1.7E: Exercises

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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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{11}\)

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

Answer: \(\displaystyle −tanh(x)sech(x)\)

Exercise \(\PageIndex{12}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{17}\)

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

Answer: \(\displaystyle 6sinh^5(x)cosh(x)\)

Exercise \(\PageIndex{18}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{25}\)

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

Answer: \(\displaystyle ln(1+cosh(x))+C\)

Exercise \(\PageIndex{26}\)

\(\displaystyle coth(x)\)

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{33}\)

\(\displaystyle tanh^{−1}(cos(x))\)

Answer: \(\displaystyle −csc(x)\)

Exercise \(\PageIndex{34}\)

\(\displaystyle e^{sinh^{−1}(x)}\)

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

Exercise \(\PageIndex{56}\)

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

Answer: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.

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: Add texts here. Do not delete this text first.