# 1.7E: Exercises

- Page ID
- 18544

## Calculus of the Hyperbolic Functions

### Exercise \(\PageIndex{1}\)

Find expressions for \(\displaystyle oshx+sinhx\) and \(\displaystyle coshx−sinhx.\) Use a calculator to graph these functions and ensure your expression is correct.

**Answer**-
\(\displaystyle e^x\) and \(\displaystyle e^{−x}\)

### Exercise \(\PageIndex{2}\)

From the definitions of \(\displaystyle cosh(x)\) and \(\displaystyle sinh(x)\), find their antiderivatives.

**Answer**-
Add texts here. Do not delete this text first.

### Exercise \(\PageIndex{3}\)

Show that \(\displaystyle cosh(x)\) and \(\displaystyle sinh(x)\) satisfy \(\displaystyle y''=y\).

**Answer**-
Answers may vary

### Exercise \(\PageIndex{4}\)

Use the quotient rule to verify that \(\displaystyle tanh(x)'=sech^2(x).\)

**Answer**-
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### Exercise \(\PageIndex{5}\)

Derive \(\displaystyle cosh^2(x)+sinh^2(x)=cosh(2x)\) from the definition.

**Answer**-
Answers may vary

### Exercise \(\PageIndex{6}\)

Take the derivative of the previous expression to find an expression for \(\displaystyle sinh(2x)\).

**Answer**-
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### Exercise \(\PageIndex{7}\)

Prove \(\displaystyle sinh(x+y)=sinh(x)cosh(y)+cosh(x)sinh(y)\) by changing the expression to exponentials.

**Answer**-
Answers may vary

### Exercise \(\PageIndex{8}\)

Take the derivative of the previous expression to find an expression for \(\displaystyle cosh(x+y).\)

**Answer**-
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**For the following exercises, find the derivatives of the given functions and graph along with the function to ensure your answer is correct.**

### Exercise \(\PageIndex{9}\)

\(\displaystyle cosh(3x+1)\)

**Answer**-
\(\displaystyle 3sinh(3x+1)\)

### Exercise \(\PageIndex{10}\)

\(\displaystyle sinh(x^2)\)

**Answer**-
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### Exercise \(\PageIndex{11}\)

\(\displaystyle \frac{1}{cosh(x)}\)

**Answer**-
\(\displaystyle −tanh(x)sech(x)\)

### Exercise \(\PageIndex{12}\)

\(\displaystyle sinh(ln(x))\)

**Answer**-
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### Exercise \(\PageIndex{13}\)

\(\displaystyle cosh^2(x)+sinh^2(x)\)

**Answer**-
\(\displaystyle 4cosh(x)sinh(x)\)

### Exercise \(\PageIndex{14}\)

\(\displaystyle cosh^2(x)−sinh^2(x)\)

**Answer**-
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### Exercise \(\PageIndex{15}\)

\(\displaystyle tanh(\sqrt{x^2+1})\)

**Answer**-
\(\displaystyle \frac{xsech^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}\)

### Exercise \(\PageIndex{16}\)

\(\displaystyle \frac{1+tanh(x)}{1−tanh(x)}\)

**Answer**-
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### Exercise \(\PageIndex{17}\)

\(\displaystyle sinh^6(x)\)

**Answer**-
\(\displaystyle 6sinh^5(x)cosh(x)\)

### Exercise \(\PageIndex{18}\)

\(\displaystyle ln(sech(x)+tanh(x))\)

**Answer**-
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**For the following exercises, find the antiderivatives for the given functions.**

### Exercise \(\PageIndex{19}\)

\(\displaystyle cosh(2x+1)\)

**Answer**-
\(\displaystyle \frac{1}{2}sinh(2x+1)+C\)

### Exercise \(\PageIndex{20}\)

\(\displaystyle tanh(3x+2)\)

**Answer**-
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### Exercise \(\PageIndex{21}\)

\(\displaystyle xcosh(x^2)\)

**Answer**-
\(\displaystyle \frac{1}{2}sinh^2(x^2)+C\)

### Exercise \(\PageIndex{22}\)

\(\displaystyle 3x^3tanh(x^4)\)

**Answer**-
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### Exercise \(\PageIndex{23}\)

\(\displaystyle cosh^2(x)sinh(x)\)

**Answer**-
\(\displaystyle \frac{1}{3}cosh^3(x)+C\)

### Exercise \(\PageIndex{24}\)

\(\displaystyle tanh^2(x)sech^2(x)\)

**Answer**-
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### Exercise \(\PageIndex{25}\)

\(\displaystyle \frac{sinh(x)}{1+cosh(x)}\)

**Answer**-
\(\displaystyle ln(1+cosh(x))+C\)

### Exercise \(\PageIndex{26}\)

\(\displaystyle coth(x)\)

**Answer**-
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### Exercise \(\PageIndex{27}\)

\(\displaystyle cosh(x)+sinh(x)\)

**Answer**-
\(\displaystyle cosh(x)+sinh(x)+C\)

### Exercise \(\PageIndex{28}\)

\(\displaystyle (cosh(x)+sinh(x))^n\)

**Answer**-
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**For the following exercises, find the derivatives for the functions.**

### Exercise \(\PageIndex{29}\)

\(\displaystyle tanh^{−1}(4x)\)

**Answer**-
\(\displaystyle \frac{4}{1−16x^2}\)

### Exercise \(\PageIndex{30}\)

\(\displaystyle sinh^{−1}(x^2)\)

**Answer**-
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### Exercise \(\PageIndex{31}\)

\(\displaystyle sinh^{−1}(cosh(x))\)

**Answer**-
\(\displaystyle \frac{sinh(x)}{\sqrt{cosh^2(x)+1}}\)

### Exercise \(\PageIndex{32}\)

\(\displaystyle cosh^{−1}(x^3)\)

**Answer**-
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### Exercise \(\PageIndex{33}\)

\(\displaystyle tanh^{−1}(cos(x))\)

**Answer**-
\(\displaystyle −csc(x)\)

### Exercise \(\PageIndex{34}\)

\(\displaystyle e^{sinh^{−1}(x)}\)

**Answer**-
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### Exercise \(\PageIndex{35}\)

\(\displaystyle ln(tanh^{−1}(x))\)

**Answer**-
\(\displaystyle −\frac{1}{(x^2−1)tanh^{−1}(x)}\)

**For the following exercises, find the antiderivatives for the functions.**

### Exercise \(\PageIndex{36}\)

\(\displaystyle ∫\frac{dx}{4−x^2}\)

**Answer**-
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### Exercise \(\PageIndex{37}\)

\(\displaystyle ∫\frac{dx}{a^2−x^2}\)

**Answer**-
\(\displaystyle \frac{1}{a}tanh^{−1}(\frac{x}{a})+C\)

### Exercise \(\PageIndex{38}\)

\(\displaystyle ∫\frac{dx}{\sqrt{x^2+1}}\)

**Answer**-
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### Exercise \(\PageIndex{39}\)

\(\displaystyle ∫\frac{xdx}{\sqrt{x^2+1}}\)

**Answer**-
\(\displaystyle \sqrt{x^2+1}+C\)

### Exercise \(\PageIndex{40}\)

\(\displaystyle ∫−\frac{dx}{x\sqrt{1−x^2}}\)

**Answer**-
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### Exercise \(\PageIndex{41}\)

\(\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−1}}\)

**Answer**-
\(\displaystyle cosh^{−1}(e^x)+C\)

### Exercise \(\PageIndex{42}\)

\(\displaystyle ∫−\frac{2x}{x^4−1}\)

**Answer**-
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**For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation \(\displaystyle dv/dt=g−v^2\).**

### Exercise \(\PageIndex{43}\)

Show that \(\displaystyle v(t)=\sqrt{g}tanh(\sqrt{gt})\) satisfies this equation.

**Answer**-
Answers may vary

### Exercise \(\PageIndex{44}\)

Derive the previous expression for \(\displaystyle v(t)\) by integrating \(\displaystyle \frac{dv}{g−v^2}=dt\).

**Answer**-
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### Exercise \(\PageIndex{45}\)

Estimate how far a body has fallen in \(\displaystyle 12\)seconds by finding the area underneath the curve of \(\displaystyle v(t)\).

**Answer**-
\(\displaystyle 37.30\)

**For the following exercises, use this scenario: A cable hanging under its own weight has a slope \(\displaystyle S=dy/dx\) that satisfies \(\displaystyle dS/dx=c\sqrt{1+S^2}\). The constant \(\displaystyle c\) is the ratio of cable density to tension.**

### Exercise \(\PageIndex{46}\)

Show that \(\displaystyle S=sinh(cx)\) satisfies this equation.

**Answer**-
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### Exercise \(\PageIndex{47}\)

Integrate \(\displaystyle dy/dx=sinh(cx)\) to find the cable height \(\displaystyle y(x)\) if \(\displaystyle y(0)=1/c\).

**Answer**-
\(\displaystyle y=\frac{1}{c}cosh(cx)\)

### Exercise \(\PageIndex{48}\)

Sketch the cable and determine how far down it sags at \(\displaystyle x=0\).

**Answer**-
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**For the following exercises, solve each problem.**

### Exercise \(\PageIndex{49}\)

A chain hangs from two posts \(\displaystyle 2\)m apart to form a catenary described by the equation \(\displaystyle y=2cosh(x/2)−1\). Find the slope of the catenary at the left fence post.

**Answer**-
\(\displaystyle −0.521095\)

### Exercise \(\PageIndex{50}\)

A chain hangs from two posts four meters apart to form a catenary described by the equation \(\displaystyle y=4cosh(x/4)−3.\) Find the total length of the catenary (arc length).

**Answer**-
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### Exercise \(\PageIndex{51}\)

A high-voltage power line is a catenary described by \(\displaystyle y=10cosh(x/10)\). Find the ratio of the area under the catenary to its arc length. What do you notice?

**Answer**-
\(\displaystyle 10\)

### Exercise \(\PageIndex{52}\)

A telephone line is a catenary described by \(\displaystyle y=acosh(x/a).\) Find the ratio of the area under the catenary to its arc length. Does this confirm your answer for the previous question?

**Answer**-
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### Exercise \(\PageIndex{53}\)

Prove the formula for the derivative of \(\displaystyle y=sinh^{−1}(x)\) by differentiating \(\displaystyle x=sinh(y).\)

(Hint: Use hyperbolic trigonometric identities.)

**Answer**-
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### Exercise \(\PageIndex{54}\)

Prove the formula for the derivative of \(\displaystyle y=cosh^{−1}(x)\) by differentiating \(\displaystyle x=cosh(y).\)

(Hint: Use hyperbolic trigonometric identities.)

**Answer**-
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### Exercise \(\PageIndex{55}\)

Prove the formula for the derivative of \(\displaystyle y=sech^{−1}(x)\) by differentiating \(\displaystyle x=sech(y).\)

(Hint: Use hyperbolic trigonometric identities.)

**Answer**-
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### Exercise \(\PageIndex{56}\)

Prove that \(\displaystyle cosh(x)+sinh(x))^n=cosh(nx)+sinh(nx).\)

**Answer**-
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### Exercise \(\PageIndex{57}\)

Prove the expression for \(\displaystyle sinh^{−1}(x).\) Multiply \(\displaystyle x=sinh(y)=(1/2)(e^y−e^{−y})\) by \(\displaystyle 2e^y\) and solve for \(\displaystyle y\). Does your expression match the textbook?

**Answer**-
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### Exercise \(\PageIndex{58}\)

Prove the expression for \(\displaystyle cosh^{−1}(x).\) Multiply \(\displaystyle x=cosh(y)=(1/2)(e^y−e^{−y})\) by \(\displaystyle 2e^y\) and solve for \(\displaystyle y\). Does your expression match the textbook?

**Answer**-
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