CalculusVolume2-SASG-02-09
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter 2
Applications of Integration
2.9 Calculus of the Hyperbolic Functions
Section Exercises
377.
[T] Find expressions for and Use a calculator to graph
these functions and ensure your expression is correct.
Answer:
379.
Show that
and satisfy
Answer: Answers may vary
381.
Derive from the definition.
Answer: Answers may vary
383.
Prove by changing the expression to exponentials.
Answer: Answers may vary
For the following exercises, find the derivatives of the given functions and graph along with
the function to ensure your answer is correct.
385.
[T]
Answer: 387.
[T]
Answer: 389.
[T]
Answer: 391.
[T]
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Answer: 393.
[T]
Answer: For the following exercises, find the antiderivatives for the given functions.
395.
Answer: 397.
Answer:
399.
Answer: 401.
Answer: ln
(
1
+
cosh
(
x
)
)
+
C
403.
Answer: For the following exercises, find the derivatives for the functions.
405.
Answer: 4
1
−
16
x
2
OpenStax Calculus Volume 2
Student Answer and Solution Guide
407.
Answer:
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
409.
Answer: 411.
Answer: For the following exercises, find the antiderivatives for the functions.
413.
Answer: 415.
Answer:
417.
Answer: For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation 419.
Show that satisfies this equation.
Answer: Answers may vary.
421.
[T] Estimate how far a body has fallen in seconds by finding the area underneath the curve of Answer:
OpenStax Calculus Volume 2
Student Answer and Solution Guide
For the following exercises, use this scenario: A cable hanging under its own weight has a slope
that satisfies The constant is the ratio of cable density to
tension.
423.
Integrate to find the cable height if Answer: For the following exercises, solve each problem.
425.
[T] A chain hangs from two posts m apart to form a catenary described by the equation Find the slope of the catenary at the left fence post.
Answer: 427.
[T] A high-voltage power line is a catenary described by Find the ratio of the area under the catenary to its arc length. What do you notice?
Answer: 429.
Prove the formula for the derivative of by differentiating (
Hint:
Use hyperbolic trigonometric identities.)
Answer: This is a proof; therefore, no answer is provided.
431.
Prove the formula for the derivative of by differentiating (
Hint:
Use hyperbolic trigonometric identities.)
Answer: This is a proof; therefore, no answer is provided.
433.
Prove the expression for Multiply by and solve for Does your expression match the textbook?
Answer: This is a proof; therefore, no answer is provided.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
Chapter Review Exercises
True or False
. Justify your answer with a proof or a counterexample.
435.
The amount of work to pump the water out of a half-full cylinder is half the amount of work to pump the water out of the full cylinder.
Answer: False
437.
The disk method can be used in any situation in which the washer method is successful at
finding the volume of a solid of revolution.
Answer: False
For the following exercises, use the requested method to determine the volume of the solid.
439.
The volume that has a base of the ellipse and cross-sections of an equilateral triangle perpendicular to the
Use the method of slicing.
Answer: 441.
and rotated around the y
-axis using the washer method
Answer: 162
π
5
For the following exercises, find
a.
the area of the region, b.
the volume of the solid when rotated around the x
-axis, and c.
the volume of the solid when rotated around the y
-axis. Use whichever method seems most appropriate to you.
443.
Answer: a.
b.
c. 445.
[T]
Answer: a.
b.
c.
447.
and Answer: a. b. c. 31
π
6
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OpenStax Calculus Volume 2
Student Answer and Solution Guide
449.
Find the mass of on a disk centered at the origin with radius
.
Answer:
451.
Find the mass and the center of mass of on the region bounded by and
Answer: Mass: , center of mass:
For the following exercises, find the requested arc lengths.
453.
The length of for from
to Answer:
√
17
+
1
8
ln
(
33
+
8
√
17
)
For the following exercises, find the surface area and volume when the given curves are revolved around the specified axis.
455.
The loudspeaker created by revolving from
to around the x
-axis.
Answer: Volume: surface area: For the following exercises, consider the Karun-3 dam in Iran. Its shape can be approximated as an isosceles triangle with height m and width m. Assume the current depth of the water is m. The density of water is kg/m
457.
You are a crime scene investigator attempting to determine the time of death of a victim. It is noon and outside and the temperature of the body is
F. You know the cooling constant is When did the victim die, assuming that a human’s temperature is
?
Answer: a.m.
OpenStax Calculus Volume 2
Student Answer and Solution Guide
For the following exercises, consider the catenoid, the only solid of revolution that has a minimal surface, or zero mean curvature. A catenoid in nature can be found when stretching soap between two rings.
459.
Find the volume of the catenoid from
that is created by rotating this curve around the x
-axis, as shown here.
Answer: This file is copyright 2016, Rice University. All Rights Reserved.