and lim f( ) L.. How can ou convince someone who has not studied limits at infinit that lim if r is a positive integer? r. If f () is a rational function, state the steps necessar to evaluate lim f( ).. a) Under what conditions does a rational function have a linear oblique asmptote? b) Eplain how to find the linear oblique asmptote of a rational function that satisfies the conditions in part a). 5. Linear oblique asmptotes to a rational function occur when the degree of the denominator is less than the degree of the numerator. Eplain wh this is so. 6. If a rational function has a linear oblique asmptote, can it also have a horizontal asmptote? Eplain. A Practise. For each of the following, state the equations of the horizontal and vertical asmptotes, if the eist. If the do not eist, eplain wh. a) b) 6. Determine whether each curve has a linear oblique asmptote. If it does, state its equation. a) f( )= + + b) = 8 c) h ( )= + + d) g ( )= 5 e) + + g ( )= f ) = + + B. Find each limit. a) lim b) c) lim 6 d) e) 5 lim 5 f ) g) lim h) i). 7 lim 75 Find each limit. a) 59 lim 7 b) lim c) lim ( )( 5 ) d) 5 lim 6 e) 6 5 lim f ) lim lim 6 lim lim lim 7 8 5 5 6 6.5 Horizontal and Oblique Asmptotes MHR 59
Appl, Solve, Communicate 5. Find the equation of the horizontal asmptote of each curve. a) f( )= b) g ( )= 5 8 + c) = d) = + 5 6 + + e) h ( )= f ) = 9 5 6. Find an equation of the oblique asmptote of each curve. a) + 5 = b) h ( )= c) + + d) 6 = = + + 5 + + e) f( )= + f) g ( )= 7. Find all horizontal and vertical asmptotes. Use them, together with the intercepts, to sketch the graph. a) g ( )= b) = + 6r + 7 c) = d) vr ()= + r 7 9t 6 e) + = f ) gt ()= t 8. Find the linear oblique asmptote of each curve and use it to help ou sketch the graph. Use a graphing calculator or graphing software to check our result. a) = + + b) g ( )= + + + + c) = d) = t t + 5+ e) st ()= f ) = t 9. Application A piece of machiner depreciates in value, V, in dollars, over time, t, in months. The value is given b t Vt () = 5 ( t + ) a) Find the value of the machiner after i) month ii) 6 months iii) ear iv) ears b) Would ou epect to find a local maimum or minimum value in the interval [, )? Eplain. c) Find lim Vt ( ). t d) Will the machiner ever have a value of $? e) In light of our result in part d), does V(t) model the value of the machiner for all time?. Inquir/Problem Solving A telecommunications compan s sales for the last ears can be n + n+ modelled b the function Sn ( )=, 7n + where S(n) represents annual sales, in millions of dollars, and n represents the number of ears since the compan s founding. a) Find lim Sn ( ). Interpret this result. n b) If the long-term average rate of inflation is.%, what is the true rate of growth of the compan?. Application A compan that installs carpet charges $6 for an area less than or equal to m and an additional $/m for an area over m. a) Find a piecewise function c() to represent the average cost, per square metre, to install square metres of carpet. b) Find lim c ( ). Eplain what this limit means. c) Graph c() for values of. d) Would ou call this compan to carpet a ver small area? Eplain.. Application A new emploee at a computer store suggests to the manager that the are not giving good enough incentives for customers who place large orders. Their current pricing is a flat rate of $ for deliver (no matter how man computers are ordered) and $ per computer. The emploee suggests a new + billing formula, C ( )=, where C() represents the average cost per computer for computers. 6 MHR Chapter 6
C a) Write a formula for A(), the average cost per computer, using the compan s current pricing. b) Graph A() and C() on the same set of aes for the domain [, ]. Which pricing formula is better for the customer who orders a large number of computers? Eplain. c) Find lim A ( ) and eplain its meaning. d) Find lim C ( ). Is this limit meaningful? Eplain. e) In light of our response to part d), should the computer compan use a different formula for orders of more than computers? Eplain.. Find the horizontal asmptotes of =. +. Find each limit. 5 a) lim b) lim( 5 ) 5. Application Two curves are said to be asmptotic if the vertical distance between the two approaches zero in some limit. For a rational function, if the degree of the numerator eceeds the degree of the denominator b more than, then the graph of the function is asmptotic to a polnomial of degree more than. a) Let f( ). Show that lim [f () ]. To which curve is the graph of f asmptotic? b) Sketch the graph of f. On the same aes, graph the polnomial function to which f is asmptotic. c) The polnomial to which a rational function is asmptotic can be determined in the same wa as linear oblique asmptotes, b dividing the numerator b the denominator and then taking an appropriate limit. Determine the polnomial function to which each rational function is asmptotic. Then, sketch the rational function and the asmptotic curve. 5 i) g ( ) ii) h ( ) d) Show that for a rational function, if the degree of the numerator eceeds the degree of the denominator b k, then the function is asmptotic to a polnomial of degree k. State an restrictions on the value of k. 6. Find lim using a) a graphing calculator or graphing software to obtain a result correct to two decimal places b) an method to obtain an answer correct to four decimal places Historical Bite: Mathematics Before Algebra In the middle ages, algebraic notation was not et invented. As an eample, Luca Pacioli (5 557) wrote in 9 on how to solve a quadratic equation: If a thing and its square equal a number, ou must square half of the thing and add it to the number. Then subtract half of the thing from the root of the total. You will have left the root plus the square. It is amazing that an problems were solved when all of the techniques had to be recorded like this. 6.5 Horizontal and Oblique Asmptotes MHR 6
6.6 Curve Sketching As modelling becomes more sophisticated in our technologicall advanced societ, more and more complicated functions are being used. Graphing technolog is usuall able to provide a good picture of the features of a graph. However, there are man occasions when a graph so produced is deceptive. For eample, the function f () 5 5 appears as shown when it is graphed in the standard window [, ], [, ]. This function actuall has a local maimum point in addition to the evident local minimum at (, ). Our knowledge of calculus will lead us to this additional information. The analtic tools that we have developed so far in this chapter complement the tools that technolog provides. Depending on the situation, it ma be better to use one or the other of technolog and analtic tools eclusivel, or else a combination of the two, to determine the important features of a function. To begin, we focus on the analtic tools. Our general strateg for sketching curves is to frame the curve (domain restrictions, asmptotes, general shape) find important points (intercepts, etrema, points of inflection) add details (smmetr, intervals of increase/decrease, concavit) sketch the curve Our goal will be to use as few tools as necessar to develop a sketch showing the important features of the function. The order and completeness of steps taken ma var with the nature of the function and the tools used. Eample Using Analtic Tools to Sketch a Graph Sketch the curve 6 9. Solution Although technolog can readil produce this graph, we use the initial eamples to illustrate the tools of calculus. Frame the curve This is a cubic polnomial whose domain is (, ). The general shape of this curve is one of two possibilities as shown. A polnomial has no asmptotes, but it is still useful to note that lim( 6 9) and lim ( 6 9). This eliminates the second possibilit for the shape of the curve. Find important points The -intercept is. (Let.) The -intercepts occur when. 6 9 ( 6 9) ( ) or 6 MHR Chapter 6
The -intercepts are and. Note that, since the factor ( ) is squared, there is a double root at, and the -values do not change sign at this point. Combining all we know so far, we get the approimate sketch at right. Net, we determine the coordinates of the local maimum and local minimum. 9 ( ) ( )( ) 6 6( ) To determine the critical numbers, set. The critical numbers are and. Just b looking at the sketches we made, we can tell that a local minimum occurs at (, ) and a local maimum occurs at (, ). We could use the second derivative test to verif this. To determine the point of inflection, note that changes sign at. Thus, (, ) is a point of inflection. Sketch the curve The graph shows the completed sketch. 6 (, ) (, ) = 6 + 9 6 8 (, ) The graph shows the completed sketch. Eample Using Analtic Tools to Sketch a Graph Sketch the graph of the function f( )= using analtic tools. Solution Frame the curve In factored form, the function is written as f( ) =. ( )( + ) Since the numerator and the denominator have no common factors, we set the denominator equal to zero to find the vertical asmptotes. ( )( ) or Since the numerator is alwas positive, the function has the same sign as the denominator. Thus, f () for (, ) and f () for (, ) and (, ). Therefore, lim lim lim lim 6.6 Curve Sketching MHR 6
Since the degree of the numerator is equal to the degree of the denominator, there is a horizontal asmptote other than. We find the horizontal asmptote b dividing each term b. lim lim The horizontal asmptote is. Find important points The -intercept is f (). The origin is also the onl -intercept. There is almost enough information to sketch the curve. We need to check for the presence of etrema and points of inflection. We need the first and second derivatives. f ( )( ) ( ) ( ) ( ) ( ) f ( ) ( ) ( ) ( )( ) ( ) ( ) 6 ( ) The critical numbers are (when ) and (when f is undefined). We alread know how f behaves at the asmptotes. To determine whether the function has a maimum or a minimum value at, note that f (), so the point (, ) is a local minimum. To determine points of inflection, we need to find where f () changes sign. If we eamine the epression for f (), we see that the onl places it can change sign are at and. Since the function is not defined at these points, there are no points of inflection. Add details Since f () f (), this is an even function. Thus, it is smmetric about the -ais. It is not necessar to determine intervals of increase/decrease or concavit when the other information we have collected is considered. Sketch the curve f( ) = 6 MHR Chapter 6
Eample Using Analsis and Technolog to Sketch a Graph + Analse the function f( )= and sketch its curve. + Solution It ma appear that this function can be easil analsed using calculus techniques. However, we shall see that the details are sufficientl comple that technolog is a good tool here. Frame the curve This is a rational function, so we set the denominator equal to zero to find the asmptotes. ( ) Therefore, and are vertical asmptotes. Since the degree of the numerator is one more than the degree of the denominator, we epect to find a linear oblique asmptote. Use long division to rewrite the function. ) = = 9 6 6 = 9 9 6 9 + Thus, f( )= +. + In the second term, the absolute value of the denominator grows much more quickl than the numerator as approaches positive or negative infinit. Thus, the linear oblique asmptote is. Add details We check the behaviour of the curve near the asmptotes using Ask mode on the TABLE SETUP screen. lim lim lim lim To determine where the graph of f is in relation to its linear oblique asmptote, test a large positive value and a large negative value. f ( ) ( ) 97. ( ) f ( ) ( ) ( ). 6.6 Curve Sketching MHR 65
At, is equal to 97. Thus, the function is above the asmptote. At, is equal to. Thus, the function is below the asmptote. Find important points There is no -intercept since is not in the domain of the function. To find the -intercepts, determine when f (). + = + Use the factor theorem. One factor is. Find the other factor using long division. + + ) + + + + + + + 9 6 Web Connection For more technological approaches to graph sketching, go to www.mcgrawhill.ca/links/caf and follow the link. 6 6 9 6 Since is not factorable over the real numbers, the onl -intercept is. To determine the critical numbers, find the first derivative. ( ) ( )( ) f ( ) ( ) 9 ( ) ( ) 6 ( ) After some algebra, f () turns out as follows. 6 ( ) f ( ) ( ) To find the critical numbers, we set f (). Thus, 6. The critical numbers can be found with the aid of technolog. 66 MHR Chapter 6
The critical numbers are approimatel 5.958 and.885. From our sketches so far, we can see that a local maimum occurs at the first critical number, and a local minimum occurs at the second critical number. The function values at these points are approimatel.9 and.9, respectivel. Thus, the local maimum occurs at approimatel (5.958,.9), and the local minimum occurs at approimatel (.885,.9). We can again use technolog to determine when the second derivative changes sign. The second derivative changes sign at the asmptotes, but these cannot be points of inflection because the function is not defined there. To determine where else the second derivative changes sign, set f (). 6 ( + + + ) = ( + ) Use the Zero operation to find the zeros. Window variables: [, ], [, ] = + + Window variables: [, ], [, ] or use the ZStandard instruction. 6 There is one point of inflection. Its -value is approimatel.757. The coordinates of the point of inflection are approimatel (.757,.). = = Sketch the curve It appears that this function would have been better handled b technolog alone. If we use the ZDecimal instruction to graph it in the friendl window, [.7,.7], [.,.], most relevant details appear. However, we would still have to calculate the coordinates for the point of inflection and the local etrema. 9 6 6 6 9 = 6.6 Curve Sketching MHR 67
Eample Using Technolog and Analsis to Sketch a Graph ( + ) Graph the function f( ) =. ( ) ( ) Solution We begin with the graphing calculator in the standard window, [, ], [, ]. This is a good start but it appears that we will have to zoom out to get the overall picture and also zoom in to get detail about critical points. To guide our zooming, we eamine the formula for f. We epect vertical asmptotes at and, because that is where the denominator is equal to. From the graph, we ( can see that and lim ( ) lim ). But what happens ( ) ( ) ( ) ( ) between and? Use the ke to find a couple of -values in this region. Clearl, the values of f are ver large in this region. We look at the graph in a region with ver large -values. The almost vertical line is a result of the graphing calculator tring to connect points across the asmptote, so we ignore it. It appears that there is a local minimum in this area. We can also determine the behaviour of the function near the asmptotes in this region: ( ) and lim ( lim ) ( ) ( ) ( ) ( ) Window variables: [,.7], [, 5] Since the degree of the denominator is greater than the degree of the numerator, the -ais is a horizontal asmptote. 68 MHR Chapter 6
Eamining the numerator, we see that and are -intercepts. Since the first factor is squared, we know that the function does not change sign at this point. The graph does cross the -ais at since the second factor, ( ), is cubed. This suggests that our curve should look roughl as shown (not to scale). ( + ) = ( ) ( ) = = Zooming b trial and error using the Zoom In and Zoom Out instructions produces the following screens that show the important details. [, ], [, ], [., ], [.5,.5] [.,.] [, ] If we need more precise information about the maimum or minimum points, or points of inflection, graphing technolog can provide approimations. The calculation of f and f b hand would be a tedious task; a computer algebra sstem would make it eas. The onl wa to show all of this information in one image is to draw a sketch b hand, such as our rough sketch above, but with significant details added. Ke Concepts Curve sketching can be performed with analtic calculus tools, with technolog, or with a combination of both. Tools should be selected carefull, but ever step should be justifiable with the tools of calculus. The tools of technolog and calculus complement each other when sketching curves and investigating important aspects of functions. A graph can be sketched analticall using the following steps: a) frame the curve b) find important points c) add details d) sketch the curve Technolog can help ou graph functions that are too complicated to graph using calculus alone. Calculus can help ou spot important features of a graph that are not initiall seen using graphing technolog. 6.6 Curve Sketching MHR 69
Communicate Your Understanding. If ou have a graphing calculator, eplain wh ou still need calculus to graph functions.. Answer each question and eplain our reasoning. a) The domain of a function is (, ). Can the function have a vertical asmptote? What about a horizontal or oblique asmptote? b) A function has a vertical asmptote. Can the domain of the function be (, )? c) A function is even. Can the function have onl one local maimum or minimum value? What if the function is odd? d) If f (a), and f (a) is a local maimum or a minimum value, can it also be a point of inflection? e) If f (a) is not defined, and f (a) is a local maimum or a minimum value, can it also be a point of inflection? f) Can a function have a maimum, a minimum, or a point of inflection at a, if a is not in the domain of f?. The sketch of a graph is given. Discuss the significance of the parts of the graph indicated. P Q R S T a b c d e f a) points P, Q, R, S, T b) intervals (, a), (a, b), (b, c), (c, d), (d, e), (e, f ), (f, ) Practise A. Match the graphs opposite with the functions below, using our knowledge of limits and asmptotes. a) f( )= b) g ( )= c) = d) f( ) = ( ) e) = f ) g) f( )= h) i) g ( )= j) q ( )= = ( ) k ( )= k) = l) f( )= ( ) 7 MHR Chapter 6
i) ii) iii) 6 iv) v) vi) vii) viii) i) 6 ) i) ii) 8 6 8 8 6 6 8 8 8 8 6.6 Curve Sketching MHR 7
B. Jessica was sketching the graph of a function but did not have time to finish it. She lost some of her notes, including the formula for the function, but still has the information below. Use the information to sketch the graph. The domain of the function is the set of all real numbers ecept and. lim f( ) lim f( ) lim f( ), f( ) as lim f( ), f( ) as The -intercept is. There are no -intercepts. Appl, Solve, Communicate. Sketch each function. Find the eact coordinates of all maimum and minimum points, and points of inflection. a) b) 5 6 c) g() ( ) d) f () 5 5 e) f) h() 5 9. Inquir/Problem Solving Sketch each function without using technolog. Following the format of the eamples, show enough steps to justif our sketch. a) = b) ft ()= + + t c) = d) = + e) = f ) + g) = h) ( ) i) lim f( ) lim f( ) = g ( )= h ( )= 5. Sketch each curve using technolog and analtic calculus methods where appropriate. For each curve, determine the equations of the asmptotes and the eact coordinates of an local etrema. a) f( )= b) f( )= + + c) = + + d) + = t e) = f ) ht ()= t g) = ( ) h) f( )= + 6. Application In order to fit properl into an architectural design, a pipe needs to be manufactured with the equation, for [, ], defining its shape. All measurements are in centimetres. Sketch a graph of the pipe s shape. 7. Application A clindrical drum is to be made to have a capacit of L. The resulting formula for the height is h, and r for the surface area of the sheet metal is 8 S r, where r is the radius in r centimetres. a) Sketch a graph of the relation between height and radius. b) Sketch a graph of the relation between surface area and radius. 8. Communication According to Coulomb s law, the force between two charged particles is directl proportional to the product of the charges and inversel proportional to the square of the distance between them. The figure shows three charges, one positive charge placed at the origin, another equal positive charge placed at, and a negative charge that can move along the -ais. The positions of the positive charges are constant. From Coulomb s law, the magnitude of the force on the negative charge is given b the equation F( ) = ( ) MODELLING M AT H 7 MHR Chapter 6
(provided the units are chosen appropriatel), where is the position of the negative charge, and. 5 a) Sketch the graph of the force function. b) Where is the force undefined? Eplain, phsicall, wh this is so. c) Where is the force? Eplain in the contet of phsics. 9. Sketch each function. a) b), f( ),, f( ),. Application i) Use technolog to obtain the graph of each function. ii) From the graph, give a rough estimate of the intervals of concavit and the location of an points of inflection. iii) Use the graph of g to give better estimates for these values. a) g() 5 b) g() 5 5 5. Application i) Use technolog to graph each function. ii) From the graph, estimate the local maimum and minimum values. iii) Find the eact values of the etrema in part a). a) h ( ) = ( ) 6 9 b) h ( )= + +. Communication Sketch a graph b hand for each function, using intercepts and asmptotes, but not derivatives. Make our sketch as accurate as possible. C a) b) c) d) ( + 5)( ) = ( ) 5 ( ) h ( ) = ( ) ( + ) ( ) = ( ) ( + ) ( ) f( ) = ( )( + ). Inquir/Problem Solving Consider the famil of ( ) ( + + k) curves defined b =. a) Using technolog, sketch the curve for five different values of k. Do the graphs have an similar features? Describe them. b) Show that is a linear oblique asmptote for ever value of k. c) For what values of k do the curves intersect the asmptote? d) Which values of k lead to two -intercepts?. Consider the function f, where f and f are both differentiable, f () for all, f () for, and f () for. Let g() f ( ). a) For which values of does g have a critical number? b) Discuss the concavit of g. 5. Find a cubic function f () a b c d that has a local maimum at (, ) and a local minimum at (, ). 6. a) Show that a cubic function has eactl one point of inflection. b) If the graph has -intercepts p, q, and r, show that the -coordinate of the point of p+ q+ r inflection is. 7. a) For which values of p does the quartic polnomial function Q() p have each number of points of inflection? i) eactl two ii) eactl one iii) zero b) Illustrate our results in part a) b sketching Q for several values of p. 6.6 Curve Sketching MHR 7
8. Consider the general cubic function, f( ) a b cd, where a. b a) Show that f( ) when =. a b) Show that the graph of f has an inflection point, and determine its coordinates. c) Determine a translation of the graph of f that brings its inflection point to the origin. One wa to do this is to introduce new coordinates X and Y, and to determine equations that relate the new coordinates to the old ones, such that, when epressed in the new coordinates, the cubic curve has its inflection point at the origin. d) The cubic curve in part c), which has its inflection point at the origin, defines a new function, Y F( X). Determine a formula for F. e) Determine the smmetr properties of the function of part d). f) Using the results of parts a) to e), make a general conclusion about the smmetr properties of all cubic functions. 9. Consider question 8. Pose and solve a related problem for quadratic functions, which culminates in a general conclusion about the smmetr properties of all quadratic functions.. a) Is it possible to generalize the analsis of questions 8 and 9 to general quartic functions? Before tackling this general problem, begin b graphing a number of specific quartic functions. Do the all share the same smmetr properties? If so what is the propert? If not, what other conclusions can ou make? b) If it is possible to state a general smmetr propert of all quartic functions, make such a conjecture and adapt the ideas of questions 8 and 9 to prove the conjecture. c) If it is not possible to make a general statement about the smmetr properties of all quartic functions, is it possible to make a statement about a large class of quartic functions? If so, determine the largest class of quartic functions that shares a nice smmetr propert. If not, eplain wh. d) If possible, etend our analsis to polnomials of higher degree. Achievement Check Knowledge/Understanding Thinking/Inquir/Problem Solving Communication Application a) Sketch a graph of a continuous function that satisfies all of the following conditions. f () for (, ) f () for (, ) f () for (, ) f () for (, ) lim f( ) f is an odd function b) Is our sketch the onl solution? Eplain. c) If there is more than one solution, what are some of the necessar features? If our solution is the onl one, which features cannot occur in an other function? 7 MHR Chapter 6
Introducing Optimization Problems 6.7 Packaging products in order to minimize the amount of materials used is a primar concern in operating a business. Furthermore, reducing packaging is an environmental concern for societ as a whole. The problem of optimal packaging can be simplified to finding the optimal dimensions for geometric figures when given certain constraints. In this section, we will put our knowledge of finding maimum and minimum values of functions to practical use. Investigate & Inquire: Optimal Dimensions of a Cereal Bo Consider a cereal bo in the shape of a rectangular prism. Assume the bo must have a capacit of 5 cm, and the thickness of the bo must be between 5 cm and cm to allow for a comfortable grasp b most people. Use a graphing calculator to determine the dimensions of the bo that require the minimum amount of materials for various fied thicknesses. Ignore an overlap in order to join the faces of the bo. There are a number of was to proceed. One strateg is as follows.. Write an equation for the surface area, A, in square centimetres, of the cereal bo, and another equation for the capacit, V, in cubic centimetres, enclosed b the bo. Given that V 5, solve the equation for V for one of the dimensions (not t) in terms of the other two. Then substitute this epression into the formula for A. The result should be an epression for A in terms of two variables, t and either w or h, depending on which ou chose to solve for. V = hwt and A = hw + ht + wt. Now choose a value, for eample t 5, for the thickness of the bo. Substitute this value into the formula for A obtained in step.. Enter the formula obtained in step into the function editor of a graphing calculator or graphing software. Using the TABLE SETUP screen or the ke, estimate the minimum value of the surface area.. Repeat steps and for several other values of t. Record our results for the minimum area in a table. Do ou notice an patterns? 5. Which value of t appears to lead to the overall minimum surface area? What does the minimum surface area appear to be? 6. What are the approimate dimensions of the bo that result in a minimum surface area and are also consistent with the stated restrictions? 7. Describe, and suggest reasons for, the patterns ou observed in the minimum surface area as ou changed the value of t. h w t 6.7 Introducing Optimization Problems MHR 75
You ma have noticed a pattern in the results of our investigation: it seems that the greater the thickness of the bo (at least for thicknesses between 5 cm and cm), the smaller the minimum surface area of the bo. This suggests that the minimum surface area of the bo, consistent with the restrictions, occurs for a thickness of cm. Using this fact, the dimensions of the optimal bo can be found using methods developed in this chapter. This is done in Eample. Eample Optimal Dimensions of a Cereal Bo A cereal bo in the shape of a rectangular prism is required to have a capacit of 5 cm, and the thickness of the bo must be cm to allow for a comfortable grasp b most people. What dimensions of the bo require the minimum amount of materials? Ignore an overlap needed to join the faces of the bo. Solution A good strateg is to sketch the situation, identifing constant given data, and introducing appropriate variables. Also, we record an relationships among constants and variables as equations. Variables: height h, width w, surface area A Constants: thickness cm, capacit 5 cm h w cm First, we write an equation for the quantit to be optimized, the surface area A. A area of front and back area of top and bottom area of left and right sides (hw) (w ) (h ) hw w h To differentiate A, we must first epress the equation for A in terms of one variable b relating h and w using given information. Since the capacit is 5, 5 h w w = 5 h Substituting this epression for w into the equation for A results in 5 A = h h h + 5 h + 5 = + + h 5 h To minimize the function A, we find the critical numbers of A. 5 A h Set the derivative equal to zero to obtain h 5 h =± 5 Since h is the height of a real object, h, we reject the negative result. Thus, h = 5. 76 MHR Chapter 6
To determine whether this value of h gives a maimum or a minimum value of A, we use the second derivative test. A h Since A for h, h = 5 gives a minimum value of A. To find the corresponding width of the bo, substitute the value of h into the equation for w in terms of h, to obtain A 5 w = 5 h 5 = A = 5 + 5 ( 5 h + h = 5 5., so the minimum surface area of the cereal bo is obtained for dimensions cm,. cm, and. cm. The graph of the function A verifies that a width of w = 5 gives a minimum, not a maimum. Eample Optimal Radius of a Can Determine the most economical shape for a can of capacit 55 cm. Solution The capacit, V, in cubic centimetres, of a clindrical can is given b V r h, where the radius, r, and the height, h, are measured in centimetres. The surface area, S, in square centimetres, is given b S rh r. We want to find the minimum value of the surface area. Since the capacit is 55 cm, we will epress the height in terms of the radius, and substitute this back into the formula for the surface area. V r h 55 r h h = 55 r S = rh+ r = r 55 r r + 7 = + r r To find the critical points of S(r), we determine when S(r). 7 S( r) r r 7 r r r 55 r 8. 5 h 6.7 Introducing Optimization Problems MHR 77 r ( h
Determine the value of S(r) on either side of the critical number.8. Test r and r. 7 S() (). 7 S( ) ( ) 589. So, S(r) is decreasing to the left of the critical number.8, and increasing to the right. Thus, S(r) is a minimum at r.8. Determine the height at this radius. 55 h = r 55 = ( 8. ) 766. Thus, the most economical clindrical can has a radius of approimatel.8 cm and a height of approimatel 7.66 cm. Note that the height is approimatel twice the radius. If we had not rounded the values, this relationship would be eact. Eample Oil Spill Containment The oil spill that is generall regarded as having caused the most serious ecological damage in the world occurred when the oil tanker Eon Valdez ran aground in 989. Over L of crude oil contaminated the sea near Alaska. To contain oil spills, rectangular booms that have a cross-link to provide stabilit are used. The cross-link joins the long sides and is parallel to the short sides. a) What is the minimum total length of boom required to enclose an oil spill covering m of water? b) Is the result of part a) different if the oil boom can onl be constructed from -m sections? Solution Identif the variables and constants in the problem and sketch a diagram. Variables: width of structure w, length of structure, total w w boom length L Constants: surface area A m Relationships: A w, L w The equation to be optimized for the total boom length L is L w. We can epress L in terms of one variable b first relating and w using the given information that A. w = = w w 78 MHR Chapter 6
We substitute this epression for into the equation for L. L = w w + = + w w Then, we determine the critical numbers. L w w w w But w, so w = 58. We use the second derivative test to determine whether this value of w gives a maimum or a minimum. L w Since L for w, this value of w gives a minimum value of L. To determine the corresponding length of the structure, we substitute the value for w into the epression for. = w 87. The minimum total boom length is L w or 59. m. The graph of L versus w verifies this result. L 5 b) For -m sections, the function modelling the length of the boom has domain {,,, }. Thus, the function consists of discrete points onl. However, the new value of w that produces the minimum length will occur near the w-value found in part a). Thus, we eamine w 5 and w 6. L(5) 55 and L(6) 59. Since the boom is constructed from -m sections, L must be a multiple of. Thus, we reject the second value of w. So, the minimum length of boom is 55 m, for which the width is 5 m and the length is m. L = w + w 5 w 6.7 Introducing Optimization Problems MHR 79
Eample Optimal Dimensions for a Silo Canada and the United States are the world s leading producers of corn, with an approimate combined annual production of million tonnes. Corn silos are usuall in the shape of a clinder surmounted b a hemisphere. If the average ield on a given farm requires that the silo contain m of corn, what dimensions of the silo would use the minimum amount of materials? Solution Variables: radius of silo, r, in metres, height of silo, h, in metres Constants: capacit of silo, V m Relationships: A area of base area of sides area of hemispherical top, V capacit of clinder capacit of hemisphere We want to optimize (minimize) the surface area, A. r A= r + rh+ (surface area of a sphere = r ) h = r + rh We can relate r and h using the given information that V. V capacit of clinder capacit of hemisphere = + = rh r capacit of a sphere r = rh+ r r h = r Substituting this epression for h in the equation for A, we have A = r + r = r + r r 5 = r + r Differentiating, we obtain A r r r r Setting A equal to zero and solving for r to find the critical numbers, we get r = r r 6 = 6 r = 5. 7588 8 MHR Chapter 6 r
Thus, the critical number is r 5. 7588. We determine the second derivative to decide whether this value of r gives a maimum or a minimum. A r Since A for r, this value of r gives a minimum value of A. To determine the corresponding height, note that, since r h = r = r 8 = r 6 / = r r = r = r r 6 = A 5, r. Thus, 8 Thus, the height of the clinder and the radius of the hemispherical top are equal for minimum surface area. The optimal dimensions of the silo have both the radius and the height equal to approimatel 5.7588 m. A = 5 r + r r 6 Ke Concepts Procedure for solving optimization problems Identif the variables and the constants in the problem and sketch a well-labelled diagram. Epress relationships among variables and constants as equations. Construct an equation for the quantit, sa Q, to be optimized. Epress the equation for Q in terms of one variable onl, b using the equations relating variables and constants. Find the critical numbers and test them. Determine the required minimum or maimum value. Check that the result satisfies an restrictions on the variables. Communicate Your Understanding. The steps required to maimize a quantit are identical to the steps required to minimize a quantit. Eplain wh. 6.7 Introducing Optimization Problems MHR 8
. What is meant b the phrase epress the quantit to be optimized in terms of one variable? If a quantit Q is originall epressed in three variables, how man equations relating given information are required to epress it in terms of one variable?. Eplain the difference between a constant and a variable dimension of a geometric figure.. Compare Eample (page 76) and its solution to actual cereal boes. Do ou think that actual cereal boes have dimensions that minimize their surface areas? If not, what other factors do ou think the manufacturers consider besides minimizing packaging costs? 5. In Eample (page 77), the most economical shape for a can is one for which the height is equal to twice the radius of the base. Few cans in the retail world have this shape. Suggest some reasons wh this is the case. B Practise. Suppose that R m n and mn. Find the value of m that minimizes R.. If K pq and p q 5, find the value of p that makes K a minimum.. If W g h and g gh 7, find the value of h that maimizes W, if g and h. Appl, Solve, Communicate. A rectangular backard plapen for a child is to be enclosed with 6 m of fleible fencing. What dimensions of the rectangle will provide the maimum area for the child to pla? 5. A rectangular corral is to be enclosed along the side of a horse barn with the barn serving as one side of the corral. What dimensions of the corral, using m of fencing, will enclose the maimum area for the horses? 6. A rectangular garden plot requires an area of m for the variet of vegetables that are to be planted. What dimensions of the plot will use the least amount of fencing to enclose the garden? 7. Communication A count fair has a holding area for the prize sheep that are entered in a contest. a) The holding area is made up of identical pens arranged in a three b four grid. If m of fencing is available, what dimensions of each pen will maimize the total holding area? b) If the pens were arranged in a two b si grid, what dimensions would maimize the holding area? c) Which arrangement would ou recommend, the three b four grid or the two b si grid? Eplain our reasoning. 8. Communication A child s pla tunnel is to be made from a -m wide sheet of cardboard. The sheet will be folded as shown. m a) Where should the fold be made in order to maimize the cross-sectional area of the tunnel? 8 MHR Chapter 6
b) Will the resulting tunnel be high and wide enough for a child to crawl through? Should the dimensions determined in part a) be modified? Eplain. 9. A tpical automotive batter has si cells, divided b walls as shown in the top-view diagram below.. An open metal bo for removing ashes from a fireplace is to be constructed from a rectangular piece of sheet metal that is m b.5 m. Squares are to be cut from each corner of the sheet metal, the sides folded upward to form the bo, and then the seams welded. What is the maimum capacit of a bo that is constructed in this wa? a) What dimensions will give the smallest total wall length including the outside perimeter if the top of each cell must have an area of 65 cm? b) A popular batter has dimensions.5 cm b 7.5 cm, with each cell having a width of.75 cm. Has the batter designer used the minimum wall length as a design consideration?. Application An eccentric architect is eperimenting with window design. To ensure adequate illumination, the area of the windows needs to be m. What dimensions will minimize the amount of outside trim required to frame the window if the window is a) a rectangle? b) an isosceles triangle? c) a rectangle surmounted b a semicircle? d) a rectangle surmounted b an equilateral triangle?. A soda cracker package (in the shape of a rectangular prism) is to be constructed with a square base. The total capacit of the package must be cm. a) What dimensions provide the minimum surface area? b) Compare the dimensions found in part a) with those of an actual soda cracker package. What factors do ou think influenced the designers of the package?. A closed displa case for student artwork, in the shape of a rectangular prism, is to be constructed from a m b m sheet of acrlic. The net used for the construction is shown. waste waste back waste top left side base right side waste waste front waste m What dimensions of the case provide the maimum capacit?. Inquir/Problem Solving The Canada Postal Guide lists the following requirements for parcels to be sent within Canada. No dimension can be more than m. m 6.7 Introducing Optimization Problems MHR 8
The length plus the girth (distance around) of the parcel must be less than m. The mass ma not eceed kg. What size rectangular bo with square ends will allow ou the largest capacit? 5. A juice manufacturer is studing the most economical shape to use for a beverage container. Each unit will contain 55 cm of juice. The manufacturer is considering a clinder versus a rectangular prism with a comfortable hand-held depth of cm. Which method of packaging, the can or the juice bo, will use the minimum amount of materials? 6. A chocolate manufacturer uses an equilateral triangular prism package. If the volume of chocolate to be contained in the package is cm, what dimensions of the package will use the minimum amount of materials? 7. A clindrical kite frame is to be constructed from a -m length of light bendable rod. The frame will be made up of two circles joined b four straight rods of equal length. In order to maimize lift, the kite frame must be constructed to maimize the volume of the clinder. Into what lengths should the pieces be cut in order to optimize the kite s flight? 8. A clindrical glass vase is to be made in order to hold large bouquets of flowers. If the capacit of the vase is to be cm, what dimensions for the vase would use the minimum amount of glass? Would these dimensions be practical? 9. Application A m-long feed trough, in the shape of an isosceles triangular prism, is to be made with steel ends and two boards cm wide. How wide should the top of the trough be to maimize the capacit of the trough? cm cm m width. Inquir/Problem Solving A heritage home features a semicircular window of radius m. A local artisan is commissioned to accent the window with a rectangular pane of stained glass. The stained glass will be attached to the inside of the window frame. What dimensions of the rectangular pane will provide the greatest possible area for the stained glass accent?. Application A landscape architect is creating a rectangular rose garden to be located in a local park. The rose garden is to have an area of 6 m and be surrounded b a lawn. The surrounding lawn is to be m wide on the north and south sides of the garden and m wide on the east and west sides. Find the dimensions of the rose garden if the total area of the garden and lawn together is to be a minimum.. The rate of blood flow through an arter of radius r is a function of the blood pressure, resistance pressure, densit of the blood, and stress factors. The flow rate, F, can be epressed as F kr cr, where k and c are constants determined b the various factors. a) Determine the maimum flow rate when the constants have values k 8 and c. b) Determine the maimum flow rate in terms of k and c. 8 MHR Chapter 6
. A plane follows a straight path described b the linear equation (all distances in kilometres) for [, ]. If the flight tower is at the origin (, ), at what point is the plane closest to the tower?. A cell culture under stressful conditions has a rate of growth of G 8t t, for t [, 8] where t is measured in hours. What is the maimum rate of growth and when is it reached? 5. The gait of an animal is a measure of how jerk or smooth the animal s motion appears while it is running. Gait, g, can be shown to be related to the power, P, necessar for the animal to run at a given speed. For an animal m long, running at a velocit of m/s, the power is given b P =. g+ + g Determine the gait that minimizes the required power for the animal to run. 6. A variation of a triathlon competition has a contestant swimming from a point, A, on one shore of a lake, to a point C on the opposite parallel shore, then running to the finish, B, further along the lakeshore. The lake is km wide and the finish line is km down the lake. If a contestant can swim at km/h and run at km/h, determine the point C that will minimize the total time for the race. km A km C? 7. The flight of a gliding bird depends upon the lift provided b the bird s outstretched wings and b the drag caused b air resistance. Drag, D, can be epressed as a function of the gliding speed, v, in metres per second: 5 D = v + 5v What speed minimizes the drag? B 8. Inquir/Problem Solving Holl and Geordie are attending an outdoor music festival where two bands are plaing on stages that are m apart. One band is three times as loud as the other. The two friends are interested in finding the quietest location along the line joining the stages. If the intensit of sound is directl proportional to the volume of the band and inversel proportional to the square of the distance from the source, find the best location. 9. The parabolic arch of a bridge over a onewa road can be described b the equation 6, where and are measured in metres. A transport truck is m wide and.5 m high. What is the maimum clearance of the truck s top corners if it passes under the bridge?. Imagine that an eploration vessel from Earth is visiting another solar sstem in the ear. The people on the vessel decide that the best place to park their vehicle is at the point on the line between two planets where the net gravitational force acting on the vessel is minimized. The magnitude of the force, F, acting on the vessel from either of the planets is F ( )= G Mm where G is a constant, M is the mass of the planet, m is the mass of the vessel, and is the distance between the planet and the vessel. If the mass of one planet is twice the mass of the other planet, where should the vessel be located to minimize the net gravitational force on it?. Application The reaction of the bod to a dose of medicine can be represented b the M function P = M k, where M is the amount of medicine absorbed into the bloodstream, in millilitres, P is the blood pressure, in millimetres of mercur, and k is a constant depending on the particular medicine. The sensitivit of the bod to a particular medicine is measured b the derivative dp. Find the dm amount of medicine to which the bod is most sensitive. 6.7 Introducing Optimization Problems MHR 85
C. Inquir/Problem Solving Two identical sodium vapour light standards A and B are located m apart in a parking lot. A light sensor is to be placed at a point P on a line l that is parallel to the line joining the light standards and at a distance of k metres from it. Intensit of light from a single source is proportional to the strength of the source and inversel proportional to the square of the distance from the source. Find the location of P so that the intensit of light is minimized using the following procedure. A P a) Find a function I() for the intensit of light at point P. Focus on the domain [, ]. b) If k 5, show that the minimum value for I() occurs at the midpoint of l. c) If k, show that I() is not minimized at the midpoint of l. d) Find the value of k, between 5 and, where the minimum illumination point abruptl changes location. e) Describe the changes in the behaviour of I() as k changes from to. Emphasize the location and tpe of the etrema. f) Use the results of part e) to help ou eplain phsicall the phenomenon in part d).. Consider a room that has a floor with dimensions m b 5 m, and walls that are m high. A spider is on one of the shorter walls, m from the floor, and m from either of the adjacent walls. It wants to get to a point on the opposite wall m from the ceiling and. m from an adjacent wall. What is the minimum distance that the spider must walk?. Find the point on the parabolic arc, with domain,, that is closest to the k k l B point (, ). Then, find the point on the arc that is most distant from the point (, ). 5. a) Find the dimensions of the largest clinder that can be inscribed in a sphere of radius k. b) Find the dimensions of the largest clinder that can be inscribed in a cone of height k and base radius k. 6. A cone-shaped paper drinking cup is made b removing a sector from a disk of radius r, and then joining the two straight edges. Find the maimum capacit of such a cup. 7. a) Determine the conditions on the function Q() for which the functions Q() and Q () have the same tpe of etremum at each of their common critical numbers. (That is, the are either both local maimum or both local minimum values, or neither of them is a local maimum or local minimum.) b) Determine the conditions on the functions P and Q for which the functions Q() and P(Q()) have the same critical numbers. c) Assume that the functions P and Q satisf the conditions of part b). Determine the conditions on the functions P and Q for which the functions Q() and P(Q()) have the same tpe of etremum at each of their common critical numbers. (Hint: First eplore some specific functions, such as P ( ) =, P ( ) =, and so on.) 8. The path of a football in an attempted field goal is given b the function.8., where, in metres, is the height above the ground and, in metres, is the horizontal distance from the point at which the ball is kicked. The centre of the horizontal bar of the goal is m awa, m above the ground. a) Does the football clear the horizontal bar of the goal posts? b) Determine the smallest distance between the ball and the bar. (Hint: The result of question 7 ma help.) 86 MHR Chapter 6
9. Communication In question 8, another wa to determine the point on the path that is closest to the goal post bar is to determine when the tangent at the point is perpendicular to the line joining the bar with the point. a) Eplain, using diagrams, wh this approach results in the minimum distance. b) Use this approach to verif the result of question 8. Achievement Check Knowledge/Understanding Inquir/Problem Solving Communication Application A container for wooden matches consists of an open-topped bo (to contain the matches) that slides into an outer bo, open at both ends. The length of the boes is fied at cm to match the length of the matches. The outer bo is designed so that one side completel overlaps for gluing. a) If the outer bo is made from a sheet of cardboard that is 6 cm b cm, what dimensions for the outer bo will maimize the capacit? b) What size should the sheet of cardboard be, to make the inner bo in this case? c) Repeat the problem, for matches of the same size, if the volume is fied at cm and ou want to minimize the amount of material to make both boes. cm 6 cm overlap Career Connection: Operations Research Operations research, sometimes called management science, is a professional area related to business and finance that relies heavil on the techniques of mathematics. Mathematical modelling on computers is a primar tool used to forecast the implications of various choices in the search for the optimum alternative. There is a ver human dimension to this mathematical problem solving, as the models constructed must function effectivel in the real world as well as being theoreticall correct. Operations researchers solve problems such as how to price the seats on an airline flight, what is the most efficient method for routing a long distance telephone call, or how a clothing manufacturer should la out a pattern to minimize wasted fabric. 6.7 Introducing Optimization Problems MHR 87
6.8 Optimization Problems in Business and Economics Common business practice is concerned with minimizing costs and maimizing profits. Man companies have teams of analsts, performing what is called operations research, whose job it is to determine optimum production levels and sales targets. In Chapter, we introduced business applications as rate problems. Here, we will appl our tools for determining maimum and minimum values to business-related problems. We use the techniques illustrated in the previous section. Eample Determining the Optimum Selling Price Market research has shown that for ever drop in price of a product there is usuall an increase in sales. Similarl, an increase in price usuall leads to a decrease in sales. A large retailer selling mountain biccles has found that, for ever $ reduction in price on the Rockhopper model, more biccles are sold per month. The Rockhopper usuall sells for $9 and at that price the store sells 5 biccles per month. a) Determine the optimum selling price in order to maimize revenue. How man biccles are sold at that price? b) Describe the shape of the graph relating price vs. number of biccles sold. c) Describe the shape of the graph relating revenue vs. number of biccles sold. Solution a) The problem requires us to maimize revenue, R, in dollars. Variables: price, p; number sold, ; revenue, R The revenue function is R number sold price p To epress R in one variable, we relate and p. The relationship is linear. We determine two points on the line, and then find the slope and the equation. p (, p) 5 9 (5, 9) 5 88 (5, 88) p m = 9 88 = 5 5 = The linear equation that relates p and is (p p ) m( ) p 9 ( 5) p (given) (after one $ reduction in price) 88 MHR Chapter 6
Substituting this epression for p into the equation for R we obtain R ( ) The derivative of the revenue function is R As usual, we set the derivative equal to zero to obtain the critical number. 7 We need the second derivative to determine whether 7 gives a maimum or a minimum value of R. R Since R is alwas negative, 7 gives a maimum value of R. The maimum revenue occurs when the number of biccles sold is 7 per month. The price at this level is p(7) $7. b) The graph relating price and number of biccles sold is a straight line with negative slope (p() ). c) The graph relating revenue and number of biccles sold is a parabola opening downward (R() ). 5 5 5 5 5 R ( ) = + p ( ) = + 5 7 9 Brief review of terminolog: If is the number of units of a commodit (its production level), then C() is the cost function, which is the total cost to produce units. p() is the demand function (also called the price function), or the price per unit that a product can be sold for, at a production level of units. R() is the revenue function, which is the total revenue obtained b selling units. P() is the profit function, which is the total profit earned b selling units. The marginal cost function is the derivative, that is, the instantaneous rate of change, of the cost function. The marginal demand, marginal revenue, and marginal profit functions are similarl defined. In practice, the cost and demand functions are onl established after etensive market research and analsis. The revenue and profit functions are as follows. revenue: R() p() (number of units price per unit) profit: P() R() C() (total revenue total cost) 6.8 Optimization Problems in Business and Economics MHR 89
Eample Maimizing Revenue and Profit The beverage industr in Canada produces over $ billion worth of product annuall. Based on a -ear stud of production costs, a winer in the Niagara region has determined that the cost of producing bottles of wine is C(). Market research shows that the demand for the wine is given b the price function p(). a) Determine the production level that maimizes the revenue. b) Determine the production level that maimizes the profit. c) Show graphicall the relationship between cost, revenue, and maimum profit. Solution a) Optimize the revenue function. R() p() (.). The derivative of the revenue function (the marginal revenue) is R(). Setting the marginal revenue equal to zero and solving for to obtain the critical number, we get.. 6 We determine the second derivative to decide whether 6 corresponds to a minimum or a maimum value of R. R(). Since R is alwas negative, 6 corresponds to a maimum value of R. A production level of 6 bottles maimizes the revenue. b) Optimize the profit function. P() R() C() (.) [. ] 8. The derivative of the profit function (the marginal profit) is P() 8.6 Setting the marginal profit function equal to zero and solving for, we get 8.6 We determine the second derivative to decide whether corresponds to a minimum or a maimum value of P. P().6 Since P is alwas negative, corresponds to a maimum value of P. Thus, a production level of about bottles maimizes profits. 9 MHR Chapter 6
c) Graphs of the cost function, revenue function, and profit function are shown. The shaded region indicates when profit is positive. The vertical dotted line shows the relationship between the maimum point on the profit curve and the vertical difference between the revenue and cost functions. Note also that the marginal cost equals the marginal revenue when the profit is a maimum; that is, the slopes of the graphs of R() and C() are equal when the difference R() C() is greatest. C ( ) = + +. R ( ) =. P ( ) = + 8. 6 For all the eamples in this section, the number of units,, is an integer. However, we are assuming that this leads to continuous (and differentiable) functions for cost, revenue, and profit. This is acceptable since regression techniques lead to continuous curves that fit the discrete points well. This closeness of fit determines how good the mathematical model of the situation is, and its value in making inferences. When calculus techniques suggest an optimal production level that is not an integer, we choose the closest convenient integer. (Sometimes we round to recognize that items are packaged in units such as hundreds.) Ke Concepts C(), the cost function, is the total cost to produce units. p(), the demand function (also called the price function), is the price per unit that a compan can sell its product for, at a production level of units. R(), the revenue function, is the total revenue obtained b selling units. revenue: R() p() (number of units price per unit) P(), the profit function, is the total profit earned selling units. profit: P() R() C() (total revenue total cost) Communicate Your Understanding. Eplain the difference between the revenue function and the profit function.. Eplain wh the production level that maimizes revenue does not necessaril maimize profit.. There is alwas an initial startup cost for an manufacturing process. What term of the cost function represents this epense? 6.8 Optimization Problems in Business and Economics MHR 9
. Is it reasonable to assume that the price function is linear? Eplain. 5. Will the revenue function alwas be quadratic? Eplain. 6. Will a cost function alwas be an increasing function for? Eplain. 7. How is marginal revenue related to the graph of the revenue function? B Practise. For the given cost and demand functions, find the production level that will maimize profit. a) C(), p() 5.5 b) C() 5 5., p() c) C() 5 5., p(). d) C()., p() 5. e) C(), p() 9 Appl, Solve, Communicate. Communication The graph shows the cost and revenue functions for a local music retailer. = C( ) = R( ) a) Identif on the graph the -value that corresponds to the maimum profit. b) Sketch a graph of the profit function, P. c) Sketch a graph of the marginal profit function, P. d) What is the meaning of P()?. Application A car manufacturing compan analsed painting costs and tabulated the following results: Number of Cars, Cost of Painting, C() ($) 5 9 75 5 a) Using quadratic regression to fit a function to the data, determine a function, C(), for the cost of painting cars. b) What is the total cost of painting cars? c) The price function for painting cars is p().. Find the revenue earned painting cars. d) Find the production level that maimizes revenue. e) Find the production level that maimizes profit.. A tetile manufacturer uses regression to determine that the cost of producing metres of woven fabric is C() 8... It forecasts that it can sell the fabric for p() 6. dollars per metre. Determine the production level that will give maimum profit. 5. Inquir/Problem Solving A large electronic retailer has been selling digital cameras for $85 each. At this price the store sells per month. From past eperience, for ever $ discount in price, the number of sales increases b 5 each month, and for ever $ increase in price, the number of sales decreases b 5 each month. a) Find the price function. b) At what price should the cameras be sold in order to maimize revenue? 9 MHR Chapter 6
c) What is the percent reduction in price that the retailer is offering the consumer at the price in part b)? 6. A professional basketball team plas in a stadium that holds spectators. With ticket prices at $6, the average attendance had been 8. When ticket prices were lowered to $55, the average attendance rose to. Based on this pattern, how should ticket prices be set to maimize revenue? 7. Application The Eco-venture charter compan offers local environmental ecursions. The fare is $5 per person if 8 to passengers sign up for the trip. The compan does not offer the trip if fewer than 8 people sign up. If more than people sign up, the fare for ever passenger is reduced b $ for each passenger in ecess of. The bus can hold onl 8 passengers. Determine the number of passengers that generates the greatest revenue for the charter compan. 8. An ice cream vendor has found that the cost of suppling cones is C() 5.. and also that the demand for the cones increases when the price drops. Approimatel 5 cones will sell per da if the are priced at $., but for ever price drop of, the number sold per da increases b. Similarl, for ever price increase of, the number sold per da decreases b. a) Determine the price of the cones that maimizes revenue. b) Determine the price of the cones that maimizes profit. 9. Inquir/Problem Solving Glnn is considering buing a truck and becoming a professional driver. The truck manufacturer indicates that he can epect his running costs when driving at v kilometres per hour to be approimated b the function cv () =. 85 +. v, where the cost, c, is in dollars per kilometre. Glnn plans to pa himself $5/h while he is driving. Find the speed that will minimize his costs for a 5-km trip.. The manager of a -room ski resort has found that, on average, 5 rooms are booked when the price is $75 per night and 6 rooms are booked when the price is $6 per night. What price should the manager set for the rooms to maimize revenue?. The ield of a crop, in tonnes, is dependent upon the time, t, in das, between planting and harvesting, according to the following formula. 8t t 5 if t[, 5] Yt () if t [, ) or t ( 5, ) On which da is the maimum ield reached, and what is the maimum ield?. Application Two buildings in a school comple need to be connected with fibre optic cable. Building A is 7 m from a roadwa and building B is m down the road. There are two choices in laing the cable. If laid underground, the cost is $/m, and if laid above ground, the cost is $5/m. The cable must be laid underground across the plaing fields. The cable will be laid from A underground to a point C on the road, and then above ground to B. a) Where should C be chosen to minimize the total cost of laing the cable? A 7 m road plaing field C m cable underground cable above ground b) What other costs might need to be considered?. Application An analst has predicted that the growth rate, as a percent, for a specific mutual fund can be modelled b the equation r 5. 6., where is the number of months and [, ]. Determine the best and worst times to invest in this mutual fund. B 6.8 Optimization Problems in Business and Economics MHR 9
. Inquir/Problem Solving A steering wheel manufacturer finds that, b manufacturing steering wheels per da, there are fied costs of $5, and $.5 in labour and materials costs per steering wheel. An agreement with the materials supplier to pa for part of the transportation costs leaves the compan with a cost of $5. The maimum capacit of the factor is 5 steering wheels. Determine the number of steering wheels that should be produced dail in order to minimize costs. 5. Inquir/Problem Solving A 5 m rectangular area of a field is to be enclosed b a fence, with a moveable inner fence built across the narrow part of the field, as shown. The perimeter fence costs $/m and the inner fence costs $/m. Determine the dimensions of the field to minimize the cost. 6. The demand function for a certain artist s print is modelled b the equation p ( ) 8, where p is the price, in dollars, when prints are produced. How man prints should be produced in order to maimize revenue? C 7. Communication The average cost is C ( ) defined as, where C() is the cost function. a) Prove that the average cost is smallest (if ever) onl when the average cost equals the marginal cost. b) Suppose that C() is the cost function for a manufacturing operation, where is measured in ten thousands of units. Is there a production level that has minimal average cost? If so, what is it? Career Connection: Economic Prediction Economics studies the wa a societ uses scarce resources, such as capital, land, labour, raw materials, and equipment, to provide goods and services. Further, economists analse the results of their research to determine the costs and benefits of making, distributing, and using resources in a particular wa. In seeking to optimize the use of these resources, economists emplo man tools that are similar to those used in calculus optimization problems. The mathematical models of economics are critical in predicting the nature and length of business ccles, the effects of a specific rate of inflation on the econom, the effects of ta legislation on unemploment levels, or the likel movement of interest rates. 9 MHR Chapter 6
Technolog Etension Racing Strateg Using TI InterActive! TM Fran is riding in a mountain bike race starting at point A and finishing at point C. The end point C is 6 km east of the starting point A, and 7 km north of the major road. Fran can ride her mountain bike at 5 km/h on the road and at 5 km/h off road. She can choose an point at which to leave the road. A D 6 The problem is to find the point D where Fran should leave the road to minimize her time. Let be the distance she travels on the road. Then, her total time is represented b the epression f( )= time on road + time off road = First, define a function of time in terms of variable. Net, define a derivative function and call it f(). The derivative function can be displaed. distance on road speed on road = + 5 + 7 + ( 6 ) 5 distance off road speed off road C B 7 To find the minimum time, set the first derivative equal to and solve for. In order to use this value later for the second derivative test, store the value in variable k. In order to use the second derivative test, define the second derivative and name it f(). Displa the epression for the second derivative. Finall, evaluate the second derivative at the value of found when the first derivative was set equal to. Since the first derivative is equal to when.75 and the second derivative is positive, there is a local minimum at.75. Therefore, Fran should ride.75 km on the road. Web Connection A trial version of TI InterActive! TM is provided b the manufacturer. Go to www.mcgrawhill.ca/links/caf and follow the instructions there. Technolog Etension: Racing Strateg Using TI InterActive! MHR 95
A second solution method is to use the spreadsheet in TI InterActive!. To access the spreadsheet option, click on the icon on the toolbar. A spreadsheet screen will open. Enter the headings as shown in the screen. You will need to make the columns wide enough to accommodate the headings. Be sure to make column F wide enough to show several decimal places. For the distance off road, use the Pthagorean relation with sides of length 7 and the value in column B. The formula is shown in the entr line in the screen below, and copied down the column. The values have all been formatted to show three digits after the decimal point. In the first column enter values for the distance that Fran travels along the road. This could var from km to 6 km. As a first step, use an increment of km. The formula to accomplish this is shown in the entr line just above the column headers in the screen below. It has been copied down to cell A9. To do this, enter the formula in cell A, grab the small square in the lower right corner of the cell and drag it down to cell A9. The Road Time in column D of this screen is found b dividing the distance in column A b the speed of 5 km/h. It is formatted to three decimal places and copied to row 9. The Road Not Used column refers to the amount of road left after Fran turns off the road. This amount is calculated b finding the difference between 6 and the road distance in column A. The formula is shown in the entr line in the screen in the net column. Cop the formula down to cell B9. In the same wa, the Off Road Time in column E is found b dividing the Distance Off Road in column C b the speed of 5 km/h and coping the formula down to row 9. 96 MHR Chapter 6
Again, the cell containing the least time has been highlighted. Using an argument similar to that in the previous spreadsheet, we can conclude that the best route for Fran to take would be somewhere between.7 km and.8 km. Based upon this information, here is one final spreadsheet going from.7 km to.8 km in steps of. km. Column F shows the total of the on-road time in column D and the off-road time in column E. These values have been formatted to eight decimal places in order to compare later values. In this screen, cell F has been highlighted to show that this cell has the smallest total time for this spreadsheet. From this spreadsheet, we get the same result,.75, as we did using the computer algebra sstem in TI InterActive! TM. This value depends upon the value found in cell A. This is not necessaril the lowest value, but it does indicate that the lowest value could be found somewhere between and, in cells A and A5, respectivel. Using this information, we recalculate the spreadsheet for values of the on-road distance starting at km and going up to km using increments of. km. Each column must be copied down to row. Practise. Fran s friend Roberta is not quite as fit as Fran, so she rides on the road at km/h and off road at km/h. At what point should Roberta leave the road to minimize her time?. Someone gave Fran the wrong information about the distance from A to B. It is actuall onl km. How does this change the solutions to the original problem and Eercise above?. Fran has decided that km/h is the fastest that she can ride off road. Using a distance from S to T of 6 km, how fast should she ride on the road so that, with the optimal strateg, she goes eactl km on the road? What if she wants to go 8 km on the road? Technolog Etension: Racing Strateg Using TI InterActive! MHR 97