3 Limits Involving Infinity: Asymptotes LIMITS INVOLVING INFINITY. 226 Chapter 3 Additional Applications of the Derivative

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1 226 Chapter 3 Additional Applications of the Derivative 52. Given the function f() , complete the following steps: (a) Graph using [, ] b [, ] and [, ] b [ 2, 2]2. (b) Fill in the following table: f() f () f () (c) Find the intercepts and the intercepts. (d) Approimate the relative maimum and relative minimum points to two decimal places. (e) Find the intervals over which f() is increasing. (f) Find the intervals over which f() is decreasing. (g) Find an inflection points. (h) Find the intervals over which the graph of f() is concave upward. (i) Find the intervals over which the graph of f() is concave downward. (j) Show that the concavit changes from upward to downward, or vice versa, when moves from a little less than the point of inflection to a little greater than the point of inflection. (k) Find the largest and smallest values for this function for Repeat Problem 52 for the function f() Limits Involving Infinit: Asmptotes LIMITS INVOLVING INFINITY So far in this chapter, ou have seen how to use the first derivative of a function f() to find where the graph of f() is rising and falling, and to use the second derivative to determine the graph s concavit. In this section, ou will see how to sketch portions of the graph of f() where or f() increase or decrease without bound. We then collect our various curve-sketching methods into a general procedure and use it to analze several eamples. First, however, we need to discuss what is meant b taking a it of f() as increases or decreases without bound. The smbol is used in mathematics to represent a quantit that becomes large (or small) beond an finite bound. It is important to remember that never represents a number. However, there are times when using the smbol in a it epresses useful information.

2 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes 227 In particular, if f() approaches the number L as increases without bound and the number M as decreases without bound, we write f() L and f() M fi fi Geometricall, this means that the graph of f() approaches the horizontal line L as increases without bound and approaches M as decreases without bound (see Figure 3.2). f() = + + c = L f() = L + f() = M = c f() = c = M FIGURE 3.2 A graphical illustration of its involving infinit. Sometimes, taking a it results in unbounded growth or decline, and we use the smbol to represent the result of such behavior. For instance, the graph in Figure 3.2 illustrates a situation in which f() and f() fi c fi c which means that f() decreases without bound as tends toward c from the left and increases without bound as tends toward c from the right. Here is an eample involving an infinite it. Find EXAMPLE as approaches 4 from the left and from the right. Solution First, note that for 2 4 the quantit f() 2 4

3 228 Chapter 3 Additional Applications of the Derivative Eplore! Refer to Eample 3.. Graph f() ( 2)/( 4) using the window [, 9.4] b [ 4, 4] to verif the it results as approaches 4 from the left and the right. Now trace f() for large positive or negative values of. What do ou observe? is negative, so as approaches 4 from the left, f() decreases without bound. We denote this fact b writing 2 fi 4 4 Likewise, as approaches 4 from the right (with 4), f() increases without bound and we write 2 fi 4 4 Limits at infinit involving powers and reciprocal powers can be computed as follows, for n : fi n and fi n More generall, the it at infinit of a polnomial is determined b the highest power term in the polnomial, which increases (or decreases) more rapidl than the other terms of lower degree. Limit at Infinit of a Polnomial If a n, then fi (a n n a n n... a a ) a n n fi That is, to find the it at infinit of a polnomial, take the it of the term of highest degree. EXAMPLE 3.2 Find ( ). fi Solution ( ) ( 3 4 ) fi fi One wa to find the it at infinit of a rational function is to first compare the degrees of the numerator and the denominator and divide numerator and denominator b raised to the smaller of these degrees. This will reduce the problem to one

4 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes 229 a in which most of the terms are of the form, which approach zero as approaches infinit. n Limit at Infinit of a Rational Function To find the it at infinit of a rational function: Step. Compare the degrees of the numerator and the denominator and divide numerator and denominator b raised to the smaller of these degrees. Step 2. Take the its of the new numerator and denominator. Here are two eamples illustrating the technique. EXAMPLE Find. fi Solution Divide the numerator and denominator b 2 to get fi fi 2 3/ / / 2/ Find. fi 3 Solution Divide numerator and denominator b to get Since EXAMPLE / fi 3 fi 3/ fi 2 2 and fi 3

5 23 Chapter 3 Additional Applications of the Derivative it follows that 3 2 fi 3 VERTICAL ASYMPTOTES 2 = 2 FIGURE 3.22 The graph of f(). 2 Limits involving infinit can be used to describe graphical features called asmptotes. In particular, the graph of a function f() is said to have a vertical asmptote at c if f() increases or decreases without bound as tends toward c, from either the right or the left. For instance, consider the rational function f() As approaches 2 from the left ( 2), the functional values decrease without bound, but the increase without bound if the approach is from the right ( 2). This behavior is illustrated in the table and demonstrated graphicall in Figure f() undefined The behavior in this eample can be summarized as follows using the one-sided it notation introduced in Section 5 of Chapter : fi 2 2 and fi 2 2 In a similar fashion, we use the it notation to define the concept of vertical asmptote. Vertical Asmptotes The line c is a vertical asmptote of the graph of f() if either or fi c fi c f() (or ) f() (or ) p() In general, a rational function R() has a vertical asmptote c whenever q() q(c) but p(c). Here is an eample of a function with a vertical asmptote.

6 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes 23 EXAMPLE 3.5 Determine all vertical asmptotes of the graph of f() Solution The numerator p() 2 9 and the denominator q() 2 3 satisf p( 3) q( 3), but fi 3 which means that the graph of f() has a hole at the point ( 3, 2) and 3 is not a vertical asmptote of the graph. However, we have p() and q(), which suggests that the ais (the vertical line ) is a vertical asmptote of the graph of f(). This asmptotic behavior is verified b noting that 2 9 fi 2 3 The graph of f() is shown in Figure fi 3 and 2 9 fi 2 3 Vertical asmptote = 3 3 = 2 9 FIGURE 3.23 The graph of f(). 2 3

7 232 Chapter 3 Additional Applications of the Derivative HORIZONTAL ASYMPTOTES In Figure 3.23, note that the graph approaches the horizontal line as increases or decreases without bound; that is, 2 9 fi 2 3 and 2 9 fi 2 3 In general, when a function f() tends toward a finite value b as either increases or decreases without bound (or both as in our eample), then the horizontal line b is called a horizontal asmptote of the graph of f(). Here is a definition. Horizontal Asmptotes The horizontal line b is called a horizontal asmptote of the graph of f() if f() b fi or f() b fi Eplore! Store the function f() in Eample 3.6 into the equation editor. Compare the graphs using both of the following two windows: the standard [, ] b [, ] and the decimal variant [ 9.4, 9.4] b [, ]. How does the graph of f() behave as ou trace out large values on the graph? EXAMPLE 3.6 Find all vertical and horizontal asmptotes of the function f() Solution Note that q() ( 3)( 5) has zeroes at 3 and 5, and since p() 3 2 has p(3) and p( 5), it follows that 3 and 5 are both vertical asmptotes of the graph of f(). Net, compute 3 2 fi and in a similar fashion, compute fi 2/ 5/ fi which tells ou that 3 is the onl horizontal asmptote. The graph of f() is shown in Figure Details of what goes into sketching the graph are discussed in Eample 3.8.

8 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes = = 5 = FIGURE 3.24 The graph of f() A GENERAL GRAPHING PROCEDURE We now have the tools we need to describe a general procedure for sketching a variet of graphs. A General Procedure for Sketching the Graph of f() Step. Find the domain of f() (that is, where f() is defined). Step 2. Find and plot all intercepts. The intercept (where ) is usuall eas to find, but the intercepts (where f() ) ma require a calculator. Step 3. Determine all vertical and horizontal asmptotes of the graph. Draw the asmptotes in a coordinate plane. Step 4. Find f () and use it to determine the critical numbers of f() and intervals of increase and decrease. Step 5. Determine all relative etrema (both coordinates). Plot each relative maimum with a cap ( ) and each relative minimum with a cup ( ). Step 6. Find f () and use it to determine intervals of concavit and points of inflection. Plot each inflection point with a twist to suggest the shape of the graph near the point. Step 7. You now have a preinar graph, with asmptotes in place, intercepts plotted, arrows indicating the direction of the graph, and caps, cups, and twists suggesting the shape at ke points. Complete the sketch b joining the plotted points in the directions indicated. Be sure to remember that the graph cannot cross a vertical asmptote. Here is a step-b-step analsis of the graph of a rational function.

9 234 Chapter 3 Additional Applications of the Derivative EXAMPLE 3.7 Sketch the graph of the function f() ( ) 2 Solution Steps and 2. The function is defined for all ecept, and the onl intercept is the origin (, ). Step 3. Since f() decreases without bound as approaches, the line is a vertical asmptote. Since fi 2 ( ) fi ( ) 2 the ais ( ) is a horizontal asmptote. Draw the lines and on a coordinate plane. Step 4. Appling the quotient rule, compute the derivative of f(): f () ( )2 () [2( )()] ( ) 4 ( ) 3 The critical numbers are (where f () ) and (where f () does not eist). Place these numbers on a number line and evaluate f () at appropriate test numbers (sa, at 2,, and 3) to obtain the arrow diagram shown. Step 5. The arrow pattern in the diagram obtained in step 4 indicates there is a relative maimum at. Since f(), we plot a cap at,. 4 4 Step 6. Appl the quotient rule again to get f () 2( 2) ( ) 4 Since f () at 2 and f () does not eist at, plot and 2 on a number line and check the sign of f () on the intervals, 2, and 2 to obtain the concavit diagram shown.

10 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes Note that the concavit changes at 2. Since f(2) 9 plot a twist at to indicate the inflection point there. 2, 2 9 Step 7. The preinar graph is shown in Figure 3.25a. Note that the vertical asmptote (dashed line) breaks the graph into two parts. Join the features in each separate part b a smooth curve to obtain the completed graph shown in Figure 3.25b. = = (, 4 ) (2, 2 9 ) (, ) (a) Preinar graph (b) Completed graph FIGURE 3.25 The graph of f(). ( ) 2 In the net eample, we use our curve sketching procedure to sketch the graph of the function in Eample 3.6. EXAMPLE 3.8 Sketch the graph of f()

11 236 Chapter 3 Additional Applications of the Derivative Solution Since ( 5)( 3), the function f() is defined ecept for 5 and 3. The onl intercept is the origin (, ). In Eample 3.6, we found that the graph of f() has vertical asmptotes 3 and 5 and the horizontal asmptote 3. Begin the preinar sketch (Figure 3.26a) b drawing the asmptotes on a coordinate plane and plotting the intercept (, ). Net, use the quotient rule to obtain f () ( 2 2 5)(6) (2 2)(3 2 ) 6( 5) ( 2 2 5) 2 ( 2 2 5) 2 The critical numbers are, 5 (where f () ), and 5 and 3 (where f () is not defined). B plotting the critical numbers on a number line and evaluating f () at appropriate test numbers (sa, at 7,,, 5, and 2), obtain the arrow diagram shown for determining intervals of increase and decrease. The arrow pattern tells us that the function f() is increasing for 5, 5, and 5 and decreasing for 3 and 3 5. There is a relative maimum at and a relative minimum at 5. Since f() and f(5) 2.8, we plot a cap at (, ) and a cup at (5, 2.8) Appl the quotient rule again to get f () 6( ) ( 2 2 5) 3 The second derivative does not eist when 5 and 3, and equals zero when (use our calculator to find the root). B plotting 5, 3, and 22.7 on a number line and evaluating f () at appropriate test numbers, obtain the concavit diagram shown. Note that the graph is concave up for 5 and for , and concave down for 5 3 and The concavit changes at all three subdivision points, but onl 22.7 corresponds to an inflection point since the other two are not in the domain of f(). We find that f(22.7) 2.83 and plot a twist at (22.7, 2.83).

12 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes The preinar graph is shown in Figure 3.26a. Note that the two vertical asmptotes break the graph into three parts. Join the features in each separate part b a smooth curve to obtain the completed graph shown in Figure 3.26b. = 5 = 3 = 5 = 3 = 3 = (a) Preinar graph (b) Completed graph 3 2 FIGURE 3.26 The graph of f() In our last eample, we use the curve sketching procedure to analze an optimization problem from economics. This eample provides a preview of the methods to be developed in the net two sections. EXAMPLE 3.9 A manufacturer estimates that when q units of a particular commodit are produced, the total cost is C(q) 3q 2 5q 75 dollars. At what level of production is the C(q) average cost A(q) minimized? q

13 238 Chapter 3 Additional Applications of the Derivative A(q) 35 Minimal average cost q 5 FIGURE 3.27 The average cost 75 A(q) 3q 5. q Solution The average cost is dollars per unit, which is continuous for q. Since onl positive values of q are meaningful in this contet, our goal is to find the smallest value of A(q) for q. Equivalentl, we want the value of q that corresponds to the lowest point on the graph of the average cost function. To sketch the graph of A(q), we need the derivative which is zero when A(q) C(q) q 3q2 5q 75 3q 5 75 q q 3 75 q 2 A (q) 3 75 q 2 q 2 25 q 5 (reject q 5) Plot the critical number q 5 on a number line and evaluate A (q) on either side (sa, at q and q 7) to obtain the arrow diagram shown for determining intervals of increase and decrease. From the diagram, we see that the average value function decreases for q 5 and increases for q 5. Thus, the smallest value of A(q) occurs when q 5 units are produced, and the minimal average cost is 75 A(5) 3(5) dollars per unit. The graph of A(q) is shown in Figure q P. R. O. B. L. E. M. S 3.3 P. R. O. B. L. E. M. S 3.3 For Problems through, find f(). fi. f() f()

14 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes f() ( 2)( 5) 4. f() ( 2 ) f() 6. f() f() 8. f() f(). f() In Problems through 6, name the vertical and horizontal asmptotes of the given graph

15 24 Chapter 3 Additional Applications of the Derivative In Problems 7 through 24, find all vertical and horizontal asmptotes of the graph of the given function f() 8. f() t 2 9. f() 2. f(t) 2 t 2 t 2 3t f(t) 22. g() t 2 5t t 23. h() 24. g(t) t 2 4 In Problems 25 through 38, sketch the graph of the given function. 25. f() f() f() f(t) 3t 4 4t f() (2 ) 2 ( 2 9) 3. f() f() 32. f() f() 34. f() f() 36. f() f() 38. f() 2 In Problems 39 through 42, diagrams indicating intervals of increase or decrease and concavit are given. Sketch a possible graph for a function with these characteristics

16 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes The function f() is differentiable for all with f () 3 ( ) 2 ( 3) (a) Find intervals of increase and decrease for f(). (b) For what values of do relative minima occur on the graph of f()? Where do relative maima occur? PROFIT 44. For the function in Problem 43, find the second derivative f (). Then determine intervals of concavit for f(). For what values of do inflection points occur? 45. Find constants A and B so that the graph of the function f() A 3 5 B will have 2 as a vertical asmptote and 4 as a horizontal asmptote. Once ou find A and B, sketch the graph of f(). 46. A patient is given an injection of a particular drug, and samples of blood are taken at regular intervals to determine the concentration of drug in the patient s sstem. It is found that the concentration of blood increases rapidl with respect to time for the first hour, after which the concentration continues to increase for three more hours, even though the rate of increase declines. After three hours, the concentration begins to decrease at a decreasing rate for one more hour, after which it graduall decreases toward zero. Sketch a possible graph for the concentration of drug C(t) as a function of time t. 47. The population of a bacterial colon increases at an increasing rate for one hour, after which it continues to increase but at a rate that graduall decreases toward zero. Sketch a possible graph for the population P(t) as a function of time t. 48. A manufacturer can produce radios at a cost of $5 apiece and estimates that if the are sold for dollars apiece, consumers will bu 2 radios a da. At what price should the manufacturer sell the radios to maimize profit?

17 242 Chapter 3 Additional Applications of the Derivative CONSUMER EXPENDITURE AVERAGE COST ANIMAL BEHAVIOR EXPERIMENTAL PSYCHOLOGY DISTRIBUTION COST 49. Suppose consumers bu (p) 6 2p units of certain commodit when the price is p dollars per unit. At what price is the total consumer ependiture F(p) p(p) the greatest? 5. Suppose the total cost in dollars of manufacturing q units is given b the function C(q) 3q 2 q 48. (a) Epress the average manufacturing cost per unit as a function of q. (b) For what value of q is the average cost the smallest? (c) For what value of q is the average cost equal to the marginal cost? Compare this value with our answer in part (b). (d) On the same set of aes graph the total cost, marginal cost, and average cost functions. 5. In some animal species, the intake of food is affected b the amount of vigilance maintained b the animal while feeding. In essence, it is hard to eat hard while watching for predators that ma eat ou. In one model,* if the animal is foraging on plants that offer a bite of size S, the intake rate of food I(S) is given b a function of the form I(S) as S c (a) Assuming that a and c are positive constants, sketch the graph of I(S). Pa special attention to the behavior of the graph as S grows large. (b) Read an article on various was that the food intake rate ma be affected b scanning for predators. Then write a paragraph on how mathematical models ma be used to stud such behavior in zoolog. The reference cited in this problem offers a good starting point. 52. To stud the rate at which animals learn, a pscholog student performed an eperiment in which a rat was sent repeatedl through a laborator maze. Suppose the time required for the rat to traverse the maze on the nth trial was approimatel 2 f(n) 3 minutes. n (a) Graph the function f(n). (b) What portion of the graph is relevant to the practical situation under consideration? (c) What happens to the graph as n increases without bound? Interpret our answer in practical terms. 53. Suppose the number of worker-hours required to distribute new telephone books to % of the households in a certain rural communit is given b the function * A. W. Willius and C. Fitzgibbon, Costs of Vigilance in Foraging Ungulates, Animal Behavior, Vol. 47, Pt. 2 (Feb. 994).

18 Chapter 3 Section 3 Limits Involving Infinit: Asmptotes 243 IMMUNIZATION INVENTORY COST MARINE BIOLOGY 6 f(). Sketch this function and specif what portion of the graph is relevant to the practical situation under 3 consideration. 54. Suppose that during a nationwide program to immunize the population against a certain form of influenza, public health officials found that the cost of inoculating 5 % of the population was approimatel f() million dollars. Sketch 2 this function and specif what portion of the graph is relevant to the practical situation under consideration. 55. A manufacturer estimates that if each shipment of raw materials contains units, the total cost of obtaining and storing the ear s suppl of raw materials will be 8, C() 2 dollars. Sketch the relevant portion of the graph of this cost function and estimate the shipment size that minimizes cost. 56. When a fish swims upstream at a speed v against a constant current v w, the energ it epends in traveling to a point upstream is given b a function of the form E(v) Cvk v v w where C and k 2 is a number that depends on the species of fish involved.* (a) Show that E(v) has eactl one critical number. Does it correspond to a relative maimum or a relative minimum? (b) Note that the critical number in part (a) depends on k. Let F(k) be the critical number. Sketch the graph of F(k). What can be said about F(k) if k is ver large? PHYSICAL CHEMISTRY 57. In phsical chemistr, it is shown that the pressure P of a gas is related to the volume V and temperature T b van der Waals equation: P a V 2 (V b) nrt where a, b, n, and R are constants. The critical temperature T c of the gas is the highest temperature at which the gaseous and liquid phases can eist as separate phases. (a) When T T c, the pressure P is given as a function P(V) of volume alone. Sketch the graph of P(V). * E. Batschelet, Introduction to Mathematics for Life Scientists, 2nd ed., New York: Springer-Verlag, 976, page 28.

19 244 Chapter 3 Additional Applications of the Derivative (b) The critical volume V c is the volume for which P (V c ) and P (V c ). Show that V c 3b. (c) Find the critical pressure P c P(V c ) and then T c in terms of a, b, n, and R. 58. Evaluate the it a n n a n n... a a fi b m m b m m... b b for constants a, a,..., a n and b, b,..., b m in each of the following cases: (a) n m (b) n m (c) n m (Note: There are two possible answers, depending on the signs of a n and b m.) 59. Let f() /3 ( 4). (a) Find f () and determine intervals of increase and decrease for f(). Locate all relative etrema on the graph of f(). (b) Find f () and determine intervals of concavit for f(). Find all inflection points on the graph of f(). (c) Find all intercepts for the graph of f(). Does the graph have an asmptotes? (d) Sketch the graph of f(). 4 Optimization 6. Repeat Problem 59 for the function f() 2/3 (2 5). 6. Repeat Problem 59 for the function f() Let f() and let g(). 2 2 (a) Use a graphing utilit to sketch the graph of f(). What happens at? (b) Sketch the graph of g(). Now what happens at? You have alread seen several situations where the methods of calculus were used to determine the largest or smallest value of a function of interest (for eample, maimum profit or minimum cost). In most such optimization problems, the goal is to find the absolute maimum or absolute minimum of a particular function on some relevant interval. The absolute maimum of a function on an interval is the largest value of the function on that interval and the absolute minimum is the smallest value. Here is a definition of absolute etrema.