3 Additional Applications of the Derivative

Transcription

1 3 Additional Applications of the Derivative 3.1 Increasing and Decreasing Functions; Relative Etrema 3.2 Concavit and Points of Inflection 3.4 Optimization Homework Problem Sets 3.1 (1, 3, 5-9, 11, 15, 17, 23, 24, 32, 35, 37, 43, 55, 56, 57, 67, 68, 71) 3.2 (5, 6, 13, 15, 16, 23, 27, 28, 30, 33, 47, 48, 53, 55, 60a, 63ab) 3.4 (1, 3, 5, 8, 17, 19, 31, 32, 45, 46, 58) mathminer.org

2 3.1 Increasing and Decreasing Functions; Relative Etrema One of our reasons for developing the derivative is to find etreme values of functions. For instance we would be interested in maimizing the profit a business earns. If the profit function happens to be quadratic, we would be able to use the verte formula. In general, there is no formula to find etreme values of functions, so we will use calculus. Before digging in, we need to go over some basic terminolog. First we will just label the following graphs, and then we will look at precise definitions. Note in the graphs above that relative maima and relative minima onl occur where there is a change in direction or turning point of the graph. But an absolute maimum or absolute minimum can occur at an tpe of point on the graph, such as a turning point or an endpoint of the graph. Also note that we often use the word local in place of relative and global in the place of absolute. Here are more formal definitions. Absolute (Global) Etrema Suppose f is defined on an interval I containing c. Then f(c) is the absolute minimum value of f on I if f(c) f() for all in I. absolute maimum value of f on I if f(c) f() for all in I. Relative (Local) Etrema A function f has a relative minimum at = c if f(c) is the minimum value of f on some open interval containing c. relative maimum at = c if f(c) is the maimum value of f on some open interval containing c. 2

3 First we will find relative etrema. These appeared where there was a change in direction of the function, so let s introduce some calculus to deal with direction. A function f is increasing if when 1 < 2, then f( 2 ) > f( 1 ). A function f is decreasing if when 1 < 2, then f( 2 ) < f( 1 ). Sketch a few tangent lines over each interval on each graph. What do ou notice? This leads to a calculus version of increasing and decreasing. Increasing/Decreasing Test If f () > 0 on an interval, then f is increasing on that interval. If f () < 0 on an interval, then f is decreasing on that interval. So what happens if f () = 0? Critical Number A value = c is called a critical number of f() if f (c) = 0 or f (c) is undefined. Eample 1 Find the intervals on which f() = is increasing or decreasing. So what do ou think happens at the critical values that define our intervals of increase and decrease? 3

4 First Derivative Test Let c be a critical number of f(). f has a relative maimum at = c if f () > 0 to the left of c and f () < 0 to the right of c. f has a relative minimum at = c if f () < 0 to the left of c and f () > 0 to the right of c. If f does not change sign at c, then f has neither a maimum nor minimum at = c. Eample 2 Find the etrema of the function f() = and sketch its graph. Eample 3 Find the etrema of the function f() = and sketch its graph. 4

5 Eample 4 A compan determines that if thousand dollars are spent on advertising a certain product, then S() units of the product will be sold, where S() = , where a. How man units will be sold if nothing is spent on advertising? What if the full budget is used? b. How much should be spent on advertising to maimize sales? c. What is the maimum sales level? Eample 5 Sketch a graph of a function that has all of the following properties: f (0) = f (1) = f (2) = 0 f () < 0 when < 0 and > 2 f () > 0 when 0 < < 1 and 1 < < 2 5

6 3.2 Concavit and Points of Inflection The first derivative has to do with the direction a function is heading. We will see here that the second derivative will have to do with the shape of the graph. A function f is concave up if f is increasing in value. A function f is concave down if f is decreasing in value. Sketch tangent lines to each curve. Notice that if a function is concave down, all its tangents are above the graph and if it is concave up, all its tangents are below the graph. The following summarizes the above. Second Derivative Test Let c be a critical number of f(). f is concave up at = c if f () > 0. There is a relative minimum at = c. f is concave down at = c if f () < 0. There is a relative maimum at = c. If f (c) = 0, this test gives no information. Eample 1 Sketch f() =

7 Inflection Point A point of inflection is a point at which the concavit of a function changes. If a function f has a point of inflection at = c, then f (c) = 0. So these are critical points of f. Eample 2 Determine the intervals where f() = is increasing or decreasing and concave up or concave down. Identif the relative etrema and an points of inflection. Sketch its graph. Eample 3 Determine the intervals where f() = is increasing or decreasing and concave up or concave down. Identif the relative etrema and an points of inflection. Sketch its graph. 7

8 Eample 4 For each of the following functions: a. Identif whether it is increasing or decreasing and whether it is concave up or down. b. Write a mathematical description for each function based on part a. Most functions look like one of the above locall, so these can be used to classif them. Eample 5 Sketch a graph of a function that has all of the following properties: f(2) = 2, f (2) = 0 f () > 0 on (, 2) and f () > 0 on (2, ) f () < 0 on (, 2) and f () > 0 on (2, ) 8

9 Eample 6 A compan estimates that when thousand dollars are spent on the marketing of a certain product, Q() = units of the product will be sold, where a. In the interval, find the amount that should be spent on marketing to maimize the number of units sold. How man units will be sold? b. Where does the graph of Q() have an inflection point? c. Graph Q(). What is the significance of the marketing ependiture that corresponds to the P.O.I.? Q() 9

10 Curve Sketching Strateg Identif the domain of f() and find an intercepts (if the -intercepts are eas to find). Identif whether or not f() has an asmptotes (use domain and Find all critical points of f(). Classif critical points as relative maima or minima. Find all points of inflection of f(). Identif intervals of increase/decrease and concavit. Sketch. lim f()). ± Eample 7 Sketch f() =

11 Eample 8 Sketch g() =

12 3.4 Optimization Note that the following function does not have an absolute maimum or absolute minimum How can we change the graph so that it does attain both an absolute maimum and an absolute minimum? Etreme Value Theorem Suppose f is continuous on the closed interval [a, b]. Then f attains both an absolute maimum and absolute minimum. These will occur at a critical number of f or at an endpoint of the interval. Eample 1 Find the absolute maimum and absolute minimum values of f() = on [ 2.5, 2]. Eample 2 A poll indicates that months after a particular candidate for public office declares her candidac, she will have the support of S() percent of the voters, where S() = 1 29 ( ) for If the election is held in November, when should the politician announce her candidac? Should she epect to win if she needs at least 50% of the vote? 12

13 Eample 3 If C(q) is the cost to produce q units of a particular commodit, then the average cost per unit is given b the function A(q) = C(q). Suppose C(q) = q 3 + 5q q a. Find A (q). b. For what values of q is A(q) increasing? For what values is it decreasing? c. For what level of production q is average cost minimized? What is the minimum average cost? What is the marginal cost at this production level? d. Sketch A(q) and C (q) on the same set of aes. What do ou notice? q 13

14 Eample 4 Suppose the price at which q units of a particular commodit can be sold is given b the demand equation p(q) = 180 2q and the total cost of producing q units is C(q) = q 3 + 5q a. Find the revenue function R(q) and the profit function P (q). b. Find the marginal revenue, cost, and profit. What do ou notice about the difference R (q) C (q)? c. For what values of q is profit increasing? For what values is it decreasing? d. For what level of production is profit maimized? What is the maimum profit? e. Sketch P (q), R (q), and C (q). What do ou notice? q 14

15 Optimization Strateg Identif what needs to be optimized (maimized or minimized) and an unknown quantities involved. Define a variable or variables. Epress a relationship between the unknown quantities with an equation or inequalit. Epress the quantit that is to be optimized in terms of one variable using the above step. Find all critical values and classif them as maimum or minimum. endpoints ma need to be tested. If working in a closed interval, the Interpret the results and answer the original question. Eample 5 There are 320 ards of fencing available to enclose a rectangular field. How should this fencing be used so that the enclosed area is as large as possible? What if a divider is added to make two enclosures? Eample 6 A cable is to be run from a power plant on one side of a river 1200 meters wide to a factor on the other side, 2000 meters downstream. The cost of running the cable under water is $25 per meter, while the cost over land is $20 per meter. What is the most economical route over which to run the cable? 15

16 Eample 7 A bo with an open top is to be constructed from a square piece of cardboard, 3 feet wide, b cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a bo can have. Eample 8 A clindrical can is to be made to hold 1 liter (equivalent to 1000 cm 3 ) of oil. Find the dimensions that will minimize the cost of the metal to manufacture the can. 16