Chapter 4
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Chapter 4
Calculus Volume 1Chapter 4
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Chapter 4
Checkpoint
4.1
$\frac{1}{72\pi}\ \text{cm/sec},$ or approximately 0.0044 cm/sec
4.2
$500\ \text{ft/sec}$
4.3
$\frac{1}{10}\ \text{rad/sec}$
4.4
$-0.61\ \text{ft/sec}$
4.5
$L(x) = 2 + \frac{1}{12}\left( {x - 8} \right);$ 2.00833
4.6
$L(x) = \text{−}x + \frac{\pi}{2}$
4.7
$L(x) = 1 + 4x$
4.8
$dy = 2xe^{x^{2}}dx$
4.9
$dy = 1.6,$ $\text{Δ}y = 1.64$
4.10
The volume measurement is accurate to within $21.6{\ \text{cm}}^{3}.$
4.11
7.6%
4.12
$x = - \frac{2}{3},$ $x = 1$
4.13
The absolute maximum is $3$ and it occurs at $x = 4.$ The absolute minimum is $-1$ and it occurs at $x = 2.$
4.14
$c = 2$
4.15
$\frac{5}{2\sqrt{2}}$ sec
4.16
$f$ has a local minimum at $-2$ and a local maximum at $3.$
4.17
$f$ has no local extrema because $f^{\prime}$ does not change sign at $x = 1.$
4.18
$f$ is concave up over the interval $\left( {\text{−}\infty,\frac{1}{2}} \right)$ and concave down over the interval $\left( {\frac{1}{2},\infty} \right)$
4.19
$f$ has a local maximum at $-2$ and a local minimum at $3.$
4.20
Both limits are $3.$ The line $y = 3$ is a horizontal asymptote.
4.21
Let $\varepsilon > 0.$ Let $N = \frac{1}{\sqrt{\varepsilon}}.$ Therefore, for all $x > N,$ we have
$\left| {3 - \frac{1}{x^{2}} - 3} \right| = \frac{1}{x^{2}} < \frac{1}{N^{2}} = \varepsilon$
Therefore, $\underset{x\rightarrow\infty}{\text{lim}}\left( {3 - 1\text{/}x^{2}} \right) = 3.$
4.22
Let $M > 0.$ Let $N = \sqrt{\frac{M}{3}}.$ Then, for all $x > N,$ we have
$3x^{2} > 3N^{2} = 3\left( \sqrt{\frac{M}{3}} \right)^{2}{}^{2} = \frac{3M}{3} = M$
4.23
$\text{−}\infty$
4.24
$\frac{3}{5}$
4.25
$\text{±}\sqrt{3}$
4.26
$\underset{x\rightarrow\infty}{\text{lim}}f(x) = \frac{3}{5},$ $\underset{x\rightarrow\text{−}\infty}{\text{lim}}f(x) = -2$
4.27
4.28
4.29
$y = \frac{3}{2}x$
4.30
The function $f$ has a cusp at $\left( {0,5} \right)$ $\underset{x\rightarrow 0^{-}}{\text{lim}}f^{\prime}(x) = \infty,$ $\underset{x\rightarrow 0^{+}}{\text{lim}}f^{\prime}(x) = \text{−}\infty.$ For end behavior, $\underset{x\rightarrow\text{±}\infty}{\text{lim}}f(x) = \text{−}\infty.$
4.31
The maximum area is $5000{\ \text{ft}}^{2}.$
4.32
$V(x) = x\left( {20 - 2x} \right)\left( {30 - 2x} \right).$ The domain is $\left\lbrack {0,10} \right\rbrack.$
4.33
$T(x) = \frac{x}{6} + \frac{\sqrt{\left( {15 - x} \right)^{2} + 1}}{2.5}$
4.34
The company should charge $\text{\$}75$ per car per day.
4.35
$A(x) = 4x\sqrt{1 - x^{2}}.$ The domain of consideration is $\left\lbrack {0,1} \right\rbrack.$
4.36
$c(x) = \frac{259.2}{x} + 0.2x^{2}$ dollars
4.37
$1$
4.38
$0$
4.39
$\underset{x\rightarrow 0^{+}}{\text{lim}}\text{cos}\mspace{2mu} x = 1.$ Therefore, we cannot apply L’Hôpital’s rule. The limit of the quotient is $\infty$
4.40
$1$
4.41
$0$
4.42
$e$
4.43
$1$
4.44
The function $2^{x}$ grows faster than $x^{100}.$
4.45
$x_{1} \approx 0.33333333,x_{2} \approx 0.347222222$
4.46
$x_{1} = 2,x_{2} = 1.75$
4.47
$x_{1} \approx - 1.842105263,x_{2} \approx - 1.772826920$
4.48
$x_{1} = 6,x_{2} = 8,x_{3} = \frac{26}{3},x_{4} = \frac{80}{9},x_{5} = \frac{242}{27};x* = 9$
4.49
$\text{−}\text{cos}\mspace{2mu} x + C$
4.50
$\frac{d}{dx}\left( {x\mspace{2mu}\text{sin}\mspace{2mu} x + \text{cos}\mspace{2mu} x + C} \right) = \text{sin}\mspace{2mu} x + x\mspace{2mu}\text{cos}\mspace{2mu} x - \text{sin}\mspace{2mu} x = x\mspace{2mu}\text{cos}\mspace{2mu} x$
4.51
$x^{4} - \frac{5}{3}x^{3} + \frac{1}{2}x^{2} - 7x + C$
4.52
$y = - \frac{3}{x} + 5$
4.53
$2.93\ \text{sec},\ 64.5\ \text{ft}$
Section 4.1 Exercises
1.
$8$
3.
$\pm \frac{13}{\sqrt{10}}$
5.
$2\sqrt{3}$ ft/sec
7.
The distance is decreasing at $390\ \text{mi/h}.$
9.
The distance between them shrinks at a rate of $\frac{1320}{13} \approx 101.5\ \text{mph}.$
11.
$\frac{9}{2}$ ft/sec
13.
It grows at a rate $\frac{4}{9}$ ft/sec
15.
The distance is increasing at $\frac{\left( {135\sqrt{26}} \right)}{26}$ ft/sec
17.
$- \frac{5}{6}$ m/sec
19.
$240\pi$ m2/sec
21.
$\frac{1}{2\sqrt{\pi}}$ cm
23.
The area is increasing at a rate $\frac{\left( {3\sqrt{3}} \right)}{8}\ \text{ft}^{2}\text{/sec.}$
25.
The depth of the water decreases at $\frac{128}{125\pi}$ ft/min.
27.
The volume is decreasing at a rate of $\frac{\left( {25\pi} \right)}{16}\text{ft}^{3}\text{/min}.$
29.
The water flows out at rate $\frac{\left( {2\pi} \right)}{5}\ \text{m}^{3}\text{/min.}$
31.
$\frac{3}{2}$ m/sec
33.
$\frac{25}{19\pi}$ ft/min
35.
$\frac{2}{45\pi}$ ft/min
37.
The angle decreases at $\frac{400}{1681}\ \text{rad/sec}.$
39.
$100\pi\text{mi/min}$
41.
The angle is changing at a rate of $\frac{11}{25}\ \text{rad/sec}.$
43.
The distance is increasing at a rate of $62.50$ ft/sec.
45.
The distance is decreasing at a rate of $11.99$ ft/sec.
Section 4.2 Exercises
47.
$f^{\prime}(a) = 0$
49.
The linear approximation exact when $y = f(x)$ is linear or constant.
51.
$L(x) = \frac{1}{2} - \frac{1}{4}(x - 2)$
53.
$L(x) = 1$
55.
$L(x) = 0$
57.
0.02
59.
$1.9996875$
61.
$0.001593$
63.
$1;$ error, $\sim 0.00005$
65.
$0.97;$ error, $\sim 0.0006$
67.
$3 - \frac{1}{600};$ error, $\sim 4.632\ \times \ 10^{-7}$
69.
$dy = \left( {\text{cos}\mspace{2mu} x - x\mspace{2mu}\text{sin}\mspace{2mu} x} \right)dx$
71.
$dy = \left( \frac{x^{2} - 2x - 2}{{(x - 1)}^{2}} \right)dx$
73.
$dy = - \frac{1}{\left( {x + 1} \right)^{2}}dx,$ $- \frac{1}{16}$
75.
$dy = \frac{9x^{2} + 12x - 2}{2\left( {x + 1} \right)^{3\text{/}2}}dx,$ $-0.1$
77.
$dy = \left( {3x^{2} + 2 - \frac{1}{x^{2}}} \right)dx,$ $0.2$
79.
$12x\mspace{2mu} dx$
81.
$4\pi r^{2}dr$
83.
$-1.2\pi{\ \text{cm}}^{3}$
85.
$-100$ ft3
Section 4.3 Exercises
91.
Answers may vary
93.
Answers will vary
95.
No; answers will vary
97.
Since the absolute maximum is the function (output) value rather than the *x* value, the answer is no; answers will vary
99.
When $a = 0$
101.
Absolute minimum at 3; Absolute maximum at −2.2; local minima at −2, 1; local maxima at −1, 2
103.
Absolute minima at −2, 2; absolute maxima at −2.5, 2.5; minima at –2, 0, and 2 (Note: –2 and 2 are both local and absolute minima); local maxima at −1, 1
105.
Answers may vary.
107.
Answers may vary.
109.
$x = 1$
111.
None
113.
$x = 0\text{;}x = \pm 2$
115.
None
117.
$x = -1,1$
119.
Absolute maximum: $x = 4,$ $y = \frac{33}{2};$ absolute minimum: $x = 1,$ $y = 3$
121.
Absolute minimum: $x = \frac{1}{2},$ $y = 4$
123.
Absolute maximum: $x = 2\pi,$ $y = 2\pi;$ absolute minimum: $x = 0,$ $y = 0$
125.
Absolute maximum: $x = -3$ and ${y = 6;};$ absolute minimum: $-1 \leq x \leq 1,$ $y = 2$
127.
Absolute maximum: $x = \frac{\pi}{4},$ $y = \sqrt{2};$ absolute minimum: $x = \frac{5\pi}{4},$ $y = \text{−}\sqrt{2}$
129.
Absolute minimum: $x = -2,$ $y = 1$
131.
Absolute minimum: $x = -3,$ $y = -135;$ local maximum: $x = 0,$ $y = 0;$ local minimum: $x = 1,$ $y = -7$
133.
Local maximum: $x = 1 - 2\sqrt{2},$ $y = 3 - 4\sqrt{2};$ local minimum: $x = 1 + 2\sqrt{2},$ $y = 3 + 4\sqrt{2}$
135.
Absolute maximum: $x = \frac{\sqrt{2}}{2},$ $y = \frac{3}{2};$ absolute minimum: $x = - \frac{\sqrt{2}}{2},$ $y = - \frac{3}{2}$
137.
Local maximum: $x = -2,$ $y = 59;$ local minimum: $x = 1,$ $y = -130$
139.
Absolute maximum: $x = 0,$ $y = 1;$ absolute minimum: $x = -2,2,$ $y = 0$
141.
$h = \frac{9245}{49}\ \text{m,}$ $t = \frac{300}{49}\ \text{s}$
143.
The absolute minimum was in 1848, when no gold was produced.
145.
Absolute minima: $x = 0,$ $x = 2,$ $y = 1;$ local maximum at $x = 1,$ $y = 2$
147.
No maxima/minima if $a$ is odd, minimum at $x = 1$ if $a$ is even
Section 4.4 Exercises
149.
One example is $\left. f(x) = \middle| x \middle| + 3,\ -2 \leq x \leq 2 \right.$
151.
Yes, but the Mean Value Theorem still does not apply
153.
Any closed interval in $\left( \text{−}\infty,0 \right)~\text{or}~(0,\infty)$
155.
Any closed interval in $\left( \text{−}\infty,-2 \right)~\text{or}~(2,\infty)$
157.
2 points
159.
5 points
161.
$c = \frac{2\sqrt{3}}{3}$
163.
$c = \frac{1}{2},1,\frac{3}{2}$
165.
$c = 1$
167.
Not differentiable
169.
Not differentiable
171.
Yes
173.
The Mean Value Theorem does not apply since the function is discontinuous at $x = \frac{1}{4},\frac{3}{4},\frac{5}{4},\frac{7}{4}.$
175.
Yes
177.
The Mean Value Theorem does not apply; discontinuous at $x = 0.$
179.
Yes
181.
The Mean Value Theorem does not apply; not differentiable at $x = 0.$
183.
$b = \text{±}2\sqrt{c}$
185.
$c = \text{±}\frac{1}{\pi}\text{cos}^{-1}\left( \frac{\sqrt{\pi}}{2} \right),$ $c = \text{±}0.1533$
187.
The Mean Value Theorem does not apply.
189.
$\frac{1}{2\sqrt{c + 1}} - \frac{2}{c^{3}} = \frac{521}{2880};$ $c = 3.133,5.867$
191.
Yes
193.
Let $s_{1}(t)$ represent the distance of Car 1 from Stoplight 1 at time $t$. Let $s_{2}(t)$ represent the distance of Car 2 from Stoplight 1 at time $t$. Then $s_{1}(0) = 0 = s_{2}(0)$. Let $t = m$ represent the time that the cars arrive at Stoplight 2. Then $s_{1}(m) = d = s_{2}(m)$ where $d$ is the distance between the stoplights.
Define a new function $f(t) = s_{1}(t) - s_{2}(t)$. Then $f(0) = 0 = f(m)$. By the Mean Value Theorem, there is a time $0 < c < m$ such that
$$f^{\prime}(c) = \frac{f(m) - f(0)}{m - 0} = \frac{0}{m} = 0.$$
At time c,
$$f^{\prime}(c) = 0$$ $$s_{1}^{\prime}(c) - s_{2}^{\prime}(c) = 0$$ $$s_{1}^{\prime}(c) = s_{2}^{\prime}(c)$$
So, at time c, Car 1 and Car 2 are traveling at the same speed.
Section 4.5 Exercises
195.
It is not a local maximum/minimum because $f^{\prime}$ does not change sign
197.
No
199.
False; for example, $y = \sqrt{x}.$
201.
Increasing for $-2 < x < -1$ and $x > 2;$ decreasing for $x < -2$ and $-1 < x < 2$
203.
Decreasing for $x < 1,$ increasing for $x > 1$
205.
Decreasing for $-2 < x < -1$ and $1 < x < 2;$ increasing for $-1 < x < 1$ and $x < -2$ and $x > 2$
207.
a\. Increasing over $-2 < x < -1,0 < x < 1,x > 2,$ decreasing over $x < -2,$ $-1 < x < 0,1 < x < 2;$ b. maxima at $x = -1$ and $x = 1,$ minima at $x = -2$ and $x = 0$ and $x = 2$
209.
a\. Increasing over $x > 0,$ decreasing over $x < 0;$ b. Minimum at $x = 0$
211.
Concave up on all $x,$ no inflection points
213.
Concave up on all $x,$ no inflection points
215.
Concave up for $x < 0$ and $x > 1,$ concave down for $0 < x < 1,$ inflection points at $x = 0$ and $x = 1$
217.
Answers will vary
219.
Answers will vary
221.
a\. Increasing over $- \frac{\pi}{2} < x < \frac{\pi}{2},$ decreasing over $x < - \frac{\pi}{2},x > \frac{\pi}{2}$ b. Local maximum at $x = \frac{\pi}{2};$ local minimum at $x = - \frac{\pi}{2}$
223.
a\. Concave up for $x > \frac{4}{3},$ concave down for $x < \frac{4}{3}$ b. Inflection point at $x = \frac{4}{3}$
225.
a\. Increasing over $x < 0$ and $x > 4,$ decreasing over $0 < x < 4$ b. Maximum at $x = 0,$ minimum at $x = 4$ c. Concave up for $x > 2,$ concave down for $x < 2$ d. Infection point at $x = 2$
227.
a\. Increasing over $x < 0$ and $x > \frac{60}{11},$ decreasing over $0 < x < \frac{60}{11}$ b. Minimum at $x = \frac{60}{11}$, local maximum at x = 0 c. Concave down for $x < \frac{54}{11},$ concave up for $x > \frac{54}{11}$ d. Inflection point at $x = \frac{54}{11}$
229.
a\. Increasing over $x > - \frac{1}{2},$ decreasing over $x < - \frac{1}{2}$ b. Minimum at $x = - \frac{1}{2}$ c. Concave up for all $x$ d. No inflection points
231.
a\. Increases over $- \frac{1}{4} < x < \frac{3}{4},$ decreases over $x > \frac{3}{4}$ and $x < - \frac{1}{4}$ b. Minimum at $x = - \frac{1}{4},$ maximum at $x = \frac{3}{4}$ c. Concave up for $- \frac{3}{4} < x < \frac{1}{4},$ concave down for $x < - \frac{3}{4}$ and $x > \frac{1}{4}$ d. Inflection points at $x = - \frac{3}{4},x = \frac{1}{4}$
233.
a\. Increasing for all $x$ b. No local minimum or maximum c. Concave up for $x > 0,$ concave down for $x < 0$ d. Inflection point at $x = 0$
235.
a\. Increasing for all $x$ where defined b. No local minima or maxima c. Concave up for $x < 1;$ concave down for $x > 1$ d. No inflection points in domain
237.
a\. Increasing over $- \frac{\pi}{4} < x < \frac{3\pi}{4},$ decreasing over $x > \frac{3\pi}{4},x < - \frac{\pi}{4}$ b. Minimum at $x = - \frac{\pi}{4},$ maximum at $x = \frac{3\pi}{4}$ c. Concave up for $- \frac{\pi}{2} < x < \frac{\pi}{2},$ concave down for $x < - \frac{\pi}{2},x > \frac{\pi}{2}$ d. Infection points at $x = \text{±}\frac{\pi}{2}$
239.
a\. Increasing over $x > 4,$ decreasing over $0 < x < 4$ b. Minimum at $x = 4$ c. Concave up for $0 < x < 8\sqrt[3]{2},$ concave down for $x > 8\sqrt[3]{2}$ d. Inflection point at $x = 8\sqrt[3]{2}$
241.
$f > 0,f^{\prime} > 0,f^{''} < 0$
243.
$f > 0,f^{\prime} < 0,f^{''} > 0$
245.
$f > 0,f^{\prime} > 0,f^{''} > 0$
247.
True, by the Mean Value Theorem
249.
True, examine derivative
Section 4.6 Exercises
251.
$x = 1$
253.
$x = -1,x = 2$
255.
$x = 0$
257.
Yes, there is a vertical asymptote
259.
Yes, there is vertical asymptote
261.
$0$
263.
$\infty$
265.
$- \frac{1}{7}$
267.
$-2$
269.
$-4$
271.
Horizontal: none, vertical: $x = 0$
273.
Horizontal: none, vertical: $x = \text{±}2$
275.
Horizontal: none, vertical: none
277.
Horizontal: $y = 0,$ vertical: $x = \text{±}1$
279.
Horizontal: $y = 0,$ vertical: $x = 0$ and $x = -1$
281.
Horizontal: $y = 1,$ vertical: $x = 1$
283.
Horizontal: none, vertical: none
285.
Answers will vary, for example: $y = \frac{2x}{x - 1}$
287.
Answers will vary, for example: $y = \frac{4x}{x + 1}$
289.
$y = 0$
291.
$\infty$
293.
$y = 3$
295.
297.
299.
301.
303.
305.
307.
$P(0) \neq 0~{and}~Q(0) = 0$
309.
$\underset{x\rightarrow 1^{-}}{\text{lim}}f(x) = {-\infty}\ \text{and}\ \underset{x\rightarrow 1^{-}}{\text{lim}}g(x) = \infty$
Section 4.7 Exercises
311.
The critical points can be the minima, maxima, or neither.
313.
False; $y = x^{2}$ has a minimum only
315.
$h = \frac{62}{3}$ in.
317.
$1$
319.
$100\ \text{ft by}\ 100\ \text{ft}$
321.
$40\ \text{ft by}\ 40\ \text{ft}$
323.
$19.73\ \text{ft}.$
325.
$84\ \text{bpm}$
327.
$T(\theta) = \frac{40\theta}{3v} + \frac{40\mspace{2mu}\text{cos}\mspace{2mu}\theta}{v}$
329.
$v = \sqrt{\frac{b}{a}}$
331.
approximately $34\ \text{mph}$
333.
$4$
335.
$0$
337.
Maximal: $x = 5,y = 5;$ minimal: $x = 0,y = 10$ and $y = 0,x = 10$
339.
Maximal: $x = 1,y = 9;$ minimal: none
341.
$\frac{4\pi}{3\sqrt{3}}$
343.
$6$
345.
$r = 2,h = 4$
347.
$\left( {2,1} \right)$
349.
$\left( {0.8351,0.6974} \right)$
351.
$\begin{matrix}
A & = & {20\left( \frac{20}{\pi + 4} \right) - 2\left( \frac{20}{\pi + 4} \right)^{2} - \frac{1}{2}\pi\left( \frac{20}{\pi + 4} \right)^{2}}
\end{matrix}$
353.
$C(x) = 5x^{2} + \frac{32}{x}$
355.
$P(x) = \left( {50 - x} \right)\left( {800 + 25x - 50} \right)$
Section 4.8 Exercises
357.
$\infty$
359.
$\frac{1}{2a}$
361.
$\frac{1}{na^{n - 1}}$
363.
Cannot apply directly; use logarithms
365.
Cannot apply directly; rewrite as $\underset{x\rightarrow 0}{\text{lim}}x^{3}$
367.
$6$
369.
$-2$
371.
$-1$
373.
$n$
375.
$- \frac{1}{2}$
377.
$\frac{1}{2}$
379.
$1$
381.
$\frac{1}{6}$
383.
$1$
385.
$0$
387.
$0$
389.
$-1$
391.
$\infty$
393.
$0$
395.
$\frac{1}{e}$
397.
$0$
399.
$1$
401.
$0$
403.
$\text{tan}(1)$
405.
$2$
Section 4.9 Exercises
407.
$F\left( x_{n} \right) = x_{n} - \frac{x_{n}{}^{3} + 2x_{n} + 1}{3x_{n}{}^{2} + 2}$
409.
$F\left( x_{n} \right) = x_{n} - \frac{e^{x_{n}}}{e^{x_{n}}}$
411.
$|c| > 0.5$ fails, $|c| \leq 0.5$ works
413.
$c = \frac{1}{f^{\prime}\left( x_{n} \right)}$
415.
a\. $x_{1} = \frac{12}{25},x_{2} = \frac{312}{625};$ b. $x_{1} = -4,x_{2} = -40$
417.
a\. $x_{1} = 1.291,x_{2} = 0.8801;$ b. $x_{1} = 0.7071,x_{2} = 1.189$
419.
a\. $x_{1} = - \frac{26}{25},x_{2} = - \frac{1224}{625};$ b. $x_{1} = 4,x_{2} = 18$
421.
a\. $x_{1} = \frac{6}{10},x_{2} = \frac{6}{10};$ b. $x_{1} = 2,x_{2} = 2$
423.
$3.1623\ \text{or}\ - 3.1623$
425.
$0,-1\ \text{or}\ 1$
427.
$0$
429.
$0.5188\ \text{or}\ - 1.2906$
431.
$0$
433.
$4.493$
435.
$0.159,3.146$
437.
We need $f$ to be twice continuously differentiable.
439.
$x = 0$
441.
$x = -1$
443.
$x = 5.619$
445.
$x = -1.326$
447.
There is no solution to the equation.
449.
It enters a cycle.
451.
$0$
453.
$-0.3513$
455.
Newton: $11$ iterations, secant: $16$ iterations
457.
Newton: three iterations, secant: six iterations
459.
Newton: five iterations, secant: eight iterations
461.
$E = 4.071$
463.
$4.394\text{\%}$
Section 4.10 Exercises
465.
$F^{\prime}(x) = 15x^{2} + 4x + 3$
467.
$F^{\prime}(x) = 2xe^{x} + x^{2}e^{x}$
469.
$F^{\prime}(x) = e^{x}$
471.
$F(x) = e^{x} - x^{3} - \text{cos}(x) + C$
473.
$F(x) = \frac{x^{2}}{2} - x - 2\mspace{2mu}\text{cos}\left( {2x} \right) + C$
475.
$F(x) = \frac{1}{2}x^{2} + 4x^{3} + C$
477.
$F(x) = \frac{2}{5}\left( \sqrt{x} \right)^{5} + C$
479.
$F(x) = \frac{3}{2}x^{2\text{/}3} + C$
481.
$F(x) = x + \text{tan}(x) + C$
483.
$F(x) = \frac{1}{3}\text{sin}^{3}(x) + C$
485.
$F(x) = - \frac{1}{2}\mspace{2mu}\text{cot}(x) - \frac{1}{x} + C$
487.
$F(x) = \text{−}\text{sec}\mspace{2mu} x - 4\mspace{2mu}\text{csc}\mspace{2mu} x + C$
489.
$F(x) = - \frac{1}{8}e^{-4x} - \text{cos}\mspace{2mu} x + C$
491.
$\text{−}\text{cos}\mspace{2mu} x + C$
493.
$3x - \frac{2}{x} + C$
495.
$\frac{8}{3}x^{3\text{/}2} + \frac{4}{5}x^{5\text{/}4} + C$
497.
$14x - \frac{2}{x} - \frac{1}{2x^{2}} + C$
499.
$f(x) = - \frac{1}{2x^{2}} + \frac{3}{2}$
501.
$f(x) = \text{sin}\mspace{2mu} x + \text{tan}\mspace{2mu} x + 1$
503.
$f(x) = - \frac{1}{6}x^{3} - \frac{2}{x} + \frac{13}{6}$
505.
Answers may vary; one possible answer is $f(x) = e^{\text{−}x}$
507.
Answers may vary; one possible answer is $f(x) = \text{−}\text{sin}\mspace{2mu} x$
509.
$5.867$ sec
511.
$7.333$ sec
513.
$13.75$ ft/sec2
515.
$F(x) = \frac{1}{3}x^{3} + 2x$
517.
$F(x) = x^{2} - \text{cos}\mspace{2mu} x + 1$
519.
$F(x) = - \frac{1}{\left( {x + 1} \right)} + 1$
521.
True
523.
False
Review Exercises
525.
True, by Mean Value Theorem
527.
True
529.
Increasing: $\left( {-2,0} \right) \cup \left( {4,\infty} \right),$ decreasing: $\left( {\text{−}\infty,-2} \right) \cup \left( {0,4} \right)$
531.
$L(x) = \frac{17}{16} + \frac{1}{2}\left( {1 + 4\pi} \right)\left( {x - \frac{1}{4}} \right)$
533.
Critical point: $x = \frac{3\pi}{4},$ absolute minimum: $x = 0,$ absolute maximum: $x = \pi$
535.
Increasing: $\left( {-1,0} \right) \cup \left( {3,\infty} \right),$ decreasing: $\left( {\text{−}\infty,-1} \right) \cup \left( {0,3} \right),$ concave up: $\left( {\text{−}\infty,\frac{1}{3}\left( {2 - \sqrt{13}} \right)} \right) \cup \left( {\frac{1}{3}\left( {2 + \sqrt{13}} \right),\infty} \right),$ concave down: $\left( {\frac{1}{3}\left( {2 - \sqrt{13}} \right),\frac{1}{3}\left( {2 + \sqrt{13}} \right)} \right)$
537.
Increasing: $\left( {\frac{1}{4},\infty} \right),$ decreasing: $\left( {0,\frac{1}{4}} \right),$ concave up: $\left( {0,\infty} \right),$ concave down: nowhere
539.
$3$
541.
$- \frac{1}{\pi}$
543.
$x_{1} = -1,x_{2} = -1$
545.
$F(x) = \frac{2x^{3\text{/}2}}{3} + \frac{1}{x} + C$
547.
Inflection points: none; critical points: $x = - \frac{1}{3};$ zeros: none; vertical asymptotes: $x = -1,$ $x = 0;$ horizontal asymptote: $y = 0$
549.
The height is decreasing at a rate of $0.125$ m/sec
551.
$x = \sqrt{ab}$ feet
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- Publisher/website: OpenStax
- Book title: Calculus Volume 1
- Publication date: Mar 30, 2016
- Location: Houston, Texas
- Book URL: https://openstax.org/books/calculus-volume-1/pages/1-introduction
- Section URL: https://openstax.org/books/calculus-volume-1/pages/chapter-4
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