Chapter 7
> 来源: OpenStax《Calculus Volume 2》| 原页: https://openstax.org/books/calculus-volume-2/pages/chapter-7
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Chapter 7
Calculus Volume 2Chapter 7
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Chapter 7
Checkpoint
7.1
7.2
$x = 2 + \frac{3}{y + 1},$ or $y = -1 + \frac{3}{x - 2}.$ This equation describes a portion of a rectangular hyperbola centered at $\left( {2,-1} \right).$
7.3
One possibility is $x(t) = t,\quad y(t) = t^{2} + 2t.$ Another possibility is $x(t) = 2t - 3,\quad y(t) = \left( {2t - 3} \right)^{2} + 2\left( {2t - 3} \right) = 4t^{2} - 8t + 3.$
There are, in fact, an infinite number of possibilities.
7.4
$x^{\prime}(t) = 2t - 4$ and $y^{\prime}(t) = 6t^{2} - 6,$ so $\frac{dy}{dx} = \frac{6t^{2} - 6}{2t - 4} = \frac{3t^{2} - 3}{t - 2}.$
This expression is undefined when $t = 2$ and equal to zero when $t = \pm 1.$
7.5
The equation of the tangent line is $y = 24x + 100.$
7.6
$\frac{d^{2}y}{dx^{2}} = \frac{3t^{2} - 12t + 3}{2\left( {t - 2} \right)^{3}}.$ Critical points $\left( {5,4} \right),\left( {-3,-4} \right),\text{and}\ \left( {-4,4} \right).$
7.7
$A = 3\pi$ (Note that the integral formula actually yields a negative answer. This is due to the fact that $x(t)$ is a decreasing function over the interval $\left\lbrack {0,2\pi} \right\rbrack;$ that is, the curve is traced from right to left.)
7.8
$s = 2\left( {10^{3\text{/}2} - 2^{3\text{/}2}} \right) \approx 57.589$
7.9
$A = \frac{\pi\left( {494\sqrt{13} + 128} \right)}{1215}$
7.10
$\left( {8\sqrt{2},\frac{5\pi}{4}} \right)$ and $\left( {-2,2\sqrt{3}} \right)$
7.11
7.12
The name of this shape is a cardioid, which we will study further later in this section.
7.13
$y = x^{2},$ which is the equation of a parabola opening upward.
7.14
Symmetric with respect to the polar axis.
7.15
$A = 3\pi\text{/}2$
7.16
$A = \frac{4\pi}{3} + 2\sqrt{3}$
7.17
$s = 3\pi$
7.18
$x = 2\left( {y + 3} \right)^{2} - 2$
7.19
$\frac{\left( {x + 1} \right)^{2}}{16} + \frac{\left( {y - 2} \right)^{2}}{9} = 1$
7.20
$\frac{\left( {y + 2} \right)^{2}}{9} - \frac{\left( {x - 1} \right)^{2}}{4} = 1.$ This is a vertical hyperbola. Asymptotes $y = -2 \pm \frac{3}{2}\left( {x - 1} \right).$
7.21
$e = \frac{c}{a} = \frac{\sqrt{74}}{7} \approx 1.229$
7.22
Here $e = 0.8$ and $p = 5.$ This conic section is an ellipse.
7.23
The conic is a hyperbola and the angle of rotation of the axes is $\theta = 22.5\text{°}.$
Section 7.1 Exercises
1.
orientation: bottom to top
3.
orientation: left to right
5.
$y = \frac{x^{2}}{4} + 1$
7.
9.
11.
13.
15.
Asymptotes are $y = x$ and $y = \text{−}x$
17.
19.
21.
$y = \frac{\sqrt{x + 1}}{2}$; domain: $x \in \left\lbrack {–1,\infty} \right).$
23.
$\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1;$ domain $x \in \left\lbrack {-4,4} \right\rbrack.$
25.
$y = 3x + 2;$ domain: all real numbers.
27.
${(x - 1)}^{2} + {(y - 3)}^{2} = 1;$ domain: $x \in \left\lbrack {0,2} \right\rbrack.$
29.
$y = \sqrt{x^{2} - 1};$ domain: $x \in \left( {-\infty,-1} \right\rbrack.$
31.
$y^{2} = \frac{1 - x}{2};$ domain: $x \in \lbrack-1,1\rbrack.$
33.
$y = \text{ln}\ x;$ domain: $x \in \lbrack 1,\infty).$
35.
$y = \text{ln}\ x;$ domain: $x \in (0,\infty).$
37.
$x^{2} + y^{2} = 4;$ domain: $x \in \lbrack-2,2\rbrack.$
39.
line
41.
parabola
43.
circle
45.
ellipse
47.
hyperbola
51.
The equations represent a cycloid.
53.
55.
22,092 meters at approximately 51 seconds.
57.
59.
61.
Section 7.2 Exercises
63.
0
65.
$\frac{-3}{5}$
67.
$\text{Slope} = 0;$ $y = 8.$
69.
Slope is undefined; $x = 2.$
71.
$\tan~t~ = ~ - 2\left( \frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}} \right),~\left( \frac{4}{\sqrt{5}},\frac{-8}{\sqrt{5}} \right)$, $\left( {\frac{- 4}{\sqrt{5}},\frac{8}{\sqrt{5}}} \right)$.
73.
No points possible; undefined expression.
75.
$y = \text{−}\left( \frac{4}{e} \right)x + 5$
77.
$y = –2x + 3$
79.
$\frac{\pi}{4},\frac{5\pi}{4},\frac{3\pi}{4},\frac{7\pi}{4}$
81.
$\frac{dy}{dx} = \text{−}\text{tan}(t)$
83.
$\frac{dy}{dx} = \frac{3}{4}$ and $\frac{d^{2}y}{dx^{2}} = 0,$ so the curve is neither concave up nor concave down at $t = 3.$ Therefore the graph is linear and has a constant slope but no concavity.
85.
$\frac{dy}{dx} = 4,\frac{d^{2}y}{dx^{2}} = -6\sqrt{3};$ the curve is concave down at $\theta = \frac{\pi}{6}.$
87.
No horizontal tangents. Vertical tangents at $(1,0),(-1,0).$
89.
$\text{−}\text{sec}^{3}\left( {\pi t} \right)$
91.
Horizontal $\left( {0,-9} \right);$ vertical $\left( {\text{±}2,-6} \right).$
93.
1
95.
0
97.
4
99.
Concave up on $t > 0.$
101.
$\frac{e^{\frac{\pi}{2}}–1}{2}$
103.
$\frac{3\pi}{2}$
105.
$6\pi a^{2}$
107.
$2\pi ab$
109.
$\frac{1}{3}\left( {2\sqrt{2} - 1} \right)$
111.
$7.075$
113.
$6a$
115.
$6\sqrt{2}$
119.
$\frac{2\pi\left( {247\sqrt{13} + 64} \right)}{1215}$
121.
59.101
123.
$\frac{8\pi}{3}\left( {17\sqrt{17} - 1} \right)$
Section 7.3 Exercises
125.
127.
129.
131.
133.
$B\begin{array}{ll}
\left( {3,\frac{\text{−}\pi}{3}} \right) & {B\left( {-3,\frac{2\pi}{3}} \right)}
\end{array}$
135.
$D\left( {5,\frac{7\pi}{6}} \right)D\left( {-5,\frac{\pi}{6}} \right)$
137.
$\left( 5,~5.356 \right)~\left( –5,~2.214 \right)$
139.
$\left( 10,~2.214 \right)\left( –10,~5.356 \right)$
141.
$\left( {2\sqrt{3},\frac{11\pi}{6}} \right)\left( {-2\sqrt{3},\frac{5\pi}{6}} \right)$
143.
$\left( \begin{array}{ll}
{\text{−}\sqrt{3},} & -1
\end{array} \right)$
145.
$\left( \begin{array}{ll}
{- \frac{\sqrt{3}}{2},} & \frac{-1}{2}
\end{array} \right)$
147.
$\left( \begin{array}{ll}
{0,} & 0
\end{array} \right)$
149.
Symmetry with respect to the *x*-axis, *y*-axis, and origin.
151.
Symmetric with respect to *x*-axis only.
153.
Symmetry with respect to *x*-axis only.
155.
Line $y = x$
157.
$y = 1$
159.
Hyperbola; polar form $r^{2}\text{cos}(2\theta) = 16$ or $r^{2} = 16\ \text{sec}\ (2\theta).$
161.
$r = \frac{2}{3\ \text{cos}\ \theta - \text{sin}\ \theta}$
163.
$x^{2} + y^{2} = 4y$
165.
$x\ \text{tan}\sqrt{x^{2} + y^{2}} = y$
167.
*y*-axis symmetry
169.
*y*-axis symmetry
171.
*x*- and *y*-axis symmetry and symmetry about the pole
173.
*x*-axis symmetry
175.
*x*- and *y*-axis symmetry and symmetry about the pole
177.
no symmetry
179.
a line
181.
183.
185.
187.
Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.
Section 7.4 Exercises
189.
$\frac{9}{2}{\int_{0}^{\pi}{\text{sin}^{2}\theta\ d\theta}}$
191.
$32{\int_{0}^{\pi\text{/}2}{\text{sin}^{2}(2\theta)d\theta}}$
193.
$\frac{1}{2}{\int_{\pi}^{2\pi}{\left( {1 - \text{sin}\ \theta} \right)^{2}d\theta}}$
195.
${\int_{\text{sin}^{-1}{({2\text{/}3})}}^{\pi\text{/}2}\left( {2 - 3\ \text{sin}\ \theta} \right)^{2}}d\theta$
197.
${\int_{\pi\text{/}3}^{\pi}{\left( {1 - 2\ \text{cos}\ \theta} \right)^{2}d\theta - {\int_{0}^{\pi\text{/}3}\left( {1 - 2\ \text{cos}\ \theta} \right)^{2}}}}d\theta$
199.
$4{\int_{0}^{\pi\text{/}3}{d\theta + 16{\int_{\pi\text{/}3}^{\pi\text{/}2}\left( {\text{cos}^{2}\theta} \right)}}}d\theta$
201.
$9\pi$
203.
$\frac{9\pi}{4}$
205.
$\frac{9\pi}{8}$
207.
$\frac{18\pi - 27\sqrt{3}}{2}$
209.
$\frac{4}{3}\left( {4\pi - 3\sqrt{3}} \right)$
211.
$\frac{3}{2}\left( {4\pi - 3\sqrt{3}} \right)$
213.
$2\pi - 4$
215.
${\int_{0}^{2\pi}\sqrt{\left( {1 + \text{sin}\ \theta} \right)^{2} + \text{cos}^{2}\theta}}d\theta$
217.
$\sqrt{2}{\int_{0}^{1}{e^{\theta}d\theta}}$
219.
$\frac{\sqrt{10}}{3}\left( {e^{6} - 1} \right)$
221.
32
223.
6.238
225.
2
227.
4.39
229.
$A = \pi\left( \frac{\sqrt{2}}{2} \right)^{2} = \frac{\pi}{2}\ \text{and}\ \frac{1}{2}{\int_{0}^{\pi}\left( {1 + 2\ \text{sin}\ \theta\ \text{cos}\ \theta} \right)}d\theta = \frac{\pi}{2}$
231.
$C = 2\pi\left( \frac{3}{2} \right) = 3\pi\ \text{and}\ {\int_{0}^{\pi}3}d\theta = 3\pi$
233.
$C = 2\pi(5) = 10\pi\ \text{and}\ {\int_{0}^{\pi}10}\ d\theta = 10\pi$
235.
$\frac{dy}{dx} = \frac{f^{\prime}(\theta)\text{sin}\ \theta + f(\theta)\ \text{cos}\ \theta}{f^{\prime}(\theta)\ \text{cos}\ \theta - f(\theta)\text{sin}\ \theta}$
237.
The slope is $\frac{1}{\sqrt{3}}.$
239.
The slope is 0.
241.
At $\left( {4,0} \right),$ the slope is undefined. At $\left( {-4,\frac{\pi}{2}} \right),$ the slope is 0.
243.
The slope is undefined at $\frac{4 + \pi}{4 - \pi}$
245.
Slope = −1.
247.
Slope is $\frac{-2}{\pi}.$
249.
Calculator answer: −0.836.
251.
Horizontal tangent at $\left( {\text{±}\sqrt{2},\frac{\pi}{6}} \right),$ $\left( {\text{±}\sqrt{2}, - \frac{\pi}{6}} \right).$
253.
Horizontal tangents at $\frac{\pi}{2},\frac{7\pi}{6},\frac{11\pi}{6}.$ Vertical tangents at $\frac{\pi}{6},\frac{5\pi}{6}$ and also at the pole $(0,0).$
Section 7.5 Exercises
255.
$y^{2} = 16x$
257.
$x^{2} = 2y$
259.
$x^{2} = -4\left( {y - 3} \right)$
261.
$\left( {x + 3} \right)^{2} = 8\left( {y - 3} \right)$
263.
$\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1$
265.
$\frac{x^{2}}{13} + \frac{y^{2}}{4} = 1$
267.
$\frac{\left( {y - 1} \right)^{2}}{16} + \frac{\left( {x + 3} \right)^{2}}{12} = 1$
269.
$\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1$
271.
$\frac{x^{2}}{25} - \frac{y^{2}}{11} = 1$
273.
$\frac{x^{2}}{7} - \frac{y^{2}}{9} = 1$
275.
$\frac{\left( {y + 2} \right)^{2}}{4} - \frac{\left( {x + 2} \right)^{2}}{32} = 1$
277.
$\frac{x^{2}}{4} - \frac{y^{2}}{32} = 1$
279.
$e = 1,$ parabola
281.
$e = \frac{1}{2},$ ellipse
283.
$e = 3,$ hyperbola
285.
$r = \frac{4}{5 + \text{cos}\ \theta}$
287.
$r = \frac{4}{1 + 2\ \text{sin}\ \theta}$
289.
291.
293.
295.
297.
299.
301.
303.
305.
307.
Hyperbola
309.
Ellipse
311.
Ellipse
313.
At the point 2.25 feet above the vertex.
315.
0.5625 feet
317.
Length is 96 feet and height is approximately 26.53 feet.
319.
$r = \frac{2.616}{1 + 0.995\ \text{cos}\ \theta}$
321.
$r = \frac{5.192}{1 + 0.0484\ \text{cos}\ \theta}$
Review Exercises
323.
True.
325.
False. Imagine $y = t + 1,$ $x = \text{−}t + 1.$
327.
$y = 1 - x^{3}$
329.
$\frac{x^{2}}{16} + {(y - 1)}^{2} = 1$
331.
Symmetric about polar axis
333.
$r^{2} = \frac{4}{\text{sin}^{2}\theta - \text{cos}^{2}\theta}$
335.
$y = \frac{3\sqrt{2}}{2} + \frac{1}{5}\left( {x + \frac{3\sqrt{2}}{2}} \right)$
337.
$\frac{e^{2}}{2}$
339.
$9\sqrt{10}$
341.
$\left( {y + 5} \right)^{2} = -8x + 32$
343.
$\frac{\left( {y + 1} \right)^{2}}{16} - \frac{\left( {x + 2} \right)^{2}}{9} = 1$
345.
$e = \frac{2}{3},$ ellipse
347.
$\frac{y^{2}}{19.03^{2}} + \frac{x^{2}}{19.63^{2}} = 1,$ $e = 0.2447$
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- Book title: Calculus Volume 2
- Publication date: Mar 30, 2016
- Location: Houston, Texas
- Book URL: https://openstax.org/books/calculus-volume-2/pages/1-introduction
- Section URL: https://openstax.org/books/calculus-volume-2/pages/chapter-7
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