
# 1.7E: Exercises


## 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.

$$\displaystyle e^x$$ and $$\displaystyle e^{−x}$$

Exercise $$\PageIndex{2}$$

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

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Exercise $$\PageIndex{3}$$

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

Exercise $$\PageIndex{4}$$

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

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Exercise $$\PageIndex{5}$$

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

Exercise $$\PageIndex{6}$$

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

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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.

Exercise $$\PageIndex{8}$$

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

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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 $$\PageIndex{9}$$

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

$$\displaystyle 3sinh(3x+1)$$

Exercise $$\PageIndex{10}$$

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

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Exercise $$\PageIndex{11}$$

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

$$\displaystyle −tanh(x)sech(x)$$

Exercise $$\PageIndex{12}$$

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

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Exercise $$\PageIndex{13}$$

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

$$\displaystyle 4cosh(x)sinh(x)$$

Exercise $$\PageIndex{14}$$

$$\displaystyle cosh^2(x)−sinh^2(x)$$

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Exercise $$\PageIndex{15}$$

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

$$\displaystyle \frac{xsech^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}$$

Exercise $$\PageIndex{16}$$

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

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Exercise $$\PageIndex{17}$$

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

$$\displaystyle 6sinh^5(x)cosh(x)$$

Exercise $$\PageIndex{18}$$

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

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

Exercise $$\PageIndex{19}$$

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

$$\displaystyle \frac{1}{2}sinh(2x+1)+C$$

Exercise $$\PageIndex{20}$$

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

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Exercise $$\PageIndex{21}$$

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

$$\displaystyle \frac{1}{2}sinh^2(x^2)+C$$

Exercise $$\PageIndex{22}$$

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

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Exercise $$\PageIndex{23}$$

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

$$\displaystyle \frac{1}{3}cosh^3(x)+C$$

Exercise $$\PageIndex{24}$$

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

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Exercise $$\PageIndex{25}$$

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

$$\displaystyle ln(1+cosh(x))+C$$

Exercise $$\PageIndex{26}$$

$$\displaystyle coth(x)$$

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Exercise $$\PageIndex{27}$$

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

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

Exercise $$\PageIndex{28}$$

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

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For the following exercises, find the derivatives for the functions.

Exercise $$\PageIndex{29}$$

$$\displaystyle tanh^{−1}(4x)$$

$$\displaystyle \frac{4}{1−16x^2}$$

Exercise $$\PageIndex{30}$$

$$\displaystyle sinh^{−1}(x^2)$$

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Exercise $$\PageIndex{31}$$

$$\displaystyle sinh^{−1}(cosh(x))$$

$$\displaystyle \frac{sinh(x)}{\sqrt{cosh^2(x)+1}}$$

Exercise $$\PageIndex{32}$$

$$\displaystyle cosh^{−1}(x^3)$$

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Exercise $$\PageIndex{33}$$

$$\displaystyle tanh^{−1}(cos(x))$$

$$\displaystyle −csc(x)$$

Exercise $$\PageIndex{34}$$

$$\displaystyle e^{sinh^{−1}(x)}$$

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Exercise $$\PageIndex{35}$$

$$\displaystyle ln(tanh^{−1}(x))$$

$$\displaystyle −\frac{1}{(x^2−1)tanh^{−1}(x)}$$

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

Exercise $$\PageIndex{36}$$

$$\displaystyle ∫\frac{dx}{4−x^2}$$

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Exercise $$\PageIndex{37}$$

$$\displaystyle ∫\frac{dx}{a^2−x^2}$$

$$\displaystyle \frac{1}{a}tanh^{−1}(\frac{x}{a})+C$$

Exercise $$\PageIndex{38}$$

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

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Exercise $$\PageIndex{39}$$

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

$$\displaystyle \sqrt{x^2+1}+C$$

Exercise $$\PageIndex{40}$$

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

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Exercise $$\PageIndex{41}$$

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

$$\displaystyle cosh^{−1}(e^x)+C$$

Exercise $$\PageIndex{42}$$

$$\displaystyle ∫−\frac{2x}{x^4−1}$$

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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.

Exercise $$\PageIndex{44}$$

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

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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 $$\displaystyle v(t)$$.

$$\displaystyle 37.30$$

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.

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Exercise $$\PageIndex{47}$$

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

$$\displaystyle y=\frac{1}{c}cosh(cx)$$

Exercise $$\PageIndex{48}$$

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

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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.

$$\displaystyle −0.521095$$

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).

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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?

$$\displaystyle 10$$

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?

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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.)

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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.)

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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.)

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Exercise $$\PageIndex{56}$$

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

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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?

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?