Chapter5 - 5.1 I Extrema and the Mean-Value Theorem 213...

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Unformatted text preview: 5.1 I Extrema and the Mean-Value Theorem 213 Assume that f is continuous on [—1, 1] and differentiable on (-1, 1), with f (O) = 2 and f’(x) = 0 for allx e (—1, 1). Find f(x). Solution Corollary 2 tells us that f (.r.) is a constant. Since we know that f (0) = 2, we have f(x)=2f0rallx E[—1.1]. I Proof of Corollary 2 Letxl, x; e (a, b),.n < x2. Then f satisfies the assumptions of the MVT on the closed interval [x1.x2]. Therefore, there exists a number c 6 (x1, x2) such that f(xz) — f(xi) = f ’(c) 12 - X1 Since f ’(c) = 0, it follows that f (x2) = f (x1). Finally, because x1, x2 are arbitrary numbers from the interval (a, b), we conclude that f is constant. I W Show that Solution sinzx + coszx = l for all x e [0. 211] This identity can be shown without calculus, but let’s see what we get if we use Corollary 2. We define f (x) = sinzx + cos2 x, 0 5 x 5 2n. Then f (x) is continuous f’(x) = 25inxcosx — Zeosxsinx = 0 Using Corollary 2 now, we conclude that f (x) is equal to a constant on [0, 2n]. To find the constant, we need only evaluate f (x) at one point in the interval. say. x = 0 We find that This proves the identity. section 5.1 Problems I 5.1.1 In Problems [—8, each function is continuous and defined on a closed interval. It therefore satisfies the assumptions of the extreme- value theorem. With the help of a graphing calculator, graph each function and locate its global extrema. (Note that a function may assume a global extremum at more than one point.) 1- ftx)=2x—1.05xsl 2. f(x)=—x2+l,—15x51 3. f(x) = sin(2x), 0 S x S 7' x 4. f(x)=cos—2—,05x52n 5- [(x)=lxl.—lsxsl 6- f(x)=(x—l)2(x+2),—25x52 7. f(.t)=e""',—15x_<_1 8. f(,t) = ln(x +1).0 s x 5 2 9- Sketch the graph of a function that is continuous on the closed interval [0, 3] and has a global maximum at the left endpoint and a global minimum at the right endpoint. 10. Sketch the graph ofa function that is continuous on the closed interval [—2, 1] and has a global maximum and a global minimum In the interior of the domain of the function. "- Sketch the graph of a function that IS continuous on the open interval (0,2) and has neither a global maximum nor a global minimum in its domain. f(0) = sin2 0 + cos2 0 =1 12. Sketch the graph of a function that is continuous on the closed interval [1. 4], except at x = 2, and has neither a global maximum nor a global minimum in its domain. - l 5.1.2 In Problems 13—18, use a graphing calculator to determine all local and global extrema of the functions on their respective domains 13. f(x) =3—x,x e [—1.3) i 14. f(x) = 5 +2x,x e (-2,1) 15. f(x) =x2 — 2,x e [—1.1] 16. f(x) = (x -— 2)2,x E [0, 3] 17. f(x) = —x2 + 1,): e [—2, 1] 18. f(x) =x2 —x,x e [0.1] In Problems 19—26, find e such that f ’(c) = 0 and determine whether f (x) has a local extremum at x = c. 19. f(x) = x2 20. f(x) = (x — 4)2 2L f(x) = —x2 22. = —(x + 3)2 ' 23. for) = x3 24. f(x) = x5 - 25. f(x) = (x +1)3 26. f(x) = —(.r — 3)-‘ ; 27. Show that f(x) = lxl has a local minimum atx = 0 but f(x) l is not differentiable at x = O. 28. Show that f(.r) = [x — 1; has a local minimum at .r = 1 but f(.r) is not differentiable at x = l. j 29. Show that fIx) = fxz — ll has local minima atx == 1 and x = ~l but f(x) is not differentiable at x = l or x = —1. 214 Chapters I Applications of Differentiation 30. Show that f (x) = —|x2 — 4| has local maxima at x = 2 and x = ——2 but f (x) is not differentiable at x = 2 or x = —2. 31. Graph 7 ‘ f(x) = l1 — lel. —15x :2 and determine all local and global extrema on [~1. 2]. 32. Graph f(x) = —|le -2l, —3 s x s 3 and determine all local and global extrema on [—3, 3]. 33. Suppose the size of a population at time t is N (t) and its growth rate is given by the logistic growth function dN N -—-'= -—— , t>0 dt rN (l K) _ where r and K are positive constants. (in) Graph the growth rate % as a function of N for r = 2 and K = 100, and find the population size for which the growth rate is maximal. (b) Show that f(N) = rN(1 -— N/K), N z 0, is differentiable for N > 0, and compute f ’(N). in (a) whenr =2andK = 100. 34. Suppose that the size of a population at time t is N (t) and its growth rate is given by the logistic growth function dN N ——= —— , [>0 d! rN<1 K) _ where r and K are positive constants. The per capita growth rate is defined by 1 dN 8‘”) = m: (a) Show that (N) - r (1 -— g " K (b) Graph g(N) as a function of N for N z 0 when r = 2 and K = 100, and find the population size for which the per capita growth rate is maximal. 1 5.1.3 35. Suppose f (x) = x2, x 6 [0,2]. (I) Find the slope of the secant line connecting the points (0. 0) and (2, 4). (b) Find a number 0 e (O, 2) such that f ’(c) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in (0, 2). 36. Suppose f(x) = 1/x,x 6 [1,2]. (9) Find the slope of the secant line connecting the points (1,1) and (2. 1/2). (b) Find a number c e (1, 2) such that f ’(c) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in (1, 2). 37. Suppose that f(x) = x2, x e [—1. 1]. (3) Find the slope of the secant line connecting the points (-1, 1) and (1, 1). (11) Find a numberc e (—1, 1) such that f’(c) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in (—1. 1). 38. Suppose that f(x) = x2 — x — 2,x e [—1, 2]. (It) Find the slope of the secant line connecting the points (—1, 0) and (2, 0). (b) Find a number c e (-1. 2) such that f’ (c) is equal to the slope of the secant line you computed in (a), and explain why such a number must exist in (—1. 2). 39. Let f (x) = x(1 ’— x). Use the MVT to find an interval that contains a number c such that f ’(c) = 0. 40. Let f (x) = 1/ (1 + x2). Use the MVT to‘find an interval that contains a number c such that f ’(c) = 0. 41. Suppose that f (x) = —-x2 + 2. Explain why there exists a point c in the interval (-1, 2) such that f ’(c) = —1. 42. Suppose that f (x) = x3. Explain why there exists a point e in the interval (—1, 1) such that f’(c) = 1. 43. Sketch the graph of a function f (x) that is continuous on the closed interval [0. 1] and differentiable on the open interval (0, 1) such that there exists exactly one point (c. f (0)) on the graph at which the slope of the tangent line is equal to the slope of the secant line connecting the points (0. f (0)) and (1. f (1)). Why can you be sure that there is such a point? 44. Sketch the graph of a function f (x) that is continuous on the closed interval [0, 1] and differentiable on the open interval (0, 1) such that there exist exactly two points (c1, f ((31)) and (c2, f (c;)) the a h at which the 310 e of the tan ent lines is e ual to the slope of the secant line connecting the points (0. f (0)) and (1, f (1)). Why can you be sure that there is at least one such point? 45. Suppose that f (x) = x2,x e [0, b]. (8) Compute the slope of the secant line through the points (a. f(a)) and (b. f(b)). (b) Find the point e e (a, b) such that the slope of the tangent line to the graph of f at (c, f (c)) is equal to the slope of the secant line determined in (a). How do you know that such a point exists? Show that c is the midpoint of the interval (a. b); that is, show that c = (a + b) /2. 46. Assume that f is continuous on (a, b] and differentiable on (a, b). Show that if f (a) < f (b), then f’ is positive at some point between a and b. 47. Assume that f is continuous on [a, b] and differentiable on (a. b). Assume further that f (a) = f (b) = 0 but f is not constant on [a, b]. Explain why there must be a point C] e (a, b) with f’(c1) > 0 and a point c; e (a, b) with f’(cz) < 0. 48. A car moves in a straight line. At time I (measured in seconds), its position (measured in meters) is 12 t =—t,0<t< 0 S() 10 _ -1 (it) Find its average velocity between t = 0 and t = 10. (b) Find its instantaneous velocity for! 6 (0,10). (c) At what time is the instantaneous velocity of the car equal to its average velocity? 49. A car moves in a straight line. At time I (measured in seconds), its position (measured in meters) is 1 t=——3,0<t<5 5() 100! _ _ (a) Find its average velocity between t = 0 and t = 5. (b) Find its instantaneous velocity for: e (O. 5). (it) At what time is the instantaneous velocity of the car equal to its average velocity? ‘ 50. Denote the population size at time t by N (t), and assume that N(0) = 50 and ldN/dtl 5 2 for allt 6 [0,5]. What can you say about N (5)? 5.2 I Monotonicity and Concavity 215 51. Denote the biomass at timet by 8(1), and assume that B(0) := 56. We have seen that 3 and [dB/dtl 5 1 for all t e [0, 3]. What can you say about 3(3)? _ u 52. SuppOse that f is differentiable for all x e R and, f(x) — foe fulnhe’rgore' mat f samfies “0) = 0 and 1 S f (x) S 2 for satisfies the differential equation al x > - (a) Use Corollary 1 of the MVT to show that _f = rflx) x 5 f (x) 3 2x (1x for all x Z 0. with f (0) = f0. This exercise will show that f (x) is in fact the (b) Use your result in (a) to explain why f (1) cannot be equal to only solution. Suppose thatr is a constant and f is a differentiable 3. function, (c) Find an upper and a lower bound for the value of f(1). __ = rflx) (5'5) 53. Suppose that f is differentiable for all x 6 R with f (2) = 3 dx and f’(x) = 0 for allx e R. Find f(x). for all x e R. and f(0) = f0. The following steps will show that 54. Suppose that f(x) = e‘L", x 6 [—2, 2]. f(x) = foe”, x 6 R: is the only solution of (5.5). (a) Show that f (—2) = f (2). (a) Define the function (b) Compute f '(x), where defined. _. _” (c) Show that there is no numberc e (—2, 2) such that f’(c) = 0. H") _ fix)" ' x E R (d) Explain why your results in (a) and (c) do not contradict Rolle’s theorem. (e) Use a graphing calculator to sketch the graph of f (x). W W differentiable for all x e R and satisfies (b) Use (a) and (5.5) to show that F ’(x) = O for all x e R. Use the product rule to show that a If“) _ f0,“ < Ix _ yIZ (53) (c) Use Corollary 2 to show that F(x) is a constant and, hence, ‘ v— . F(x)=F(0)=fu. for all x, y e R, then f (x) is constant. [Hmt. Show that (5.3) (d) Showman (C) implies that implies that lim w = o (5.4) fl, = my” x-oy x — and use the definition of the derivative to interpret the left~hand and therefore, side of (5.4).] f(x) = foe” I 5.2 Monotonicity and Concavity Fish are indeterminate growers; they increase in body size throughout their life. How- ever, as they become older, they grow proportionately more slowly. Their growth is often described mathematically by the von Bertalanffy equation, which fits a large number of both freshwater and marine fishes This equation is given by L(x) = Loo _ (Loo— Lo)?“ where L(x) denotes the length of the fish at age x, L0 the length at age 0, and Lao the asymptotic maximum attainable length. We assume that Loo > L0. K is related to how quickly the fish grows. Figure 5.20 shows examples for two different values of K; Leo and L0 are the same in both cases We see from the graphs that for larger K , the asymptotic length Loo is approached more quickly. The fact that fish increase their body size throughout their life can be expressed mathematically by the first derivative of the function L(x). Looking at the graph, we see that L(x) is an increasing function of x: The tangent line at any point of the graph has a positive slope, or, equivalently, L’(x) > 0. We can compute L’(x) = 1mm — Low“ Since Loo > L0 (by assumption) and r“ > 0 (this holds for all x, regardless of K), we see that, indeed, L'(x) > O. The graph of L’(x) is shown in Figure 5.21. The graph of L’(.t) shows that L’(x) is a decreasing function ofx: Although fish increase their body size throughout their life, they do so at a rate that decreases with age. Mathematically, this relationship can be expressed with the second derivative of W 222 Chapter5 I Applicationsrof Differentiation "Section 5.2 Problems-2'- ! 5.2.1 and 5.2.2 In Problems 1—20, determine where each function is increasing, decreasing, concave up, and concave down. With the help of a graphing calculator, sketch the graph of each function and label the intervals where it is increasing, decreasing, concave up, and concave down. Make sure that your graphs and your calculations agree. 1.y=3x—x2.xeR 2.y=x2+5x.xeR 3.y=x2+x—4,xeR 4.‘y=x2—x+3,xeR S. y = —§x3+ §x2 —3x+4,x E R (3) Graph the growth rate if} as a function of N for r = 3 and K =10. I ' (b) The function f(N) = rN(1—- N /K), N z 0, is differentiable for N > 0. Compute f’(N), and determine where the function f (N) is increasing and where it is decreasing. 27. Logistic Growth Suppose that the size of a population at time t is N (t.)'and the growth rate of the population is given by the logistic growth function d—IY—=rN(1—E), t_>_0 6.y=(x—-2)3+3,xER dt K _ l 3 . 7' y = x + I" Z "‘1 8' y ‘ (3x 5' 1) / ‘x e R where r and K are positive constants The per capita growth rate 1 ‘ ' d n d b = _ 10. = is e ne y 9'y x’x’é0 y x2+3 ldN ' N 5 3(N)=N§T=’ 1'; 11. (x2 +1)‘/3,x e R 12. y = ,x 5e 2 x ‘ 2 (a) Graph g(N) as a function of N for N z 0 when r = 3 and x2 K =10 _ = , _ 14. = , z 0 ' . . . , 13 y (1 +x)2 x 9!: l y x2+t x (h) The function g(N) =r(1—N/K),N a 0.18d1fferenttabletor 16. y=cos[7r(x2—1)],25x§3 l7. y=e‘.xeR 18. y=lnx,x>0 19. y = e"2/2, x e R 21. Sketch the graph of (a) a function that is increasing at an accelerating rate; and (h) a function that is increasing at a decelerating rate. (c) Assume that your functions in (a) and (b) are twice differen- tiable. Explain in each case how you could check the respective properties by using the first and the second derivatives Which of the functions is concave up, and which is concave down? 22. Show that if f (x) is the linear function y = mx + b. then increases in f (x) are proportional to increases in x. That is, if we increase x by Ax, then f (x) increases by the same amount Ay, regardless of the value of x. Compute Ay as a function of Ax. 23. We frequently must solve equations of the form f (x) = 0. When f is a continuous function on [a, b] and f (a) and f (b) have opposite signs, the intermediate-value theorem guarantees that there exists at least one solution of the equation f (x) = 0 in [a, b]. (a) Explain in words why there exists exactly one solution in (a, b) if, in addition, f is differentiable in (a, b) and f ’(x) is either strictly positive or strictly negative throughout (a, b). (b) Use the result in (a) to show that 20.y= xeR 1+e'x’ x3—4x+1=0 has exactly one solution in {—1. 1]. 24. First-Derivative Test for Monotonicity Suppose that f is continuous on [a,b] and differentiable on (a.b). Show that if f’(x) < Ofor allx e (a, b), then f is decreasing on [a, bl. 25. Second-DerivativeTestforConcavity Suppose that f is twice differentiable on an open interval I. Show that if f”(x) < 0. then f is concave down. 26. Suppose the size of a population at time t is N(tjt. and the growth rate of the population is given by the logistic growth function where r and K are positive constants. “W is increasing and where it is decreasing. 28. Resource-Dependent Growth The growth rate of a plant depends on the amount of resources available. A simple and frequently used model for resource-dependent growth is the Monod model, according to which the growth rate is equal to (IR R = , R > 0 f ( ) k + R _ where R denotes the resource level and a and k are positive constants. When is the growth rate increasing? When is it decreasing? 29. Population Growth Suppose that the growth rate of a population is given by f(N) = N (1 - where N is the size of the population. K is a positive constan‘ denoting the carrying capacity, and 0 is a parameter greater that 1. Find f ’(N ), and determine where the growth rate is increasing and where it is decreasing. l 30. Predation Spruce budworms are a major pest that defoliate balsam fir. They are preyed upon by birds. A model for the pe capita predation rate is given by aN k.2 + N2 where N denotes the density of spruce budworms and a an k are positive constants. Find f ’(N), and determine where th predation rate is increasing and where it is decreasing. 31. Host—Parasitoid Interactions Parasitoids are insects that la their eggs in, on, or close to other (host) insects. Parasitoid larva then devour the host insect. The likelihood of escaping parasitisr may depend on parasitoid density. One model expressing th dependence sets the probability of escaping parasitism equal to f(N)= [(P) = e“"" where P is the parasitoid density and a is a positive constan Determine whether the probability of escaping parasitisr increases or decreases with parasitoid density. 32, Host—Parasitoid Interactions As an alternative to the model set forth in Problem 31, another model sets the probability of escaping parasitism equal to -k ftP)=(l+%) where P is the parasitoid density and a and k are positive con- stants. Determine whether the probability of escaping parasitism increases or decreases with parasitoid density. 33. Tree Growth Suppose that the height y in feet of a tree as a function of the age x in years of the tree is given by y = 11713—10“, x > 0 (a) Show that the height of the tree increases with age. What is the maximum attainable height? (1;) Where is the graph of height versus age concave up, and where is it concave down? (c) Use a graphing calculator to sketch the graph of height versus age. (d) Use a graphing calculator to verify that the rate of growth is greatest at the point where the graph in (0) changes concavity. gies: polycarpy, in which reproduction occurs repeatedly during the lifetime of the organism, and monocarpy, in which repro- duction occurs only once during the lifetime of the organism. (Bamboo. for instance, is a monocarpic plant.) The following quote is taken from Iwasa et al. (1995): The optimal strategy is polycarpy (repeated repro- duction) if reproductive success increases with the investment at a decreasing rate, [or] monocarpy (“big bang“ reproduction) or intermittent reproduction if the reproductive success increases at an increasing rate. (a) Sketch the graph of reproductive success as a function of reproductive investment for the cases of (i) polycarpy and (ii) monocarpy. (b) Given that the second derivative describes whether a curve bends upward or downward, explain the preceding quote in terms of the second derivative of the reproductive success function. 35. Pollinator Visits Assume that the formula (Iwasa et al., 1995) X(F) = cFV where c is a positive constant, expresses the relationship between the number of flowers on a plant, F, and the average number of pollinator visits, X(F). Find the range ofvalues for the parameter y such that the average number of pollinator visits to a plant increases with the number of flowers F but the rate of increase decreases with F. Explain your answer in terms of appropriate derivatives of the function X(F). 36. Pollinator Visits Assume that the dependence of the average number of pollinator visits to a plant, X, on the number of flowers, F , is given by X (F) = CF V where y is a positive constant less than 1 and c is a positive constant (Iwasa et al., 1995). How does the average number of pollen grains exported per flower, E(F), change with the number of flowers on the plant, F, if E(F) is proportional to 1— exp [— X(F)] F Where It is a positive constant? 5.2 I Monotonicity and Concavity 223 37. Population Size Denote the size of a population by N (t), and assume that N (t) satisfies where a is a positive constant. (a) Show that the nontrivial equilibrium N “ satisfies e—rIN' = N‘ (b) Assume now that the nontrivial equilibrium N ‘ is a function of the parameter a. Use implicit differentiation to show that N‘ is a decreasing function of a. 38. Population Size Denote the size of a population by N (r), and assume that N (t) satisfies dN N —-= —— —Nl N dz (1 K) n where K is a positive constant. (a) Show that if K > 1, then there exists a nontrivial equilibrium N‘ > 0 that satisfies 1 - —-- = l N. K n (b) Assume now that the nontrivial equilibrium N‘ is a function of the parameter K. Use implicit differentiation to show that N ‘ is an increasing function of K. 39. lntraspecific Competition (Adapted from Bellows, I981) Suppose that a study plot contains N annual plants, each of which produces S seeds that are sown within the same plot. The number of surviving plants in the next year is given by NS A(N) —- ————1 + (“NV (5.6) for some positive constants a and b. This mathematical model incorporates density-dependent mortality: The greater the num- ber of plants in the plot, the lower is the number of surviving offspring per plant, which is given by A(N)/N and is called the net reproductive rate. (it) Use calculus to show that A(N )/ N is a decreasing function of N (b) The following quantity, called the k—value, can be used to quantify the effects of intraspecific competition (i.e., competition between individuals of the same species): k = log [initial density] — log [final density] Here, “log” denotes the logarithm to base 10. The initial density is the product of the number of plants (N) and the number of seeds each plant produces (S). The final density is given by (5.6). Use the expression for k and (5.6) to show that NS k = |og[NS] — log’:l + (ah/V] = log[l + (uN)"] We typically plot k versus log N; the slope of the resulting curve is then used to quantify the effects of competition. (i) Show that (I log N l dN ‘ N ln 10 where ln denotes the natural logarithm. 224 Chapter5 I Applications of Differentiation (ii) Show that dk dk b = 10 N — = —— dlogN a“ ) dN 1 + (am-b "1 Find (n) l. dk NE"... d log N (iv) Show that if dk < 1 dlogN then A(N) is increasing, whereas if dk > 1 dlogN then A(N) is decreasing. [Hint Compute A’(N).] Explain in words what the two inequalities mean with respect to varying the initial density of seeds and observing the number of surviving plants the next year. (Hint: The first case is called undercompensation and the second case is called avercompensation.) (v) The case dk =1 is referred to as exact compensation. Suppose that you plot k versus log N and observe that, over a certain range of values of N, the slope of the resulting curve is equal to 1. Explain what this means. 40. (Adapted from Reiss, I989) Suppose that the rate at which body weight W changes with age x is -—— or W“ dx where a is some species-specific positive constant. (a) The relative growth rate (percentage weight gained per unit of time) is defined as (5-7) 1 dW W dx What is the relationship between the relative growth rate and body weight? For which values of a is the relative growth rate increasing, and for which values is it decreasing? (b) As fish grow larger, their weight increases each day but the relative growth rate decreases. If the rate of growth is described by (5.7), what values of a can you exclude on the basis of your results in (8)? Explain how the increase in percentage weight (relative to the current body weight) differs for juvenile fish and for adult fish. 41. Allometric Growth Allometric equations describe the seal- ing relationship between two measurements, such as tree height versus tree diameter or skull length versus backbone length. These equations are often of the form Y = bx" (5.8) where b is some positive constant and a is a constant that can be positive, negative, or zero. (a) Assume that X and Y are body measurements (and therefore positive) and that their relationship is described by an allometric equation of the form (5.8). For what values of a is Y an increasing function of X, but one such that the ratio Y/X decreases with increasing X? Is Y concave up or concave down in this case? (b) In vertebrates, we typically find [skull length] or [body length]“ for some a e (0, 1). Use your answer in (a) to explain what this means for skull length versus body length in juveniles versus adults; that is, at which developmental stage do vertebrates have larger skulls relative to their body length? hydrogen ions, denoted by [IF], and is defined as pH = -— log[H+] Use calculus to decide whether the pH value of a solution increases or decreases as the concentration of H“ increases. 43. Allometric Growth The differential equation d 1:1,! x dx describes allometric growth, where k is a positive constant. Assume that x and y are both positive variables and that y = f (x) is twice differentiable. Use implicit differentiation to determine for which values of k the function y = f (x) is concave up. 44. Population Size Let N (r) denote the population size at time t, and assume that N (r) is twice differentiable and satisfies the differential equation dN dr — r where r is a real number. Differentiate the differential equation with respect to t, and state whether N (t) is concave up or down. I 5.3 Extrema, Inflection Points, and Graphing l 5.3.] Extrema If f is a continuous function on the closed interval [a, b], then f has a global maximum and a global minimum in [a, b]. This is the content of the extreme-value theorem, which is an existence result: It tells us only that global extrema exist under certain conditions, but it does not tell us how to find them. Our strategy for finding global extrema in the case where f is a continuous function defined on a closed interval will be, first, to identify all local extrema of the function and, then, to select the global extrema from the set of local extrema. , If f is a continuous function defined on an open interval or half-open interval, the . existence of global extrema is no longer guaranteed, and we must compare the local extrema with the behavior of the function near the open boundaries of the domain. ; (Sec Example 5 in Section 5.1.) In particular, if f (x) is defined on R, we need to ‘ MM l l 5.3 a Extrema, Inflection Points, and Graphing 235 We find that f (x) is concave up forx < —1 and x > 1 and is concave down for —1 < x < 1. There are two inflection points, one atx = —1, namely, (—1,e‘1/2), and the other at x = 1, namely, (1, e"'/2). There are no other inflection pointS, since f "(.t) is defined for all x e R. STEP 4. We have lim f (x) = 0 and .I-O-W lim f(x) = 0 .r-> +00 This shows that y = 0 is a horizontal asymptote. The graph of f (x) is shown in Figure 5.49. ; l Inflection point (_1’ e-l/2) ~5 —4 —3 —2 Figure 5.49 The graph of f(x) = e"z/Z. ‘ Section 5.3 Problems 1 5.3.1 Find the local maxima and minima of each of the functions in Problems 1—16. Determine whether each function has absolute maxima and minima and find their coordinates. For each function, find the intervals on which itis increasing and the intervals on which it is decreasing. l.y=(2—.t)2,—25x53 2.y=‘/x-1,15.\152 3.y=ln(2x—1),15x52 4.y=lnfil-,x>0 5-y=xe"‘.05x51 6.y=|16—x2l,-55x58 7-y=(.r—l)3+l,xeR 8.y=x3—3x+1.xeR 9- y=cos(nx2),—15x_<_l 10. y = sin[27r(x —3)],2 5 x 5 3 11- y=e""',xeR -12. y=e“z/‘,xek 13°y=§x3+%x2—6x+2.xeR 14-y=x2(1—x),xeR 15. y=(x—1)”3.XER l6.y=‘/l+.r2.xeR 17. [This problem illustrates the fact that f’(c) = 0 is not a sufficient condition for the existence of a local extremum of a differentiable function] Show that the function f(x) = x3 has a horizontal tangent at x = 0; that is, show that f'(0) = 0, but f(x) does not change sign at x = 0 and. hence, f(x) does not have a local extrcmum at x = 0. 18. Suppose that f(x) is twice differentiable on R, with f(x) > t) forx e R. Show that if f(x) has a local maximum at x = c, then 80') = ln f(x) also has a local maximum at x = c. f (x) Max (0. 1) Inflection point (I. e‘ “2) 0 l 2 3 4 5 x t I 5.3.2 In Problems 19—24, determine all inflection points. 19. f(x)=x3—2,x ER 21. f(x) = e"“2.x 3 0 7r 7: 23. =1 ’—— _ f(x) anx 2 <x< 2 20. f(x)=(x—3)5,0€R 22. f(x)=xe",x 20 24. f(x) =lnx+%,x >0 25. [This problem illustrates the fact that f”(c) = 0 is not a sufficient condition for an inflection point of a twice-differentiable function] Show that the function f(x) = x‘ has f”(0) = 0 but that f”(x) does not change sign at x = 0 and, hence, f(x) does not have an inflection point at x = 0. 26. Logistic Equation Suppose that the size of a population at time t is denoted by N (t) and satisfies 100 N! =---——— () 1+3e'2' for! 2 0. (a) Show that N(O) = 25. (b) Show that N(t) is strictly increasing. (c) Show that lim N(!) = 100 t [600 (d) Show that N(t) has an inflection point when N(t) = SO—that , is. when the size of the population is at half its limiting value. ‘4 (C) Use your results in (a)—(d) to sketch the graph of N(r). . _ ,, __._ .. fl. *.__4_*w,~h___~_~_____________—————__h___________ 236 Chapters I Applications of Differentiation 1 5.3.3 Find the local maxima and minima of the functions in Problems 27-34. Determine whether the fiinctions have absolute maxima and minima, and, if so, find their coordinates Find inflection points. Find the intervals on which the function is increasing, on which it is decreasing, on which it is concave up, and on which it is concave down. Sketch the graph of each function. 27. y=§g3_zxZ—6x+2for—25x_<_5 28' y=x‘-2x2,X€R 29. y=|x2-9|,-4Sx55 30. y = ./|xl,x ER 31, y =x+cosx,x ER 71' yr) 32.y—tanx—x,xe 2,2 x2 — 1 33- y = x2 ls-x e R 34. 'y =ln(x2+1),x ER 35. Let x f(x) = _ , x 751 (a) Show that lim f(x) = 1 x—>—oo and lim f(x) = 1 x-H-co That is, show that y = 1 is a horizontal asymptote of the curve X x—l ' (b) Show that lim f(x) = -oo x—vl" and lim f(x) = +00 x->l+ That is, show that x = 1 is a vertical asymptote of the curve y = :i-r- . (c) Determine where f (x) is increasing and where it is decreasing. Does f (x) have local extrema? ' (d) Determine where f (x) is concave up and where it is concave down. Does f (x) have inflection points? (e) Sketch the graph of f (x) together with its asymptotes 36. Let 2 f(x)-- x2_1. x95 1.1 (a) Show that lim f(x) = 0 x—»+oo and lim f(x) = 0 That is, show that y = 0 is a horizontal asymptote of f (x). (b) Show that lim f(x) = —00 .r-s—l" and lim f(x) = +00 .r—v—l“ and that lim f(x) = +00 .t—rl' and lim f (x) = —oo x—vfi' That is, show that x = -—1 and x = 1 are vertical asymptotes of f (x)- (c) Determine where f (x) is increasing and where it is decreasing. Does f (x) have local extrema? (d) Determine where f (x) is concave up and where it is concave down. Does f (x) have inflection points? (e) Sketch the graph of f (it) together with its asymptotes. 37. Let 2 2x —-5 f(x)- x+2v (a) Show that x = —2 is a vertical asymptote. (b) Determine where f(x) is increasing and where it is decreasing. Does f (x) have local extrema? (c) Determine where f (x) is concave up and where it is concave down. Does f (x) have inflection points? ((1) Since the degree of the numerator is one higher than the degree of the denominator, f (x) has an oblique asymptote. Find xaé—Z wwa (e) Sketch the graph of f (x) together with its asymptotes 38. Let ‘ sin x f(X)=Ty x950 (a) Show that y = 0 is a horizontal asymptote. (is) Since f (x) is not defined at x = 0, does this mean that f (x) has a vertical asymptote at x = 0? Find lim,_,0+ f (x) and lim‘_,(,_ f(x). (c) Use a graphing calculator to sketch the graph of f (x). 39. Let x2 f(x)=1+x2. (:1) Determine where f (x) is increasing and where it is decreasing. (b) Where is the function concave up and where is it concave down? Find all inflection points of f (x). (c) Find limHioo f (x) and decide whether f (x) has a horizontal asymptote. (d) Sketch the graph of f (x) together with its asymptotes and; inflection points (if they exist). 40. Let xeR xk f(x)=1+xk. where k is a positive integer greater than 1. (:1) Determine where f (x) is increasing and where it is decreasing. x20 (b) Where is the function concave up and where is it concave; down? Find all inflection points of f (x). (c) Find lim,_,°° f (x) and decide whether f (x) has a horizontal asymptote. (d) Sketch the graph of f (x) together with its asymptotes and inflection points (if they exist). 41. Let x a+x' f(x):- where a is a positive constant. , (a) Determine where f(x) is increasing and where it is decreasing. ‘ x30 s \ t a .5 5.4 I Optimization 237 (b) Where is the function concave up and where is it concave 43. Population Growth Suppose that the growth rate of a down? Find all inflection points of f (x). population is given by (c) Find limHoe f (x) and decide. whether f (x) has a horizontal 9 asymptote. _ _ d (d) Sketch the graph of [(x) together with its asymptotes and fuv) " N <1 (K N Z 0 inflection points (if they exist). 42- Let where N is the size of the population, K is a positive constant _ 2 denotin the car in ca acit ,and9isa arameter reater than foo—1+ x.xeR 2; we 9 y P g e- 1. Find the population size for which the growth rate is maximal. (:1) Determine where f(x) is increasing and where it is , 44. Predation Rate Spruce budworms are a major pest that decreasmg' defoliate balsam fir. They are preyed upon by birds. A model for (b) Where is the function concave up and where [S it concave the per capita predation rate is given by down? Find all inflection points of fix). (c) Find limb.” f (x) and decide whether f (x) has a horizontal aN asymptote. f(N) ___ k2 + N: (d) Find lim,,_°o f (x) and decide whether f (x) has a horizontal asymptote. where N denotes the density of spruce budworm and a and k are (9) Sketch the graph of f (x) together with its asymptotes and positive constants. For which density of spruce budworms is the inflection points (if they exist). per capita predation rate maximal? There are many situations in which we wish to maximize or minimize certain quantities. For instance, in a chemical reaction, you might wish to know under which conditions the reaction rate is maximized. In an agricultural setting, you might be interested in finding the amount of fertilizer that would maximize the yield of some crops. In a medical setting, you might wish to optimize the dosage of a drug for m Chemical Reaction Consider the chemical reaction A + B —> AB In Example 5 of Subsection 1 2 2, we found that the reaction rate 15 given by the function R(x) = k(a — x)(b —— x), 0 5 x 5 min(a. b) Rm where x is the concentration of the product AB and min(a b) denotes the minimum 25 Absolute maximum of the two values of a and b. The constants a and b are the concentrations of the 20 (0, 20) reactants A and B at the beginning of the reaction. To be concrete, we choose k = fir) —- 2. a = 2, and b = 5. Then R(x)=2(2——x)(5—x) forOSxSZ 0 05 I 15 2 2 5 (See Figure 5.50.) ' I . i It We are interested In finding the concentration 1: that maxnnizes the reaction rate, Figure 5.59 The chemical reaction this is the absolute maximum of R(x). Since R(x) is differentiable on (0‘ 2), we can ?::)R(x2)(12n Exagple 1j Tgl:gra5h20f find all local extrema on (0. 2) by investigating the first derivative. To compute the . = — I — x ‘ _ x _ , I l . I has an absolute maximum at (0' 20). first derivative of R(,r), we mumply R”) out. R(.r)=20— 14x+2x2 torosx 52 Differentiating with respect to x yields R’(x) = ~14+4x forO < x < 2 5.4 l Optimization 243 we conclude that there is a local maximum at 2 = k. To see whether it is a global maximum, we compare w(k) with w(O) and lim,‘_,c,o w(x). We have R x2 _R x xk2+x2‘ k2+x2 mm = 5for) = x so R w(O) = 0 w(k) = — lim w(x) = 0 2k x—mo Hence, 2 = k is where the absolute maximum occurs; for our choice of f (x) = Fifi, the optimal clutch size Nap. satisfies Nap, = R/ k. [Other choices of f (x) would give . a different result] There is a geometric way of finding i. Since for) = E x it follows that the tangent line at ()2, f (2)) has slope ff) . This line can be obtained by drawing a straight line through the origin that just touches the ra h of = Section 5.4 Problems 1 . Find the smallest perimeter possible fora rectangle whose area 6. Find the largest possible area of a right triangle whose is 25 in.2. hypotenuse is 4 cm'long. 2. Show that, among all rectangles with a given perimeter, the 7. Suppose that a and b are the side lengths in a right triangle square has the largest area. whose hypotenuse is 5 cm long. What is the largest perimeter 3. A rectangle has its base on the x-axis and its upper two vertices POSS'ble? on the parabola y = 3 — x2, as shown in Figure 5.56. What is the 8. Su ose that nd b are the side len ths 'n a i ht t ia le largest area the rectangle can have? pp 0 a g 1 T g r “8 whose hypotenuse is 10 cm long. Show that the area of the triangle is largest when a = b. 9. A rectangle has its base on the x-axis, its lower left corner at (0, 0), and its upper right corner on the curve y = l/x. What is the smallest perimeter the rectangle can have? 10. A rectangle has its base on the x-axis and its upper left and right corners on the curve y = J4 — x2, as shown in Figure 5.57. The left and the right corners are equidistant from the vertical axis What is the largest area the rectangle can have? -l Filure 5.56 The graph of y = 3 — x2 together with the inscribed rectangle in Problem 3. 4- A rectangular study area is to be enclosed by a fence and divided into two equal parts, with the fence running along the division parallel to one of the sides If the total area is 384 ftz, find the dimensions of the study area that will minimize the total length 0f the fence. How much fencing will be required? 5- A rectangular field is bounded on one side by a river and on ihe other three sides by a fence. Find the dimensions of the field I 2 l 2 ' hat will maximize the enclosed area ifthe fence has a total length fig”? 5-57 The graph 0f y = (4 — x ) / loge'hc" Wlth the )f320 n. inscribed rectangle in Problem 10. s 244 Chapter5 I Applications of Differentiation 11. Denote by (x. y) a point on the straight line y = 4 - 3x. (See Figure 5.58.) )’ 14 4—3):— Figure 5.58 The graph of y = 4 — Sr in Problem 11. (a) Show that the distance from (x, y) to the origin is given by [(x) = ,/x2 + (4 — 3x)z (b) Give the coordinates of the point on the line y = 4 — 3x that is closest to the origin. (Hint: Find x so that the distance you computed in (a) is minimized.) (c) Show that the square of the distance between the point (x, y) on the line and the origin is given by g(1=)=[f(x)]2 = x2 + (4 4 3oz and find the minimum of g(x). Show that this minimum agrees with your answer in (b). 12. How close does the line y = 1 + 2x come to the origin? 13. How close does the curve y = 1/x come to the origin? (Hint: Find the point on the curve that minimizes the square of the distance between the origin and the point on the curve. If you use the square of the distance instead of the distance, you avoid dealing with square roots) 14. How close does the circle with radius fl and center (2, 2) come to the origin. 15. Show that if f (x) is a positive twice-differentiable function that has a local minimum at x = c, then g(x) = [f (x)]2 has a local minimum at x = c as well. 16. Show that if f (x) is a differentiable function with f (x) < 0 for all x e R and with a local maximum at x = c, then g(x) = [f (x)]2 has a local minimum at x = c. 17. Find the dimensions of a right circular cylindrical can (with bottom and top closed) that has a volume of 1 liter and that minimizes the amount of material used. (Note: One liter corresponds to 1000 cm’.) 18. Find the dimensions of a right circular cylinder that is open on the top, is closed on the bottom, holds 1 liter, and uses the least amount of material. 19. A circular sector with radius r and angle 9 has area A. Find r and 6 so that the perimeter is smallest when (a) A = 2 and (b) A = 10. (Note: A = %r20, and the length of the arc s = r9, when 0 is measured in radians; see Figure 5.59.) 4 s Figure 5.59 The circular sector in Problems 19 and 20. 20. A circular sector with radius r and angle 0 has area A. Find r and 9 so that the perimeter is smallest for a given area A. (Note: A = %r29, and the length of the arc s =‘ r8, when 6 is measured in radians; see Figure 5.59.) 21. Repeat Example 4 under the assumption that the top of the can is made out of aluminum that is three times as thick as the aluminum used for the wall and the bottom. b = 20 and ab is a maximum. ' 23. Find two numbers a and b such that a — b = 4 and ab is a minimum. 24. Classical Model of Viability Selection Consider a population of diploid organisms (i.e., each individual carries two copies of each chromosome). Genes reside on chromosomes, and we call the location of a gene on a chromosome a locus. Different versions of the same gene are called alleles. Let us examine the case of one locus with two possible alleles, A1 and A2. Since the individuals are diploid, the following types, called genotypes, may occur: AlAl, AlAz, and AzAz (where A1A2 and A2A1 are considered to be equivalent). If two parents mate and produce an offspring, the offspring receives one gene from each parent. If mating is random, then we can imagine all genes being put into one big gene pool from which we choose two genes at random. If we assume that the frequency of A1 in the population is p and the frequency of A2 is q = 1 —- p, then the combination A1A1 is picked with probability p2, the combination A1A2 with probability 2pq (the factor 2 appears because A, can come from either the father or the mother), and the combination MA: with probability q’. We assume that the survival chances of offspring depend on their genotypes. We define the quantities w“, mm, and 1.022 to describe the differential survival chances of the types AIA‘, A1 A2, and A2A2, respectively. The ratio A1A12A1A2:A2A2 among adults is given by pzquZquntqzwzz The average fitness of this population is defined as 'u‘) = pzwn + 2qu12 + (12102 We will investigate the preceding function. Since q = 1 — p, w is a function of p only; specifically, W) = Pzwu + 2:20 — p)w12 + (1 — P)2w22 for 0 5 p 51. We consider the following three cases: (i) Directional selection: w” > wn > wzz (ii) Overdominance: wlz > w“, wzz (iii) Underdominance: w; < w“, wzz “W Wwfiuhw.akm.mfln . mes...“ (a) Show that 71K?) = Pziwn - 21012 + 1022) 2P(w12 —- wzz) + W2: and graph '11“ p) for each of the three cases, where we choose the parameters as follows: (i) w“ = 1, 1012 = 0.7, 1022 = 0.3 w“ = 0.7, [1112 = 1, L022 = 0,3 w“ = 1, w); = 1022 = 0.7 (b) Show that 3:7; = 2170011 - 2w12 + 1022) + 2(w12 - mu) (c) Find the global maximum of 'tiJ‘( p) in each of the three cases considered in (a). (Note that the global maximum may occur at the boundary of the domain of E.) (d) We can show that under a certain mating scheme the gene frequencies change until E reaches its global maximum. Assume that this is the case, and state what the equilibrium frequency will be for each of the three cases considered in (a). 25. Continuation of Problem 94 from Section 4.3 We discussed the properties of hatchin ffsrin er unit time w t in the spec1es eu eroacty us coqui. ” e function w(t) was given by where f (t) is the proportion of offspring that survive if t is the time spent brooding and where C is the cost associated with the time spent searching for other mates. We assume now that f (t), t z 0, is twice differentiable and concave down with f(0) = 0 and 0 s f 5 1. The optimal brooding time is defined as the time that maximizes w(t). (a) Show that the optimal brooding time can be obtained by finding the point on the curve f (t) where the line through (—C. 0) is tangential to the curve f (t). (b) Use the procedure in (a) to find the optimal brooding time for f (t) = if; and C = 2. Determine the equation of the line through (—2, O) that is tangential to the curve f (I) = 1'1, and graph both f (t) and the tangent together. 26. Optimal Age of Reproduction (from Raff} I992) Semelparous organisms breed only once during their lifetime. Examples of this type of reproduction can be found in Pacific salmon and bamboo. The per capita rate of increase, r, can be I 5.5 L’Hospital’s Rule 5.5 I L'Hospital’s Rule 245 thought of as a measure of reproductive fitness. The greater the value of r, the more offspring an individual produces. The intrinsic rate of increase is typically a function of age x. Models for age- structured populations of semelparous organisms predict that the intrinsic rate of increase as a function of x is given by in [l(x)m(x)] X r(x) :- where 1(x) is the probability of surviving to age x and m(x) is the number of female offspring at age x. The optimal age of reproduction is the age x that maximizes r(x). (in) Find the optimal age of reproduction for [(x) = e‘” and m(x) = bxc where a, b, and c are positive constants (b) Use a graphing calculator to sketch the graph of r(x) when a = 0.1, b = 4, andc = 0.9. 27. Optimal Age at First Reproduction (from Lloy 1987) u. 00’! ;. ‘5 ~-- ore anouceuringt exr ifetime. Consider a model in which the intrinsic rate of increase, r, depends on the age of first reproduction, denoted by x, and satisfies the equation e—x(r(x)+L)(1 _ e—h)3c 1 __ e—(r(x)+L) = 1 where k, L, and c are positive constants describing the life history of the organism. The optimal age of first reproduction is the age J: for which r(x) is maximized. Since we cannot separate r(x) in the preceding equation, we must use implicit differentiation to find a candidate for the optimal age of reproduction. (3) Find an equation for 5f. [Hint Take logarithms of both sides of (5.13) before differentiating with respect to x.] (b) Set 3-; = 0 and show that this gives r(x) = 1 — e‘k' — L [To find the candidate for the optimal age x, you would need to substitute for r(x) in (5.13) and solve the equation numerically. Then you would still need to check that this solution actually gives you the absolute maximum. It can, in fact, be done] ‘ Guillaume Francois l’Hospital was born in France in 1661. He became interested in calculus around 1690, when articles on the new calculus by Leibniz and the Bernoulli brothers began to appear. Johann Bernoulli was in Paris in 1691, and l’Hospital asked Bernoulli to teach him some calculus. Bernoulli left Paris a year later, but continued to provide l‘Hospital with new material on calculus. Bernoulli received a monthly salary for his service and agreed that he would not give anyone else access to the material. Once l’Hospital thought he understood the material well enough, he decided to write a book on the subject, which was published under his name and met with great success Bernoulli was not particularly happy about this development, as his contributions were hardly acknowledged in the book; l’Hospital perhaps felt that because he had paid for the course material, he had a right to publish it. 252 Chapter5 I Applications of Differentiation The limit is now of the form 00 - 0 (since In tan § = 1111 = 0). We evaluate the limit by writing it in the form % and then applying l’Hospital’s rule: l, (t (2):) 1 t ) 1_ lntanx rm ln tanx 1m an - n anx = 1m = 1 mar/4)- x->(n/4)- 53-2—25 war/4)“ “‘9” Since d seczx cosx 1 —lntanx= — =-,—-——— dx tanx cos x smx smx cosx and d ._ —— cot(2x) = —(csc2(2x)) -2 = —.-—-— dx sm2(2x) it follows that 1 . 2 ‘ lnlanx , sinxcosx , s1n (2.x) hm — = 11m —-——é——— = 11111 —,—-——- x-e(n/4)- C0t(2x) x—»(n/4)" Tin—(2‘) X_.,(n/4)— “2 smx cosx y (tanx)“"(7")—- 1 = —1 Therefore, lim (tan x)‘““""" = exp lim (tan(2x)ln tanx) x—>(n['4)‘ x->(7r/4)‘ 0 % 1 = exp [—1] = e" Figure 5.63 The graph of y = (taan‘n‘m- The graph of f (x) = (tan x)““‘(7-") is shown in Figure 5.63. I Section 5.5 Problems ' 2 Use I'Hospital’s rule to find the limits in Problems 1—50. -, ex _ 1 _ x _ x_ 2 x — 25 x — 2 . e"r — 1 — x 2 1. 1' 2, 1‘ 19. hm —-—— 20. lim———— 12 x '- 5 XL“; x2 - 4 x—90 x2 x—bO x3 2 lnx 2 7 3. lim 3x +5x-2 4. “m x+3 21. lim( 2) 22.11m5; x—e-Z x + 2 x..._3 X2 + 2x —' 3 x"°° xt x—vcoe: 1 anx —— / ._ _ /—"‘ 23. lim 24. lim , s. iii—2 6. §___2:x..+—9 x-t()[/2)‘ seczx x—vO Sin-x NO x H0 2" 25. lim n“ 26. lim x2e“ 7 l, sinx _ xsinx Hm 4”“ - x13 x cos x 8. 113; m 27. lim x5e" 28. lim x”e", n e N X—im I—‘m _ 1 _ cosx sin(% —- x) 29. lim film: 30. lim x2 lnx 9. 11m m— 10. —- x—~0* 140+ “'0 “‘“fl cosx 31. lim x5 lnx 32. lim x" lnx,n e N , _,o+ ~O+ 11. lirn ———‘/E— 12 lim 19$ 3 n x ,, x—>0+ “10’ +1) x—eoo J; 33. lim — x) secx 34. lim (1 — x) tan l l x—>(n/2)' x—el‘ 13. lim "( '1’) 14. lim mu" ’0 , 1 1 x...” x ,4“, lnx 35. 11m fisin — 36. lim .1:2 sin 7 21 _ 1 5‘ _ 1 x—ooo X x—boo x 15. lim 3‘ _1 16- lim 7‘ _1 37. lim (cotx —- cscx) 38. lim (x —— ‘/x2 —- 1) .r—vO x->0 3—.01- x—wo 3“ ¢ 1 -x — 1 [7. lim 18. lim 2 1 39. lim - 40. lim ( 1 — .1-90 2’ _ 1 x—eO 5‘ —' 1 x—+0+ x...0+ sinzx x 5.6 I Difference Equations: Stability (Optional) 253 41. lim x" 42. lim x“"’ 63. For p > 0, determine the values of p for which the following x—-0+ “0+ limit is either 1 or 00 or a constant that is neither 1 nor oo: 43. lim x1“ 44. mm cum 0 . x—voo 3"” ‘i' _) i. X x ' —> XP " - 3 46 lim 1+ 3) ’ °° 1ft 45' ,‘L"; (I + x ' a...” x 64. Show that l 2 I ‘ 3 1 lim xpe“ = O r 47. lim (1 —- -) 48. 11m (1+ Pm, lv ‘*’°° x x "’°° for any positive number p. Graph f (x) = xPe“, x > O, for i . x - 3/x p = 1/2, 1. and 2. Since f(x) = xl’e“ = xP/e’, the limiting 0. 1 2x . . . - ‘ 49' ‘13:, (1 +x) 5 Aim“ )) behavior (hmHm :7, = 0) shows that the exponential function ; grows faster than any power of x as x -» 00. ff 3 Find the limits in Problems 51—60. Be sure to check whether you 65 Show mm 5 i can apply I'Hmpital's rule before you evaluate the limit. ' In x g 1 ‘ lim -— = 0 t. 51. lim xe“ 52. lim 5- H” x, x->0 15’0" x for any number p > O. This shows that the logarithmic function ~ 53. “m (tan x + sec x) 54 um tanx grows more slowly than any positive power of x as xi 00: mi x—bOr/Z)‘ “(n/2r 1 + 5‘30" 66. When l’HOSpital introduced indeterminate limits in his i; 55 1- x2 __ 1 ' 56. r 1 _ cos x textbook, his first example was . . rm ,_ xiii} x + 1 x->° secx _ ‘/2a3x — x4 - a 3 azx . 11:7 r » . m in; ; 1m ;‘7 “a “‘V‘m‘s 7'“, _.— —. x . . . . . " m " 0+ x where a Is a posrtive constant. ('Ihrs example was communicated 59 “m x3: ' 60 “m (x + 1) to him by Bernoulli.) Show that this limit is equal to (16/9)a. ' H0+ was x + 2 67. The height y in feet of a tree as a function of the tree’s age x 61. Use l’Hospital’s rule to find in years is given by ax _ 1 y =121e‘m‘ forx > 0 lim H0 bx " 1 (2) Determine (1) the rate of growth when x —> 0‘L and (2) the h b 0 limit of the height asx —+ 00. W cm a’ ' > I ' (b) Find the age at which the growth rate is maximal. 62' use I Hosp“! 8 rule to find (c) Show that the height of the tree is an increasing function of I ' c 1 age. At what age is the height increasing at an accelerating rate 1"“ (1 + ;) and at what age at a decelerating rate? 1“” (d) Sketch the graph of both the height and the rate of growth of where c is a constant. the tree as functions of age. I 5.6 Difference Equations: Stability (Optional) In. Chapter 2, we introduced difference equations and saw that first-order difference equations can be described by recursions of the form x‘ = f(x*) (5.15) and has the property that if x0 = x", then x, = x" fort 1,2,3, . . .. We also saw in a number of applications that, under certain conditions, x, converged to the fixed point as r —> 00 even if x0 7-": x". However, back then, we were not able to predict when this behavior would occur. ll 260 Chapters I Applications of Differentiation Using the product rule and the chain rule, we find that f’(N) = exp [R (1 — + N exp [R (1 _ (-7121) =[R (1 — a] <1 — a) Now, f’(0) = eR > 1 for R > 0, so N * = 0 is unstable. Since f’(K)=1-R andlf’(K)| = II — R| <1if—l < 1 — R < lorO < R < 2,weconclude that N “ = K is locally stable ifO < R < 2. We can say abit more now: If0 < R < 1, then N ‘ = K is approached without oscillations, since f’(K) > 0;if1 < R < 2, N ‘ = K is approached with oscillations, since f ’(K ) < 0. I I 5.6.1 1. Assume a discrete-time population whose size at generation I + 1 is related to the size of the population at generation 2 by N,“ = (1.03)N,, t: 0,1, 2, . .. (a) If N0 = 10. how large will the population be at generation I = 5? (b) How many generations will it take for the population size to reach double the size at generation 0? 2. Suppose a discrete-time population evolves according to N,+1=(0.9)N,, 1: 0,1,2... (3) If N0 = 50, how large will the population be at generation I = 6? (b) After how many generations will the size of the population be one-quarter of its original size? (c) What will happen to the population in the long run—that is, as: —> oo? 3. Assume the discrete-time population model N,+1=bN,, t=0,1,2... Assume also that the population increases by 2% each generation. (2) Determine b. (b) Find the size of the population at generation 10 when N0 := 20. (c) After how many generations will the population size have doubled? 4 Assume the discrete-time population model Nl+i=leu t=0.l,2.... Assume also that the population decreases by 3% each generation. (a) Determine b. (b) Find the size of the population at generation 10 when M, = 50. (c) How long will it take until the population is one-half its original size? 5. Assume the discrete-time population model Np+l=be, t=0,1.2.... Assume that the population increases by x% each generation. (2) Determine b. (b) After how many generations will the population size have doubled? Compute the doubling time for x = 0.1, 0.5, 1, 2, 5, and 10. 6. (:1) Find all equilibria of N,+1=1.3N;, t=0,1,2,... (b) Use cobwebbing to determine the stability of the equilibria you found in (a). 7. (3) Find all equilibria of N,+1=0.9N., 1: 0,1,2, (b) Use cobwebbing to determine the stability of the equilibria you found in (a). 8. (in) Find all equilibria of Nf+l=Nh ‘=0,1,2.... (b) How will the population size N, change over time, starting at time 0 with No? l 5.6.2 9. Use the stability criterion to characterize the stability of the equilibria of 2 2 2 1r+l = ' — "xrv 3 3 10. Use the stability criterion to characterize the stability of the equilibria of 150.13.... 3 2 Xl+i =§xr2_'5‘. t=0.1.2.... 11. use 111 e stability criterion to characterize the stability of the equilibria of It , t=0,l,2.... 0.5+x, xH-l = 12. use the stability criterion to characterize the stability of the equilibria of X! 0.3 + x, ' l=0,1,2.... xH-l = 13. (it) Use the stability criterion to characterize the stability of the equilibria of (b) Use cobwebbing to decide to which value x, converges as t—> 00 if (i) x9 = 0.5 and (ii) x0 = 2. 14. (it) Use the stability criterion to characterize the stability of the equilibria of . l_ I t+ (b) Use cobwebbing to decide to which value x, converges as t —+ 00 if (i) x0 = 0.5 and (ii) x0 = 3. 1 5.6.3 15. Ricker’s curve is given by R(P) = aPe'“ for P z 0, where P denotes the size of the parental stock and R(P) the number of recruits. The parameters at and ,3 are positive constants. (a) Show that R(O) = Oand R(P) > Ofor P > 0. (b) Find lim R(P) P—ooo (c) For what size of the parental stock is the number of recruits maximal? (d) Does R(P) have inflection points? If so, find them. (e) Sketch the graph of f(x) when (I ‘= 2 and fl = 1/2. 16. Suppose that the size of a fish population at generation t is given by 1v,+1 =1.5N,e'°‘°°'"’ fort=0,1,2, (3) Assume that N0 = 100. Find the size of the fish population at generation: for t = 1, 2. . . . , 20. (b) Assume that N0 = 800. Find the size of the fish population at generation t fort = l, 2. . . . .20. (c) Determine all fixed points. On the basis of your computations in (a) and (b), make a guess as to what will happen to the population in the long run, starting from (i) N0 = 100 and “D N0 = 800. ((1) Use the cobwebbing method to illustrate your answer in (a). (9) Explain why the dynamical system converges to the nontrivial ‘ixed point. 5.6 I Difference Equations: Stability (Optional) 261 17. Suppose that the size of a fish population at generation t is given by IV”>1 = ION'e-OIHN, fort =0.1,2,.... (a) Assume that N0 = 100. Find the size of the fish population at generation t fort = l. 2, . . . . 20. (b) Show that if M; = 100ln 10. then N, = lOOln 10 for t = 1.2, 3, . . .; that is, show that N' = [OOln 10 is a nontrivial fixed point, or equilibrium. How would you find N‘? Are there any other equilibria? (c) On the basis of your computations in (a), make a prediction about the long-term behavior of the fish population when M, = 100. How does your answer compare with that in (b)? (d) Use the cobwebbing method to illustrate your answer in (c). In Problems 18—20, consider the following discrete-time dynamical system, which is called the discrete logistic model and which models the size of a population over time: N N,+1=N,[1+R(1~T$)] ort=0.l.2,.... 18. (It) Find all equilibria when R = 0.5. (b) Investigate the system when N0 = 10 and describe what you see. 19. (It) Find all equilibria when R = 1.5. (b) Investigate the system when M; = 10 and describe what you see. 20. (a) Find all equilibria when R = 2.5. (b) Investigate the system when N0 = 10 and describe what you see. In Problems 21—22, we investigate the canonical discrete-time logistic growth model le = rxt(1 -xt) fort=0.1.2,.... 21. Show that for r > 1, there are two fixed points. For which values of r is the nonzero fixed point locally stable? 22. Use a calculator or a spreadsheet to simulate the canon- ical discrete-time logistic growth model with x0 = 0.1 for t = 0, 1,2. . . . , 100, and describe the behavior when (a) r = 3.20 (b) r = 3.52 (c) r = 3.80 (d) r = 3.83 (e) r = 3.828 In Problems 23—25, we consider density-dependent population growth models of the form Nl+l = R(NI)NI The function R( N ) describes the per capita growth. Various forms have been considered. For each function R(N), find all nontrivial fixed points N‘ (i.e., N‘ > 0) and determine the stability as a fitnclion of the parameter values. We assume that the fitnclion parameters are r > 0, K > O, and y > 1. 23. R N = N'”)’ 24. N = ——5— ( ) r m ) 1+N/K 25. R(N) “NI—W) 266 Chapter 5 I Applications of Differentiation compute x1, we find that Figure 5.78 The graph of f (x) in Example 5 together with the first two approximations in Figure 5.78. ‘_ Section 5.7 Problems 1. Use the Newton—Raphson method to find a numerical, approximation to the solution of x2 — 7 = 0 that is correct to six decimal places. 2. Use the Newton-Raphson method to find a numerical approximation to the solution of e“‘ =x that is correct to six decimal places. 3. Use the Newton-Raphson method to find a numerical approximation to the solution of x2+lnx=0 that is correct to six decimal places 4. The equation x2 — 5 = 0 has two solutions Use the Newton—Raphson method to approxi. mate the two solutions. 5. Use the Newton—Raphson method to solve the equation sinx -1x " 2 in the interval (0. 7r). 6. Let ‘/x —-1 forx 21 —-‘/1-x forx 51 (a) Show that if you use the Newton—Raphson method to solve f(x) = 0, then the following statement holds: If x0 = l + h, then x. =1 —h,andifx0=1—h.thenx1 =1+h. (b) Does the Newton—Raphson method converge? Use a graph to explain what happens. f(x) = Successive values are collected in the following list: 3 x1 = (—0.7) —- = 8.225 x; = 6.184 X7 = 1.613 x3 = 4.659 x8 = 1.306 X4 = 3.521 X9 = 1.115 X5 = 2.678 x10 = 1.024 x6 = 2.059 x11 = 1.001 We conclude that the method converges to the root r = 1. The situation is illustrated { 7. In Example 4, we discussed the case of finding the root 0- x”3 = 0. (a) Given x0, find a formula for |x,.|. (b) Find lim lxn| Il—VW (c) Graph f (x) = 1:”3 and illustrate what happens when you apply the N ewton—Raphson method. 8. In Example 5. we considered the equation x4—x2=0 (a) What happens if you choose 1 x0 = —‘2—\/§ in the Newton-Raphson method? Give a graphical illustration. (b) Repeat the procedure in (a) for x0 = —0.71. and compare your result with the result we obtained in Example 5 when x0 = —-O.70. Give a graphical illustration and explain it in words. What happens when 10 = —0.6? (This is an example in which small changes in the initial value can drastically change the outcome.) 9. Use the Newton~Raphson method to find a numerical approximation to the solution of xz—16=0 when your initial guess is (a) x0 = 3 and (b) x0 = 4. 10. Suppose that you wish to use the Newton—Raphson method to solve f (x) = 0 numerically. It just so happens that your initial guess x0 satisfies f(xo) = 0. What happens to subsequent iterations? Give a graphical illustration of your results. [Assume that f’(xo) 9E 0.] W’ 272 Chapter5 l Applications of Differentiation Section 5.8 Problems ' ~ ‘ . . . d In Problems 1—40, find the general antidertvattve of the given 51. g): = ,(1 _ 1)., Z 0 52_ 3:: = 120 _ 12),, Z 0 function. t d 1- f(X)‘4"2"‘ 2'f(")=2“5x2 53. 41””sz ‘ 54.1: —e'3’.t20 3, f0.) 3: x2 + 3x — 4 4. f(x) = 3x2 — x4 dt 7 dt 5'f(x)=x‘43x2+1 6'f(x)=2x3+x2-5§ 4 55. d—y=sin(zrs).0'5351 7. f(x)==4x3—2x+3 8.f(x)=x—2xz—3x ~4x ds 1 1 2 2 3 dy 9.f(x)=l+;+;3 10. f(x)=x -;5+-;5 56.E;=cos(2ns),05s51 1 _ 3 _1_ dy x 11. f(x)=1-; 12-f(x)—x *x3 57. ——=sec2(—).-—1<x<1 1 dx 2 13. (x) = 14. f(x) = f 1+x5 1+x1 58.31=1+sec2(;-),—1<x<1 x = 4 — 16. = x7 + — 15. fix) 5x + x4 fix) ’57 In Problems 59-72, solve the initial-value problem. 1 1 dy 17. f(x)=1+2x 18~ f(x)=1+3x 59. a;=3x7',forxszithy=1whcnx=0 19. f(x) = {3* dy J", 60. — = —,forx 20withy =2whenx =0 1 3 23. = — 24. (x) = — f(X) eh f r" 61. f1—?'—=2‘/3t',forx ZOWithy=2whenx=1 25. f (x) = sin(2x) 26. f (x) = cos(3x) dx d 1 . 27. f(x) = sin +cos 62. 3% = fixer): a 1 With y = 3 whenx = 4 28- f(x) = «18(5) — sir! (i) 63. ‘iI-V- : 1 fort > 1 with N0) = 10 5 5 d: t’ ‘ 7r :1 29. f(X)=2§in(-x)—3COS(—X> fl: ’ f t>0 =2 2 2 - dt —-—H._2,or _ w1 () Jr 71 . = —3 ' — 4 -- d 30 fl") sm(3x)+ C°S( 4x) 65. —av:I—=e',fort20withW(0)=1 31. f(x) = sec2(2x) 32. f(x) = sec2(-4x) dW _ = -3: - ___ 33- = sec: 34. = secZ (_ 66. d! e , fort Z 0 With 2 dW _3 , 35. f(x) = ——-—-—sec:o::osx 36. f(x) = sinzx + coszx 67' '2’; = e 2 f0“ ->- 0 mm Wm) = 2/3 37. = x-7 + 3x5 + sin(2x) 68. ii“: = e—St’ for, Z O with = 1 t 38. f(x) = 2r” + soc2 (IT 2 3 69. :1}— : sin(7rt), fort _>_ 0 with T(0) = 3 39. f(x) =sec2(3x — 1) + x ; dT ' 70. —— = t ,f t > 0 ' = 40' fix) =5e3x _Sec2(x _3) dt cos(1r ) or _ With T(0) 3 In Problems 41—46, assume that a is a positiveconstant. Find the 71. d_)' = C” + 9‘ for x > 0 with y = 0 when x = 0 general antiderivative of the given function. dx 2 ' _ e(a+1)x . 2 2 dN 41- f(x) = 42- f(x) = sm (0 x +1) 72. 7“— : 1“”, fort > Owith N(0) = 60 43. fix) ___ 1 44. f“) ____ a 73. Suppose that the length of a certain organism at age J: is given ax + 3 a + x by L(x), which satisfies the differential equation 45. f(x) = x0“ — a"+2 46. f(x) = 5i- d], 20 T ___ e--0.lirV x a 0 In Problems 47—58, find the general solution of the difierentia! x equation. Find L(x) if the limiting length Lm is given by dy 2 dy 2 47.Z;=;—-x,x>0 4s.a=F-x3.x>0 L..=xli»n;L(x)=25 d)’ —4: 49. Z; =X(1+X).X >0 50. I; =5 ix >0 How big is the organism at agcx =0? 74. Fish are indeterminate growers; that is, their length L(x) increases with age 1: throughout their lifetime. If we plot the growth rate dL/dx versus age x on semilog paper, a straight line with negative slope results. Set up a differential equation that relates growth rate and age. Solve this equation under the assumption that L(0) = 5, L(1) = 10, and lim L(x) = 20 x—voo Graph the solution L(x) as a function of x. 75. An object is dropped from a height of 100 ft. Its acceleration is 32 ft/sz. When will the object hit the ground. and what will its speed be at impact? 76. Suppose that the growth rate of a population at time t undergoes seasonal fluctuations according to I = 3 sin(27rt) Chapters I Review Problems 273 where t is measured in years and N (t) denotes the size of the population at time r. If N (0) = 10 (measured in thousands), find an expression for N (I). How are the seasonal fluctuations in the growth rate reflected in the population size? 77. Suppose that the amount of water contained in a plant at time t is denoted by V(r). Due to evaporation, V(t) changes over time. Suppose that the change in volume at time I, measured over a 24-hour period, is proportional to 2(24 - I), measured in grams per hour. To offset the water loss, you water the plant at a constant rate of 4 grams of water per hour. (3) Explain why d V. __ = ._ _ 4 d! at(24 t) + 0 5 t s 24. for some positive constant a, describes this situation. (h) Determine the constant a for which the net water loss over a 24-hour period is equal to 0. : Chapter 5 Key Terms _ Discuss the following definitions and 9. Concavity: concave up and concave l7. Asymptotes: horizontal, vertical, and . I v " I p o I" 1. Global or absolute extrema 10. Concavity and the second derivative 18. Using calculus to graph functions 2. Local or relative extrema: local 11. Diminishing return ‘ I , minimum and local minimum 12. Candidates for local extrema 19' L Hospnal s mle 3~ The °Xtreme"’31“e theorem ~ 13. Monotonicity and local extrema 20. Dynamical systems: cobwebbing 4. Fermat's theorem 5. Mean-value theorem 6. Rolle’s theorem 7. Increasing and decreasing function 8. Monotonicity and the first derivative extrema 15. Inflection points derivative 14. The second-derivative test for local 16. Inflection points and the second 21. Stability of equilibria 22. Newton—Raphson method for finding roots 23. Antiderivative , Chapter 5 Review Problems . I 1. Suppose that f(x) = xe“. x z 0 (a) Show that f(0) = 0, f(x) > 0 forx > O, and lim f(x) = 0 mac (b) Find local and absolute extrema. (c) Find inflection points. (d) Use the foregoing information to graph f (x). 2. Suppose that f(x) =xlnx. x > 0 (1!) Define f(x) at x = 0 so that f(x) is cOntinuous for all x 2 0. (b) Find extrema and inflection points. (c) Graph f(x). 3. In Review Problem 17 of Chapter 2 we introduced the hyper- bolic functions e‘ - e“ sinhx = 2 . x E R e" + e“ coshx = 2 , x 6 R e‘ — e“ tanhx = . x e R ex + 6—: (a) Show that f (x) = tanhx, x e R, is a strictly increasing function on R. Evaluate lim tanh x [-4 —w and lim tanh Jr 1-500 (b) Use your results in (a) to explain why f (x) = tanh x, x e R, is invertible, and show that its inverse function f " (x) = tanh-1x is given by 1 + x l—x 1 "l __ f (x)_2]n What is the domain of f"(x)? (c) Show that d ,1 1 3f (r\I)—1_x2 (d) Use your result in (c) and the facts that sinh x tanhx = cosh x and cosh2 x — sinh2 x = l to show that d tanhx 1 dx — coshzx . .0--. .______._.._._.._._:,___‘ __._..~ a.‘__‘_‘..'. .4 -A_A._-,_ . a .Wa—u' s ’4s.wuf‘h‘ft¥ii3'§f_miiem - “a 274 Chapter 5 I Applications of Differentiation 4. Let x 1 + e“ ’ (a) Show that y = 0 is a horizontal asymptote as x 4 -oo. f(x)= XER (b) Show that y = x is an oblique asymptote as x —> +oo. (c) Show that 1 + e‘”r (1 + x) (1 + r02 (d) Use your result in (c) to show that f (x) has exactly one local extremum at x = c, where c satisfies the equation f’(x) = 1+c+e‘=0 [Hinc Use your result in (c) to show that f ’(x) =-. 0 if and only if 1+ e“(1 +x) = 0. Let g(x) = 1 + e""(1+ x). Show that g(x) is strictly increasing for x < 0, that g(0) > 0, and g(-2) < 0. This implies that g(x) = 0 has exactly one solution on (—2. 0). Since g(—2) < 0 and g(x) is strictly increasing for x < 0, there are no solutions of g(x) = 0 for x < —2. Furthermore, g(x) > 0 for x > 0; hence, there are no solutions of g(x) = 0 for x > 0.] .- . v numerically . With the help of a calculator, find a numerical approximation to c. [Hint From (d), you know that c e (-2, 0).] (0 Show that f (x) < 0forx < 0. [This implies that, forx < 0, the graph of f (x) is below the horizontal asymptote y = 0.] (3) Show thatx — f (x) > 0 forx > 0. [This implies that, for x > 0. the graph of f (x) is below the oblique asymptote y = x.] (in) Use your results in (a)-(g) and the fact that f (O) =- 0 and f’(0) = 5 to sketch the graph of f (x). 5. Recruitment Model Ricker’s curve describes the relationship between the size of the parental stock of some fish and the number of recruits. If we denote the size of the parental stock by P and the number of recruits by R, then Ricker’s curve is given by R(P) = aPe'” for P z o where a and ,6 are positive constants. [Note that R(O) = 0; that is. without parents there are no offspring. Furthermore, R(P) > 0 when P > 0.] ' We are interested in the size P of the parental stock that maximizes the number R(P) of recruits. Since R(P) is differentiable, we can use its first derivative to solve this problem. (2) Use the product rule to show that, for P > 0, R’(P) = ore-“(1 — are) R”(P) = —ape""’(2 — M) (b) Show that R’(P) = 0 if P = l/fi and that R"(1/fi) < 0. This shows that R(P) has a local maximum at P = %. Show that R(l/fl) = %e‘1 > 0. (c) To find the global maximum, you need to check R(O) and limb,“ R(P). Show that R(O) = 0 and lim R( P) = 0 P—om and that this implies that there is a global maximum at P = 1/,‘3. (d) Show that R(P) has an inflection point at P = 2/13. (e) Sketch the graph of R(P) fora = 2 and fl = l. 6. Gompertz Growth Model The Gompertz growth curve is sometimes used to study the growth of populations. Its properties are quite similar to the properties of the logistic growth curve. The Gompertz growth curve is given by N(t) = K exp[-—ae‘b'] fort z 0, where K and b are positive constants. (s) Show that N(O) = K e‘“ and, hence, a —- n 70; if N0 = N (0). (b) Show that y = K is a horizontal asymptote and that N (r) < KifNo < K,N(r) = KifNo= K,andN(t) > KifNo > K. (1:) Show that 2—1:] = bN(an — lnN) and dZN dN (d) Use your results in (b) and (c) to show that N (t) is strictly o z I A. ..-I l _ I' A. (e) When does N (t), I _>_ 0, have an inflection point? Discuss its ' concavity. (1') Graph N(t) when K = 100 and b = 1 if (i) N0 = 20, (ii) N0 = 70, and (iii) N0 = 150, and compare your graphs with your answers in (b)—(e). 7. Monod Growth Model The Monod growth curve is given by ex f(x) = k +x for x z 0, where c and k are positive constants. The equation can be used to describe the specific growth rate of a species as a function of a resource level x. (a) Show that y = c is a horizontal asymptote for x —+ 00. The constant c is called the saturation value. (b) Show that f (x), x z 0, is strictly increasing and concave down. Explain why this implies that the saturation value is equal to the maximal specific growth rate. (c) Show that if x = k, then f (x) is equal to half the saturation value. (For this reason, the constant k is called the half-saturation constant.) (d) Sketch a graph of f (x) for k = 2 and c = 5, clearly marking the saturation value and the half-saturation constant. Compare this graph with one where k = 3 and c = 5. (e) Without graphing the three curves, explain how you can use the saturation value and the half-saturation constant to decide quickly that 10x 10x 8x > > 3 + x 5 + x 5 + x forx z 0. 8. Logistic Growth The logistic growth curve is given by N (t) = K 1+ (Nin -1)e-~ for r z 0, where K, No, and r are positive constants and N (t) denotes the population size at time t. (a) Show that N(O) = N0 and that y = asymptote as! ~> 00. (b) Show that N(!) < K ifNo < K, N(t) = K ifNo = K, and N(r) > KifNo > K. K is a horizontal (c) Show that and (d) Use your results in (b) and (c) to show that N (t) is strictly increasing if No < K and strictly decreasing if No > K. (e) Show that if No < K/Z, then N0), 1 2 0, has exactly one inflection point (t‘, N(t‘)), with r‘ > O and HE N(t)_2» (i.e., half the carrying capacity). What happensif K /2 < No < K 7 What if M; > K? Where is the function N (t), t a 0, concave up, and where is it concave down? (f) Sketch the graphs of N (1) fort z 0 when (i) K=100,No=10,r=1 (ii) K = 100,N0 =70,r = 1 (ill) K = 100, N0 = 150, r = 1 point clearly if it exists. 9. Genetics A population is said to be in Hardy-Weinberg equilibrium, with respect to a single gene with two alleles A and a, if the three genotypes AA, Aa, and aa have respective frequencies PM = 92.1%: = 29(1—9),and pa, = (1 «0)2 for someB e [0.1]. Suppose that we take a random sample of size n from a population. We can show that the probability of observing n1 individuals of type AA, n2 individuals of type Aa. and 113 individuals of type aa is given by ' n! n1 n3 n1 “17 P Pad nllnzlngl M A" where n! = n(n — 1)(n - 2) ~--3 - 2 - 1 (read “It factorial”). Here, n1 + n; + n3 = n. This probability depends on 0. There is a method, called the maximum likelihood method, that can be used to estimate 9. The principle is simple: We find the value of 0 that maximizes the probability of the observed data. Since the coefficient n! m ! n2! 713! does not depend on 0. we need only maximize Lo) = p11. p22 p23 (a) Suppose m = 8, n; = 6, and n; = 3. Compute L(6). (b) Show that if L(6) is maximal for 9 = (5 (read “theta hat”), a then in L(9) is also maximal for 0 = 6. (c) Use your result in (b) to find the value 5 that maximizes L09) for the data given in (a). The numberé is the maximum likelihood estimate. 10. Cell Volume Suppose the volume of a cell is increasing at a constant rate of 10"2 ch/s. (a) If V(t) denotes the cell volume at time I, set up an initial—value )roblem that describes this situation if the initial volume is 10'10 r J .m . Chapters II Review Problems 275 (b) Solve the initial-value problem given in (a), and determine the volume of the cell after 10 seconds. ' 11-. Drug Concentration Suppose the concentration 0(1) of a drug in the bloodstream at time t satisfies E = _0-1e—0.3l d! fort 2 0. (a) Solve the differential equation under the assumption that there will eventually be no trace of the drug in the blood. (b) How long does it take until the concentration reaches half its initial value? 12. Resource-Limited Growth Sterner (1997) investigated the effect of food quality on zooplankton dynamics In his model, zooplankton may be limited by either carbon (C) or phosphorus (P). He argued that when food quantity is low, demand for carbon increases relative to demand for phosphorus in order to to Satisfy basic metabolic requirements and that there should be a curve separating C- and P-limited growth when food quantity C,- (measured in amount of carbon per liter) is graphed as a function of the C: P ratio of the food, = C : . e . . e uation or the curve se aratin the two re 'ons: M‘MSkercfi‘ffie‘fis‘pecmonzontal asymptotes. Mark the inflection q p g .9 c = a. F Czar: “Cg _ pzf Here, m denotes the respiration rate, 5 the ingestion rate, and ac (0}!) the assimilation rate of carbon (phosphorus). Cz and P; are, respectively, the carbon and the phosphorus content of the zooplankton. (a) Show that the graph of y = Cp( f) approaches the horizontal. liney=fi3asf+oo. ' . (b) The graph of CF (f) has a vertical asymptote. Let f = CFIPp (the OP ratio of the food). Show that the vertical asymptote is at (c) Sketch a graph of Cp( f ) as a function of f. (d) The graph of Cp( f ) separates C-limited (below the curve) from P-Iimited (above the curve) growth. Explain why this graph indicates that when food quantity is low, the demand for carbon relative to phosphorus increases. 13. Velocity and Distance Neglecting air resistance, the height (in meters) of an object thrown vertically from the ground with initial velocity on is given by 1 h“) = v0! — iglz where g = 9.81m/s2 is the earth’s gravitational constant and t is the time (in seconds) elapsed since the object was released (it) Find the time at which the object reaches its maximum height. (h) Find the maximum height. (c) Find the velocity of the object at the time it reaches its maximum height. (d) At what time t > 0 will the object reach the initial height again? ...
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This note was uploaded on 02/17/2012 for the course MATH 17A taught by Professor Lyles during the Winter '08 term at UC Davis.

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Chapter5 - 5.1 I Extrema and the Mean-Value Theorem 213...

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