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7 INVERSE FUNCTIONS:Exponential, Logarithmic, and Inverse Trigonometric Functions
7.1 Inverse Functions
1. (a) See Definition 1.
(b) It must pass the Horizontal Line Test.
3. f is not one-to-one because 2 6= 6, but f(2) = 2.0 = f(6).
5. No horizontal line intersects the graph of f more than once. Thus, by the Horizontal Line Test, f is one-to-one.
7. The horizontal line y = 0 (the x-axis) intersects the graph of f in more than one point. Thus, by the Horizontal Line Test,
f is not one-to-one.
9. The graph of f(x) = x2 − 2x is a parabola with axis of symmetry x = − b
2a= − −2
2(1)= 1. Pick any x-values equidistant
from 1 to find two equal function values. For example, f(0) = 0 and f(2) = 0, so f is not one-to-one.
11. g(x) = 1/x. x1 6= x2 ⇒ 1/x1 6= 1/x2 ⇒ g (x1) 6= g (x2), so g is one-to-one.
Geometric solution: The graph of g is the hyperbola shown in Figure 14 in Section 1.2. It passes the Horizontal Line Test,
so g is one-to-one.
13. h(x) = 1 + cosx is not one-to-one since h(a+ 2πn) = h(a) for any real number a and any integer n.
15. A football will attain every height h up to its maximum height twice: once on the way up, and again on the way down. Thus,
even if t1 does not equal t2, f(t1) may equal f(t2), so f is not 1-1.
17. Since f(2) = 9 and f is 1-1, we know that f−1(9) = 2. Remember, if the point (2, 9) is on the graph of f , then the point
(9, 2) is on the graph of f−1.
19. h(x) = x+√x ⇒ h0(x) = 1 + 1/(2
√x ) > 0 on (0,∞). So h is increasing and hence, 1-1. By inspection,
h(4) = 4 +√4 = 6, so h−1(6) = 4.
21. We solve C = 59 (F − 32) for F : 9
5C = F − 32 ⇒ F = 95C + 32. This gives us a formula for the inverse function, that
is, the Fahrenheit temperature F as a function of the Celsius temperature C. F ≥ −459.67 ⇒ 95C + 32 ≥ −459.67 ⇒
95C ≥ −491.67 ⇒ C ≥ −273.15, the domain of the inverse function.
23. y = f(x) = 3− 2x ⇒ 2x = 3− y ⇒ x =3− y
2. Interchange x and y: y =
3− x
2. So f−1(x) = 3− x
2.
25. f(x) =√10− 3x ⇒ y =
√10− 3x (y ≥ 0) ⇒ y2 = 10− 3x ⇒ 3x = 10− y2 ⇒ x = − 1
3y2 + 10
3 .
Interchange x and y: y = − 13x2 + 10
3. So f−1(x) = − 1
3x2 + 10
3. Note that the domain of f−1 is x ≥ 0.
245
246 ¤ CHAPTER 7 INVERSE FUNCTIONS
27. For f(x) = 1−√x1 +
√x
, the domain is x ≥ 0. f(0) = 1 and as x increases, y decreases. As x →∞,
1−√x1 +
√x· 1/
√x
1/√x=1/√x− 1
1/√x+ 1
→ −11= −1, so the range of f is −1 < y ≤ 1. Thus, the domain of f−1 is −1 < x ≤ 1.
y =1−√x1 +
√x
⇒ y(1 +√x ) = 1−√x ⇒ y + y
√x = 1−√x ⇒ √
x+ y√x = 1− y ⇒
√x(1 + y) = 1− y ⇒ √
x =1− y
1 + y⇒ x =
1− y
1 + y
2
. Interchange x and y: y =1− x
1 + x
2
. So
f−1(x) =1− x
1 + x
2
with −1 < x ≤ 1.
29. y = f(x) = x4 + 1 ⇒ y − 1 = x4 ⇒ x = 4√y − 1 (not ± since
x ≥ 0). Interchange x and y : y = 4√x− 1. So f−1(x) = 4
√x− 1. The
graph of y = 4√x− 1 is just the graph of y = 4
√x shifted right one unit.
From the graph, we see that f and f−1 are reflections about the line y = x.
31. Reflect the graph of f about the line y = x. The points (−1,−2), (1,−1),(2, 2), and (3, 3) on f are reflected to (−2,−1), (−1, 1), (2, 2), and (3, 3)
on f−1.
33. (a) x1 6= x2 ⇒ x31 6= x32 ⇒ f(x1) 6= f(x2), so f is one-to-one.
(b) f 0(x) = 3x2 and f(2) = 8 ⇒ f−1(8) = 2, so (f−1)0(8) = 1/f 0(f−1(8)) = 1/f 0(2) = 112 .
(c) y = x3 ⇒ x = y1/3. Interchanging x and y gives y = x1/3,
so f−1(x) = x1/3. Domain f−1 = range(f) = R.
Range(f−1) = domain(f) = R.
(d) f−1(x) = x1/3 ⇒ (f−1)0(x) = 13x−2/3 ⇒
(f−1)0(8) = 13
14= 1
12as in part (b).
(e)
35. (a) Since x ≥ 0, x1 6= x2 ⇒ x21 6= x22 ⇒ 9− x21 6= 9− x22 ⇒ f(x1) 6= f(x2), so f is 1-1.
(b) f 0(x) = −2x and f(1) = 8 ⇒ f−1(8) = 1, so (f−1)0(8) = 1
f 0(f−1(8))=
1
f 0(1)=
1
−2 = −1
2.
SECTION 7.1 INVERSE FUNCTIONS ¤ 247
(c) y = 9− x2 ⇒ x2 = 9− y ⇒ x =√9− y.
Interchange x and y: y =√9− x, so f−1(x) =
√9− x.
Domain(f−1) = range (f) = [0, 9].
Range(f−1) = domain (f) = [0, 3].
(d) (f−1)0(x) = −1/ 2√9− x ⇒ (f−1)0(8) = − 12
as in part (b).
(e)
37. f(0) = 4 ⇒ f−1(4) = 0, and f(x) = 2x3 + 3x2 + 7x+ 4 ⇒ f 0(x) = 6x2 + 6x+ 7 and f 0(0) = 7.
Thus, (f−1)0(4) = 1
f 0(f−1(4))=
1
f 0(0)=1
7.
39. f(0) = 3 ⇒ f−1(3) = 0, and f(x) = 3 + x2 + tan(πx/2) ⇒ f 0(x) = 2x+ π2sec2(πx/2) and
f 0(0) = π2 · 1 = π
2 . Thus, f−10(3) = 1/f 0 f−1(3) = 1/f 0(0) = 2/π.
41. f(4) = 5 ⇒ f−1(5) = 4. Thus, (f−1)0(5) = 1
f 0(f−1(5))=
1
f 0(4)=
1
2/3=3
2.
43. We see that the graph of y = f(x) =√x3 + x2 + x+ 1 is increasing, so f is 1-1.
Enter x = y3 + y2 + y + 1 and use your CAS to solve the equation for y.
Using Derive, we get two (irrelevant) solutions involving imaginary expressions,
as well as one which can be simplified to the following:
y = f−1(x) = − 3√46
3√D − 27x2 + 20− 3
√D + 27x2 − 20 + 3
√2
where D = 3√3√27x4 − 40x2 + 16.
Maple and Mathematica each give two complex expressions and one real expression, and the real expression is equivalent
to that given by Derive. For example, Maple’s expression simplifies to 16
M2/3 − 8− 2M1/3
2M1/3, where
M = 108x2 + 12√48− 120x2 + 81x4 − 80.
45. (a) If the point (x, y) is on the graph of y = f(x), then the point (x− c, y) is that point shifted c units to the left. Since f is
1-1, the point (y, x) is on the graph of y = f−1(x) and the point corresponding to (x− c, y) on the graph of f is
(y, x− c) on the graph of f−1. Thus, the curve’s reflection is shifted down the same number of units as the curve itself is
shifted to the left. So an expression for the inverse function is g−1(x) = f−1(x)− c.
(b) If we compress (or stretch) a curve horizontally, the curve’s reflection in the line y = x is compressed (or stretched)
vertically by the same factor. Using this geometric principle, we see that the inverse of h(x) = f(cx) can be expressed as
h−1(x) = (1/c) f−1(x).
248 ¤ CHAPTER 7 INVERSE FUNCTIONS
7.2 Exponential Functions and Their Derivatives
1. (a) f(x) = ax, a > 0 (b) R
(c) (0,∞) (d) See Figures 6(c), 6(b), and 6(a), respectively.
3. All of these graphs approach 0 as x→−∞, all of them pass through the point
(0, 1), and all of them are increasing and approach∞ as x→∞. The larger the
base, the faster the function increases for x > 0, and the faster it approaches 0 as
x→−∞.
5. The functions with bases greater than 1 (3x and 10x) are increasing, while those
with bases less than 1 13
x and 110
x are decreasing. The graph of 13
x is the
reflection of that of 3x about the y-axis, and the graph of 110
x is the reflection of
that of 10x about the y-axis. The graph of 10x increases more quickly than that of
3x for x > 0, and approaches 0 faster as x→−∞.
7. We start with the graph of y = 4x (Figure 3) and then
shift 3 units downward. This shift doesn’t affect the
domain, but the range of y = 4x − 3 is (−3,∞) .There is a horizontal asymptote of y = −3.
y = 4x y = 4x − 3
9. We start with the graph of y = 2x (Figure 3),
reflect it about the y-axis, and then about the
x-axis (or just rotate 180◦ to handle both
reflections) to obtain the graph of y = −2−x.
In each graph, y = 0 is the horizontal
asymptote.y = 2x y = 2−x y = −2−x
SECTION 7.2 EXPONENTIAL FUNCTIONS AND THEIR DERIVATIVES ¤ 249
11. We start with the graph of y = ex (Figure 12 ) and reflect about the y-axis to get the graph of y = e−x. Then we compress the
graph vertically by a factor of 2 to obtain the graph of y = 12e−x and then reflect about the x-axis to get the graph of
y = − 12e−x. Finally, we shift the graph upward one unit to get the graph of y = 1− 1
2e−x.
13. (a) To find the equation of the graph that results from shifting the graph of y = ex 2 units downward, we subtract 2 from the
original function to get y = ex − 2.
(b) To find the equation of the graph that results from shifting the graph of y = ex 2 units to the right, we replace x with x− 2in the original function to get y = e(x−2).
(c) To find the equation of the graph that results from reflecting the graph of y = ex about the x-axis, we multiply the original
function by −1 to get y = −ex.
(d) To find the equation of the graph that results from reflecting the graph of y = ex about the y-axis, we replace x with −x in
the original function to get y = e−x.
(e) To find the equation of the graph that results from reflecting the graph of y = ex about the x-axis and then about the
y-axis, we first multiply the original function by −1 (to get y = −ex) and then replace x with −x in this equation to
get y = −e−x.
15. (a) The denominator 1 + ex is never equal to zero because ex > 0, so the domain of f(x) = 1/(1 + ex) is R.
(b) 1− ex = 0 ⇔ ex = 1 ⇔ x = 0, so the domain of f(x) = 1/(1− ex) is (−∞, 0) ∪ (0,∞).
17. Use y = Cax with the points (1, 6) and (3, 24). 6 = Ca1 C = 6a
and 24 = Ca3 ⇒ 24 =6
aa3 ⇒
4 = a2 ⇒ a = 2 [since a > 0] and C = 62= 3. The function is f(x) = 3 · 2x.
19. 2 ft = 24 in, f(24) = 242 in = 576 in = 48 ft. g(24) = 224 in = 224/(12 · 5280) mi ≈ 265 mi
21. The graph of g finally surpasses that of f at x ≈ 35.8.
23. limx→∞
(1.001)x =∞ by (3), since 1.001 > 1.
250 ¤ CHAPTER 7 INVERSE FUNCTIONS
25. Divide numerator and denominator by e3x: limx→∞
e3x − e−3x
e3x + e−3x= lim
x→∞1− e−6x
1 + e−6x=1− 01 + 0
= 1
27. Let t = 3/(2− x). As x→ 2+, t→−∞. So limx→2+
e3/(2−x) = limt→−∞
et = 0 by (10).
29. Since −1 ≤ cosx ≤ 1 and e−2x > 0, we have −e−2x ≤ e−2x cosx ≤ e−2x. We know that limx→∞
(−e−2x) = 0 and
limx→∞
e−2x = 0, so by the Squeeze Theorem, limx→∞
(e−2x cosx) = 0.
31. By the Product Rule, f(x) = (x3 + 2x)ex ⇒
f 0(x) = (x3 + 2x)(ex)0 + ex(x3 + 2x)0 = (x3 + 2x)ex + ex(3x2 + 2)
= ex[(x3 + 2x) + (3x2 + 2)] = ex(x3 + 3x2 + 2x+ 2)
33. By (9), y = eax3 ⇒ y0 = eax
3 d
dx(ax3) = 3ax2eax
3.
35. f(u) = e1/u ⇒ f 0(u) = e1/u · d
du
1
u= e1/u
−1u2
=−1u2
e1/u
37. By (9), F (t) = et sin 2t ⇒ F 0(t) = et sin 2t(t sin 2t)0 = et sin 2t(t · 2 cos 2t+ sin 2t · 1) = et sin 2t(2t cos 2t+ sin 2t)
39. y =√1 + 2e3x ⇒ y0 =
1
2(1 + 2e3x)−1/2
d
dx(1 + 2e3x) =
1
2√1 + 2e3x
(2e3x · 3) = 3e3x√1 + 2e3x
41. y = eex ⇒ y0 = ee
x · d
dx(ex) = ee
x · ex or eex+x
43. By the Quotient Rule, y = aex + b
cex + d⇒
y0 =(cex + d)(aex)− (aex + b)(cex)
(cex + d)2=(acex + ad− acex − bc)ex
(cex + d)2=(ad− bc)ex
(cex + d)2.
45. y = cos 1− e2x
1 + e2x⇒
y0 = − sin 1− e2x
1 + e2x· d
dx
1− e2x
1 + e2x= − sin 1− e2x
1 + e2x· (1 + e2x)(−2e2x)− (1− e2x)(2e2x)
(1 + e2x)2
= − sin 1− e2x
1 + e2x· −2e
2x (1 + e2x) + (1− e2x)
(1 + e2x)2= − sin 1− e2x
1 + e2x· −2e
2x(2)
(1 + e2x)2=
4e2x
(1 + e2x)2· sin 1− e2x
1 + e2x
47. y = e2x cosπx ⇒ y0 = e2x(−π sinπx) + (cosπx)(2e2x) = e2x(2 cosπx− π sinπx).
At (0, 1), y0 = 1(2− 0) = 2, so an equation of the tangent line is y − 1 = 2(x− 0), or y = 2x+ 1.
49. d
dxex
2y =d
dx(x+ y) ⇒ ex
2y(x2y0 + y · 2x) = 1 + y0 ⇒ x2ex2yy0 + 2xyex
2y = 1 + y0 ⇒
x2ex2yy0 − y0 = 1− 2xyex2y ⇒ y0(x2ex
2y − 1) = 1− 2xyex2y ⇒ y0 =1− 2xyex2yx2ex2y − 1
SECTION 7.2 EXPONENTIAL FUNCTIONS AND THEIR DERIVATIVES ¤ 251
51. y = ex + e−x/2 ⇒ y0 = ex − 12e−x/2 ⇒ y00 = ex + 1
4e−x/2, so
2y00 − y0 − y = 2 ex + 14e−x/2 − ex − 1
2e−x/2 − ex + e−x/2 = 0.
53. y = erx ⇒ y0 = rerx ⇒ y00 = r2erx, so if y = erx satisfies the differential equation y00 + 6y0 + 8y = 0,
then r2erx + 6rerx + 8erx = 0; that is, erx(r2 + 6r + 8) = 0. Since erx > 0 for all x, we must have r2 + 6r + 8 = 0,
or (r + 2)(r + 4) = 0, so r = −2 or −4.
55. f(x) = e2x ⇒ f 0(x) = 2e2x ⇒ f 00(x) = 2 · 2e2x = 22e2x ⇒f 000(x) = 22 · 2e2x = 23e2x ⇒ · · · ⇒ f (n)(x) = 2ne2x
57. (a) f(x) = ex + x is continuous on R and f(−1) = e−1 − 1 < 0 < 1 = f(0), so by the Intermediate Value Theorem,
ex + x = 0 has a root in (−1, 0).
(b) f(x) = ex + x ⇒ f 0(x) = ex + 1, so xn+1 = xn − exn + xnexn + 1
. Using x1 = −0.5, we get x2 ≈ −0.566311,
x3 ≈ −0.567143 ≈ x4, so the root is −0.567143 to six decimal places.
59. (a) limt→∞
p(t) = limt→∞
1
1 + ae−kt=
1
1 + a · 0 = 1, since k > 0 ⇒ −kt→ −∞ ⇒ e−kt → 0.
(b) p(t) = (1 + ae−kt)−1 ⇒ dp
dt= −(1 + ae−kt)−2(−kae−kt) = kae−kt
(1 + ae−kt)2
(c)From the graph of p(t) = (1 + 10e−0.5t)−1, it seems that p(t) = 0.8
(indicating that 80% of the population has heard the rumor) when
t ≈ 7.4 hours.
61. f(x) = x− ex ⇒ f 0(x) = 1− ex = 0 ⇔ ex = 1 ⇔ x = 0. Now f 0(x) > 0 for all x < 0 and f 0(x) < 0 for all
x > 0, so the absolute maximum value is f(0) = 0− 1 = −1.
63. f(x) = xe−x2/8, [−1, 4]. f 0(x) = x · e−x2/8 · (−x
4) + e−x
2/8 · 1 = e−x2/8(−x2
4+ 1). Since e−x
2/8 is never 0,
f 0(x) = 0 ⇒ −x2/4 + 1 = 0 ⇒ 1 = x2/4 ⇒ x2 = 4 ⇒ x = ±2, but −2 is not in the given interval, [−1, 4].
f(−1) = −e−1/8 ≈ −0.88, f(2) = 2e−1/2 ≈ 1.21, and f(4) = 4e−2 ≈ 0.54. So f(2) = 2e−1/2 is the absolute maximum
value and f(−1) = −e−1/8 is the absolute minimum value.
65. (a) f(x) = (1− x)e−x ⇒ f 0(x) = (1− x)(−e−x) + e−x(−1) = e−x(x− 2) > 0 ⇒ x > 2, so f is increasing on
(2,∞) and decreasing on (−∞, 2).
(b) f 00(x) = e−x(1) + (x− 2)(−e−x) = e−x(3− x) > 0 ⇔ x < 3, so f is CU on (−∞, 3) and CD on (3,∞).
(c) f 00 changes sign at x = 3, so there is an IP at 3,−2e−3 .
252 ¤ CHAPTER 7 INVERSE FUNCTIONS
67. y = f(x) = e−1/(x+1) A. D = {x | x 6= −1} = (−∞,−1) ∪ (−1,∞) B. No x-intercept; y-intercept = f(0) = e−1
C. No symmetry D. limx→±∞
e−1/(x+1) = 1 since −1/(x+ 1)→ 0, so y = 1 is a HA. limx→−1+
e−1/(x+1) = 0 since
−1 /(x+ 1) → −∞, limx→−1−
e−1/(x+1) = ∞ since −1/(x+ 1) → ∞, so x = −1 is a VA.
E. f 0(x) = e−1/(x+1)/(x+ 1)2 ⇒ f 0(x) > 0 for all x except 1, so
f is increasing on (−∞,−1) and (−1,∞). F. No extreme values
G. f 00(x) = e−1/(x+1)
(x+ 1)4+
e−1/(x+1)(−2)(x+ 1)3
= −e−1/(x+1)(2x+ 1)(x+ 1)4
⇒
f 00(x) > 0 ⇔ 2x+ 1 < 0 ⇔ x < − 12
, so f is CU on (−∞,−1)and −1,− 1
2, and CD on − 1
2,∞ . f has an IP at − 1
2, e−2 .
H.
69. S(t) = Atpe−kt with A = 0.01, p = 4, and k = 0.07. We will find the
zeros of f 00 for f(t) = tpe−kt.
f 0(t) = tp(−ke−kt) + e−kt(ptp−1) = e−kt(−ktp + ptp−1)
f 00(t) = e−kt(−kptp−1 + p(p− 1)tp−2) + (−ktp + ptp−1)(−ke−kt)= tp−2e−kt[−kpt+ p(p− 1) + k2t2 − kpt]
= tp−2e−kt(k2t2 − 2kpt+ p2 − p)
Using the given values of p and k gives us f 00(t) = t2e−0.07t(0.0049t2 − 0.56t+ 12). So S00(t) = 0.01f 00(t) and its zeros
are t = 0 and the solutions of 0.0049t2 − 0.56t+ 12 = 0, which are t1 = 2007≈ 28.57 and t2 = 600
7≈ 85.71.
At t1 minutes, the rate of increase of the level of medication in the bloodstream is at its greatest and at t2 minutes, the rate of
decrease is the greatest.
71. f(x) = ex3−x → 0 as x→ −∞, and
f(x)→∞ as x→∞. From the graph,
it appears that f has a local minimum of
about f(0.58) = 0.68, and a local
maximum of about f(−0.58) = 1.47.
To find the exact values, we calculate
f 0(x) = 3x2 − 1 ex3−x, which is 0 when 3x2 − 1 = 0 ⇔ x = ± 1√
3. The negative root corresponds to the local
maximum f − 1√3= e(−1/
√3)3− (−1/√3) = e2
√3/9, and the positive root corresponds to the local minimum
f 1√3= e(1/
√3)3− (1/
√3) = e−2
√3/9. To estimate the inflection points, we calculate and graph
f 00(x) =d
dx3x2 − 1 ex
3−x = 3x2 − 1 ex3−x 3x2 − 1 + ex
3−x(6x) = ex3−x 9x4 − 6x2 + 6x+ 1 .
From the graph, it appears that f 00(x) changes sign (and thus f has inflection points) at x ≈ −0.15 and x ≈ −1.09. From the
graph of f , we see that these x-values correspond to inflection points at about (−0.15, 1.15) and (−1.09, 0.82).
SECTION 7.2 EXPONENTIAL FUNCTIONS AND THEIR DERIVATIVES ¤ 253
73. Let u = −3x. Then du = −3 dx, so 5
0e−3x dx = − 1
3
−150
eu du = − 13 [e
u]−150 = − 13e−15 − e0 = 1
31− e−15 .
75. Let u = 1 + ex. Then du = ex dx, so ex√1 + ex dx =
√udu = 2
3u3/2 +C = 2
3 (1 + ex)3/2 + C.
77. (ex + e−x)2 dx = (e2x + 2 + e−2x) dx = 12e2x + 2x− 1
2e−2x +C
79. sinx ecos x dxu = cosx,
du = − sinx dx = eu (−du) = −eu +C = −ecos x + C
81. Let u =√x. Then du = 1
2√xdx, so e
√x
√xdx = 2 eu du = 2eu + C = 2e
√x +C.
83. Area = 1
0e3x − ex dx = 1
3e3x − ex
1
0= 1
3e3 − e − 1
3− 1 = 1
3e3 − e+ 2
3≈ 4.644
85. V =1
0π(ex)2 dx = π
1
0e2x dx = 1
2π e2x1
0= π
2e2 − 1
87. (a) erf(x) = 2√π
x
0
e−t2
dt ⇒x
0
e−t2
dt =
√π
2erf(x). By Property 5 of definite integrals in Section 5.2,
b
0e−t
2dt =
a
0e−t
2dt+
b
ae−t
2dt, so
b
a
e−t2
dt =b
0
e−t2
dt−a
0
e−t2
dt =
√π
2erf(b)−
√π
2erf(a) = 1
2
√π [erf(b)− erf(a)].
(b) y = ex2
erf(x) ⇒ y0 = 2xex2
erf(x) + ex2
erf 0(x) = 2xy + ex2 · 2√
πe−x
2
[by FTC1] = 2xy +2√π
.
89. We use Theorem 7.1.7. Note that f(0) = 3 + 0 + e0 = 4, so f−1(4) = 0. Also f 0(x) = 1 + ex. Therefore,
f−1 0(4) =
1
f 0(f−1(4))=
1
f 0(0)=
1
1 + e0=1
2.
91. From the graph, it appears that f is an odd function (f is undefined for x = 0).
To prove this, we must show that f(−x) = −f(x).
f(−x) = 1− e1/(−x)
1 + e1/(−x)=1− e(−1/x)
1 + e(−1/x)=1− 1
e1/x
1 +1
e1/x
· e1/x
e1/x=
e1/x − 1e1/x + 1
= −1− e1/x
1 + e1/x= −f(x)
so f is an odd function.
93. (a) Let f(x) = ex − 1− x. Now f(0) = e0 − 1 = 0, and for x ≥ 0, we have f 0(x) = ex − 1 ≥ 0. Now, since f(0) = 0 and
f is increasing on [0,∞), f(x) ≥ 0 for x ≥ 0 ⇒ ex − 1− x ≥ 0 ⇒ ex ≥ 1 + x.
(b) For 0 ≤ x ≤ 1, x2 ≤ x, so ex2 ≤ ex [since ex is increasing]. Hence [from (a)] 1 + x2 ≤ ex
2 ≤ ex.
So 43=
1
01 + x2 dx ≤ 1
0ex
2dx ≤ 1
0ex dx = e− 1 < e ⇒ 4
3≤ 1
0ex
2dx ≤ e.
254 ¤ CHAPTER 7 INVERSE FUNCTIONS
95. (a) By Exercise 93(a), the result holds for n = 1. Suppose that ex ≥ 1 + x+x2
2!+ · · ·+ xk
k!for x ≥ 0.
Let f(x) = ex − 1− x− x2
2!− · · ·− xk
k!− xk+1
(k + 1)!. Then f 0(x) = ex − 1− x− · · ·− xk
k!≥ 0 by assumption. Hence
f(x) is increasing on (0,∞). So 0 ≤ x implies that 0 = f(0) ≤ f(x) = ex − 1− x− · · ·− xk
k!− xk+1
(k + 1)!, and hence
ex ≥ 1 + x+ · · ·+ xk
k!+
xk+1
(k + 1)!for x ≥ 0. Therefore, for x ≥ 0, ex ≥ 1 + x+
x2
2!+ · · ·+ xn
n!for every positive
integer n, by mathematical induction.
(b) Taking n = 4 and x = 1 in (a), we have e = e1 ≥ 1 + 12+ 1
6+ 1
24= 2.7083 > 2.7.
(c) ex ≥ 1 + x+ · · ·+ xk
k!+
xk+1
(k + 1)!⇒ ex
xk≥ 1
xk+
1
xk−1+ · · ·+ 1
k!+
x
(k + 1)!≥ x
(k + 1)!.
But limx→∞
x
(k + 1)!=∞, so lim
x→∞ex
xk=∞.
7.3 Logarithmic Functions
1. (a) It is defined as the inverse of the exponential function with base a, that is, loga x = y ⇔ ay = x.
(b) (0,∞) (c) R (d) See Figure 1.
3. (a) log5 125 = 3 since 53 = 125. (b) log31
27= −3 since 3−3 = 1
33=1
27.
5. (a) log5 125= log5 5
−2 = −2 by (2). (b) eln 15 = 15 by (8).
7. (a) log2 6− log2 15 + log2 20 = log2( 615 ) + log2 20 [by Law 2]
= log2(615· 20) [by Law 1]
= log2 8, and log2 8 = 3 since 23 = 8.
(b) log3 100− log3 18− log3 50 = log310018
− log3 50 = log3 10018·50
= log3(19), and log3
19= −2 since 3−2 = 1
9.
9. log2x3y
z2= log2(x
3y)− log2 z2 = log2 x3 + log2 y − log2 z2 = 3 log2 x+ log2 y − 2 log2 z
[assuming that the variables are positive]
11. ln(uv)10 = 10 ln(uv) = 10(lnu+ ln v) = 10 lnu+ 10 ln v
13. log10 a− log10 b+ log10 c = log10a
b+ log10 c = log10
a
b· c = log10
ac
b
15. ln 5 + 5 ln 3 = ln 5 + ln 35 [by Law 3]
= ln(5 · 35) [by Law 1]
= ln1215
SECTION 7.3 LOGARITHMIC FUNCTIONS ¤ 255
17. ln(1 + x2) + 12lnx− ln sinx = ln(1 + x2) + lnx1/2 − ln sinx = ln[(1 + x2)
√x ]− ln sinx = ln (1 + x2)
√x
sinx
19. (a) log12 e =ln e
ln 12=
1
ln 12≈ 0.402430
(b) log6 13.54 =ln 13.54
ln 6≈ 1.454240
(c) log2 π =lnπ
ln 2≈ 1.651496
21. To graph these functions, we use log1.5 x =lnx
ln 1.5and log50 x =
lnx
ln 50.
These graphs all approach−∞ as x→ 0+, and they all pass through the
point (1, 0). Also, they are all increasing, and all approach∞ as x→∞.
The functions with larger bases increase extremely slowly, and the ones with
smaller bases do so somewhat more quickly. The functions with large bases
approach the y-axis more closely as x→ 0+.
23. (a) Shift the graph of y = log10 x five units to the left to
obtain the graph of y = log10(x+5). Note the vertical
asymptote of x = −5.
y = log10 x y = log10(x+ 5)
(b) Reflect the graph of y = lnx about the x-axis to obtain
the graph of y = − lnx.
y = lnx y = − lnx
25. (a) 2 lnx = 1 ⇒ lnx = 12⇒ x = e1/2 =
√e
(b) e−x = 5 ⇒ −x = ln 5 ⇒ x = − ln 5
27. (a) 2x−5 = 3 ⇔ log2 3 = x− 5 ⇔ x = 5 + log2 3.
Or: 2x−5 = 3 ⇔ ln 2x−5 = ln3 ⇔ (x− 5) ln 2 = ln 3 ⇔ x− 5 = ln 3
ln 2⇔ x = 5 +
ln 3
ln 2
(b) lnx+ ln(x− 1) = ln(x(x− 1)) = 1 ⇔ x(x− 1) = e1 ⇔ x2 − x− e = 0. The quadratic formula (with a = 1,
b = −1, and c = −e) gives x = 121±√1 + 4e , but we reject the negative root since the natural logarithm is not
defined for x < 0. So x = 121 +
√1 + 4e .
29. 3xex + x2ex = 0 ⇔ xex(3 + x) = 0 ⇔ x = 0 or −3
31. ln(lnx) = 1 ⇔ eln(ln x) = e1 ⇔ lnx = e1 = e ⇔ eln x = ee ⇔ x = ee
33. e2x − ex − 6 = 0 ⇔ (ex − 3)(ex + 2) = 0 ⇔ ex = 3 or −2 ⇒ x = ln 3 since ex > 0.
256 ¤ CHAPTER 7 INVERSE FUNCTIONS
35. (a) e2+5x = 100 ⇒ ln e2+5x = ln 100 ⇒ 2 + 5x = ln 100 ⇒ 5x = ln 100− 2 ⇒x = 1
5(ln 100− 2) ≈ 0.5210
(b) ln(ex − 2) = 3 ⇒ ex − 2 = e3 ⇒ ex = e3 + 2 ⇒ x = ln(e3 + 2) ≈ 3.0949
37. (a) ex < 10 ⇒ ln ex < ln 10 ⇒ x < ln 10 ⇒ x ∈ (−∞, ln 10)
(b) lnx > −1 ⇒ elnx > e−1 ⇒ x > e−1 ⇒ x ∈ (1/e,∞)
39. 3 ft = 36 in, so we need x such that log2 x = 36 ⇔ x = 236 = 68,719,476,736. In miles, this is
68,719,476,736 in · 1 ft12 in
· 1 mi5280 ft
≈ 1,084,587.7 mi.
41. If I is the intensity of the 1989 San Francisco earthquake, then log10(I/S) = 7.1 ⇒log10(16I/S) = log10 16 + log10(I/S) = log10 16 + 7.1 ≈ 8.3.
43. (a) n = 100 · 2t/3 ⇒ n
100= 2t/3 ⇒ log2
n
100=
t
3⇒ t = 3 log2
n
100. Using formula (7), we can write this
as t = f−1(n) = 3 · ln(n/100)ln 2
. This function tells us how long it will take to obtain n bacteria (given the number n).
(b) n = 50,000 ⇒ t = f−1(50,000) = 3 · ln50,000100
ln 2= 3
ln 500
ln 2≈ 26.9 hours
45. Let t = x2 − 9. Then as x→ 3+, t→ 0+, and limx→3+
ln(x2 − 9) = limt→0+
ln t = −∞ by (8).
47. limx→0
ln(cosx) = ln 1 = 0. [ln(cosx) is continuous at x = 0 since it is the composite of two continuous functions.]
49. limx→∞
[ln(1 + x2)− ln(1 + x)] = limx→∞
ln1 + x2
1 + x= ln lim
x→∞1 + x2
1 + x= ln lim
x→∞
1x+ x
1x+ 1
=∞, since the limit in
parentheses is∞.
51. f(x) = log10(x2 − 9). Df = x | x2 − 9 > 0 = x |x| > 3 = (−∞,−3) ∪ (3,∞)
53. (a) For f(x) =√3− e2x, we must have 3− e2x ≥ 0 ⇒ e2x ≤ 3 ⇒ 2x ≤ ln 3 ⇒ x ≤ 1
2ln 3.
Thus, the domain of f is (−∞, 12ln 3].
(b) y = f(x) =√3− e2x [note that y ≥ 0] ⇒ y2 = 3− e2x ⇒ e2x = 3− y2 ⇒ 2x = ln(3− y2) ⇒
x = 12ln(3− y2). Interchange x and y: y = 1
2ln(3− x2). So f−1(x) = 1
2ln(3− x2). For the domain of f−1, we must
have 3− x2 > 0 ⇒ x2 < 3 ⇒ |x| < √3 ⇒ −√3 < x <√3 ⇒ 0 ≤ x <
√3 since x ≥ 0. Note that the
domain of f−1, [0,√3 ), equals the range of f .
55. y = ln(x+ 3) ⇒ ey = eln(x+3) = x+ 3 ⇒ x = ey − 3.
Interchange x and y: the inverse function is y = ex − 3.
57. y = f(x) = ex3 ⇒ ln y = x3 ⇒ x = 3
√ln y. Interchange x and y: y = 3
√lnx. So f−1(x) = 3
√lnx.
SECTION 7.4 DERIVATIVES OF LOGARITHMIC FUNCTIONS ¤ 257
59. y = log10 1 +1
x⇒ 10y = 1 +
1
x⇒ 1
x= 10y − 1 ⇒ x =
1
10y − 1 .
Interchange x and y: y =1
10x − 1 is the inverse function.
61. f(x) = e3x − ex ⇒ f 0(x) = 3e3x − ex. Thus, f 0(x) > 0 ⇔ 3e3x > ex ⇔ 3e3x
ex>
ex
ex⇔ 3e2x > 1 ⇔
e2x > 13⇔ 2x > ln 1
3= − ln 3 ⇔ x > − 1
2ln 3, so f is increasing on − 1
2ln 3,∞ .
63. y = f(x) = (x2 − 2)e−x ⇒ y0 = (x2 − 2)(−e−x) + e−x(2x) = e−x(−x2 + 2x+ 2) ⇒y00 = e−x(−2x+ 2) + (−x2 + 2x+ 2)(−e−x) = e−x(x2 − 4x) = xe−x(x− 4).y00 > 0 ⇒ x < 0 or x > 4 ⇒ f is CU on (−∞, 0) and (4,∞).
65. (a) We have to show that −f(x) = f(−x).
−f(x) = − ln x+√x2 + 1 = ln x+
√x2 + 1
−1= ln
1
x+√x2 + 1
= ln1
x+√x2 + 1
· x−√x2 + 1
x−√x2 + 1 = lnx−√x2 + 1x2 − x2 − 1 = ln
√x2 + 1− x = f(−x)
Thus, f is an odd function.
(b) Let y = ln x+√x2 + 1 . Then ey = x+
√x2 + 1 ⇔ (ey − x)2 = x2 + 1 ⇔ e2y − 2xey + x2 = x2 + 1 ⇔
2xey = e2y − 1 ⇔ x =e2y − 12ey
= 12(ey − e−y). Thus, the inverse function is f−1(x) = 1
2(ex − e−x).
67. x1/ lnx = 2 ⇒ ln(x1/ lnx) = ln(2) ⇒ 1
lnx· lnx = ln 2 ⇒ 1 = ln 2, a contradiction, so the given equation has no
solution. The function f(x) = x1/ lnx = (eln x)1/ ln x = e1 = e for all x > 0, so the function f(x) = x1/ lnx is the constant
function f(x) = e.
69. (a) Let ε > 0 be given. We need N such that |ax − 0| < ε when x < N . But ax < ε ⇔ x < loga ε. Let N = loga ε.
Then x < N ⇒ x < loga ε ⇒ |ax − 0| = ax < ε, so limx→−∞
ax = 0.
(b) Let M > 0 be given. We need N such that ax > M when x > N . But ax > M ⇔ x > logaM . Let N = logaM .
Then x > N ⇒ x > logaM ⇒ ax > M , so limx→∞
ax =∞.
71. ln(x2 − 2x− 2) ≤ 0 ⇒ 0 < x2 − 2x− 2 ≤ 1. Now x2 − 2x− 2 ≤ 1 gives x2 − 2x− 3 ≤ 0 and hence
(x− 3)(x+ 1) ≤ 0. So −1 ≤ x ≤ 3. Now 0 < x2 − 2x− 2 ⇒ x < 1−√3 or x > 1 +√3. Therefore,
ln(x2 − 2x− 2) ≤ 0 ⇔ −1 ≤ x < 1−√3 or 1 +√3 < x ≤ 3.
7.4 Derivatives of Logarithmic Functions
1. The differentiation formula for logarithmic functions, d
dx(loga x) =
1
x ln a, is simplest when a = e because ln e = 1.
3. f(x) = sin(lnx) ⇒ f 0(x) = cos(lnx) · d
dxlnx = cos(lnx) · 1
x=cos(lnx)
x
258 ¤ CHAPTER 7 INVERSE FUNCTIONS
5. f(x) = log2(1− 3x) ⇒ f 0(x) =1
(1− 3x) ln 2d
dx(1− 3x) = −3
(1− 3x) ln 2 or 3
(3x− 1) ln 2
7. f(x) = 5√lnx = (lnx)1/5 ⇒ f 0(x) = 1
5(lnx)−4/5
d
dx(lnx) =
1
5(lnx)4/5· 1x=
1
5x 5 (lnx)4
9. f(x) = sinx ln(5x) ⇒ f 0(x) = sinx · 15x· ddx(5x)+ ln(5x) · cosx = sinx · 5
5x+cosx ln(5x) =
sinx
x+cosx ln(5x)
11. F (t) = ln (2t+ 1)3
(3t− 1)4 = ln(2t+ 1)3 − ln(3t − 1)4 = 3 ln(2t+ 1)− 4 ln(3t− 1) ⇒
F 0(t) = 3 · 1
2t+ 1· 2− 4 · 1
3t− 1 · 3 =6
2t+ 1− 12
3t− 1 , or combined, −6(t+ 3)(2t+ 1)(3t− 1) .
13. g(x) = ln x√x2 − 1 = lnx + ln(x2 − 1)1/2 = lnx + 1
2ln(x2 − 1) ⇒
g0(x) =1
x+1
2· 1
x2 − 1 · 2x =1
x+
x
x2 − 1 =x2 − 1 + x · xx(x2 − 1) =
2x2 − 1x(x2 − 1)
15. f(u) = lnu
1 + ln(2u)⇒
f 0(u) =[1 + ln(2u)] · 1
u − lnu · 12u · 2
[1 + ln(2u)]2=
1u [1 + ln(2u)− lnu][1 + ln(2u)]2
=1 + (ln 2 + lnu)− lnu
u[1 + ln(2u)]2=
1 + ln 2
u[1 + ln(2u)]2
17. h(t) = t3 − 3t ⇒ h0(t) = 3t2 − 3t ln 3
19. y = ln 2− x− 5x2 ⇒ y0 =1
2− x− 5x2 · (−1− 10x) =−10x− 12− x− 5x2 or 10x+ 1
5x2 + x− 2
21. y = ln(e−x + xe−x) = ln(e−x(1 + x)) = ln(e−x) + ln(1 + x) = −x+ ln(1 + x) ⇒
y0 = −1 + 1
1 + x=−1− x+ 1
1 + x= − x
1 + x
23. y = 2x log10√x = 2x log10 x
1/2 = 2x · 12 log10 x = x log10 x ⇒ y0 = x · 1
x ln 10+ log10 x · 1 =
1
ln 10+ log10 x
Note: 1
ln 10=ln e
ln 10= log10 e, so the answer could be written as 1
ln 10+ log10 x = log10 e+ log10 x = log10 ex.
25. Using Formula 7 and the Chain Rule, y = 5−1/x ⇒ y0 = 5−1/x(ln 5) −1 · −x−2 = 5−1/x(ln 5)/x2
27. y = x2 ln(2x) ⇒ y0 = x2 · 12x· 2 + ln(2x) · (2x) = x + 2x ln(2x) ⇒
y00 = 1 + 2x · 12x· 2 + ln(2x) · 2 = 1 + 2 + 2 ln(2x) = 3 + 2 ln(2x)
29. y = ln x+√1 + x2 ⇒
y0 =1
x+√1 + x2
d
dxx+
√1 + x2 =
1
x+√1 + x2
1 + 12(1 + x2)−1/2(2x)
=1
x+√1 + x2
1 +x√1 + x2
=1
x+√1 + x2
·√1 + x2 + x√1 + x2
=1√1 + x2
⇒
y00 = − 12 (1 + x2)−3/2(2x) =
−x(1 + x2)3/2
SECTION 7.4 DERIVATIVES OF LOGARITHMIC FUNCTIONS ¤ 259
31. f(x) = x
1− ln(x− 1) ⇒
f 0(x) =[1− ln(x− 1)] · 1− x · −1
x− 1[1− ln(x− 1)]2 =
(x− 1)[1− ln(x− 1)] + x
x− 1[1− ln(x− 1)]2 =
x− 1− (x− 1) ln(x− 1) + x
(x− 1)[1− ln(x− 1)]2
=2x− 1− (x− 1) ln(x− 1)(x− 1)[1− ln(x− 1)]2
Dom(f) = {x | x− 1 > 0 and 1− ln(x− 1) 6= 0} = {x | x > 1 and ln(x− 1) 6= 1}= x | x > 1 and x− 1 6= e1 = {x | x > 1 and x 6= 1 + e} = (1, 1 + e) ∪ (1 + e,∞)
33. f(x) = ln(x2 − 2x) ⇒ f 0(x) =1
x2 − 2x (2x− 2) =2(x− 1)x(x− 2) .
Dom(f) = {x | x(x− 2) > 0} = (−∞, 0) ∪ (2,∞).
35. f(x) = lnx
1 + x2⇒ f 0(x) =
(1 + x2)1
x− (lnx)(2x)
(1 + x2)2, so f 0(1) = 2(1)− 0(2)
22=2
4=1
2.
37. y = ln xex2
= lnx+ ln ex2
= lnx+ x2 ⇒ y0 =1
x+ 2x. At (1, 1), the slope of the tangent line is
y0(1) = 1 + 2 = 3, and an equation of the tangent line is y − 1 = 3(x− 1), or y = 3x− 2.
39. f(x) = sinx+ lnx ⇒ f 0(x) = cosx+ 1/x.
This is reasonable, because the graph shows that f increases when f 0 is
positive, and f 0(x) = 0 when f has a horizontal tangent.
41. y = (2x+ 1)5(x4 − 3)6 ⇒ ln y = ln (2x+ 1)5(x4 − 3)6 ⇒ ln y = 5 ln(2x+ 1) + 6 ln(x4 − 3) ⇒1
yy0 = 5 · 1
2x+ 1· 2 + 6 · 1
x4 − 3 · 4x3 ⇒
y0 = y10
2x+ 1+
24x3
x4 − 3 = (2x+ 1)5(x4 − 3)6 10
2x+ 1+
24x3
x4 − 3 .
[The answer could be simplified to y0 = 2(2x+ 1)4(x4 − 3)5(29x4 + 12x3 − 15), but this is unnecessary.]
43. y = sin2 x tan4 x
(x2 + 1)2⇒ ln y = ln(sin2 x tan4 x) − ln(x2 + 1)2 ⇒
ln y = ln(sinx)2 + ln(tanx)4 − ln(x2 + 1)2 ⇒ ln y = 2 ln |sinx|+ 4 ln |tanx|− 2 ln(x2 + 1) ⇒1
yy0 = 2 · 1
sinx· cosx+ 4 · 1
tanx· sec2 x− 2 · 1
x2 + 1· 2x ⇒ y0 =
sin2 x tan4 x
(x2 + 1)22 cotx+
4 sec2 x
tanx− 4x
x2 + 1
45. y = xx ⇒ ln y = lnxx ⇒ ln y = x lnx ⇒ y0/y = x(1/x) + (lnx) · 1 ⇒ y0 = y(1 + lnx) ⇒y0 = xx(1 + lnx)
260 ¤ CHAPTER 7 INVERSE FUNCTIONS
47. y = x sin x ⇒ ln y = lnx sinx ⇒ ln y = sinx lnx ⇒ y0
y= (sinx) · 1
x+ (lnx)(cosx) ⇒
y0 = ysinx
x+ lnx cosx ⇒ y0 = x sin x
sinx
x+ lnx cosx
49. y = (cosx)x ⇒ ln y = ln(cosx)x ⇒ ln y = x ln cosx ⇒ 1
yy0 = x · 1
cosx· (− sinx) + ln cosx · 1 ⇒
y0 = y ln cosx− x sinx
cosx⇒ y0 = (cosx)x(ln cosx− x tanx)
51. y = (tanx)1/x ⇒ ln y = ln(tanx)1/x ⇒ ln y =1
xln tanx ⇒
1
yy0 =
1
x· 1
tanx· sec2 x+ ln tanx · − 1
x2⇒ y0 = y
sec2 x
x tanx− ln tanx
x2⇒
y0 = (tanx)1/xsec2 x
x tanx− ln tanx
x2or y0 = (tanx)1/x · 1
xcscx secx− ln tanx
x
53. y = ln(x2 + y2) ⇒ y0 =1
x2 + y2d
dx(x2 + y2) ⇒ y0 =
2x+ 2yy0
x2 + y2⇒ x2y0 + y2y0 = 2x+ 2yy0 ⇒
x2y0 + y2y0 − 2yy0 = 2x ⇒ (x2 + y2 − 2y)y0 = 2x ⇒ y0 =2x
x2 + y2 − 2y
55. f(x) = ln(x− 1) ⇒ f 0(x) =1
(x− 1) = (x− 1)−1 ⇒ f 00(x) = −(x− 1)−2 ⇒ f 000 (x) = 2(x− 1)−3 ⇒
f (4)(x) = −2 · 3(x− 1)−4 ⇒ · · · ⇒ f (n)(x) = (−1)n−1 · 2 · 3 · 4 · · · · · (n− 1)(x− 1)−n = (−1)n−1 (n− 1)!(x− 1)n
57. From the graph, it appears that the curves y = (x− 4)2 and y = lnx intersect
just to the left of x = 3 and to the right of x = 5, at about x = 5.3. Let
f(x) = lnx− (x− 4)2. Then f 0(x) = 1/x− 2(x− 4), so Newton’s Method
says that xn+1 = xn − f(xn)/f0(xn) = xn − lnxn − (xn − 4)2
1/xn − 2(xn − 4) . Taking
x0 = 3, we get x1 ≈ 2.957738, x2 ≈ 2.958516 ≈ x3, so the first root is 2.958516, to six decimal places. Taking x0 = 5, we
get x1 ≈ 5.290755, x2 ≈ 5.290718 ≈ x3, so the second (and final) root is 5.290718, to six decimal places.
59. f(x) = lnx√x
⇒ f 0(x) =
√x (1/x)− (lnx)[1/(2√x )]
x=2− lnx2x3/2
⇒
f 00(x) =2x3/2(−1/x)− (2− lnx)(3x1/2)
4x3=3 lnx− 84x5/2
> 0 ⇔ lnx > 83⇔ x > e8/3, so f is CU on (e8/3,∞)
and CD on (0, e8/3). The inflection point is e8/3, 83e−4/3 .
61. y = f(x) = ln(sinx)
A. D = {x in R | sinx > 0} =∞
n=−∞(2nπ, (2n+ 1)π) = · · · ∪ (−4π,−3π) ∪ (−2π,−π) ∪ (0, π) ∪ (2π, 3π) ∪ · · ·
B. No y-intercept; x-intercepts: f(x) = 0 ⇔ ln(sinx) = 0 ⇔ sinx = e0 = 1 ⇔ x = 2nπ + π2
for each
SECTION 7.4 DERIVATIVES OF LOGARITHMIC FUNCTIONS ¤ 261
integer n. C. f is periodic with period 2π. D. limx→(2nπ)+
f(x) = −∞ and limx→[(2n+1)π]−
f(x) = −∞, so the lines
x = nπ are VAs for all integers n. E. f 0(x) = cosx
sinx= cotx, so f 0(x) > 0 when 2nπ < x < 2nπ + π
2for each
integer n, and f 0(x) < 0 when 2nπ + π2< x < (2n+ 1)π. Thus, f is increasing on 2nπ, 2nπ + π
2and
decreasing on 2nπ + π2 , (2n+ 1)π for each integer n.
F. Local maximum values f 2nπ + π2= 0, no local minimum.
G. f 00(x) = − csc2 x < 0, so f is CD on (2nπ, (2n+ 1)π) for
each integer n. No IP
H.
63. y = f(x) = ln(1 + x2) A. D = R B. Both intercepts are 0. C. f(−x) = f(x), so the curve is symmetric about the
y-axis. D. limx→±∞
ln(1 + x2) =∞, no asymptotes. E. f 0(x) = 2x
1 + x2> 0 ⇔
x > 0, so f is increasing on (0,∞) and decreasing on (−∞, 0) .
F. f(0) = 0 is a local and absolute minimum.
G. f 00(x) = 2(1 + x2)− 2x(2x)(1 + x2)2
=2(1− x2)
(1 + x2)2> 0 ⇔
|x| < 1, so f is CU on (−1, 1), CD on (−∞,−1) and (1,∞). IP
(1, ln 2) and (−1, ln 2).
H.
65. We use the CAS to calculate f 0(x) = 2 + sinx+ x cosx
2x+ x sinxand
f 00(x) =2x2 sinx+ 4 sinx− cos2 x+ x2 + 5
x2(cos2 x− 4 sinx− 5) . From the graphs, it
seems that f 0 > 0 (and so f is increasing) on approximately the intervals
(0, 2.7), (4.5, 8.2) and (10.9, 14.3). It seems that f 00 changes sign
(indicating inflection points) at x ≈ 3.8, 5.7, 10.0 and 12.0.
Looking back at the graph of f(x) = ln(2x+ x sinx), this implies that the inflection points have approximate coordinates
(3.8, 1.7), (5.7, 2.1), (10.0, 2.7), and (12.0, 2.9).
67. (a) Using a calculator or CAS, we obtain the model Q = abt with a ≈ 100.0124369 and b ≈ 0.000045145933.
(b) Use Q0(t) = abt ln b (from Formula 7) with the values of a and b from part (a) to get Q0(0.04) ≈ −670.63 μA.
The result of Example 2 in Section 2.1 was −670 μA.
69.4
2
3
xdx = 3
4
2
1
xdx = 3 ln |x| 4
2= 3(ln 4− ln 2) = 3 ln 4
2= 3 ln 2
71.2
1
dt
8− 3t = −13ln |8− 3t|
2
1
= −13ln 2− −1
3ln 5 =
1
3(ln 5− ln 2) = 1
3ln5
2
Or: Let u = 8 − 3t. Then du = −3 dt, so2
1
dt
8− 3t =2
5
− 13du
u= −1
3ln |u|
2
5
= −13ln 2− −1
3ln 5 =
1
3(ln 5− ln 2) = 1
3ln5
2.
262 ¤ CHAPTER 7 INVERSE FUNCTIONS
73.e
1
x2 + x+ 1
xdx =
e
1
x+ 1 +1
xdx = 1
2x2 + x+ lnx
e
1= 1
2e2 + e+ 1 − 1
2+ 1 + 0
= 12e2 + e− 1
2
75. Let u = lnx. Then du = dx
x⇒ (lnx)2
xdx = u2 du =
1
3u3 + C =
1
3(lnx)3 +C.
77. sin 2x
1 + cos2 xdx = 2
sinx cosx
1 + cos2 xdx = 2I. Let u = cosx. Then du = − sinxdx, so
2I = −2 udu
1 + u2= −2 · 12 ln(1 + u2) + C = − ln(1 + u2) +C = − ln(1 + cos2 x) + C.
Or: Let u = 1 + cos2 x.
79.2
1
10t dt =10t
ln 10
2
1
=102
ln 10− 101
ln 10=100− 10ln 10
=90
ln 10
81. (a) d
dx(ln |sinx|+ C) =
1
sinxcosx = cotx
(b) Let u = sinx. Then du = cosxdx, so cotxdx =cosx
sinxdx =
du
u= ln |u|+C = ln |sinx|+C.
83. The cross-sectional area is π 1/√x+ 1
2= π/(x + 1). Therefore, the volume is
1
0
π
x+ 1dx = π[ln(x+ 1)]10 = π(ln 2− ln 1) = π ln 2.
85. W =V2V1
P dV =1000
600
C
VdV = C
1000
600
1
VdV = C ln |V | 1000
600= C(ln 1000− ln 600) = C ln 1000
600= C ln 5
3.
Initially, PV = C, where P = 150 kPa and V = 600 cm3, so C = (150)(600) = 90,000. Thus,
W = 90, 000 ln 53≈ 45, 974 kPa · cm3 = 45, 974(103 Pa)(10−6 m3) = 45, 974 Pa·m3 = 45, 974 N·m [Pa = N/m2]
= 45, 974 J
87. f(x) = 2x+ lnx ⇒ f 0(x) = 2 + 1/x. If g = f−1, then f(1) = 2 ⇒ g(2) = 1, so
g0(2) = 1/f 0(g(2)) = 1/f 0(1) = 13 .
89. The curve and the line will determine a region when they intersect at two or
more points. So we solve the equation x/(x2 + 1) = mx ⇒ x = 0 or
mx2 +m− 1 = 0 ⇒ x = 0 or x =± −4(m)(m− 1)
2m= ± 1
m− 1.
Note that if m = 1, this has only the solution x = 0, and no region is
determined. But if 1/m− 1 > 0 ⇔ 1/m > 1 ⇔ 0 < m < 1, then there are two solutions. [Another way of seeing this
SECTION 7.2* THE NATURAL LOGARITHMIC FUNCTION ¤ 263
is to observe that the slope of the tangent to y = x/(x2 + 1) at the origin is y0 = 1 and therefore we must have 0 < m < 1.]
Note that we cannot just integrate between the positive and negative roots, since the curve and the line cross at the origin.
Since mx and x/(x2 + 1) are both odd functions, the total area is twice the area between the curves on the interval
0, 1/m− 1 . So the total area enclosed is
2
√1/m−1
0
x
x2 + 1−mx dx= 2 1
2ln(x2 + 1)− 1
2mx2
√1/m−1
0
= ln1
m− 1 + 1 −m
1
m− 1 − (ln 1− 0)
= ln1
m+m− 1 = m− lnm− 1
91. If f(x) = ln (1 + x), then f 0(x) =1
1 + x, so f 0(0) = 1.
Thus, limx→0
ln(1 + x)
x= lim
x→0
f(x)
x= lim
x→0
f(x)− f(0)
x− 0 = f 0(0) = 1.
7.2* The Natural Logarithmic Function
1. ln r2
3√s= ln r2 − ln 3√s = 2 ln r − (ln 3 + ln s1/2) = 2 ln r − ln 3− 1
2 ln s
3. ln(uv)10 = 10 ln(uv) = 10(lnu+ ln v) = 10 lnu+ 10 ln v
5. ln 5 + 5 ln 3 = ln 5 + ln 35 [by Law 3]
= ln(5 · 35) [by Law 1]
= ln1215
7. ln(1 + x2) + 12lnx− ln sinx = ln(1 + x2) + lnx1/2 − ln sinx = ln[(1 + x2)
√x ]− ln sinx = ln (1 + x2)
√x
sinx
9. Reflect the graph of y = lnx about the x-axis to obtain
the graph of y = − lnx.
y = lnx y = − lnx
11.
y = lnx y = ln(x+ 3)
13. Let t = x2 − 9. Then as x→ 3+, t→ 0+, and limx→3+
ln(x2 − 9) = limt→0+
ln t = −∞ by (4).
264 ¤ CHAPTER 7 INVERSE FUNCTIONS
15. f(x) =√x lnx ⇒ f 0(x) =
1
2√xlnx+
√x
1
x=lnx+ 2
2√x
17. f(x) = sin(lnx) ⇒ f 0(x) = cos(lnx) · d
dxlnx = cos(lnx) · 1
x=cos(lnx)
x
19. f(x) = 5√lnx = (lnx)1/5 ⇒ f 0(x) = 1
5 (lnx)−4/5 d
dx(lnx) =
1
5(lnx)4/5· 1x=
1
5x 5 (lnx)4
21. f(x) = sinx ln(5x) ⇒ f 0(x) = sinx · 15x· ddx(5x)+ ln(5x) · cosx = sinx · 5
5x+cosx ln(5x) =
sinx
x+cosx ln(5x)
23. g(x) = ln a− x
a+ x= ln(a− x)− ln(a+ x) ⇒
g0(x) =1
a− x(−1)− 1
a+ x=−(a+ x)− (a− x)
(a− x)(a+ x)=
−2aa2 − x2
25. F (t) = ln (2t+ 1)3
(3t− 1)4 = ln(2t+ 1)3 − ln(3t − 1)4 = 3 ln(2t+ 1)− 4 ln(3t− 1) ⇒
F 0(t) = 3 · 1
2t+ 1· 2− 4 · 1
3t− 1 · 3 =6
2t+ 1− 12
3t− 1 , or combined, −6(t+ 3)(2t+ 1)(3t− 1) .
27. g(x) = ln x√x2 − 1 = lnx + ln(x2 − 1)1/2 = lnx + 1
2 ln(x2 − 1) ⇒
g0(x) =1
x+1
2· 1
x2 − 1 · 2x =1
x+
x
x2 − 1 =x2 − 1 + x · xx(x2 − 1) =
2x2 − 1x(x2 − 1)
29. f(u) = lnu
1 + ln(2u)⇒
f 0(u) =[1 + ln(2u)] · 1
u− lnu · 1
2u· 2
[1 + ln(2u)]2=
1u[1 + ln(2u)− lnu][1 + ln(2u)]2
=1 + (ln 2 + lnu)− lnu
u[1 + ln(2u)]2=
1 + ln 2
u[1 + ln(2u)]2
31. y = ln 2− x− 5x2 ⇒ y0 =1
2− x− 5x2 · (−1− 10x) =−10x− 12− x− 5x2 or 10x+ 1
5x2 + x− 2
33. y = tan [ln(ax+ b)] ⇒ y0 = sec2[ln(ax+ b)] · 1
ax+ b· a = sec2[ln(ax+ b)]
a
ax+ b
35. y = x2 ln(2x) ⇒ y0 = x2 · 12x· 2 + ln(2x) · (2x) = x + 2x ln(2x) ⇒
y00 = 1 + 2x · 12x· 2 + ln(2x) · 2 = 1 + 2 + 2 ln(2x) = 3 + 2 ln(2x)
37. f(x) = x
1− ln(x− 1) ⇒
f 0(x) =[1− ln(x− 1)] · 1− x · −1
x− 1[1− ln(x− 1)]2 =
(x− 1)[1− ln(x− 1)] + x
x− 1[1− ln(x− 1)]2 =
x− 1− (x− 1) ln(x− 1) + x
(x− 1)[1− ln(x− 1)]2
=2x− 1− (x− 1) ln(x− 1)(x− 1)[1− ln(x− 1)]2
Dom(f) = {x | x− 1 > 0 and 1− ln(x− 1) 6= 0} = {x | x > 1 and ln(x− 1) 6= 1}= x | x > 1 and x− 1 6= e1 = {x | x > 1 and x 6= 1 + e} = (1, 1 + e) ∪ (1 + e,∞)
SECTION 7.2* THE NATURAL LOGARITHMIC FUNCTION ¤ 265
39. f(x) =√1− lnx is defined ⇔ x > 0 [so that lnx is defined] and 1 − lnx ≥ 0 ⇔
x > 0 and lnx ≤ 1 ⇔ 0 < x ≤ e, so the domain of f is (0, e]. Now
f 0(x) =1
2(1− lnx)−1/2 · d
dx(1− lnx) = 1
2√1− lnx · − 1
x=
−12x√1− lnx .
41. f(x) = lnx
1 + x2⇒ f 0(x) =
(1 + x2)1
x− (lnx)(2x)
(1 + x2)2, so f 0(1) = 2(1)− 0(2)
22=2
4=1
2.
43. f(x) = sinx+ lnx ⇒ f 0(x) = cosx+ 1/x.
This is reasonable, because the graph shows that f increases when f 0 is
positive, and f 0(x) = 0 when f has a horizontal tangent.
45. y = sin(2 lnx) ⇒ y0 = cos(2 lnx) · 2x
. At (1, 0), y0 = cos 0 · 21= 2, so an equation of the tangent line is
y − 0 = 2 · (x− 1), or y = 2x− 2.
47. y = ln(x2 + y2) ⇒ y0 =1
x2 + y2d
dx(x2 + y2) ⇒ y0 =
2x+ 2yy0
x2 + y2⇒ x2y0 + y2y0 = 2x+ 2yy0 ⇒
x2y0 + y2y0 − 2yy0 = 2x ⇒ (x2 + y2 − 2y)y0 = 2x ⇒ y0 =2x
x2 + y2 − 2y
49. f(x) = ln(x− 1) ⇒ f 0(x) =1
(x− 1) = (x− 1)−1 ⇒ f 00(x) = −(x− 1)−2 ⇒ f 000 (x) = 2(x− 1)−3 ⇒
f (4)(x) = −2 · 3(x− 1)−4 ⇒ · · · ⇒ f (n)(x) = (−1)n−1 · 2 · 3 · 4 · · · · · (n− 1)(x− 1)−n = (−1)n−1 (n− 1)!(x− 1)n
51. From the graph, it appears that the curves y = (x− 4)2 and y = lnx intersect
just to the left of x = 3 and to the right of x = 5, at about x = 5.3. Let
f(x) = lnx− (x− 4)2. Then f 0(x) = 1/x− 2(x− 4), so Newton’s Method
says that xn+1 = xn − f(xn)/f0(xn) = xn − lnxn − (xn − 4)2
1/xn − 2(xn − 4) . Taking
x0 = 3, we get x1 ≈ 2.957738, x2 ≈ 2.958516 ≈ x3, so the first root is 2.958516, to six decimal places. Taking x0 = 5, we
get x1 ≈ 5.290755, x2 ≈ 5.290718 ≈ x3, so the second (and final) root is 5.290718, to six decimal places.
53. y = f(x) = ln(sinx)
A. D = {x in R | sinx > 0} =∞
n=−∞(2nπ, (2n+ 1)π) = · · · ∪ (−4π,−3π) ∪ (−2π,−π) ∪ (0, π) ∪ (2π, 3π) ∪ · · ·
B. No y-intercept; x-intercepts: f(x) = 0 ⇔ ln(sinx) = 0 ⇔ sinx = e0 = 1 ⇔ x = 2nπ + π2
for each
266 ¤ CHAPTER 7 INVERSE FUNCTIONS
integer n. C. f is periodic with period 2π. D. limx→(2nπ)+
f(x) = −∞ and limx→[(2n+1)π]−
f(x) = −∞, so the lines
x = nπ are VAs for all integers n. E. f 0(x) = cosx
sinx= cotx, so f 0(x) > 0 when 2nπ < x < 2nπ + π
2for each
integer n, and f 0(x) < 0 when 2nπ + π2 < x < (2n+ 1)π. Thus, f is increasing on 2nπ, 2nπ + π
2and
decreasing on 2nπ + π2, (2n+ 1)π for each integer n.
F. Local maximum values f 2nπ + π2= 0, no local minimum.
G. f 00(x) = − csc2 x < 0, so f is CD on (2nπ, (2n+ 1)π) for
each integer n. No IP
H.
55. y = f(x) = ln(1 + x2) A. D = R B. Both intercepts are 0. C. f(−x) = f(x), so the curve is symmetric about the
y-axis. D. limx→±∞
ln(1 + x2) =∞, no asymptotes. E. f 0(x) = 2x
1 + x2> 0 ⇔
x > 0, so f is increasing on (0,∞) and decreasing on (−∞, 0) .
F. f(0) = 0 is a local and absolute minimum.
G. f 00(x) = 2(1 + x2)− 2x(2x)(1 + x2)2
=2(1− x2)
(1 + x2)2> 0 ⇔
|x| < 1, so f is CU on (−1, 1), CD on (−∞,−1) and (1,∞). IP
(1, ln 2) and (−1, ln 2).
H.
57. We use the CAS to calculate f 0(x) = 2 + sinx+ x cosx
2x+ x sinxand
f 00(x) =2x2 sinx+ 4 sinx− cos2 x+ x2 + 5
x2(cos2 x− 4 sinx− 5) . From the graphs, it
seems that f 0 > 0 (and so f is increasing) on approximately the intervals
(0, 2.7), (4.5, 8.2) and (10.9, 14.3). It seems that f 00 changes sign
(indicating inflection points) at x ≈ 3.8, 5.7, 10.0 and 12.0.
Looking back at the graph of f(x) = ln(2x+ x sinx), this implies that the inflection points have approximate coordinates
(3.8, 1.7), (5.7, 2.1), (10.0, 2.7), and (12.0, 2.9).
59. y = (2x+ 1)5(x4 − 3)6 ⇒ ln y = ln (2x+ 1)5(x4 − 3)6 ⇒ ln y = 5 ln(2x+ 1) + 6 ln(x4 − 3) ⇒1
yy0 = 5 · 1
2x+ 1· 2 + 6 · 1
x4 − 3 · 4x3 ⇒
y0 = y10
2x+ 1+
24x3
x4 − 3 = (2x+ 1)5(x4 − 3)6 10
2x+ 1+
24x3
x4 − 3 .
[The answer could be simplified to y0 = 2(2x+ 1)4(x4 − 3)5(29x4 + 12x3 − 15), but this is unnecessary.]
SECTION 7.2* THE NATURAL LOGARITHMIC FUNCTION ¤ 267
61. y = sin2 x tan4 x
(x2 + 1)2⇒ ln y = ln(sin2 x tan4 x) − ln(x2 + 1)2 ⇒
ln y = ln(sinx)2 + ln(tanx)4 − ln(x2 + 1)2 ⇒ ln y = 2 ln |sinx|+ 4 ln |tanx|− 2 ln(x2 + 1) ⇒1
yy0 = 2 · 1
sinx· cosx+ 4 · 1
tanx· sec2 x− 2 · 1
x2 + 1· 2x ⇒ y0 =
sin2 x tan4 x
(x2 + 1)22 cotx+
4 sec2 x
tanx− 4x
x2 + 1
63.4
2
3
xdx = 3
4
2
1
xdx = 3 ln |x| 4
2= 3(ln 4− ln 2) = 3 ln 4
2= 3 ln 2
65.2
1
dt
8− 3t = −13ln |8− 3t|
2
1
= −13ln 2− −1
3ln 5 =
1
3(ln 5− ln 2) = 1
3ln5
2
Or: Let u = 8 − 3t. Then du = −3 dt, so2
1
dt
8− 3t =2
5
− 13 du
u= −1
3ln |u|
2
5
= −13ln 2− −1
3ln 5 =
1
3(ln 5− ln 2) = 1
3ln5
2.
67.e
1
x2 + x+ 1
xdx =
e
1
x+ 1 +1
xdx = 1
2x2 + x+ lnx
e
1= 1
2e2 + e+ 1 − 1
2+ 1 + 0
= 12e
2 + e− 12
69. Let u = lnx. Then du = dx
x⇒ (lnx)2
xdx = u2 du =
1
3u3 + C =
1
3(lnx)3 +C.
71. sin 2x
1 + cos2 xdx = 2
sinx cosx
1 + cos2 xdx = 2I. Let u = cosx. Then du = − sinxdx, so
2I = −2 udu
1 + u2= −2 · 1
2ln(1 + u2) + C = − ln(1 + u2) +C = − ln(1 + cos2 x) + C.
Or: Let u = 1 + cos2 x.
73. (a) d
dx(ln |sinx|+ C) =
1
sinxcosx = cotx
(b) Let u = sinx. Then du = cosxdx, so cotxdx =cosx
sinxdx =
du
u= ln |u|+C = ln |sinx|+C.
75. The cross-sectional area is π 1/√x+ 1
2= π/(x + 1). Therefore, the volume is
1
0
π
x+ 1dx = π[ln(x+ 1)]10 = π(ln 2− ln 1) = π ln 2.
77. W =V2V1
P dV =1000
600
C
VdV = C
1000
600
1
VdV = C ln |V | 1000
600= C(ln 1000− ln 600) = C ln 1000
600= C ln 5
3.
Initially, PV = C, where P = 150 kPa and V = 600 cm3, so C = (150)(600) = 90,000. Thus,
W = 90, 000 ln 53≈ 45, 974 kPa · cm3 = 45, 974(103 Pa)(10−6 m3) = 45, 974 Pa·m3 = 45, 974 N·m [Pa = N/m2]
= 45, 974 J
79. f(x) = 2x+ lnx ⇒ f 0(x) = 2 + 1/x. If g = f−1, then f(1) = 2 ⇒ g(2) = 1, so
g0(2) = 1/f 0(g(2)) = 1/f 0(1) = 13
.
268 ¤ CHAPTER 7 INVERSE FUNCTIONS
81. (a) We interpret ln 1.5 as the area under the curve y = 1/x from x = 1 to
x = 1.5. The area of the rectangle BCDE is 12· 23= 1
3. The area of the
trapezoid ABCD is 12 · 12 1 + 2
3= 5
12 . Thus, by comparing areas, we
observe that 13< ln 1.5 < 5
12.
(b) With f(t) = 1/t, n = 10, and ∆t = 0.05, we have
ln 1.5 =1.5
1(1/t) dt ≈ (0.05)[f(1.025) + f(1.075) + · · ·+ f(1.475)]
= (0.05) 11.025 +
11.075 + · · ·+ 1
1.475≈ 0.4054
83. The area of Ri is 1
i+ 1and so 1
2+1
3+ · · ·+ 1
n<
n
1
1
tdt = lnn.
The area of Si is 1i
and so 1 + 1
2+ · · ·+ 1
n− 1 >n
1
1
tdt = lnn.
85. The curve and the line will determine a region when they intersect at two or
more points. So we solve the equation x/(x2 + 1) = mx ⇒ x = 0 or
mx2 +m− 1 = 0 ⇒ x = 0 or x =± −4(m)(m− 1)
2m= ± 1
m− 1.
Note that if m = 1, this has only the solution x = 0, and no region is
determined. But if 1/m− 1 > 0 ⇔ 1/m > 1 ⇔ 0 < m < 1, then there are two solutions. [Another way of seeing this
is to observe that the slope of the tangent to y = x/(x2 + 1) at the origin is y0 = 1 and therefore we must have 0 < m < 1.]
Note that we cannot just integrate between the positive and negative roots, since the curve and the line cross at the origin.
Since mx and x/(x2 + 1) are both odd functions, the total area is twice the area between the curves on the interval
0, 1/m− 1 . So the total area enclosed is
2
√1/m−1
0
x
x2 + 1−mx dx= 2 1
2ln(x2 + 1)− 1
2mx2
√1/m−1
0
= ln1
m− 1 + 1 −m
1
m− 1 − (ln 1− 0)
= ln1
m+m− 1 = m− lnm− 1
SECTION 7.3* THE NATURAL EXPONENTIAL FUNCTION ¤ 269
87. If f(x) = ln (1 + x), then f 0(x) =1
1 + x, so f 0(0) = 1.
Thus, limx→0
ln(1 + x)
x= lim
x→0
f(x)
x= lim
x→0
f(x)− f(0)
x− 0 = f 0(0) = 1.
7.3* The Natural Exponential Function
1. The function value at x = 0 is 1 and the slope at x = 0 is 1.
3. (a) e−2 ln 5 = eln 5−2 (4)= 5−2 =
1
52=1
25(b) ln ln ee
10 (5)= ln(e10)
(5)= 10
5. (a) 2 lnx = 1 ⇒ lnx = 12 ⇒ x = e1/2 =
√e (b) e−x = 5 ⇒ −x = ln 5 ⇒ x = − ln 5
7. (a) e3x+1 = k ⇔ 3x+ 1 = ln k ⇔ x = 13(lnk − 1)
(b) lnx+ ln(x− 1) = ln(x(x− 1)) = 1 ⇔ x(x− 1) = e1 ⇔ x2 − x− e = 0. The quadratic formula (with a = 1,
b = −1, and c = −e) gives x = 121±√1 + 4e , but we reject the negative root since the natural logarithm is not
defined for x < 0. So x = 121 +
√1 + 4e .
9. 3xex + x2ex = 0 ⇔ xex(3 + x) = 0 ⇔ x = 0 or −3
11. e2x − ex − 6 = 0 ⇔ (ex − 3)(ex + 2) = 0 ⇔ ex = 3 or −2 ⇒ x = ln 3 since ex > 0.
13. (a) e2+5x = 100 ⇒ ln e2+5x = ln 100 ⇒ 2 + 5x = ln 100 ⇒ 5x = ln 100− 2 ⇒x = 1
5(ln 100− 2) ≈ 0.5210
(b) ln(ex − 2) = 3 ⇒ ex − 2 = e3 ⇒ ex = e3 + 2 ⇒ x = ln(e3 + 2) ≈ 3.0949
15. (a) ex < 10 ⇒ ln ex < ln 10 ⇒ x < ln 10 ⇒ x ∈ (−∞, ln 10)
(b) lnx > −1 ⇒ elnx > e−1 ⇒ x > e−1 ⇒ x ∈ (1/e,∞)
17.
y = ex y = e−x
270 ¤ CHAPTER 7 INVERSE FUNCTIONS
19. We start with the graph of y = ex (Figure 2) and reflect about the y-axis to get the graph of y = e−x. Then we compress the
graph vertically by a factor of 2 to obtain the graph of y = 12e−x and then reflect about the x-axis to get the graph of
y = − 12e−x. Finally, we shift the graph upward one unit to get the graph of y = 1− 1
2e−x.
21. (a) For f(x) =√3− e2x, we must have 3− e2x ≥ 0 ⇒ e2x ≤ 3 ⇒ 2x ≤ ln 3 ⇒ x ≤ 1
2 ln 3.
Thus, the domain of f is (−∞, 12 ln 3].
(b) y = f(x) =√3− e2x [note that y ≥ 0] ⇒ y2 = 3− e2x ⇒ e2x = 3− y2 ⇒ 2x = ln(3− y2) ⇒
x = 12 ln(3− y2). Interchange x and y: y = 1
2 ln(3− x2). So f−1(x) = 12 ln(3− x2). For the domain of f−1, we must
have 3− x2 > 0 ⇒ x2 < 3 ⇒ |x| < √3 ⇒ −√3 < x <√3 ⇒ 0 ≤ x <
√3 since x ≥ 0. Note that the
domain of f−1, [0,√3 ), equals the range of f .
23. y = ln(x+ 3) ⇒ ey = eln(x+3) = x+ 3 ⇒ x = ey − 3.
Interchange x and y: the inverse function is y = ex − 3.
25. y = f(x) = ex3 ⇒ ln y = x3 ⇒ x = 3
√ln y. Interchange x and y: y = 3
√lnx. So f−1(x) = 3
√lnx.
27. Divide numerator and denominator by e3x: limx→∞
e3x − e−3x
e3x + e−3x= lim
x→∞1− e−6x
1 + e−6x=1− 01 + 0
= 1
29. Let t = 3/(2− x). As x→ 2+, t→−∞. So limx→2+
e3/(2−x) = limt→−∞
et = 0 by (6).
31. Since −1 ≤ cosx ≤ 1 and e−2x > 0, we have −e−2x ≤ e−2x cosx ≤ e−2x. We know that limx→∞
(−e−2x) = 0 and
limx→∞
e−2x = 0, so by the Squeeze Theorem, limx→∞
(e−2x cosx) = 0.
33. By the Product Rule, f(x) = (x3 + 2x)ex ⇒
f 0(x) = (x3 + 2x)(ex)0 + ex(x3 + 2x)0 = (x3 + 2x)ex + ex(3x2 + 2)
= ex[(x3 + 2x) + (3x2 + 2)] = ex(x3 + 3x2 + 2x+ 2)
35. By (9), y = eax3 ⇒ y0 = eax
3 d
dx(ax3) = 3ax2eax
3.
37. f(u) = e1/u ⇒ f 0(u) = e1/u · d
du
1
u= e1/u
−1u2
=−1u2
e1/u
39. By (9), F (t) = et sin 2t ⇒ F 0(t) = et sin 2t(t sin 2t)0 = et sin 2t(t · 2 cos 2t+ sin 2t · 1) = et sin 2t(2t cos 2t+ sin 2t)
SECTION 7.3* THE NATURAL EXPONENTIAL FUNCTION ¤ 271
41. y =√1 + 2e3x ⇒ y0 =
1
2(1 + 2e3x)−1/2
d
dx(1 + 2e3x) =
1
2√1 + 2e3x
(2e3x · 3) = 3e3x√1 + 2e3x
43. y = eex ⇒ y0 = ee
x · d
dx(ex) = ee
x · ex or eex+x
45. By the Quotient Rule, y = aex + b
cex + d⇒
y0 =(cex + d)(aex)− (aex + b)(cex)
(cex + d)2=(acex + ad− acex − bc)ex
(cex + d)2=(ad− bc)ex
(cex + d)2.
47. y = cos 1− e2x
1 + e2x⇒
y0 = − sin 1− e2x
1 + e2x· d
dx
1− e2x
1 + e2x= − sin 1− e2x
1 + e2x· (1 + e2x)(−2e2x)− (1− e2x)(2e2x)
(1 + e2x)2
= − sin 1− e2x
1 + e2x· −2e
2x (1 + e2x) + (1− e2x)
(1 + e2x)2= − sin 1− e2x
1 + e2x· −2e
2x(2)
(1 + e2x)2=
4e2x
(1 + e2x)2· sin 1− e2x
1 + e2x
49. y = e2x cosπx ⇒ y0 = e2x(−π sinπx) + (cosπx)(2e2x) = e2x(2 cosπx− π sinπx).
At (0, 1), y0 = 1(2− 0) = 2, so an equation of the tangent line is y − 1 = 2(x− 0), or y = 2x+ 1.
51. d
dxex
2y =d
dx(x+ y) ⇒ ex
2y(x2y0 + y · 2x) = 1 + y0 ⇒ x2ex2yy0 + 2xyex
2y = 1 + y0 ⇒
x2ex2yy0 − y0 = 1− 2xyex2y ⇒ y0(x2ex
2y − 1) = 1− 2xyex2y ⇒ y0 =1− 2xyex2yx2ex2y − 1
53. y = ex + e−x/2 ⇒ y0 = ex − 12e−x/2 ⇒ y00 = ex + 1
4e−x/2, so
2y00 − y0 − y = 2 ex + 14e−x/2 − ex − 1
2e−x/2 − ex + e−x/2 = 0.
55. y = erx ⇒ y0 = rerx ⇒ y00 = r2erx, so if y = erx satisfies the differential equation y00 + 6y0 + 8y = 0,
then r2erx + 6rerx + 8erx = 0; that is, erx(r2 + 6r + 8) = 0. Since erx > 0 for all x, we must have r2 + 6r + 8 = 0,
or (r + 2)(r + 4) = 0, so r = −2 or −4.
57. f(x) = e2x ⇒ f 0(x) = 2e2x ⇒ f 00(x) = 2 · 2e2x = 22e2x ⇒f 000(x) = 22 · 2e2x = 23e2x ⇒ · · · ⇒ f (n)(x) = 2ne2x
59. (a) f(x) = ex + x is continuous on R and f(−1) = e−1 − 1 < 0 < 1 = f(0), so by the Intermediate Value Theorem,
ex + x = 0 has a root in (−1, 0).
(b) f(x) = ex + x ⇒ f 0(x) = ex + 1, so xn+1 = xn − exn + xnexn + 1
. Using x1 = −0.5, we get x2 ≈ −0.566311,
x3 ≈ −0.567143 ≈ x4, so the root is −0.567143 to six decimal places.
61. (a) limt→∞
p(t) = limt→∞
1
1 + ae−kt=
1
1 + a · 0 = 1, since k > 0 ⇒ −kt→ −∞ ⇒ e−kt → 0.
(b) p(t) = (1 + ae−kt)−1 ⇒ dp
dt= −(1 + ae−kt)−2(−kae−kt) = kae−kt
(1 + ae−kt)2
272 ¤ CHAPTER 7 INVERSE FUNCTIONS
(c)From the graph of p(t) = (1 + 10e−0.5t)−1, it seems that p(t) = 0.8
(indicating that 80% of the population has heard the rumor) when
t ≈ 7.4 hours.
63. f(x) = x− ex ⇒ f 0(x) = 1− ex = 0 ⇔ ex = 1 ⇔ x = 0. Now f 0(x) > 0 for all x < 0 and f 0(x) < 0 for all
x > 0, so the absolute maximum value is f(0) = 0− 1 = −1.
65. f(x) = xe−x2/8, [−1, 4]. f 0(x) = x · e−x2/8 · (−x
4) + e−x
2/8 · 1 = e−x2/8(−x2
4+ 1). Since e−x
2/8 is never 0,
f 0(x) = 0 ⇒ −x2/4 + 1 = 0 ⇒ 1 = x2/4 ⇒ x2 = 4 ⇒ x = ±2, but −2 is not in the given interval, [−1, 4].f(−1) = −e−1/8 ≈ −0.88, f(2) = 2e−1/2 ≈ 1.21, and f(4) = 4e−2 ≈ 0.54. So f(2) = 2e−1/2 is the absolute maximum
value and f(−1) = −e−1/8 is the absolute minimum value.
67. (a) f(x) = (1− x)e−x ⇒ f 0(x) = (1− x)(−e−x) + e−x(−1) = e−x(x− 2) > 0 ⇒ x > 2, so f is increasing on
(2,∞) and decreasing on (−∞, 2).
(b) f 00(x) = e−x(1) + (x− 2)(−e−x) = e−x(3− x) > 0 ⇔ x < 3, so f is CU on (−∞, 3) and CD on (3,∞).
(c) f 00 changes sign at x = 3, so there is an IP at 3,−2e−3 .
69. y = f(x) = e−1/(x+1) A. D = {x | x 6= −1} = (−∞,−1) ∪ (−1,∞) B. No x-intercept; y-intercept = f(0) = e−1
C. No symmetry D. limx→±∞
e−1/(x+1) = 1 since −1/(x+ 1)→ 0, so y = 1 is a HA. limx→−1+
e−1/(x+1) = 0 since
−1 /(x+ 1) → −∞, limx→−1−
e−1/(x+1) = ∞ since −1/(x+ 1) → ∞, so x = −1 is a VA.
E. f 0(x) = e−1/(x+1)/(x+ 1)2 ⇒ f 0(x) > 0 for all x except 1, so
f is increasing on (−∞,−1) and (−1,∞). F. No extreme values
G. f 00(x) = e−1/(x+1)
(x+ 1)4+
e−1/(x+1)(−2)(x+ 1)3
= −e−1/(x+1)(2x+ 1)(x+ 1)4
⇒
f 00(x) > 0 ⇔ 2x+ 1 < 0 ⇔ x < − 12
, so f is CU on (−∞,−1)and −1,− 1
2, and CD on − 1
2,∞ . f has an IP at − 1
2, e−2 .
H.
71. S(t) = Atpe−kt with A = 0.01, p = 4, and k = 0.07. We will find the
zeros of f 00 for f(t) = tpe−kt.
f 0(t) = tp(−ke−kt) + e−kt(ptp−1) = e−kt(−ktp + ptp−1)
f 00(t) = e−kt(−kptp−1 + p(p− 1)tp−2) + (−ktp + ptp−1)(−ke−kt)= tp−2e−kt[−kpt+ p(p− 1) + k2t2 − kpt]
= tp−2e−kt(k2t2 − 2kpt+ p2 − p)
Using the given values of p and k gives us f 00(t) = t2e−0.07t(0.0049t2 − 0.56t+ 12). So S00(t) = 0.01f 00(t) and its zeros
SECTION 7.3* THE NATURAL EXPONENTIAL FUNCTION ¤ 273
are t = 0 and the solutions of 0.0049t2 − 0.56t+ 12 = 0, which are t1 = 2007 ≈ 28.57 and t2 = 600
7 ≈ 85.71.
At t1 minutes, the rate of increase of the level of medication in the bloodstream is at its greatest and at t2 minutes, the rate of
decrease is the greatest.
73. f(x) = ex3−x → 0 as x→ −∞, and
f(x)→∞ as x→∞. From the graph,
it appears that f has a local minimum of
about f(0.58) = 0.68, and a local
maximum of about f(−0.58) = 1.47.
To find the exact values, we calculate
f 0(x) = 3x2 − 1 ex3−x, which is 0 when 3x2 − 1 = 0 ⇔ x = ± 1√
3. The negative root corresponds to the local
maximum f − 1√3= e(−1/
√3)3− (−1/√3) = e2
√3/9, and the positive root corresponds to the local minimum
f 1√3= e(1/
√3)3− (1/
√3) = e−2
√3/9. To estimate the inflection points, we calculate and graph
f 00(x) =d
dx3x2 − 1 ex
3−x = 3x2 − 1 ex3−x 3x2 − 1 + ex
3−x(6x) = ex3−x 9x4 − 6x2 + 6x+ 1 .
From the graph, it appears that f 00(x) changes sign (and thus f has inflection points) at x ≈ −0.15 and x ≈ −1.09. From the
graph of f , we see that these x-values correspond to inflection points at about (−0.15, 1.15) and (−1.09, 0.82).
75. Let u = −3x. Then du = −3 dx, so 5
0e−3x dx = − 1
3
−150
eu du = − 13[eu]−150 = − 1
3e−15 − e0 = 1
31− e−15 .
77. Let u = 1 + ex. Then du = ex dx, so ex√1 + ex dx =
√udu = 2
3u3/2 +C = 2
3(1 + ex)3/2 + C.
79. (ex + e−x)2 dx = (e2x + 2 + e−2x) dx = 12e2x + 2x− 1
2e−2x +C
81. sinx ecos x dxu = cosx,
du = − sinx dx = eu (−du) = −eu +C = −ecos x + C
83. Let u =√x. Then du = 1
2√xdx, so e
√x
√xdx = 2 eu du = 2eu + C = 2e
√x +C.
85. Area = 1
0e3x − ex dx = 1
3e3x − ex
1
0= 1
3e3 − e − 1
3− 1 = 1
3e3 − e+ 2
3≈ 4.644
87. V =1
0π(ex)2 dx = π
1
0e2x dx = 1
2π e2x
1
0= π
2e2 − 1
89. (a) erf(x) = 2√π
x
0
e−t2
dt ⇒x
0
e−t2
dt =
√π
2erf(x). By Property 5 of definite integrals in Section 5.2,
b
0e−t
2
dt =a
0e−t
2
dt+b
ae−t
2
dt, so
b
a
e−t2
dt =b
0
e−t2
dt−a
0
e−t2
dt =
√π
2erf(b)−
√π
2erf(a) = 1
2
√π [erf(b)− erf(a)].
(b) y = ex2
erf(x) ⇒ y0 = 2xex2
erf(x) + ex2
erf 0(x) = 2xy + ex2 · 2√
πe−x
2
[by FTC1] = 2xy +2√π
.
274 ¤ CHAPTER 7 INVERSE FUNCTIONS
91. We use Theorem 7.1.7. Note that f(0) = 3 + 0 + e0 = 4, so f−1(4) = 0. Also f 0(x) = 1 + ex. Therefore,
f−1 0(4) =
1
f 0(f−1(4))=
1
f 0(0)=
1
1 + e0=1
2.
93. From the graph, it appears that f is an odd function (f is undefined for x = 0).
To prove this, we must show that f(−x) = −f(x).
f(−x) = 1− e1/(−x)
1 + e1/(−x)=1− e(−1/x)
1 + e(−1/x)=1− 1
e1/x
1 +1
e1/x
· e1/x
e1/x=
e1/x − 1e1/x + 1
= −1− e1/x
1 + e1/x= −f(x)
so f is an odd function.
95. Using the second law of logarithms and Equation 5, we have ln(ex/ey) = ln ex − ln ey = x− y = ln(ex− y). Since ln is a
one-to-one function, it follows that ex/ey = ex− y .
97. (a) Let f(x) = ex − 1− x. Now f(0) = e0 − 1 = 0, and for x ≥ 0, we have f 0(x) = ex − 1 ≥ 0. Now, since f(0) = 0 and
f is increasing on [0,∞), f(x) ≥ 0 for x ≥ 0 ⇒ ex − 1− x ≥ 0 ⇒ ex ≥ 1 + x.
(b) For 0 ≤ x ≤ 1, x2 ≤ x, so ex2 ≤ ex [since ex is increasing]. Hence [from (a)] 1 + x2 ≤ ex
2 ≤ ex.
So 43=
1
01 + x2 dx ≤ 1
0ex
2
dx ≤ 1
0ex dx = e− 1 < e ⇒ 4
3≤ 1
0ex
2
dx ≤ e.
99. (a) By Exercise 97(a), the result holds for n = 1. Suppose that ex ≥ 1 + x+x2
2!+ · · ·+ xk
k!for x ≥ 0.
Let f(x) = ex − 1− x− x2
2!− · · ·− xk
k!− xk+1
(k + 1)!. Then f 0(x) = ex − 1− x− · · ·− xk
k!≥ 0 by assumption. Hence
f(x) is increasing on (0,∞). So 0 ≤ x implies that 0 = f(0) ≤ f(x) = ex − 1− x− · · ·− xk
k!− xk+1
(k + 1)!, and hence
ex ≥ 1 + x+ · · ·+ xk
k!+
xk+1
(k + 1)!for x ≥ 0. Therefore, for x ≥ 0, ex ≥ 1 + x+
x2
2!+ · · ·+ xn
n!for every positive
integer n, by mathematical induction.
(b) Taking n = 4 and x = 1 in (a), we have e = e1 ≥ 1 + 12+ 1
6+ 1
24= 2.7083 > 2.7.
(c) ex ≥ 1 + x+ · · ·+ xk
k!+
xk+1
(k + 1)!⇒ ex
xk≥ 1
xk+
1
xk−1+ · · ·+ 1
k!+
x
(k + 1)!≥ x
(k + 1)!.
But limx→∞
x
(k + 1)!=∞, so lim
x→∞ex
xk=∞.
SECTION 7.4* GENERAL LOGARITHMIC AND EXPONENTIAL FUNCTIONS ¤ 275
7.4* General Logarithmic and Exponential Functions
1. (a) ax = ex ln a
(b) The domain of f(x) = ax is R.
(c) The range of f(x) = ax [a 6= 1] is (0,∞).(d) (i) See Figure 1. (ii) See Figure 3. (iii) See Figure 2.
3. 5√7 = (eln 5)
√7 = e
√7 ln 5 5. (cosx)x = (eln cosx)x = ex ln(cos x).
7. (a) log5 125 = 3 since 53 = 125. (b) log31
27= −3 since 3−3 = 1
33=1
27.
9. (a) log2 6− log2 15 + log2 20 = log2( 615 ) + log2 20 [by Law 2]
= log2(615· 20) [by Law 1]
= log2 8, and log2 8 = 3 since 23 = 8.
(b) log3 100− log3 18− log3 50 = log310018
− log3 50 = log3 10018·50
= log3(19 ), and log3
19= −2 since 3−2 = 1
9 .
11. All of these graphs approach 0 as x→−∞, all of them pass through the point
(0, 1), and all of them are increasing and approach∞ as x→∞. The larger the
base, the faster the function increases for x > 0, and the faster it approaches 0 as
x→−∞.
13. (a) log12 e =ln e
ln 12=
1
ln 12≈ 0.402430
(b) log6 13.54 =ln 13.54
ln 6≈ 1.454240
(c) log2 π =lnπ
ln 2≈ 1.651496
15. To graph these functions, we use log1.5 x =lnx
ln 1.5and log50 x =
lnx
ln 50.
These graphs all approach−∞ as x→ 0+, and they all pass through the
point (1, 0). Also, they are all increasing, and all approach∞ as x→∞.
The functions with larger bases increase extremely slowly, and the ones with
smaller bases do so somewhat more quickly. The functions with large bases
approach the y-axis more closely as x→ 0+.
17. Use y = Cax with the points (1, 6) and (3, 24). 6 = Ca1 C = 6a
and 24 = Ca3 ⇒ 24 =6
aa3 ⇒
4 = a2 ⇒ a = 2 [since a > 0] and C = 62 = 3. The function is f(x) = 3 · 2x.
19. (a) 2 ft = 24 in, f(24) = 242 in = 576 in = 48 ft. g(24) = 224 in = 224/(12 · 5280) mi ≈ 265 mi
(b) 3 ft = 36 in, so we need x such that log2 x = 36 ⇔ x = 236 = 68,719,476,736. In miles, this is
68,719,476,736 in · 1 ft12 in
· 1 mi5280 ft
≈ 1,084,587.7 mi.
276 ¤ CHAPTER 7 INVERSE FUNCTIONS
21. limx→∞
(1.001)x =∞ by Figure 1, since 1.001 > 1.
23. limt→∞
2−t2
= limu→−∞
2u [where u = −t2] = 0
25. h(t) = t3 − 3t ⇒ h0(t) = 3t2 − 3t ln 3
27. Using Formula 4 and the Chain Rule, y = 5−1/x ⇒ y0 = 5−1/x(ln 5) −1 · −x−2 = 5−1/x(ln 5)/x2
29. f(u) = (2u + 2−u)10 ⇒
f 0(u) = 10(2u + 2−u)9d
du(2u + 2−u) = 10(2u + 2−u)9 2u ln 2 + 2−u ln 2 · (−1)
= 10 ln 2(2u + 2−u)9(2u − 2−u)
31. f(x) = log2(1− 3x) ⇒ f 0(x) =1
(1− 3x) ln 2d
dx(1− 3x) = −3
(1− 3x) ln 2 or 3
(3x− 1) ln 2
33. y = 2x log10√x = 2x log10 x
1/2 = 2x · 12log10 x = x log10 x ⇒ y0 = x · 1
x ln 10+ log10 x · 1 =
1
ln 10+ log10 x
Note: 1
ln 10=ln e
ln 10= log10 e, so the answer could be written as 1
ln 10+ log10 x = log10 e+ log10 x = log10 ex.
35. y = xx ⇒ ln y = lnxx ⇒ ln y = x lnx ⇒ y0/y = x(1/x) + (lnx) · 1 ⇒ y0 = y(1 + lnx) ⇒y0 = xx(1 + lnx)
37. y = x sin x ⇒ ln y = lnx sinx ⇒ ln y = sinx lnx ⇒ y0
y= (sinx) · 1
x+ (lnx)(cosx) ⇒
y0 = ysinx
x+ lnx cosx ⇒ y0 = x sin x
sinx
x+ lnx cosx
39. y = (cosx)x ⇒ ln y = ln(cosx)x ⇒ ln y = x ln cosx ⇒ 1
yy0 = x · 1
cosx· (− sinx) + ln cosx · 1 ⇒
y0 = y ln cosx− x sinx
cosx⇒ y0 = (cosx)x(ln cosx− x tanx)
41. y = (tanx)1/x ⇒ ln y = ln(tanx)1/x ⇒ ln y =1
xln tanx ⇒
1
yy0 =
1
x· 1
tanx· sec2 x+ ln tanx · − 1
x2⇒ y0 = y
sec2 x
x tanx− ln tanx
x2⇒
y0 = (tanx)1/xsec2 x
x tanx− ln tanx
x2or y0 = (tanx)1/x · 1
xcscx secx− ln tanx
x
43. y = 10x ⇒ y0 = 10x ln 10, so at (1, 10), the slope of the tangent line is 101 ln 10 = 10 ln 10, and its equation is
y − 10 = 10 ln 10(x− 1), or y = (10 ln 10)x+ 10(1− ln 10).
45.2
1
10t dt =10t
ln 10
2
1
=102
ln 10− 101
ln 10=100− 10ln 10
=90
ln 10
SECTION 7.4* GENERAL LOGARITHMIC AND EXPONENTIAL FUNCTIONS ¤ 277
47. log10 x
xdx =
(lnx)/(ln 10)
xdx =
1
ln 10
lnx
xdx. Now put u = lnx, so du = 1
xdx, and the expression becomes
1
ln 10udu =
1
ln 1012u2 + C1 =
1
2 ln 10(lnx)2 + C.
Or: The substitution u = log10 x gives du = dx
x ln 10and we get log10 x
xdx = 1
2ln 10(log10 x)
2 + C.
49. Let u = sin θ. Then du = cos θ dθ and 3sin θ cos θ dθ = 3udu =3u
ln 3+ C =
1
ln 33sin θ +C.
51. A =0
−1(2x − 5x) dx+
1
0
(5x − 2x) dx = 2x
ln 2− 5x
ln 5
0
−1+
5x
ln 5− 2x
ln 2
1
0
=1
ln 2− 1
ln 5− 1/2
ln 2− 1/5
ln 5+
5
ln 5− 2
ln 2− 1
ln 5− 1
ln 2
=16
5 ln 5− 1
2 ln 2
53. We see that the graphs of y = 2x and y = 1 + 3−x intersect at x ≈ 0.6. We
let f(x) = 2x − 1− 3−x and calculate f 0(x) = 2x ln 2 + 3−x ln 3, and
using the formula xn+1 = xn − f(xn) /f0(xn) (Newton’s Method), we get
x1 = 0.6, x2 ≈ x3 ≈ 0.600967. So, correct to six decimal places, the root
occurs at x = 0.600967.
55. y = log10 1 +1
x⇒ 10y = 1 +
1
x⇒ 1
x= 10y − 1 ⇒ x =
1
10y − 1 .
Interchange x and y: y =1
10x − 1 is the inverse function.
57. If I is the intensity of the 1989 San Francisco earthquake, then log10(I/S) = 7.1 ⇒log10(16I/S) = log10 16 + log10(I/S) = log10 16 + 7.1 ≈ 8.3.
59. We find I with the loudness formula from Exercise 58, substituting I0 = 10−12 and L = 50:
50 = 10 log10I
10−12⇔ 5 = log10
I
10−12⇔ 105 =
I
10−12⇔ I = 10−7 watt/m2. Now we differentiate L with
respect to I: L = 10 log10I
I0⇒ dL
dI= 10
1
(I/I0) ln 10
1
I0=
10
ln 10
1
I. Substituting I = 10−7, we get
L0 (50) =10
ln 10
1
10−7=108
ln 10≈ 4.34× 107 dB
watt/m2.
61. (a) Using a calculator or CAS, we obtain the model Q = abt with a ≈ 100.0124369 and b ≈ 0.000045145933.
(b) Use Q0(t) = abt ln b (from Formula 4) with the values of a and b from part (a) to get Q0(0.04) ≈ −670.63 μA.
The result of Example 2 in Section 2.1 was −670 μA.
63. Using Definition 1 and the second law of exponents for ex, we have ax− y = e(x− y) ln a = ex ln a− y ln a =ex ln a
ey ln a=
ax
ay.
278 ¤ CHAPTER 7 INVERSE FUNCTIONS
65. Let loga x = r and loga y = s. Then ar = x and as = y.
(a) xy = aras = ar+ s ⇒ loga (xy) = r + s = loga x+ loga y
(b) x
y=
ar
as= ar− s ⇒ loga
x
y= r − s = loga x− loga y
(c) xy = (ar)y = ary ⇒ loga (xy) = ry = y loga x
7.5 Exponential Growth and Decay
1. The relative growth rate is 1P
dP
dt= 0.7944, so dP
dt= 0.7944P and, by Theorem 2, P (t) = P (0)e0.7944t = 2e0.7944t.
Thus, P (6) = 2e0.7944(6) ≈ 234.99 or about 235 members.
3. (a) By Theorem 2, P (t) = P (0)ekt = 100ekt. Now P (1) = 100ek(1) = 420 ⇒ ek = 420100
⇒ k = ln 4.2.
So P (t) = 100e(ln 4.2)t = 100(4.2)t.
(b) P (3) = 100(4.2)3 = 7408.8 ≈ 7409 bacteria
(c) dP/dt = kP ⇒ P 0(3) = k · P (3) = (ln 4.2) 100(4.2)3 [from part (a)] ≈ 10,632 bacteria/hour
(d) P (t) = 100(4.2)t = 10,000 ⇒ (4.2)t = 100 ⇒ t = (ln 100)/(ln 4.2) ≈ 3.2 hours
5. (a) Let the population (in millions) in the year t be P (t). Since the initial time is the year 1750, we substitute t− 1750 for t in
Theorem 2, so the exponential model gives P (t) = P (1750)ek(t−1750). Then P (1800) = 980 = 790ek(1800−1750) ⇒980790 = ek(50) ⇒ ln 980
790 = 50k ⇒ k = 150 ln
980790 ≈ 0.0043104. So with this model, we have
P (1900) = 790ek(1900−1750) ≈ 1508 million, and P (1950) = 790ek(1950−1750) ≈ 1871 million. Both of these
estimates are much too low.
(b) In this case, the exponential model gives P (t) = P (1850)ek(t−1850) ⇒ P (1900) = 1650 = 1260ek(1900−1850) ⇒ln 1650
1260= k(50) ⇒ k = 1
50ln 1650
1260≈ 0.005393. So with this model, we estimate
P (1950) = 1260ek(1950−1850) ≈ 2161 million. This is still too low, but closer than the estimate of P (1950) in part (a).
(c) The exponential model gives P (t) = P (1900)ek(t−1900) ⇒ P (1950) = 2560 = 1650ek(1950−1900) ⇒ln 2560
1650 = k(50) ⇒ k = 150 ln
25601650 ≈ 0.008785. With this model, we estimate
P (2000) = 1650ek(2000−1900) ≈ 3972 million. This is much too low. The discrepancy is explained by the fact that the
world birth rate (average yearly number of births per person) is about the same as always, whereas the mortality rate
(especially the infant mortality rate) is much lower, owing mostly to advances in medical science and to the wars in the first
part of the twentieth century. The exponential model assumes, among other things, that the birth and mortality rates will
remain constant.
7. (a) If y = [N2O5] then by Theorem 2, dydt= −0.0005y ⇒ y(t) = y(0)e−0.0005t = Ce−0.0005t.
(b) y(t) = Ce−0.0005t = 0.9C ⇒ e−0.0005t = 0.9 ⇒ −0.0005t = ln 0.9 ⇒ t = −2000 ln 0.9 ≈ 211 s
SECTION 7.5 EXPONENTIAL GROWTH AND DECAY ¤ 279
9. (a) If y(t) is the mass (in mg) remaining after t years, then y(t) = y(0)ekt = 100ekt.
y(30) = 100e30k = 12 (100) ⇒ e30k = 1
2 ⇒ k = −(ln 2)/30 ⇒ y(t) = 100e−(ln 2)t/30 = 100 · 2−t/30
(b) y(100) = 100 · 2−100/30 ≈ 9.92 mg
(c) 100e−(ln 2)t/30 = 1 ⇒ −(ln 2)t/30 = ln 1100 ⇒ t = −30 ln 0.01ln 2 ≈ 199.3 years
11. Let y(t) be the level of radioactivity. Thus, y(t) = y(0)e−kt and k is determined by using the half-life:
y(5730) = 12y(0) ⇒ y(0)e−k(5730) = 1
2y(0) ⇒ e−5730k = 12 ⇒ −5730k = ln 1
2 ⇒ k = − ln12
5730=ln 2
5730.
If 74% of the 14C remains, then we know that y(t) = 0.74y(0) ⇒ 0.74 = e−t(ln 2)/5730 ⇒ ln 0.74 = − t ln 2
5730⇒
t = −5730(ln 0.74)ln 2
≈ 2489 ≈ 2500 years.
13. (a) Using Newton’s Law of Cooling, dTdt
= k(T − Ts), we have dT
dt= k(T − 75). Now let y = T − 75, so
y(0) = T (0)− 75 = 185− 75 = 110, so y is a solution of the initial-value problem dy/dt = ky with y(0) = 110 and by
Theorem 2 we have y(t) = y(0)ekt = 110ekt.
y(30) = 110e30k = 150− 75 ⇒ e30k = 75110
= 1522
⇒ k = 130ln 15
22, so y(t) = 110e
130t ln( 1522 ) and
y(45) = 110e4530
ln( 1522 ) ≈ 62◦F. Thus, T (45) ≈ 62 + 75 = 137◦F.
(b) T (t) = 100 ⇒ y(t) = 25. y(t) = 110e130t ln( 1522 ) = 25 ⇒ e
130t ln( 1522 ) = 25
110⇒ 1
30t ln 15
22= ln 25
110⇒
t =30 ln 25
110
ln 1522
≈ 116 min.
15. dT
dt= k(T − 20). Letting y = T − 20, we get dy
dt= ky, so y(t) = y(0)ekt. y(0) = T (0)− 20 = 5− 20 = −15, so
y(25) = y(0)e25k = −15e25k, and y(25) = T (25)− 20 = 10− 20 = −10, so −15e25k = −10 ⇒ e25k = 23
. Thus,
25k = ln 23
and k = 125 ln
23
, so y(t) = y(0)ekt = −15e(1/25) ln(2/3)t. More simply, e25k = 23 ⇒ ek = 2
3
1/25 ⇒
ekt = 23
t/25 ⇒ y(t) = −15 · 23
t/25.
(a) T (50) = 20 + y(50) = 20− 15 · 23
50/25= 20− 15 · 2
3
2= 20− 20
3 = 13.3̄◦C
(b) 15 = T (t) = 20 + y(t) = 20− 15 · 23
t/25 ⇒ 15 · 23
t/25= 5 ⇒ 2
3
t/25= 1
3 ⇒
(t/25) ln 23= ln 1
3⇒ t = 25 ln 1
3ln 2
3≈ 67.74 min.
17. (a) Let P (h) be the pressure at altitude h. Then dP/dh = kP ⇒ P (h) = P (0)ekh = 101.3ekh.
P (1000) = 101.3e1000k = 87.14 ⇒ 1000k = ln 87.14101.3
⇒ k = 11000
ln 87.14101.3
⇒
P (h) = 101.3 e1
1000h ln( 87.14101.3 ), so P (3000) = 101.3e3 ln(
87.14101.3 ) ≈ 64.5 kPa.
(b) P (6187) = 101.3 e61871000
ln( 87.14101.3 ) ≈ 39.9 kPa
280 ¤ CHAPTER 7 INVERSE FUNCTIONS
19. (a) Using A = A0 1 +r
n
nt
with A0 = 3000, r = 0.05, and t = 5, we have:
(i) Annually: n = 1; A = 3000 1 + 0.051
1·5= $3828.84
(ii) Semiannually: n = 2; A = 3000 1 + 0.052
2·5= $3840.25
(iii) Monthly: n = 12; A = 3000 1 + 0.0512
12·5= $3850.08
(iv) Weekly: n = 52; A = 3000 1 + 0.0552
52·5= $3851.61
(v) Daily: n = 365; A = 3000 1 + 0.05365
365·5= $3852.01
(vi) Continuously: A = 3000e(0.05)5 = $3852.08
(b) dA/dt = 0.05A and A(0) = 3000.
7.6 Inverse Trigonometric Functions
1. (a) sin−1√32
= π3
since sin π3=√32
and π3
is in −π2, π2
.
(b) cos−1(−1) = π since cosπ = −1 and π is in [0, π].
3. (a) arctan 1 = π4
since tan π4= 1 and π
4is in −π
2, π2
.
(b) sin−1 1√2= π
4 since sin π4 =
1√2
and π4 is in −π
2 ,π2
.
5. (a) In general, tan(arctanx) = x for any real number x. Thus, tan(arctan 10) = 10.
(b) sin−1 sin 7π3
= sin−1 sin π3= sin−1
√32= π
3since sin π
3=√32
and π3
is in −π2, π2
.
[Recall that 7π3= π
3+ 2π and the sine function is periodic with period 2π.]
7. Let θ = sin−1 23
.
Then tan sin−1 23
= tan θ =2√5
.
9. Let θ = tan−1√2. Then
sin 2 tan−1√2 = sin(2θ) = 2 sin θ cos θ
= 2
√2√3
1√3
=2√2
3
11. Let y = sin−1 x. Then −π2 ≤ y ≤ π
2 ⇒ cos y ≥ 0, so cos(sin−1 x) = cos y = 1− sin2 y = √1− x2.
13. Let y = tan−1 x. Then tan y = x, so from the triangle we see that
sin(tan−1 x) = sin y =x√1 + x2
.
SECTION 7.6 INVERSE TRIGONOMETRIC FUNCTIONS ¤ 281
15. The graph of sin−1 x is the reflection of the graph of
sinx about the line y = x.
17. Let y = cos−1 x. Then cos y = x and 0 ≤ y ≤ π ⇒ − sin y dy
dx= 1 ⇒
dy
dx= − 1
sin y= − 1
1− cos2 y = −1√1− x2
. [Note that sin y ≥ 0 for 0 ≤ y ≤ π.]
19. Let y = cot−1 x. Then cot y = x ⇒ − csc2 y dy
dx= 1 ⇒ dy
dx= − 1
csc2 y= − 1
1 + cot2 y= − 1
1 + x2.
21. Let y = csc−1 x. Then csc y = x ⇒ − csc y cot y dy
dx= 1 ⇒
dy
dx= − 1
csc y cot y= − 1
csc y csc2 y − 1 = −1
x√x2 − 1 . Note that cot y ≥ 0 on the domain of csc−1 x.
23. y = tan−1√x ⇒ y0 =
1
1 +√x
2 ·d
dx
√x =
1
1 + x12x−1/2 =
1
2√x (1 + x)
25. y = sin−1(2x + 1) ⇒y0 =
1
1− (2x+ 1)2 ·d
dx(2x+ 1) =
1
1− (4x2 + 4x+ 1) · 2 =2√−4x2 − 4x =
1√−x2 − x
27. G(x) =√1− x2 arccosx ⇒ G0(x) =
√1− x2 · −1√
1− x2+ arccosx · 1
2(1− x2)−1/2(−2x) = −1− x arccosx√
1− x2
29. y = cos−1(e2x) ⇒ y0 = − 1
1− (e2x)2 ·d
dx(e2x) = − 2e2x√
1− e4x
31. y = arctan(cos θ) ⇒ y0 =1
1 + (cos θ)2(− sin θ) = − sin θ
1 + cos2 θ
33. h(t) = cot−1(t) + cot−1(1/t) ⇒
h0(t) = − 1
1 + t2− 1
1 + (1/t)2· ddt
1
t= − 1
1 + t2− t2
t2 + 1· − 1
t2= − 1
1 + t2+
1
t2 + 1= 0.
Note that this makes sense because h(t) = π
2for t > 0 and h(t) = 3π
2for t < 0.
35. y = arccos b+ a cosx
a+ b cosx⇒
y0 = − 1
1− b+ a cosx
a+ b cosx
2
(a+ b cosx)(−a sinx)− (b+ a cosx)(−b sinx)(a+ b cosx)2
=1√
a2 + b2 cos2 x− b2 − a2 cos2 x
(a2 − b2) sinx
|a+ b cosx| =1√
a2 − b2√1− cos2 x
(a2 − b2) sinx
|a+ b cosx| =
√a2 − b2
|a+ b cosx|sinx
|sinx|
But 0 ≤ x ≤ π, so |sinx| = sinx. Also a > b > 0 ⇒ b cosx ≥ −b > −a, so a+ b cosx > 0. Thus y0 =√a2 − b2
a+ b cosx.
282 ¤ CHAPTER 7 INVERSE FUNCTIONS
37. g(x) = cos−1(3 − 2x) ⇒ g0(x) = − 1
1− (3− 2x)2(−2) = 2
1− (3− 2x)2 .
Domain(g) = {x | −1 ≤ 3− 2x ≤ 1} = {x | −4 ≤ −2x ≤ −2} = {x | 2 ≥ x ≥ 1} = [1, 2].
Domain(g0) = x | 1− (3− 2x)2 > 0 = x | (3− 2x)2 < 1 = {x | |3− 2x| < 1}= {x | −1 < 3− 2x < 1} = {x | −4 < −2x < −2} = {x | 2 > x > 1} = (1, 2)
39. g(x) = x sin−1x
4+√16− x2 ⇒ g0(x) = sin−1
x
4+
x
4 1− (x/4)2 −x√
16− x2= sin−1
x
4⇒
g0(2) = sin−1 12= π
6
41. f(x) =√1− x2 arcsinx ⇒ f 0(x) =
√1− x2 · 1√
1− x2+ arcsinx · 1
21− x2
−1/2(−2x) = 1− x arcsinx√
1− x2
Note that f 0 = 0 where the graph of f has a horizontal tangent. Also note
that f 0 is negative when f is decreasing and f 0 is positive when f is
increasing.
43. limx→−1+
sin−1 x = sin−1(−1) = −π2
45. Let t = ex. As x→∞, t→∞. limx→∞
arctan(ex) = limt→∞
arctan t = π2
by (8).
47. From the figure, tanα = 5
xand tanβ = 2
3− x. Since
α+ β + θ = 180◦ = π, θ = π − tan−1 5
x− tan−1 2
3− x⇒
dθ
dx= − 1
1 +5
x
2 − 5
x2− 1
1 +2
3− x
2
2
(3− x)2
=x2
x2 + 25· 5x2− (3− x)2
(3− x)2 + 4· 2
(3− x)2.
Now dθ
dx= 0 ⇒ 5
x2 + 25=
2
x2 − 6x+ 13 ⇒ 2x2 + 50 = 5x2 − 30x+ 65 ⇒
3x2 − 30x+ 15 = 0 ⇒ x2 − 10x+ 5 = 0 ⇒ x = 5± 2√5. We reject the root with the + sign, since it is larger
than 3. dθ/dx > 0 for x < 5− 2√5 and dθ/dx < 0 for x > 5− 2√5, so θ is maximized when
|AP | = x = 5− 2√5 ≈ 0.53.
49. dx
dt= 2 ft/s, sin θ = x
10⇒ θ = sin−1
x
10, dθdx
=1/10
1− (x/10)2 ,
dθ
dt=
dθ
dx
dx
dt=
1/10
1− (x/10)2 (2) rad/s, dθ
dt x=6
=2/10
1− (6/10)2 rad/s = 14
rad/s
SECTION 7.6 INVERSE TRIGONOMETRIC FUNCTIONS ¤ 283
51. y = f(x) = sin−1(x/(x+ 1)) A. D = {x | −1 ≤ x/(x+ 1) ≤ 1}. For x > −1 we have −x− 1 ≤ x ≤ x+ 1 ⇔2x ≥ −1 ⇔ x ≥ − 1
2, so D = − 1
2,∞ . B. Intercepts are 0 C. No symmetry
D. limx→∞
sin−1x
x+ 1= lim
x→∞sin−1
1
1 + 1/x= sin−1 1 = π
2, so y = π
2is a HA.
E. f 0(x) = 1
1− [x/(x+ 1)]2(x+ 1)− x
(x+ 1)2=
1
(x+ 1)√2x+ 1
> 0,
so f is increasing on − 12,∞ . F. No local maximum or minimum,
f − 12= sin−1 (−1) = −π
2is an absolute minimum
G. f 00(x) = −√2x+ 1 + (x+ 1)/
√2x+ 1
(x+ 1)2(2x+ 1)
= − 3x+ 2
(x+ 1)2(2x+ 1)3/2< 0 on D, so f is CD on − 1
2,∞ .
H.
53. y = f(x) = x− tan−1 x A. D = R B. Intercepts are 0 C. f(−x) = −f(x), so the curve is symmetric about the
origin. D. limx→∞
(x− tan−1 x) =∞ and limx→−∞
(x− tan−1 x) = −∞, no HA.
But f(x)− x− π2= − tan−1 x+ π
2 → 0 as x→∞, and
f(x)− x+ π2= − tan−1 x− π
2 → 0 as x→−∞, so y = x± π2 are
slant asymptotes. E. f 0(x) = 1− 1
x2 + 1=
x2
x2 + 1> 0, so f is
increasing on R. F. No extrema
G. f 00(x) = (1 + x2)(2x)− x2(2x)
(1 + x2)2=
2x
(1 + x2)2> 0 ⇔ x > 0, so
f is CU on (0,∞), CD on (−∞, 0). IP at (0, 0).
H.
55. f(x) = arctan(cos(3 arcsinx)). We use a CAS to compute f 0 and f 00, and to graph f , f 0, and f 00:
From the graph of f 0, it appears that the only maximum occurs at x = 0 and there are minima at x = ±0.87. From the graph
of f 00, it appears that there are inflection points at x = ±0.52.
57. f(x) = 2 + x2
1 + x2=1 + (1 + x2)
1 + x2=
1
1 + x2+ 1 ⇒ F (x) = tan−1 x+ x+C
59.√3/2
1/2
6√1− t2
dt = 6
√3/2
1/2
1√1− t2
dt = 6 sin−1 t√3/2
1/2= 6 sin−1
√32
− sin−1 12
= 6 π3− π
6= 6 π
6= π
284 ¤ CHAPTER 7 INVERSE FUNCTIONS
61. Let u = 4x. Then du = 4 dx, so√3/4
0
dx
1 + 16x2=1
4
√3
0
1
1 + u2du = 1
4tan−1 u
√3
0= 1
4tan−1
√3− tan−1 0 = 1
4π3− 0 = π
12.
63. Let u = 1 + x2. Then du = 2xdx, so
1 + x
1 + x2dx =
1
1 + x2dx+
x
1 + x2dx = tan−1 x+
12du
u= tan−1 x+ 1
2ln|u|+C
= tan−1 x+ 12 ln 1 + x2 + C = tan−1 x+ 1
2 ln(1 + x2) +C [since 1 + x2 > 0].
65. Let u = sin−1 x. Then du = 1√1− x2
dx, so dx√1− x2 sin−1 x
=1
udu = ln |u|+C = ln sin−1 x +C.
67. Let u = t3. Then du = 3t2 dt and t2√1− t6
dt =13du√
1− u2= 1
3sin−1 u+C = 1
3sin−1(t3) + C.
69. Let u =√x. Then du = dx
2√x
and dx√x (1 + x)
=2 du
1 + u2= 2 tan−1 u+ C = 2 tan−1
√x+ C.
71. Let u = x/a. Then du = dx/a, so dx√a2 − x2
=dx
a 1− (x/a)2 =du√1− u2
= sin−1 u+C = sin−1x
a+C.
73. The integral represents the area below the curve y = sin−1 x on the interval
x ∈ [0, 1]. The bounding curves are y = sin−1 x ⇔ x = sin y, y = 0 and
x = 1. We see that y ranges between sin−1 0 = 0 and sin−1 1 = π2
. So we have to
integrate the function x = 1− sin y between y = 0 and y = π2 :
1
0sin−1 xdx = π/2
0(1− sin y) dy = π
2 + cosπ2− (0 + cos 0) = π
2 − 1.
75. (a) arctan 12+ arctan 1
3= arctan
12+ 1
3
1− 12· 13
= arctan 1 =π
4
(b) 2 arctan 13+ arctan 1
7= arctan 1
3+ arctan 1
3+ arctan 1
7= arctan
13+ 1
3
1− 13· 13
+ arctan 17
= arctan 34 + arctan
17 = arctan
34+ 1
7
1− 34 · 17
= arctan 1 =π
4
77. Let f(x) = 2 sin−1 x− cos−1(1− 2x2). Then f 0(x) = 2√1− x2
− 4x
1− (1− 2x2)2=
2√1− x2
− 4x
2x√1− x2
= 0
[since x ≥ 0]. Thus f 0(x) = 0 for all x ∈ [0, 1). Thus f(x) = C. To find C let x = 0. Thus
2 sin−1(0)− cos−1(1) = 0 = C. Therefore we see that f(x) = 2 sin−1 x− cos−1(1− 2x2) = 0 ⇒
2 sin−1 x = cos−1(1− 2x2).
SECTION 7.7 HYPERBOLIC FUNCTIONS ¤ 285
79. y = sec−1 x ⇒ sec y = x ⇒ sec y tan ydy
dx= 1 ⇒ dy
dx=
1
sec y tan y. Now tan2 y = sec2 y − 1 = x2 − 1, so
tan y = ±√x2 − 1. For y ∈ 0, π2 , x ≥ 1, so sec y = x = |x| and tan y ≥ 0 ⇒ dy
dx=
1
x√x2 − 1 =
1
|x|√x2 − 1 .
For y ∈ π2, π , x ≤ −1, so |x| = −x and tan y = −√x2 − 1 ⇒
dy
dx=
1
sec y tan y=
1
x −√x2 − 1 =1
(−x)√x2 − 1 =1
|x|√x2 − 1
7.7 Hyperbolic Functions
1. (a) sinh 0 = 12(e0 − e0) = 0 (b) cosh 0 = 1
2(e0 + e0) = 1
2(1 + 1) = 1
3. (a) sinh(ln 2) = eln 2 − e−ln 2
2=
eln 2 − (eln 2)−12
=2− 2−12
=2− 1
2
2=3
4
(b) sinh 2 = 12(e2 − e−2) ≈ 3.62686
5. (a) sech 0 = 1
cosh 0=1
1= 1 (b) cosh−1 1 = 0 because cosh 0 = 1.
7. sinh(−x) = 12[e−x − e−(−x)] = 1
2(e−x − ex) = − 1
2(e−x − ex) = − sinhx
9. coshx+ sinhx = 12(ex + e−x) + 1
2(ex − e−x) = 1
2(2ex) = ex
11. sinhx cosh y + coshx sinh y = 12(ex − e−x) 1
2(ey + e−y) + 1
2(ex + e−x) 1
2(ey − e−y)
= 14[(ex+y + ex−y − e−x+y − e−x−y) + (ex+y − ex−y + e−x+y − e−x−y)]
= 14 (2e
x+y − 2e−x−y) = 12 [e
x+y − e−(x+y)] = sinh(x+ y)
13. Divide both sides of the identity cosh2 x− sinh2 x = 1 by sinh2 x:
cosh2 x
sinh2 x− sinh2 x
sinh2 x=
1
sinh2 x⇔ coth2 x− 1 = csch2 x.
15. Putting y = x in the result from Exercise 11, we have
sinh 2x = sinh(x+ x) = sinhx coshx+ coshx sinhx = 2 sinhx coshx.
17. tanh(lnx) = sinh(lnx)
cosh(lnx)=(eln x − e− lnx)/2
(eln x + e− lnx)/2=
x− (eln x)−1x+ (eln x)−1
=x− x−1
x+ x−1=
x− 1/xx+ 1/x
=(x2 − 1)/x(x2 + 1)/x
=x2 − 1x2 + 1
19. By Exercise 9, (coshx+ sinhx)n = (ex)n = enx = coshnx+ sinhnx.
21. sechx = 1
coshx⇒ sechx =
1
5/3=3
5.
cosh2 x− sinh2 x = 1 ⇒ sinh2 x = cosh2 x− 1 = 53
2 − 1 = 169 ⇒ sinhx = 4
3 [because x > 0].
cschx =1
sinhx⇒ cschx =
1
4/3=3
4.
tanhx =sinhx
coshx⇒ tanhx =
4/3
5/3=4
5.
cothx =1
tanhx⇒ cothx =
1
4/5=5
4.
286 ¤ CHAPTER 7 INVERSE FUNCTIONS
23. (a) limx→∞
tanhx = limx→∞
ex − e−x
ex + e−x· e−x
e−x= lim
x→∞1− e−2x
1 + e−2x=1− 01 + 0
= 1
(b) limx→−∞
tanhx = limx→−∞
ex − e−x
ex + e−x· e
x
ex= lim
x→−∞e2x − 1e2x + 1
=0− 10 + 1
= −1
(c) limx→∞
sinhx = limx→∞
ex − e−x
2=∞
(d) limx→−∞
sinhx = limx→−∞
ex − e−x
2= −∞
(e) limx→∞
sechx = limx→∞
2
ex + e−x= 0
(f ) limx→∞
cothx = limx→∞
ex + e−x
ex − e−x· e−x
e−x= lim
x→∞1 + e−2x
1− e−2x=1 + 0
1− 0 = 1 [Or: Use part (a)]
(g) limx→0+
cothx = limx→0+
coshx
sinhx=∞, since sinhx→ 0 through positive values and coshx→ 1.
(h) limx→0−
cothx = limx→0−
coshx
sinhx= −∞, since sinhx→ 0 through negative values and coshx→ 1.
(i) limx→−∞
cschx = limx→−∞
2
ex − e−x= 0
25. Let y = sinh−1 x. Then sinh y = x and, by Example 1(a), cosh2 y − sinh2 y = 1 ⇒ [with cosh y > 0]
cosh y = 1 + sinh2 y =√1 + x2. So by Exercise 9, ey = sinh y + cosh y = x+
√1 + x2 ⇒ y = ln x+
√1 + x2 .
27. (a) Let y = tanh−1 x. Then x = tanh y = sinh y
cosh y=(ey − e−y)/2(ey + e−y)/2
· ey
ey=
e2y − 1e2y + 1
⇒ xe2y + x = e2y − 1 ⇒
1+ x = e2y − xe2y ⇒ 1+ x = e2y(1− x) ⇒ e2y =1 + x
1− x⇒ 2y = ln
1 + x
1− x⇒ y = 1
2 ln1 + x
1− x.
(b) Let y = tanh−1 x. Then x = tanh y, so from Exercise 18 we have
e2y =1 + tanh y
1− tanh y =1 + x
1− x⇒ 2y = ln
1 + x
1− x⇒ y = 1
2 ln1 + x
1− x.
29. (a) Let y = cosh−1 x. Then cosh y = x and y ≥ 0 ⇒ sinh ydy
dx= 1 ⇒
dy
dx=
1
sinh y=
1
cosh2 y − 1=
1√x2 − 1 [since sinh y ≥ 0 for y ≥ 0]. Or: Use Formula 4.
(b) Let y = tanh−1 x. Then tanh y = x ⇒ sech2 ydy
dx= 1 ⇒ dy
dx=
1
sech2y=
1
1− tanh2 y =1
1− x2.
Or: Use Formula 5.
(c) Let y = csch−1 x. Then csch y = x ⇒ − csch y coth y dy
dx= 1 ⇒ dy
dx= − 1
csch y coth y. By Exercise 13,
coth y = ± csch2 y + 1 = ±√x2 + 1. If x > 0, then coth y > 0, so coth y =√x2 + 1. If x < 0, then coth y < 0,
so coth y = −√x2 + 1. In either case we have dy
dx= − 1
csch y coth y= − 1
|x|√x2 + 1 .
SECTION 7.7 HYPERBOLIC FUNCTIONS ¤ 287
(d) Let y = sech−1 x. Then sech y = x ⇒ − sech y tanh y dy
dx= 1 ⇒
dy
dx= − 1
sech y tanh y= − 1
sech y 1− sech2y= − 1
x√1− x2
. [Note that y > 0 and so tanh y > 0.]
(e) Let y = coth−1 x. Then coth y = x ⇒ − csch2 y dy
dx= 1 ⇒ dy
dx= − 1
csch2 y=
1
1− coth2 y =1
1− x2
by Exercise 13.
31. f(x) = x sinhx− coshx ⇒ f 0(x) = x (sinhx)0 + sinhx · 1− sinhx = x coshx
33. h(x) = ln(coshx) ⇒ h0(x) =1
coshx(coshx)0 =
sinhx
coshx= tanhx
35. y = ecosh 3x ⇒ y0 = ecosh 3x · sinh 3x · 3 = 3ecosh 3x sinh 3x
37. f(t) = sech2(et) = [sech(et)]2 ⇒f 0(t) = 2[sech(et)] [sech(et)]0 = 2 sech(et) − sech(et) tanh(et) · et = −2et sech2(et) tanh(et)
39. y = arctan(tanhx) ⇒ y0 =1
1 + (tanhx)2(tanhx)0 =
sech2 x
1 + tanh2 x
41. G(x) = 1− coshx1 + coshx
⇒
G0(x) =(1 + coshx)(− sinhx)− (1− coshx)(sinhx)
(1 + coshx)2=− sinhx− sinhx coshx− sinhx+ sinhx coshx
(1 + coshx)2
=−2 sinhx
(1 + coshx)2
43. y = tanh−1√x ⇒ y0 =
1
1−√x
2 ·1
2x−1/2 =
1
2√x (1− x)
45. y = x sinh−1(x/3) − √9 + x2 ⇒
y0 = sinh−1x
3+ x
1/3
1 + (x/3)2− 2x
2√9 + x2
= sinh−1x
3+
x√9 + x2
− x√9 + x2
= sinh−1x
3
47. y = coth−1√x2 + 1 ⇒ y0 =
1
1− (x2 + 1)2x
2√x2 + 1
= − 1
x√x2 + 1
49. As the depth d of the water gets large, the fraction 2πdL
gets large, and from Figure 3 or Exercise 23(a), tanh 2πd
L
approaches 1. Thus, v = gL
2πtanh
2πd
L≈ gL
2π(1) =
gL
2π.
51. (a) y = 20 cosh(x/20)− 15 ⇒ y0 = 20 sinh(x/20) · 120= sinh(x/20). Since the right pole is positioned at x = 7,
we have y0(7) = sinh 720≈ 0.3572.
(b) If α is the angle between the tangent line and the x-axis, then tanα = slope of the line = sinh 720 , so
α = tan−1 sinh 720
≈ 0.343 rad ≈ 19.66◦. Thus, the angle between the line and the pole is θ = 90◦ − α ≈ 70.34◦.
288 ¤ CHAPTER 7 INVERSE FUNCTIONS
53. (a) y = A sinhmx + B coshmx ⇒ y0 = mA coshmx +mB sinhmx ⇒y00 = m2A sinhmx+m2B coshmx = m2(A sinhmx+B coshmx) = m2y
(b) From part (a), a solution of y00 = 9y is y(x) = A sinh 3x+B cosh 3x. So −4 = y(0) = A sinh 0 +B cosh 0 = B, so
B = −4. Now y0(x) = 3A cosh 3x− 12 sinh 3x ⇒ 6 = y0(0) = 3A ⇒ A = 2, so y = 2 sinh 3x− 4 cosh 3x.
55. The tangent to y = coshx has slope 1 when y0 = sinhx = 1 ⇒ x = sinh−1 1 = ln 1 +√2 , by Equation 3.
Since sinhx = 1 and y = coshx = 1 + sinh2 x, we have coshx =√2. The point is ln 1 +
√2 ,
√2 .
57. Let u = coshx. Then du = sinhxdx, so sinhx cosh2 xdx = u2 du = 13u
3 + C = 13 cosh
3 x+C.
59. Let u =√x. Then du = dx
2√x
and sinh√x√
xdx = sinhu · 2 du = 2 coshu+C = 2 cosh
√x+C.
61. coshx
cosh2 x− 1 dx =coshx
sinh2 xdx =
coshx
sinhx· 1
sinhxdx = cothx cschxdx = − cschx+ C
63. Let t = 3u. Then dt = 3 du and6
4
1√t2 − 9 dt =
2
4/3
1√9u2 − 9 3 du =
2
4/3
du√u2 − 1 = cosh−1 u
2
4/3= cosh−1 2− cosh−1 4
3or
= cosh−1 u2
4/3= ln u+
√u2 − 1
2
4/3= ln 2 +
√3 − ln 4 +
√7
3= ln
6 + 3√3
4 +√7
65. Let u = ex. Then du = ex dx and ex
1− e2xdx =
du
1− u2= tanh−1 u+ C = tanh−1(ex) + C
or 12ln
1 + ex
1− ex+ C .
67. (a) From the graphs, we estimate
that the two curves y = cosh 2x
and y = 1 + sinhx intersect at
x = 0 and at x = a ≈ 0.481.
(b) We have found the two roots of the equation cosh 2x = 1 + sinhx to be x = 0 and x = a ≈ 0.481. Note from the first
graph that 1 + sinhx > cosh 2x on the interval (0, a), so the area between the two curves is
A=a
0(1 + sinhx− cosh 2x) dx = x+ coshx− 1
2sinh 2x
a
0
= [a+ cosh a− 12sinh 2a]− [0 + cosh 0− 1
2sinh 0] ≈ 0.0402
SECTION 7.8 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 289
69. If aex + be−x = α cosh(x+ β) [or α sinh(x+ β)], then
aex + be−x = α2ex+β ± e−x−β = α
2exeβ ± e−xe−β = α
2eβ ex ± α
2e−β e−x. Comparing coefficients of ex
and e−x, we have a = α2eβ (1) and b = ±α
2e−β (2). We need to find α and β. Dividing equation (1) by equation (2)
gives us ab= ±e2β ⇒ ( ) 2β = ln ±a
b⇒ β = 1
2ln ±a
b. Solving equations (1) and (2) for eβ gives us
eβ =2a
αand eβ = ± α
2b, so 2a
α= ± α
2b⇒ α2 = ±4ab ⇒ α = 2
ñab.
( ) If ab> 0, we use the + sign and obtain a cosh function, whereas if a
b< 0, we use the − sign and obtain a sinh
function.
In summary, if a and b have the same sign, we have aex + be−x = 2√ab cosh x+ 1
2 lnab
, whereas, if a and b have the
opposite sign, then aex + be−x = 2√−ab sinh x+ 1
2ln −a
b.
7.8 Indeterminate Forms and L'Hospital's Rule
Note: The use of l’Hospital’s Rule is indicated by an H above the equal sign: H=
1. (a) limx→a
f(x)
g(x)is an indeterminate form of type 0
0.
(b) limx→a
f(x)
p(x)= 0 because the numerator approaches 0 while the denominator becomes large.
(c) limx→a
h(x)
p(x)= 0 because the numerator approaches a finite number while the denominator becomes large.
(d) If limx→a
p(x) =∞ and f(x)→ 0 through positive values, then limx→a
p(x)
f(x)=∞. [For example, take a = 0, p(x) = 1/x2,
and f(x) = x2.] If f(x)→ 0 through negative values, then limx→a
p(x)
f(x)= −∞. [For example, take a = 0, p(x) = 1/x2,
and f(x) = −x2.] If f(x)→ 0 through both positive and negative values, then the limit might not exist. [For example,
take a = 0, p(x) = 1/x2, and f(x) = x.]
(e) limx→a
p(x)
q(x)is an indeterminate form of type ∞∞ .
3. (a) When x is near a, f(x) is near 0 and p(x) is large, so f(x)− p(x) is large negative. Thus, limx→a
[f(x)− p(x)] = −∞.
(b) limx→a
[ p(x)− q(x)] is an indeterminate form of type∞−∞.
(c) When x is near a, p(x) and q(x) are both large, so p(x) + q(x) is large. Thus, limx→a
[p(x) + q(x)] =∞.
5. This limit has the form 00
. We can simply factor and simplify to evaluate the limit.
limx→1
x2 − 1x2 − x
= limx→1
(x+ 1)(x− 1)x(x− 1) = lim
x→1
x+ 1
x=1 + 1
1= 2
7. This limit has the form 00
. limx→1
x9 − 1x5 − 1
H= lim
x→1
9x8
5x4=9
5limx→1
x4 =9
5(1) =
9
5
290 ¤ CHAPTER 7 INVERSE FUNCTIONS
9. This limit has the form 00 . lim
x→(π/2)+
cosx
1− sinxH= lim
x→(π/2)+
− sinx− cosx = lim
x→(π/2)+tanx = −∞.
11. This limit has the form 00
. limt→0
et − 1t3
H= lim
t→0
et
3t2=∞ since et → 1 and 3t2 → 0+ as t→ 0.
13. This limit has the form 00 . lim
x→0
tan px
tan qxH= lim
x→0
p sec2 px
q sec2 qx=
p(1)2
q(1)2=
p
q
15. This limit has the form ∞∞ . lim
x→∞lnx√x
H= lim
x→∞1/x
12x−1/2 = lim
x→∞2√x= 0
17. limx→0+
[(lnx)/x] = −∞ since lnx→ −∞ as x→ 0+ and dividing by small values of x just increases the magnitude of the
quotient (lnx)/x. L’Hospital’s Rule does not apply.
19. This limit has the form ∞∞ . lim
x→∞ex
x3H= lim
x→∞ex
3x2H= lim
x→∞ex
6xH= lim
x→∞ex
6=∞
21. This limit has the form 00
. limx→0
ex − 1− x
x2H= lim
x→0
ex − 12x
H= lim
x→0
ex
2=1
2
23. This limit has the form 00
. limx→0
tanhx
tanxH= lim
x→0
sech 2x
sec2 x=sech2 0
sec2 0=1
1= 1
25. This limit has the form 00 . lim
t→0
5t − 3tt
H= lim
t→0
5t ln 5− 3t ln 31
= ln 5− ln 3 = ln 53
27. This limit has the form 00
. limx→0
sin−1 xx
H= lim
x→0
1/√1− x2
1= lim
x→0
1√1− x2
=1
1= 1
29. This limit has the form 00 . lim
x→0
1− cosxx2
H= lim
x→0
sinx
2xH= lim
x→0
cosx
2=1
2
31. limx→0
x+ sinx
x+ cosx=0 + 0
0 + 1=0
1= 0. L’Hospital’s Rule does not apply.
33. This limit has the form 00 . lim
x→1
1− x+ lnx
1 + cosπxH= lim
x→1
−1 + 1/x−π sinπx
H= lim
x→1
−1/x2−π2 cosπx =
−1−π2 (−1) = −
1
π2
35. This limit has the form 00
. limx→1
xa − ax+ a− 1(x− 1)2
H= lim
x→1
axa−1 − a
2(x− 1)H= lim
x→1
a(a− 1)xa−22
=a(a− 1)
2
37. This limit has the form 00
. limx→0
cosx− 1 + 12x
2
x4H= lim
x→0
− sinx+ x
4x3H= lim
x→0
− cosx+ 112x2
H= lim
x→0
sinx
24xH= lim
x→0
cosx
24=1
24
39. This limit has the form ∞ · 0.
limx→∞
x sin(π/x) = limx→∞
sin(π/x)
1/xH= lim
x→∞cos(π/x)(−π/x2)
−1/x2 = π limx→∞
cos(π/x) = π(1) = π
41. This limit has the form ∞ · 0. We’ll change it to the form 00
.
limx→0
cot 2x sin 6x = limx→0
sin 6x
tan 2xH= lim
x→0
6 cos 6x
2 sec2 2x=6(1)
2(1)2= 3
43. This limit has the form∞ · 0. limx→∞
x3e−x2= lim
x→∞x3
ex2H= lim
x→∞3x2
2xex2= lim
x→∞3x
2ex2H= lim
x→∞3
4xex2= 0
SECTION 7.8 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 291
45. This limit has the form 0 · (−∞).
limx→1+
lnx tan(πx/2) = limx→1+
lnx
cot(πx/2)H= lim
x→1+
1/x
(−π/2) csc2(πx/2) =1
(−π/2)(1)2 = −2
π
47. This limit has the form∞−∞.
limx→1
x
x− 1 −1
lnx= lim
x→1
x lnx− (x− 1)(x− 1) lnx
H= lim
x→1
x(1/x) + lnx− 1(x− 1)(1/x) + lnx = lim
x→1
lnx
1− (1/x) + lnxH= lim
x→1
1/x
1/x2 + 1/x· x
2
x2= lim
x→1
x
1 + x=
1
1 + 1=1
2
49. We will multiply and divide by the conjugate of the expression to change the form of the expression.
limx→∞
√x2 + x− x = lim
x→∞
√x2 + x− x
1·√x2 + x+ x√x2 + x+ x
= limx→∞
x2 + x − x2√x2 + x+ x
= limx→∞
x√x2 + x+ x
= limx→∞
1
1 + 1/x+ 1=
1√1 + 1
=1
2
As an alternate solution, write√x2 + x− x as
√x2 + x−√x2, factor out
√x2, rewrite as ( 1 + 1/x− 1)/(1/x), and
apply l’Hospital’s Rule.
51. The limit has the form∞−∞ and we will change the form to a product by factoring out x.
limx→∞
(x− lnx) = limx→∞
x 1− lnx
x=∞ since lim
x→∞lnx
xH= lim
x→∞1/x
1= 0.
53. y = xx2 ⇒ ln y = x2 lnx, so lim
x→0+ln y = lim
x→0+x2 lnx = lim
x→0+
lnx
1/x2H= lim
x→0+
1/x
−2/x3 = limx→0+
−12x2 = 0 ⇒
limx→0+
xx2= lim
x→0+eln y = e0 = 1.
55. y = (1− 2x)1/x ⇒ ln y =1
xln(1− 2x), so lim
x→0ln y = lim
x→0
ln(1− 2x)x
H= lim
x→0
−2/(1− 2x)1
= −2 ⇒
limx→0
(1− 2x)1/x = limx→0
eln y = e−2.
57. y = 1 +3
x+5
x2
x
⇒ ln y = x ln 1 +3
x+5
x2⇒
limx→∞
ln y = limx→∞
ln 1 +3
x+5
x2
1/xH= lim
x→∞
− 3
x2− 10
x31 +
3
x+5
x2
−1/x2 = limx→∞
3 +10
x
1 +3
x+5
x2
= 3,
so limx→∞
1 +3
x+5
x2
x
= limx→∞
eln y = e3.
59. y = x1/x ⇒ ln y = (1/x) lnx ⇒ limx→∞
ln y = limx→∞
lnx
xH= lim
x→∞1/x
1= 0 ⇒
limx→∞
x1/x = limx→∞
eln y = e0 = 1
61. y = (4x+ 1)cot x ⇒ ln y = cotx ln(4x+ 1), so limx→0+
ln y = limx→0+
ln(4x+ 1)
tanxH= lim
x→0+
4
4x+ 1sec2 x
= 4 ⇒
limx→0+
(4x+ 1)cotx = limx→0+
eln y = e4.
292 ¤ CHAPTER 7 INVERSE FUNCTIONS
63. y = (cosx)1/x2 ⇒ ln y =
1
x2ln cosx ⇒
limx→0+
ln y = limx→0+
ln cosx
x2H= lim
x→0+
− tanx2x
H= lim
x→0+
− sec2 x2
= −12
⇒
limx→0+
(cosx)1/x2
= limx→0+
eln y = e−1/2 = 1/√e
65. From the graph, if x = 500, y ≈ 7.36. The limit has the form 1∞.
Now y = 1 +2
x
x
⇒ ln y = x ln 1 +2
x⇒
limx→∞
ln y = limx→∞
ln(1 + 2/x)
1/xH= lim
x→∞
1
1 + 2/x− 2
x2
−1/x2
= 2 limx→∞
1
1 + 2/x= 2(1) = 2 ⇒
limx→∞
1 +2
x
x
= limx→∞
eln y = e2 [≈ 7.39]
67. From the graph, it appears that limx→0
f(x)
g(x)= lim
x→0
f 0(x)g0(x)
= 0.25.
We calculate limx→0
f(x)
g(x)= lim
x→0
ex − 1x3 + 4x
H= lim
x→0
ex
3x2 + 4=1
4.
69. y = f(x) = xe−x A. D = R B. Intercepts are 0 C. No symmetry
D. limx→∞
xe−x = limx→∞
x
exH= lim
x→∞1
ex= 0, so y = 0 is a HA. lim
x→−∞xe−x = −∞
E. f 0(x) = e−x − xe−x = e−x(1− x) > 0 ⇔ x < 1, so f is increasing
on (−∞, 1) and decreasing on (1,∞) . F. Absolute and local maximum
value f(1) = 1/e. G. f 00(x) = e−x (x− 2) > 0 ⇔ x > 2, so f is CU
on (2,∞) and CD on (−∞, 2). IP at 2, 2/e2
H.
71. y = f(x) = xe−x2
A. D = R B. Intercepts are 0 C. f(−x) = −f(x), so the curve is symmetric
about the origin. D. limx→±∞
xe−x2= lim
x→±∞x
ex2H= lim
x→±∞1
2xex2= 0, so y = 0 is a HA.
E. f 0(x) = e−x2 − 2x2e−x2 = e−x
2(1− 2x2) > 0 ⇔ x2 < 1
2⇔ |x| < 1√
2, so f is increasing on − 1√
2, 1√
2
and decreasing on −∞,− 1√2
and 1√2,∞ . F. Local maximum value f 1√
2= 1/
√2e, local minimum
value f − 1√2= −1/√2e G. f 00(x) = −2xe−x2(1− 2x2)− 4xe−x2 = 2xe−x2(2x2 − 3) > 0 ⇔
SECTION 7.8 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 293
x > 32 or − 3
2 < x < 0, so f is CU on 32 ,∞ and
− 32 , 0 and CD on −∞,− 3
2and 0, 3
2.
IP are (0, 0) and ± 32,± 3
2e−3/2 .
H.
73. y = f(x) = x− ln(1 + x) A. D = {x | x > −1} = (−1,∞) B. Intercepts are 0 C. No symmetry
D. limx→−1+
[x− ln(1 + x)] =∞, so x = −1 is a VA. limx→∞
[x− ln(1 + x)] = limx→∞
x 1− ln(1 + x)
x=∞,
since limx→∞
ln(1 + x)
xH= lim
x→∞1/(1 + x)
1= 0.
E. f 0(x) = 1− 1
1 + x=
x
1 + x> 0 ⇔ x > 0 since x+ 1 > 0.
So f is increasing on (0,∞) and decreasing on (−1, 0).F. f(0) = 0 is an absolute minimum.
G. f 00(x) = 1/(1 + x)2 > 0, so f is CU on (−1,∞).
H.
75. (a) f(x) = x−x
(b) y = f(x) = x−x. We note that ln f(x) = lnx−x = −x lnx = − lnx1/x
, so
limx→0+
ln f(x)H= lim
x→0+− 1/x
−x−2 = limx→0+
x = 0. Thus limx→0+
f(x) = limx→0+
eln f(x) = e0 = 1.
(c) From the graph, it appears that there is a local and absolute maximum of about f(0.37) ≈ 1.44. To find the exact value, we
differentiate: f(x) = x−x = e−x lnx ⇒ f 0(x) = e−x lnx −x 1
x+ lnx(−1) = −x−x(1 + lnx). This is 0 only
when 1 + lnx = 0 ⇔ x = e−1. Also f 0(x) changes from positive to negative at e−1. So the maximum value is
f(1/e) = (1/e)−1/e = e1/e.
(d) We differentiate again to get
f 00(x) = −x−x(1/x) + (1 + lnx)2(x−x) = x−x[(1 + lnx)2 − 1/x]
From the graph of f 00(x), it seems that f 00(x) changes from negative to
positive at x = 1, so we estimate that f has an IP at x = 1.
294 ¤ CHAPTER 7 INVERSE FUNCTIONS
77. (a) f(x) = x1/x
(b) Recall that ab = eb ln a. limx→0+
x1/x = limx→0+
e(1/x) ln x. As x→ 0+, lnxx→ −∞, so x1/x = e(1/x) lnx → 0. This
indicates that there is a hole at (0, 0). As x→∞, we have the indeterminate form∞0. limx→∞
x1/x = limx→∞
e(1/x) ln x,
but limx→∞
lnx
xH= lim
x→∞1/x
1= 0, so lim
x→∞x1/x = e0 = 1. This indicates that y = 1 is a HA.
(c) Estimated maximum: (2.72, 1.45). No estimated minimum. We use logarithmic differentiation to find any critical
numbers. y = x1/x ⇒ ln y =1
xlnx ⇒ y0
y=1
x· 1x+ (lnx) − 1
x2⇒ y0 = x1/x
1− lnxx2
= 0 ⇒
lnx = 1 ⇒ x = e. For 0 < x < e, y0 > 0 and for x > e, y0 < 0, so f(e) = e1/e is a local maximum value. This
point is approximately (2.7183, 1.4447), which agrees with our estimate.
(d) From the graph, we see that f 00(x) = 0 at x ≈ 0.58 and x ≈ 4.37. Since f 00
changes sign at these values, they are x-coordinates of inflection points.
79. If c < 0, then limx→−∞
f(x) = limx→−∞
xe−cx = limx→−∞
x
ecxH= lim
x→−∞1
cecx= 0, and lim
x→∞f(x) =∞.
If c > 0, then limx→−∞
f(x) = −∞, and limx→∞
f(x)H= lim
x→∞1
cecx= 0.
If c = 0, then f(x) = x, so limx→±∞
f(x) = ±∞, respectively.
So we see that c = 0 is a transitional value. We now exclude the case c = 0, since we know how the function behaves
in that case. To find the maxima and minima of f , we differentiate: f(x) = xe−cx ⇒f 0(x) = x(−ce−cx) + e−cx = (1− cx)e−cx. This is 0 when 1− cx = 0 ⇔ x = 1/c. If c < 0 then this
represents a minimum value of f(1/c) = 1/(ce), since f 0(x) changes from negative to positive at x = 1/c;
and if c > 0, it represents a maximum value. As |c| increases, the maximum or
minimum point gets closer to the origin. To find the inflection points, we
differentiate again: f 0(x) = e−cx(1− cx) ⇒f 00(x) = e−cx(−c) + (1− cx)(−ce−cx) = (cx− 2)ce−cx. This changes sign
when cx− 2 = 0 ⇔ x = 2/c. So as |c| increases, the points of inflection get
closer to the origin.
SECTION 7.8 INDETERMINATE FORMS AND L’HOSPITAL’S RULE ¤ 295
81. limx→∞
x√x2 + 1
H= lim
x→∞1
12(x2 + 1)−1/2(2x)
= limx→∞
√x2 + 1
x. Repeated applications of l’Hospital’s Rule result in the
original limit or the limit of the reciprocal of the function. Another method is to try dividing the numerator and denominator
by x: limx→∞
x√x2 + 1
= limx→∞
x/x
x2/x2 + 1/x2= lim
x→∞1
1 + 1/x2=1
1= 1
83. limE→0+
P (E) = limE→0+
eE + e−E
eE − e−E− 1
E
= limE→0+
E eE + e−E − 1 eE − e−E
(eE − e−E)E= lim
E→0+
EeE +Ee−E − eE + e−E
EeE −Ee−Eform is 0
0
H= lim
E→0+
EeE + eE · 1 +E −e−E + e−E · 1− eE + −e−EEeE + eE · 1− [E(−e−E) + e−E · 1]
= limE→0+
EeE −Ee−E
EeE + eE +Ee−E − e−E= lim
E→0+
eE − e−E
eE +eE
E+ e−E − e−E
E
[divide by E]
=0
2 + L, where L = lim
E→0+
eE − e−E
Eform is 0
0
H= lim
E→0+
eE + e−E
1=1 + 1
1= 2
Thus, limE→0+
P (E) =0
2 + 2= 0.
85. First we will find limn→∞
1 +r
n
nt
, which is of the form 1∞. y = 1 +r
n
nt
⇒ ln y = nt ln 1 +r
n, so
limn→∞
ln y = limn→∞
nt ln 1 +r
n= t lim
n→∞ln(1 + r/n)
1/nH= t lim
n→∞−r/n2
(1 + r/n)(−1/n2) = t limn→∞
r
1 + i/n= tr ⇒
limn→∞
y = ert. Thus, as n→∞, A = A0 1 +r
n
nt
→ A0ert.
87. Both numerator and denominator approach 0 as x→ 0, so we use l’Hospital’s Rule (and FTC1):
limx→0
S(x)
x3= lim
x→0
x
0sin(πt2/2)dt
x3H= lim
x→0
sin(πx2/2)
3x2H= lim
x→0
πx cos(πx2/2)
6x=
π
6· cos 0 = π
6
89. Since f(2) = 0, the given limit has the form 00 .
limx→0
f(2 + 3x) + f(2 + 5x)
xH= lim
x→0
f 0(2 + 3x) · 3 + f 0(2 + 5x) · 51
= f 0(2) · 3 + f 0(2) · 5 = 8f 0(2) = 8 · 7 = 56
91. Since limh→0
[f(x+ h)− f(x− h)] = f(x)− f(x) = 0 (f is differentiable and hence continuous) and limh→0
2h = 0, we use
l’Hospital’s Rule:
limh→0
f(x+ h)− f(x− h)
2hH= lim
h→0
f 0(x+ h)(1)− f 0(x− h)(−1)2
=f 0(x) + f 0(x)
2=2f 0(x)2
= f 0(x)
f(x+ h)− f(x− h)
2his the slope of the secant line between
(x− h, f(x− h)) and (x+ h, f(x+ h)). As h→ 0, this line gets closer
to the tangent line and its slope approaches f 0(x).
296 ¤ CHAPTER 7 INVERSE FUNCTIONS
93. limx→∞
ex
xnH= lim
x→∞ex
nxn−1H= lim
x→∞ex
n(n− 1)xn−2H= · · · H
= limx→∞
ex
n!=∞
95. limx→0+
xα lnx = limx→0+
lnx
x−αH= lim
x→0+
1/x
−αx−α− 1 = limx→0+
xα
−α = 0 since α > 0.
97. Let the radius of the circle be r. We see that A(θ) is the area of the whole figure (a sector of the circle with radius 1), minus
the area of4OPR. But the area of the sector of the circle is 12r2θ (see Reference Page 1), and the area of the triangle is
12r |PQ| = 1
2r(r sin θ) = 1
2r2 sin θ. So we have A(θ) = 1
2r2θ − 1
2r2 sin θ = 1
2r2(θ − sin θ). Now by elementary
trigonometry, B(θ) = 12 |QR| |PQ| = 1
2 (r − |OQ|) |PQ| = 12 (r − r cos θ)(r sin θ) = 1
2r2(1− cos θ) sin θ.
So the limit we want is
limθ→0+
A(θ)
B(θ)= lim
θ→0+
12r
2(θ − sin θ)12r2(1− cos θ) sin θ
H= lim
θ→0+
1− cos θ(1− cos θ) cos θ + sin θ (sin θ)
= limθ→0+
1− cos θcos θ − cos2 θ + sin2 θ
H= lim
θ→0+
sin θ
− sin θ − 2 cos θ (− sin θ) + 2 sin θ (cos θ)
= limθ→0+
sin θ
− sin θ + 4 sin θ cos θ = limθ→0+
1
−1 + 4 cos θ =1
−1 + 4 cos 0 =1
3
99. (a) We show that limx→0
f(x)
xn= 0 for every integer n ≥ 0. Let y = 1
x2. Then
limx→0
f(x)
x2n= lim
x→0
e−1/x2
(x2)n= lim
y→∞yn
eyH= lim
y→∞nyn−1
eyH= · · · H
= limy→∞
n!
ey= 0 ⇒
limx→0
f(x)
xn= lim
x→0xn
f(x)
x2n= lim
x→0xn lim
x→0
f(x)
x2n= 0. Thus, f 0(0) = lim
x→0
f(x)− f(0)
x− 0 = limx→0
f(x)
x= 0.
(b) Using the Chain Rule and the Quotient Rule we see that f (n)(x) exists for x 6= 0. In fact, we prove by induction that for
each n ≥ 0, there is a polynomial pn and a non-negative integer kn with f (n)(x) = pn(x)f(x)/xkn for x 6= 0. This is
true for n = 0; suppose it is true for the nth derivative. Then f 0(x) = f(x)(2/x3), so
f (n+1)(x) = xkn [p0n(x) f(x) + pn(x) f0(x)]− knx
kn−1pn(x) f(x) x−2kn
= xknp0n(x) + pn(x)(2/x3)− knx
kn−1pn(x) f(x)x−2kn
= xkn+3p0n(x) + 2pn(x)− knxkn+2 pn(x) f(x)x
−(2kn+3)
which has the desired form.
Now we show by induction that f (n) (0) = 0 for all n. By part (a), f 0(0) = 0. Suppose that f (n)(0) = 0. Then
f (n+1)(0) = limx→0
f (n)(x)− f (n)(0)
x− 0 = limx→0
f (n)(x)
x= lim
x→0
pn(x) f(x)/xkn
x= lim
x→0
pn(x) f(x)
xkn+1
= limx→0
pn(x) limx→0
f(x)
xkn+1= pn(0) · 0 = 0
CHAPTER 7 REVIEW ¤ 297
7 Review
1. (a) A function f is called a one-to-one function if it never takes on the same value twice; that is, if f(x1) 6= f(x2) whenever
x1 6= x2. (Or, f is 1-1 if each output corresponds to only one input.)
Use the Horizontal Line Test: A function is one-to-one if and only if no horizontal line intersects its graph more thanonce.
(b) If f is a one-to-one function with domain A and range B, then its inverse function f−1 has domain B and range A and is
defined by
f−1(y) = x ⇔ f(x) = y
for any y in B. The graph of f−1 is obtained by reflecting the graph of f about the line y = x.
(c) (f−1)0(a) = 1
f 0(f−1(a))
2. (a) The function f(x) = ex has domain R and range (0,∞).
(b) The function f(x) = lnx has domain (0,∞) and range R.
(c) The graphs are reflections of one another about the line y = x. See Figure 7.3.3 or Figure 7.3*.1.
(d) loga x =lnx
ln a
3. (a) The inverse sine function f(x) = sin−1 x is defined as follows:
sin−1 x = y ⇔ sin y = x and −π
2≤ y ≤ π
2
Its domain is −1 ≤ x ≤ 1 and its range is −π
2≤ y ≤ π
2.
(b) The inverse cosine function f(x) = cos−1 x is defined as follows:
cos−1 x = y ⇔ cos y = x and 0 ≤ y ≤ π
Its domain is −1 ≤ x ≤ 1 and its range is 0 ≤ y ≤ π.
(c) The inverse tangent function f(x) = tan−1 x is defined as follows:
tan−1 x = y ⇔ tan y = x and −π
2< y <
π
2
Its domain is R and its range is −π
2< y <
π
2.
4. sinhx = ex − e−x
2, coshx = ex + e−x
2, tanhx = sinhx
coshx=
ex − e−x
ex + e−x
5. (a) y = ex ⇒ y0 = ex (b) y = ax ⇒ y0 = ax lna
(c) y = lnx ⇒ y0 = 1/x (d) y = loga x ⇒ y0 = 1/(x ln a)
(e) y = sin−1 x ⇒ y0 = 1/√1− x2 (f ) y = cos−1 x ⇒ y0 = −1/√1− x2
(g) y = tan−1 x ⇒ y0 = 1/(1 + x2) (h) y = sinhx ⇒ y0 = coshx
298 ¤ CHAPTER 7 INVERSE FUNCTIONS
(i) y = coshx ⇒ y0 = sinhx ( j) y = tanhx ⇒ y0 = sech2 x
(k) y = sinh−1 x ⇒ y0 = 1/√1 + x2 (l) y = cosh−1 x ⇒ y0 = 1/
√x2 − 1
(m) y = tanh−1 x ⇒ y0 = 1/(1− x2)
6. (a) See Definition 7.2.7 (or 7.2*.5).
(b) e = limx→0
(1 + x)1/x
(c) The differentiation formula for y = ax [y0 = ax ln a] is simplest when a = e because ln e = 1.
(d) The differentiation formula for y = loga x [y0 = 1/(x ln a)] is simplest when a = e because ln e = 1.
7. (a) dy
dt= ky
(b) The equation in part (a) is an appropriate model for population growth, assuming that there is enough room and nutrition to
support the growth.
(c) If y(0) = y0, then the solution is y(t) = y0ekt.
8. (a) See l’Hospital’s Rule and the three notes that follow it in Section 7.8.
(b) Write fg as f
1/gor g
1/f.
(c) Convert the difference into a quotient using a common denominator, rationalizing, factoring, or some other method.
(d) Convert the power to a product by taking the natural logarithm of both sides of y = fg or by writing fg as eg ln f .
1. True. If f is one-to-one, with domain R, then f−1(f(6)) = 6 by the first cancellation equation [see (4) in Section 7.1].
3. False. For example, cos π2 = cos −π2
, so cosx is not 1-1.
5. True, since lnx is an increasing function on (0,∞).
7. True. We can divide by ex since ex 6= 0 for every x.
9. False. Let x = e. Then (lnx)6 = (ln e)6 = 16 = 1, but 6 lnx = 6 ln e = 6 · 1 = 6 6= 1 = (lnx)6.
11. False. ln 10 is a constant, so its derivative is 0.
13. False. The “−1” is not an exponent; it is an indication of an inverse function.
15. True. See Figure 2 in Section 7.7.
17. True. 16
2(1/x) dx = lnx
16
2= ln 16− ln 2 = ln 16
2= ln 8 = ln 23 = 3 ln 2
CHAPTER 7 REVIEW ¤ 299
1. No. f is not 1-1 because the graph of f fails the Horizontal Line Test.
3. (a) f−1(3) = 7 since f(7) = 3. (b) (f−1)0(3) = 1
f 0(f−1(3))=
1
f 0(7)=1
8
5.
y = 5x − 1
7. Reflect the graph of y = lnx about the x-axis to obtain the graph
of y = − lnx.
y = lnx y = − lnx
9.
y = 2arctanx
11. (a) e2 ln 3 = (eln 3)2 = 32 = 9 (b) log10 25 + log10 4 = log10(25 · 4) = log10 100 = log10 102 = 2
13. lnx = 13⇔ loge x =
13⇒ x = e1/3
15. eex= 17 ⇒ ln ee
x= ln 17 ⇒ ex = ln 17 ⇒ ln ex = ln(ln 17) ⇒ x = ln ln 17
17. ln(x+ 1) + ln(x− 1) = 1 ⇒ ln [(x+ 1)(x− 1)] = 1 ⇒ ln(x2 − 1) = ln e ⇒ x2 − 1 = e ⇒x2 = e+ 1 ⇒ x =
√e+ 1 since ln(x− 1) is defined only when x > 1.
19. tan−1 x = 1 ⇒ tan tan−1 x = tan 1 ⇒ x = tan 1 (≈ 1.5574)
21. f(t) = t2 ln t ⇒ f 0(t) = t2 · 1t+ (ln t)(2t) = t+ 2t ln t or t(1 + 2 ln t)
23. h(θ) = etan 2θ ⇒ h0(θ) = etan 2θ · sec2 2θ · 2 = 2 sec2(2θ) etan 2θ
25. y = ln |sec 5x+ tan 5x| ⇒
y0 =1
sec 5x+ tan 5x(sec 5x tan 5x · 5 + sec2 5x · 5) = 5 sec 5x (tan 5x+ sec 5x)
sec 5x+ tan 5x= 5 sec 5x
27. y = ecx(c sinx− cosx) ⇒ y0 = cecx(c sinx− cosx) + ecx(c cosx+ sinx) = (c2 + 1)ecx sinx
29. y = ln(sec2 x) = 2 ln |secx| ⇒ y0 = (2/ secx)(secx tanx) = 2 tanx
300 ¤ CHAPTER 7 INVERSE FUNCTIONS
31. y = e1/x
x2⇒ y0 =
x2(e1/x)0 − e1/x x20
(x2)2=
x2(e1/x)(−1/x2)− e1/x(2x)
x4=−e1/x(1 + 2x)
x4
33. y = 3x ln x ⇒ y0 = 3x ln x(ln 3)d
dx(x lnx) = 3x lnx(ln 3) x · 1
x+ lnx · 1 = 3x ln x(ln 3)(1 + lnx)
35. H(v) = v tan−1 v ⇒ H0(v) = v · 1
1 + v2+ tan−1 v · 1 = v
1 + v2+ tan−1 v
37. y = x sinh(x2) ⇒ y0 = x cosh(x2) · 2x+ sinh(x2) · 1 = 2x2 cosh(x2) + sinh(x2)
39. y = ln sinx− 12 sin
2 x ⇒ y0 =1
sinx· cosx− 1
2 · 2 sinx · cosx = cotx− sinx cosx
41. y = ln 1
x+
1
lnx= lnx−1 + (lnx)−1 = − lnx+ (lnx)−1 ⇒ y0 = −1 · 1
x+ (−1)(lnx)−2 · 1
x= − 1
x− 1
x (lnx)2
43. y = ln(cosh 3x) ⇒ y0 = (1/ cosh 3x)(sinh 3x)(3) = 3 tanh 3x
45. y = cosh−1(sinhx) ⇒ y0 = (coshx)/ sinh2 x− 1
47. y = cos e√tan 3x ⇒
y0 = − sin e√tan 3x · e
√tan 3x
0= − sin e
√tan 3x e
√tan 3x · 1
2(tan 3x)−1/2 · sec2(3x) · 3
=−3 sin e
√tan 3x e
√tan 3x sec2(3x)
2√tan 3x
49. f(x) = eg(x) ⇒ f 0(x) = eg(x)g0(x)
51. f(x) = ln |g(x)| ⇒ f 0(x) =1
g(x)g0(x) =
g0(x)g(x)
53. f(x) = 2x ⇒ f 0(x) = 2x ln 2 ⇒ f 00(x) = 2x(ln 2)2 ⇒ · · · ⇒ f (n)(x) = 2x(ln 2)n
55. We first show it is true for n = 1: f 0(x) = ex + xex = (x+ 1)ex. We now assume it is true for n = k:
f (k)(x) = (x+ k)ex. With this assumption, we must show it is true for n = k + 1:
f (k+1)(x) =d
dxf (k)(x) =
d
dx[(x+ k)ex] = ex + (x+ k)ex = [x+ (k + 1)] ex.
Therefore, f (n)(x) = (x+ n)ex by mathematical induction.
57. y = (2+ x)e−x ⇒ y0 = (2+ x)(−e−x) + e−x · 1 = e−x[−(2 + x) + 1] = e−x(−x− 1). At (0, 2), y0 = 1(−1) = −1,
so an equation of the tangent line is y − 2 = −1(x− 0), or y = −x+ 2.
59. y = [ln(x+ 4)]2 ⇒ y0 = 2[ln(x+ 4)]1 · 1
x+ 4· 1 = 2 ln(x+ 4)
x+ 4and y0 = 0 ⇔ ln(x+ 4) = 0 ⇔
x+ 4 = e0 ⇒ x+ 4 = 1 ⇔ x = −3, so the tangent is horizontal at the point (−3, 0).
CHAPTER 7 REVIEW ¤ 301
61. (a) The line x− 4y = 1 has slope 14 . A tangent to y = ex has slope 1
4 when y0 = ex = 14 ⇒ x = ln 1
4 = − ln 4.
Since y = ex, the y-coordinate is 14
and the point of tangency is − ln 4, 14
. Thus, an equation of the tangent line
is y − 14= 1
4(x+ ln 4) or y = 1
4x+ 1
4(ln 4 + 1).
(b) The slope of the tangent at the point (a, ea) is d
dxex
x= a
= ea. Thus, an equation of the tangent line is
y − ea = ea(x− a). We substitute x = 0, y = 0 into this equation, since we want the line to pass through the origin:
0− ea = ea(0− a) ⇔ −ea = ea(−a) ⇔ a = 1. So an equation of the tangent line at the point (a, ea) = (1, e)
is y − e = e(x− 1) or y = ex.
63. limx→∞
e−3x = 0 since −3x→−∞ as x→∞ and limt→−∞
et = 0.
65. Let t = 2/(x− 3). As x→ 3−, t→ −∞. limx→3−
e2/(x−3) = limt→−∞
et = 0
67. Let t = sinhx. As x→ 0+, t→ 0+. limx→0+
ln(sinhx) = limt→0+
ln t = −∞
69. limx→∞
(1 + 2x)/2x
(1− 2x)/2x = limx→∞
1/2x + 1
1/2x − 1 =0 + 1
0− 1 = −1
71. This limit has the form 00 . lim
x→0
tanπx
ln(1 + x)H= lim
x→0
π sec2 πx
1/(1 + x)=
π · 121/1
= π
73. This limit has the form 00 . lim
x→0
e4x − 1− 4xx2
H= lim
x→0
4e4x − 42x
H= lim
x→0
16e4x
2= lim
x→08e4x = 8 · 1 = 8
75. This limit has the form∞ · 0. limx→∞
x3e−x = limx→∞
x3
exH= lim
x→∞3x2
exH= lim
x→∞6x
exH= lim
x→∞6
ex= 0
77. This limit has the form∞−∞.
limx→1+
x
x− 1 −1
lnx= lim
x→1+
x lnx− x+ 1
(x− 1) lnxH= lim
x→1+
x · (1/x) + lnx− 1(x− 1) · (1/x) + lnx = lim
x→1+
lnx
1− 1/x+ lnxH= lim
x→1+
1/x
1/x2 + 1/x=
1
1 + 1=1
2
79. y = f(x) = tan−1(1/x) A. D = {x | x 6= 0} B. No intercept C. f (−x) = −f(x), so the curve is symmetric
about the origin. D. limx→±∞
tan−1(1/x) = tan−1 0 = 0, so y = 0 is a HA. limx→0+
tan−1(1/x) = π2
and
limx→0−
tan−1(1/x) = −π2
since 1x→ ±∞ as x→ 0±.
E. f 0(x) = 1
1 + (1/x)2−1/x2 =
−1x2 + 1
⇒ f 0(x) < 0, so f is
decreasing on (−∞, 0) and (0,∞) . F. No maximum nor minimum
H.
G. f 00(x) = 2x
(x2 + 1)2> 0 ⇔ x > 0, so f is CU on (0,∞) and CD on (−∞, 0).
302 ¤ CHAPTER 7 INVERSE FUNCTIONS
81. y = f(x) = x lnx A. D = (0,∞) B. No y-intercept; x-intercept 1. C. No symmetry D. No asymptote
[Note that the graph approaches the point (0, 0) as x→ 0+.]
E. f 0(x) = x(1/x) + (lnx)(1) = 1 + lnx, so f 0(x)→−∞ as x→ 0+ and
f 0(x)→∞ as x→∞. f 0(x) = 0 ⇔ lnx = −1 ⇔ x = e−1 = 1/e.
f 0(x) > 0 for x > 1/e, so f is decreasing on (0, 1/e) and increasing on
(1/e,∞). F. Local minimum: f(1/e) = −1/e. No local maximum.
G. f 00(x) = 1/x, so f 00(x) > 0 for x > 0. The graph is CU on (0,∞) and
there is no IP.
H.
83. y = f(x) = xe−2x A. D = R B. y-intercept: f(0) = 0; x-intercept: f(x) = 0 ⇔ x = 0
C. No symmetry D. limx→∞
xe−2x = limx→∞
x
e2xH= lim
x→∞1
2e2x= 0, so y = 0 is a HA.
E. f 0(x) = x(−2e−2x) + e−2x(1) = e−2x(−2x+ 1) > 0 ⇔ −2x+ 1 > 0 ⇔ x < 12
and f 0(x) < 0 ⇔ x > 12
,
so f is increasing on −∞, 12
and decreasing on 12,∞ . F. Local maximum value f 1
2= 1
2e−1 = 1/(2e);
no local minimum value
G. f 00(x) = e−2x(−2) + (−2x+ 1)(−2e−2x)= 2e−2x[−1− (−2x+ 1)] = 4 (x− 1) e−2x.
f 00(x) > 0 ⇔ x > 1 and f 00(x) < 0 ⇔ x < 1, so f is
CU on (1,∞) and CD on (−∞, 1). IP at (1, f(1)) = (1, e−2)
H.
85. From the graph, we estimate the points of inflection to be about (±0.82, 0.22).f(x) = e−1/x
2 ⇒ f 0(x) = 2x−3e−1/x2 ⇒
f 00(x) = 2[x−3(2x−3)e−1/x2
+ e−1/x2
(−3x−4)] = 2x−6e−1/x2 2− 3x2 .
This is 0 when 2− 3x2 = 0 ⇔ x = ± 23
, so the inflection points
are ± 23, e−3/2 .
87. s(t) = Ae−ct cos(ωt+ δ) ⇒v(t) = s0(t) = A{e−ct [−ω sin(ωt+ δ)] + cos(ωt+ δ)(−ce−ct)} = −Ae−ct [ω sin(ωt+ δ) + c cos(ωt+ δ)] ⇒a(t) = v0(t) = −A{e−ct[ω2 cos(ωt+ δ)− cω sin(ωt+ δ)] + [ω sin(ωt+ δ) + c cos(ωt+ δ)](−ce−ct)}
= −Ae−ct[ω2 cos(ωt+ δ)− cω sin(ωt+ δ)− cω sin(ωt+ δ)− c2 cos(ωt+ δ)]
= −Ae−ct[(ω2 − c2) cos(ωt+ δ)− 2cω sin(ωt+ δ)] = Ae−ct[(c2 − ω2) cos(ωt+ δ) + 2cω sin(ωt+ δ)]
89. (a) y(t) = y(0)ekt = 200ekt ⇒ y(0.5) = 200e0.5k = 360 ⇒ e0.5k = 1.8 ⇒ 0.5k = ln 1.8 ⇒k = 2 ln 1.8 = ln(1.8)2 = ln 3.24 ⇒ y(t) = 200e(ln 3.24)t = 200(3.24)t
(b) y(4) = 200(3.24)4 ≈ 22,040 bacteria
CHAPTER 7 REVIEW ¤ 303
(c) y0(t) = 200(3.24)t · ln 3.24, so y0(4) = 200(3.24)4 · ln 3.24 ≈ 25,910 bacteria per hour
(d) 200(3.24)t = 10,000 ⇒ (3.24)t = 50 ⇒ t ln 3.24 = ln 50 ⇒ t = ln 50/ ln 3.24 ≈ 3.33 hours
91. Let P (t) = 64
1 + 31e−0.7944t=
A
1 +Bect= A(1 +Bect)−1, where A = 64, B = 31, and c = −0.7944.
P 0(t) = −A(1 +Bect)−2(Bcect) = −ABcect(1 +Bect)−2
P 00(t) = −ABcect[− 2(1 +Bect)−3(Bcect)] + (1 +Bect)−2(−ABc2ect)
= −ABc2ect(1 +Bect)−3[−2Bect + (1 +Bect)] = −ABc2ect(1−Bect)
(1 +Bect)3
The population is increasing most rapidly when its graph changes from CU to CD; that is, when P 00(t) = 0 in this case.
P 00(t) = 0 ⇒ Bect = 1 ⇒ ect =1
B⇒ ct = ln
1
B⇒ t =
ln(1/B)
c=ln(1/31)
−0.7944 ≈ 4.32 days. Note that
P1
cln1
B=
A
1 +Bec(1/c) ln(1/B)=
A
1 +Beln(1/B)=
A
1 +B(1/B)=
A
1 + 1=
A
2, one-half the limit of P as t→∞.
93. Let u = −2y2. Then du = −4y dy and 1
0ye−2y
2dy =
−20
eu − 14du = − 1
4eu
−20= − 1
4(e−2 − 1) = 1
4(1− e−2).
95. Let u = ex, so du = ex dx. When x = 0, u = 1; when x = 1, u = e. Thus,1
0
ex
1 + e2xdx =
e
1
1
1 + u2du = arctanu
e
1= arctan e− arctan 1 = arctan e− π
4.
97. Let u =√x. Then du = dx
2√x
⇒ e√x
√xdx = 2 eu du = 2eu +C = 2e
√x + C.
99. Let u = x2 + 2x. Then du = (2x + 2) dx = 2(x + 1) dx and
x+ 1
x2 + 2xdx =
12du
u= 1
2ln |u|+C = 1
2ln x2 + 2x +C.
101. Let u = ln(cosx). Then du =− sinxcosx
dx = − tanxdx ⇒
tanx ln(cosx) dx = − udu = − 12u
2 +C = − 12 [ ln(cosx)]
2 + C.
103. Let u = tan θ. Then du = sec2 θ dθ and 2tan θ sec2 θ dθ = 2udu =2u
ln 2+ C =
2tan θ
ln 2+ C.
105. 1− x
x
2
dx =1
x− 1
2
dx =1
x2− 2
x+ 1 dx = − 1
x− 2 ln |x|+ x+C
107. cosx ≤ 1 ⇒ ex cosx ≤ ex ⇒ 1
0ex cosxdx ≤ 1
0ex dx = ex
1
0= e− 1
109. f(x) =√x
1
es
sds ⇒ f 0(x) =
d
dx
√x
1
es
sds =
e√x
√x
d
dx
√x =
e√x
√x
1
2√x=
e√x
2x
111. fave =1
4− 14
1
1
xdx =
1
3ln |x| 4
1=1
3[ln 4− ln 1] = 1
3ln 4
304 ¤ CHAPTER 7 INVERSE FUNCTIONS
113. V =1
0
2πx
1 + x4dx by cylindrical shells. Let u = x2 ⇒ du = 2xdx. Then
V =1
0
π
1 + u2du = π tan−1 u
1
0= π tan−1 1− tan−1 0 = π
π
4=
π2
4.
115. f(x) = lnx+ tan−1 x ⇒ f(1) = ln 1 + tan−1 1 = π4⇒ g π
4= 1 [where g = f−1].
f 0(x) =1
x+
1
1 + x2, so g0 π
4=
1
f 0(1)=
1
3/2=2
3.
117. We find the equation of a tangent to the curve y = e−x, so that we can find the
x- and y-intercepts of this tangent, and then we can find the area of the triangle.
The slope of the tangent at the point a, e−a is given by d
dxe−x
x=a
= −e−a,
and so the equation of the tangent is y − e−a = −e−a(x− a) ⇔
y = e−a(a− x+ 1).
The y-intercept of this line is y = e−a(a− 0 + 1) = e−a(a+ 1). To find the x-intercept we set y = 0 ⇒
e−a(a− x+ 1) = 0 ⇒ x = a+ 1. So the area of the triangle is A(a) = 12e−a(a+ 1) (a+ 1) = 1
2e−a(a+ 1)2. We
differentiate this with respect to a: A0(a) = 12e−a(2)(a+ 1) + (a+ 1)2e−a(−1) = 1
2e−a 1− a2 . This is 0
at a = ±1, and the root a = 1 gives a maximum, by the First Derivative Test. So the maximum area of the triangle is
A(1) = 12e−1(1 + 1)2 = 2e−1 = 2/e.
119. limx→−1
F (x) = limx→−1
bx+1 − ax+1
x+ 1H= lim
x→−1bx+1 ln b− ax+1 ln a
1= ln b− ln a = F (−1), so F is continuous at −1.
121. Using FTC1, we differentiate both sides of the given equation, x
0f(t) dt = xe2x +
x
0e−tf(t) dt, and get
f(x) = e2x + 2xe2x + e−xf(x) ⇒ f(x) 1− e−x = e2x + 2xe2x ⇒ f(x) =e2x(1 + 2x)
1− e−x.
PROBLEMS PLUS1. Let y = f(x) = e−x
2
. The area of the rectangle under the curve from−x to x is A(x) = 2xe−x2
where x ≥ 0. We maximize
A(x): A0(x) = 2e−x2 − 4x2e−x2 = 2e−x2 1− 2x2 = 0 ⇒ x = 1√
2. This gives a maximum since A0(x) > 0
for 0 ≤ x < 1√2
and A0(x) < 0 for x > 1√2
. We next determine the points of inflection of f(x). Notice that
f 0(x) = −2xe−x2 = −A(x). So f 00(x) = −A0(x) and hence, f 00(x) < 0 for − 1√2< x < 1√
2and f 00(x) > 0 for x < − 1√
2
and x > 1√2
. So f(x) changes concavity at x = ± 1√2
, and the two vertices of the rectangle of largest area are at the inflection
points.
3. Consider the statement that dn
dxn(eax sin bx) = rneax sin(bx + nθ). For n = 1,
d
dx(eax sin bx) = aeax sin bx+ beax cos bx, and
reax sin(bx+ θ) = reax[sin bx cos θ + cos bx sin θ] = reaxa
rsin bx+
b
rcos bx = aeax sin bx+ beax cos bx
since tan θ = b
a⇒ sin θ =
b
rand cos θ = a
r. So the statement is true for n = 1.
Assume it is true for n = k. Then
dk+1
dxk+1(eax sin bx) =
d
dxrkeax sin(bx+ kθ) = rkaeax sin(bx+ kθ) + rkeaxb cos(bx+ kθ)
= rkeax[a sin(bx+ kθ) + b cos(bx+ kθ)]
But
sin[bx+ (k + 1)θ] = sin[(bx+ kθ) + θ] = sin(bx+ kθ) cos θ + sin θ cos(bx+ kθ) = ar sin(bx+ kθ) + b
r cos(bx+ kθ).
Hence, a sin(bx+ kθ) + b cos(bx+ kθ) = r sin[bx+ (k + 1)θ]. So
dk+1
dxk+1(eax sin bx) = rkeax[a sin(bx+kθ)+b cos(bx+kθ)] = rkeax[r sin(bx+(k+1)θ)] = rk+1eax[sin(bx+(k+1)θ)].
Therefore, the statement is true for all n by mathematical induction.
5. We first show that x
1 + x2< tan−1 x for x > 0. Let f(x) = tan−1 x− x
1 + x2. Then
f 0(x) =1
1 + x2− 1(1 + x2)− x(2x)
(1 + x2)2=(1 + x2)− (1− x2)
(1 + x2)2=
2x2
(1 + x2)2> 0 for x > 0. So f(x) is increasing
on (0,∞). Hence, 0 < x ⇒ 0 = f(0) < f(x) = tan−1 x− x
1 + x2. So x
1 + x2< tan−1 x for 0 < x. We next show
that tan−1 x < x for x > 0. Let h(x) = x− tan−1 x. Then h0(x) = 1− 1
1 + x2=
x2
1 + x2> 0. Hence, h(x) is increasing
on (0,∞). So for 0 < x, 0 = h(0) < h(x) = x− tan−1 x. Hence, tan−1 x < x for x > 0, and we conclude thatx
1 + x2< tan−1 x < x for x > 0.
7. By the Fundamental Theorem of Calculus, f(x) = x
1
√1 + t3 dt ⇒ f 0(x) =
√1 + x3 > 0 for x > −1.
So f is increasing on (−1,∞) and hence is one-to-one. Note that f(1) = 0, so f−1(1) = 0 ⇒(f−1)0(0) = 1/f 0(1) = 1√
2.
305
306 ¤ CHAPTER 7 PROBLEMS PLUS
9. If L = limx→∞
x+ a
x− a
x
, then L has the indeterminate form 1∞, so
lnL = limx→∞
lnx+ a
x− a
x
= limx→∞
x lnx+ a
x− a= lim
x→∞ln(x+ a)− ln(x− a)
1/xH= lim
x→∞
1
x+ a− 1
x− a−1/x2
= limx→∞
(x− a)− (x+ a)
(x+ a)(x− a)· −x
2
1= lim
x→∞2ax2
x2 − a2= lim
x→∞2a
1− a2/x2= 2a
Hence, lnL = 2a, so L = e2a. From the original equation, we want L = e1 ⇒ 2a = 1 ⇒ a = 12 .
11. Both sides of the inequality are positive, so cosh(sinhx) < sinh(coshx)
⇔ cosh2(sinhx) < sinh2(coshx) ⇔ sinh2(sinhx) + 1 < sinh2(coshx)
⇔ 1 < [sinh(coshx)− sinh(sinhx)][sinh(coshx) + sinh(sinhx)]
⇔ 1 < sinhex + e−x
2− sinh ex − e−x
2sinh
ex + e−x
2+ sinh
ex − e−x
2
⇔ 1 < [2 cosh(ex/2) sinh(ex/2)][2 sinh(ex/2) cosh(ex/2)] [use the addition formulas and cancel]
⇔ 1 < [2 sinh(ex/2) cosh(ex/2)][2 sinh(ex/2) cosh(ex/2)] ⇔ 1 < sinh ex sinh e−x,
by the half-angle formula. Now both ex and e−x are positive, and sinh y > y for y > 0, since sinh 0 = 0 and
(sinh y − y)0 = cosh y − 1 > 0 for x > 0, so 1 = exe−x < sinh ex sinh e−x. So, following this chain of reasoning
backward, we arrive at the desired result.
13. Let f(x) = e2x and g(x) = k√x [k > 0]. From the graphs of f and g,
we see that f will intersect g exactly once when f and g share a tangent
line. Thus, we must have f = g and f 0 = g0 at x = a.
f(a) = g(a) ⇒ e2a = k√a ( )
and f 0(a) = g0(a) ⇒ 2e2a =k
2√a
⇒ e2a =k
4√a
.
So we must have k√a =
k
4√a
⇒√a
2
=k
4k⇒ a = 1
4. From ( ), e2(1/4) = k 1/4 ⇒
k = 2e1/2 = 2√e ≈ 3.297.
15. Suppose that the curve y = ax intersects the line y = x. Then ax0 = x0 for some x0 > 0, and hence a = x1/x00 . We find the
maximum value of g(x) = x1/x, x > 0, because if a is larger than the maximum value of this function, then the curve y = ax
does not intersect the line y = x. g0(x) = e(1/x) ln x − 1
x2lnx+
1
x· 1x
= x1/x1
x2(1− lnx). This is 0 only where
x = e, and for 0 < x < e, f 0(x) > 0, while for x > e, f 0(x) < 0, so g has an absolute maximum of g(e) = e1/e. So if
y = ax intersects y = x, we must have 0 < a ≤ e1/e. Conversely, suppose that 0 < a ≤ e1/e. Then ae ≤ e, so the graph of
y = ax lies below or touches the graph of y = x at x = e. Also a0 = 1 > 0, so the graph of y = ax lies above that of y = x
at x = 0. Therefore, by the Intermediate Value Theorem, the graphs of y = ax and y = x must intersect somewhere between
x = 0 and x = e.
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