4.3 How derivatives affect the shape of a graph. The first derivative test and the second derivative test.

Transcription

1 Chapter 4: Applications of Differentiation In this chapter we will cover: 41 Maimum and minimum values The critical points method for finding etrema 43 How derivatives affect the shape of a graph The first derivative test and the second derivative test 44 Indeterminate forms and l Hospital s Rule 45 Summary of curve sketching 47 Optimization problems 49 Antiderivatives 410 Chapter Review 41 Maimum and minimum values The critical points method for finding etrema Motivation: Finding etrema (maimum and minimum) values of a function is an etremely important application in mathematics (in particular, in our class it is an application of the derivative) For eample, problems of maimizing profit, minimizing cost, maimizing (or minimizing) areas or volumes and many other similar problems (see for eample, the problems in section 47) are such typical applications of the methods which we will learn in this chapter An entire branch of mathematics called optimization is dedicated to finding etrema of functions Goals: define points of etrema (local and global) ; develop the critical point method for finding global etrema of a function; I Definition of points of etrema The etreme value theorem: Questions: What are points of etrema for a function? What are maimum and minimum points for a function? Do all functions have etrema (over an interval )? 1

2 Definition 1: A Global (absolute) etrema: Consider a function f : D R and c D Then f ( is a: global (or absolute) maimum value of ( f on D if (1) f f ( for D is called the - value where f ( achieves its global maimum) ; global (or absolute) minimum value of ( ( (in this case the point c f on D if () f f ( for D is called the - value where f ( achieves its global minimum) ; ( (in this case the point c Eample 1: For eample, in Figure 1, we see that the global maimum of f ( is 5 achieved for c 3and its global minimum is achieved for c 6 B Local etrema: f ( is a: local maimum value of f ( on D if 0such that a (3) f f ( for c a, c a c, in this case c is called the value where f ( achieves its local maimum ) ; local minimum value of f ( on D if 0such that Eample : ( (that is, near a (4) f f ( for c a, c a in this case c is called the value where f ( achieves its local minimum ) ; ( (that is, near c, For eample, in Figure, we see that f ( has a global maimum at d : f (d), a global minimum at a: f (a), and local minima at c and e, and local maima at b and d Note that, in general, a global maimum is the greatest local maimum (ecept when it is an endpoint) and a global minimum is the smallest local minimum (again, ecept when it is an endpoint) See also Figure 3 where a few local minima and maima are displayed

3 Note again that endpoints (of the interval where the function is considered) can be global etrema, but not local etrema, since they would not satisfy a condition of the type (3) (or (4)) Therefore, local etrema are always inside the interval Figure 4: Consider also Figure 4, where the function the interval 1,4 For this eample, identify: f 4 3 ( is graphed on local minima: local maima: global minimum: global maimum: Now that we can identify from a graph points of etrema, let us try to answer the following essential question: Do all functions defined over a certain interval have a local (or global) maimum and a local (or global) minimum over that interval? Eample : a) Let 3 f :,,, given by : f ( what are its local minima and maima? b) Let 3 f : 1,1 1,1, given by : f ( what are its local minima and maima? Let 3 f : 1,1 1,1, given by : f ( what are its local minima and maima? d) Let f :, 0,, given by : f ( what are its local minima and maima? We see that some of these functions (such as the function given in have local etrema, while (most of) the others do not The following important theorem establishes which functions are guaranteed to have an etrema over their corresponding interval Theorem 1 (Etreme value theorem): Consider a continuous function f a, b R minimum value f (d) at some points c d a, b : Then f ( achieves a global maimum value f ( and a global, This theorem is given without proof, since the proof is quite advanced (based on Cantor s completeness aiom) Draw a few typical figures of a continuous function on a closed interval to convince yourself of its validity Then look again at the functions in Eample in the light of this theorem 3

4 Remark 1: 1 Note that we need two conditions to be verified in order for global etrema to eist: the function f ( needs to be continuous and the interval where the function is considered needs to be closed Otherwise, we may well have that the function does not have a (local or global) etremum on the interval ; The functions for which we want to find etrema in this chapter almost always satisfy these two conditions (that is they are continuous functions considered on a closed interval) and therefore global etrema are guaranteed to eist for these functions Figure 5: A function which satisfies the EVT Figure 6: A function which does not satisfy the conditions of the conditions of EVT: it has a global minimum but no global maimum, this is in agreement with the theorem II Fermat s theorem Finding points of global etrema for a function: Therefore, so far we have learned how to identify the local and global etrema, and conditions under which such etrema are guaranteed to eist Net, we consider the problem of finding such etremum values for a given function (epression) for which the graph is not given Note (see Figure 10 in the tetbook) that for most functions local etrema occur at stationary points of the function (that is points where f '( 0 ) although this is not always the case However, these points are good candidates for etrema, as the following theorem states: Theorem (Fermat): If f : a, b R has a local maimum or a local minimum at c a, b then f '( 0 So: If f : a, b R and c a, b and f ( is a local etrema and f '( eists, then f '( 0 (If interested, see the proof of this theorem in the tetbook) and if f '( eists, Note that f '( 0 does not in general imply that f ( is a local etrema as Eample (a to showed However, stationary points of f ( are valuable candidates for etrema of f ( Eample 3: Consider : 1,1 0,1 f given by: f ( In this case, the local etrema are again not points where f '( 0 (0 is the local minimum and f '(0) does not eist) 4

5 Remark : However, for a function f : a, b R which is continuous on b c a, b and f '( eists f '( 0 (from Fermat s theorem) ; OR c a, b and f '( does not eist (as in eample 3 above) ; OR c is either a or b (the endpoints of the interval ) a,, if c [ a, b] is an etrema, then either: These 3 possibilities are the only 3 possibilities for an etrema of a function f a, b R : This is the basis of the following definition and of the Critical point method which follows Definition (critical points) : A critical point (read a valuable candidate of an etrema) of a function f : a, b R is a value c a, b such that either f '( 0 or f '( does not eist Note, that after we find the critical values of a given function, the only other candidates of an etrema for the function are the endpoints of the interval 3/5 Eample 4: Consider f ( 4 Determine the critical points of f ( The reasoning outlines in Remark above gives us the procedure of finding the global etrema of any continuous function defined on a closed interval: f : a, b R This procedure is oulined in the Critical Point Method below: Critical Point Method for finding the global etrema of a continuous function f a, b R : : Step 1: Find all critical values of f ( (that is values c where either f '( 0 or f '( f ( for all of these critical values Step : Find f ( a) and f ( b) Step 3: The largest of the values calculated in Steps 1 and is the global maimum of ( the smallest of these values is the global minimum of ( f on b a, does not eist) Find f on b a, ; Eample 5: 3 1 a) Find the absolute etrema of f ( 3 1 for 4 5

6 Do problems 47 and 48 from Eercise set 41 in the tetbook Homework: problems 1,,3,5,6,7,11,14,15,19,3,5,8,30,3,34,38,49,50,51,5,54,56,57 from problem set How derivatives affect the shape of a graph The first derivative test and the second derivative test Goals: prove the First Derivative Test and learn how to use it to find all local etrema of a function f ( on a given interval; prove the Second Derivative Test and learn how to use it to find all local etrema of a function f ( on a given interval; I The first derivative test : In order to develop this test, we pose the question: What does f '( say about f (? To answer this question, remember that in section 30 we introduced f '( as the slope of the tangent line to G f at The basis of the first derivative test is given by the following: Theorem 1: Consider f ( differentiable on an interval I a) If f '( 0 on I then ( b) If f '( 0 on I then ( f is strictly increasing on I (that is : f f any I 1 1 1, for ) f is strictly decreasing on I (that is : f f any I 1 1 1, for ) Proof: Follows easily using the mean value theorem from f ( Use also your visual intuition to see why this theorem must be true This theorem allows us to construct a table of monotonicity for a function f (, as long as we can determine the sign of its derivative 6

7 This table of monotonicity will allow us to find easily all local and global etrema of a function f ( on an interval I Eample 1: 4 3 Consider f ( on I [3,3] Find where f ( is increasing and where it is decreasing on I, then construct a monotonicity table and use this table to determine all points of etrema of f ( on I The method shown in Eample 1 suggests the following theorem: Theorem (The First Derivative Test): Consider f : D R, f ( continuous on D, and let cd a critical number of f ( Then : a) If f '( changes sign at c from positive to negative (as we move from left to right) then f ( has a local maimum at c b) If f '( changes sign at c from negative to positive (as we move from left to right) then f ( has a local minimum at c Proof: Is immediate, if we construct a monotonicity table for f ( around c Eample : Do problems 9 and 10 in the Eercise set 43 II The second derivative test : In order to develop this test, we pose the question: What does f ''( say about f (? Figure 1: a concave up function (Figure 1a) and a concave down function (Figure 1b) Definition 1: a) If '( b) If '( f is increasing on some interval I (so if ''( 0 f is decreasing on some interval I (so if ''( 0 f on I), then f ( is called concave up on I; f on I), then f ( is called concave down on I 7

8 A typical shape of a concave up function is (as shown in Figure 1a) For this type of function, f '( is increasing (so f ''( 0 ) A typical shape of a concave down function is (as shown in Figure 1b) For this type of function, f '( is decreasing (so f ''( 0) Look also at Figure 7 in the tetbook and determine the regions where f ( is concave up and where it is concave down Convince yourself that on these regions f '( has the epected monotonicity behavior (so f ''( has the epected sign) Therefore, the sign of f ''( provides the concavity of the function f ( Definition : A point P(, f ( ) on G is called an inflection point of a continuous function f ( if f ( changes concavity at P f (or, in other words, if f ''( changes sign at P) Therefore, a point where f '( changes sign is a point of local etremum for f (, and a point where f ''( changes sign is a point of inflection for f ( Eample 3: Draw a monotonicity table and a concavity table (that is a table which shows the sign of f ''( on D f ) for the function f ( Use this information to sketch the graph of f ( Theorem (Second Derivative Test): Consider f : D R, f ( continuous on D, and let cd a stationary point of f (, that is let f '( 0 a) If f ''( 0, then f ( has a local minimum at c; b) If f ''( 0, then f ( has a local maimum at c Proof: a) Use the definition of a concave up function (see Definition1a) with the fact that f '( 0 to determine a monotonicity table of f ( near c, which shows that c is a point of local minimum for f ( b) Similarly Eample 4: Problem 19 from Eercise Set 43 8

9 Comparison of the 3 methods for finding etrema: So far we have developed the 3 main methods for finding the points of etrema of a function f ( : the critical point method (in section 41), the first derivative test (Theorem in section 43) and the second derivative test (Theorem 3 in section 43) A comparison of the advantages and disadvantages of these tests reveal that the first derivative test is usually the best (preferred) test to use when determining the points of etrema of a function since: For the Critical Point Method, only the global etrema are found No information is known about the other critical points (that is, if they are local etrema or no etrema at all) Also, this method is quite tedious and long The First Derivative Test determine all local etrema and the entire monotonicity behavior of f ( on I It is the best since it provides a complete information about the monotonicity of the function f ( if we can determine the sign of f '( ; The Second Derivative Test determines the local etrema but only for stationary points It does not work for the singular points of f ( or for endpoints Therefore, when possible, it is preferred to use the First Derivative Test when determining the etrema of a function Eample 5: Problem 1 from Eercise Set 43 Homework: Problems 1,4,5,7,8,10,11,13,16,0,1,33,36,39 and 45 from Eercise Set Indeterminate forms and l Hospital s Rule In this section we turn back to calculating its of functions So far we know how to calculate its of the form e and but how do we calculate 3 ln( ln( its like: 1, or? 1 1 l Hospital s Rule, given by the net theorem, provides a general and simple yet effective method to solve such its Theorem 1 (l Hospital s Rule): Consider f ( and g( differentiable and g '( 0 near a point a (ecept possibly at a ) Suppose that f ( 0 and g( 0 a a (such that f ( 0 g ( ) 0 ) a or that f ( (or ) and g( (or ) Then a f ( a g( f '( a g'( a f ( (such that ( ) ) a g f '( when eists (or when it is or ) a g'( 9

10 Proof: Figure 1: For the actual proof (in a simplified case), let us assume that f ( a) g( a) 0, that f '( and g'( are continuous near a, and that g' ( 0 near a f ( f ( a) f '( a) a ( ) ( ) ( ) In this case: a f f a f g'( a) g( g( a) a g( g( a) a g( a a For the more general proof (based on the mean value theorem) see Appendi F QED Eample 1: Using l Hospital s Rule (make sure that you check all the conditions before applying this rule, and especially the 0 f '( condition of a case of study of the type or 0 and that eist), calculate the following its: a g'( a) ln( b) 1 1 ln( 1 d) e e e) f) ln( 3 g) sin( 0 tan( h) 0 3 i) ln( 0 1 j) tan( cos( 10

11 k) 0 0 (recall that there are 3 eponential cases of study : 0, and 1 0 notation (1) L g( f ( and applying ln to the equality (1) 0 which are solved by using the l) Problems 15, 19 and 0 from the Eercise set 44 Homework: Problems 1,,3,4,5,8,10,1,1,,3,5,8,3,40,4,43,44,49,5,55,56,63,71,7,73 and 86 from the Eercise Set Summary of curve sketching We can now use many of the tools we have studied so far in Calculus, especially the table of monotonicity, the table of concavity and the asymptotes to develop a systematic and effective method for sketching the graph of any function f ( There are many important reasons for sketching the graph of a particular function by hand versus graphing it by calculator: even the best graphing devices have to be used intelligently and the graphs produced have to be fully understood; it is very important when graphing a given function to choose an appropriate viewing rectangle to avoid getting a misleading graph (see for eample the graphs of f ( 3,, f sin150 f ( f ( 8, ( and other eamples studied in section 14) Notions of Calculus (and algebra) are necessary to determine the domains and ranges of these functions (and therefore the appropriate drawing rectangles); using Calculus we discover the most important characteristics of a given function (such as asymptotes, minima, maima and points of inflection) which can be easily overlooked or at best very hard to find accurately using only a graphical device For eample, 3 Figure 1 shows the graph of f ( It appears quite reasonable, it is the graph of a cubic, similar with the graph of 3 f (, apparently with no maimum or minimum value over this interval However, with the use of Calculus (the first derivative test), we find a maimum of f ( at 075and a minimum at 1 Calculus also insures that there are no other points of etrema for this function (if D R ) 3 Figure 1: The graph of f ( Figure : The graph of Figure 1 zoomed in 11

12 Therefore, there should be a close interaction between Calculus and graphing devices, as each can be of assistance to other: Calculus can provide the rigorous and accurate framework for finding the domain, range, asymptotes and all points of etrema of a given function, while a calculator (or another graphing device) can be used for carrying on necessary calculation or for checking our work) Guideline (steps) for sketching the graph of a generic function f ( : 1 Find the domain D of f ( (the set of values where f ( is well defined ) by imposing conditions for epressions in ; Find O: by solving f ( 0 Find Oy: by calculating f (0) ; 3 (often optional) check for symmetries: with the y ais : f ( is symmetric with the y ais if f ( f ( f (, D with origin: f ( is symmetric with the origin if ( check if f ( is periodic : is even, that is if f is odd, that is if f ( f (, D f ( is periodic if pr such that f p f (, D The smallest such p is called the fundamental period of f ( If f ( is periodic with a fundamental period of T, then it needs to be graphed only on an interval of length T, typically on 0,T 4 Find the asymptotes: a) HA : at (if in D): Calculate f ( at (if in D): Calculate f ( b) VA are lines a (points ecluded from the domain D) such that either f ( (or ) f ( (or ) (or both) a a or 5 Draw the monotonicity and the concavity table of f ( (that is, determine the signs of f '( and f ''( on D ) and use the information from 1-4 above to complete this table 6 Sketch the graph of f ( using the information from the table of monotonicity and concavity drawn in 5 Eample 1: Use the steps outlined above to sketch the graphs of: 3 a) f ( b) f ( 3 f ( d) 1 1 f ( e Homework: Problems 1,3,9,10,11,13,16,,4,30,33,37,47 and 5 from Eercise Set 45 1

13 47 Optimization problems : In this section we solve practical problems in which we use the methods studied in sections 41 and 43 to find the etreme values of a practical quantity of interest There are many problems of this type, for eample: a business person who wants to minimize costs and to maimize profit, a traveler wants to minimize his/her traveling time, or practical problems which come from physical or mathematical principles (such as the problem which generates the geodesic curves (the shortest curve between two points on a given surface), Fermat s principle of light which states that light follows the path which takes the least time See also the My dog knows calculus problem (discussed here: and mentioned in section 11) In this section we solve problems such as these, and also problems in which we are interested in maimizing areas, volumes, and profits and in minimizing distances, times and costs The problems in this section are typically word problems, somewhat similar with the related rates problems, studied in section 39, but with the notable difference that this time our main goal is to find the etreme values of a practical quantity of interest (usually called the objective function), and not a derivative It is important to follow the following guidelines when solving the optimization problems in this section: 1 Understand the problem: read the problem carefully until it is clearly understood Determine the objective function and the variables that this function depends on Draw a diagram: In most problems it is useful to draw a diagram to determine the given and the required quantities and the relation between these 3 Introduce notation: Assign a symbol for the objective function (let us call it Q ) and for other unknown quantities in the diagram (for eample, call these a, b, c,, y,) Label the diagram with these symbols 4 Epress Q in terms of these symbols: a, b, c,, y, Our final preinary goal is to epress Q in terms of one unknown variable only 5 Use the diagram (or equations/relations stated in the problem) to find relations between the unknown independent variables such that Q depends on one independent (unknown) variable only Set then Q f ( Find the domain of this function D 6 Use the methods studied in sections 41 and 43 (usually the First Derivative Test) to find the absolute minimum or the absolute maimum (as required) of Q f ( on D Eamples : 1 A farmer has 400 feet of fence and he wants to fence off a rectangular field What are the dimensions of the field which has the largest area? 13

14 A cylindrical can is to be made to hold 1000 cm 3 (1L) of oil, as in the Figure 3 below What are the dimensions (find r and h) which will minimize the cost of the metal to manufacture the can? 3 Find the point on the parabola y which is closest to the point ( 1,4) Hint: it is easier here to minimize d, y,(1,4) instead of, y d,(1,4) 14

15 4 See and solve the My dog knows calculus problem at Homework: Problems 1,,4,5,7,8,10,1,13,15,18,19,1,4,6,9,30,34,38,48,5,69 and 79 from Eercise Set Anti-derivatives: f ( h) f ( In Chapter 3 we defined f '( for a continuous function f (, when this it eists, and h0 h developed methods to calculate the derivative f '( quickly for many functions, using a table of derivatives Anti-derivatives: Sometimes we are interested in the inverse problem: If f( is seen as a rate of change (the derivative) of some unknown function, what is that unknown function? In other words, the derivative of which function produces the known (given) function f (? This type of problem is often met and very important in science For eample, a physicist often knows the force (the acceleration) of a particle and wants to find the position An engineer who can measure the variable rate at which water is leaking from a tank wants to know the amount leaked over a certain time period Definition: Consider a continuous function f : D R An anti-derivative of f( is a function Fsuch ( ) that () F '( f ( NOTE: Fis ( ) also denoted by f ( d Based on this definition and on the table of derivatives, we can find the anti-derivative of many functions 15

16 Eample 1: Find d, d, d n, sin( d NOTE: An anti-derivative is not unique, but unique up to an arbitrary constant If Fis ( ) an anti-derivative of f( ), then F( C (where C is an arbitrary constant) represents the entire family of anti-derivatives of f( ) (or the general anti-derivative of f( )) Operations: f ( g( d f ( d g( d f ( d f ( d f ( g( d f ( d g( d (true?) Look at the table of anti-derivatives below, understand and memorize these formulas Figure 1: Anti-derivative of basic functions Eample : Find all functions 5 g( such that g'( 4sin( 5 (In other words, calculate 4 sin( d ) 16

17 Eample 3: Find all functions F( such that F'( e 01 and F ( 0) (In other words calculate the particular anti-derivative F e 01 ( d such that F ( 0) Eample 4: Find F ( such that F' '( and F ( 0) 4, F(1) 1 Eample 5: Do Eamples 6 and 7 from section 49 in the tetbook Homework: Problems 1,4,5,8,10,1,14,16,,3,6,37,41 and 49 from Eercise Set 49 17