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Unformatted text preview: 5.1 I Extrema and the MeanValue 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” xoy 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 Problems2' ! 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 intermediatevalue 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. FirstDerivative 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. SecondDerivativeTestforConcavity 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. ResourceDependent Growth The growth rate of a plant
depends on the amount of resources available. A simple and frequently used model for resourcedependent 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 densitydependent 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 + (amb
"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 speciesspeciﬁc positive constant. (a) The relative growth rate (percentage weight gained per unit
of time) is deﬁned as (57) 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 extremevalue
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 halfopen 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 .IOW 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’ el/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=‘/x1,15.\152
3.y=ln(2x—1),15x52 4.y=lnﬁl,x>0 5y=xe"‘.05x51 6.y=16—x2l,55x58
7y=(.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 14y=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 twicedifferentiable
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
2734. 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=x29,4Sx55 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 lsx 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
xHco 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 = :ir . (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
.rs—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 xaxis 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 xaxis, 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 xaxis 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”? 557 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 twicedifferentiable 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) 532—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 —,———
xe(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 +5x2 4. “m x+3 21. lim( 2) 22.11m5;
x—eZ x + 2 x..._3 X2 + 2x —' 3 x"°° xt x—vcoe: 1
anx ——
/ ._ _ /—"‘ 23. lim 24. lim ,
s. iii—2 6. §___2:x..+—9 xt()[/2)‘ seczx x—vO Sinx
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 — .190 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 ﬁrstorder 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)=1R 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 discretetime 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 discretetime 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 onequarter of its original size? (c) What will happen to the population in the long run—that is,
as: —> oo? 3. Assume the discretetime 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 discretetime 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 onehalf its
original size? 5. Assume the discretetime 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, xHl = 12. use the stability criterion to characterize the stability of the
equilibria of X!
0.3 + x, ' l=0,1,2.... xHl = 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'eOIHN,
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 longterm 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 discretetime 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 discretetime
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 discretetime 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 densitydependent 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 NewtonRaphson method to ﬁnd a numerical
approximation to the solution of e“‘ =x that is correct to six decimal places. 3. Use the NewtonRaphson 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
—‘/1x 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 NewtonRaphson 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+x25§ 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; 12f(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 5972, solve the initialvalue 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. ‘iIV : 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. = x7 + 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 ___ e0.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=Fx3.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
24hour 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
24hour 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. Meanvalue theorem 6. Rolle’s theorem 7. Increasing and decreasing function
8. Monotonicity and the ﬁrst derivative extrema
15. Inﬂection points derivative 14. The secondderivative 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
1500 (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) = tanh1x
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 halfsaturation
constant.) (d) Sketch a graph of f (x) for k = 2 and c = 5, clearly marking
the saturation value and the halfsaturation 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 halfsaturation 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 HardyWeinberg
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 initialvalue 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 = _01e—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. ResourceLimited 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 Plimited 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 Climited (below the curve)
from PIimited (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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 Winter '08
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