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Mathematics LibreTexts

1.7E: Exercises

  • Page ID
    18544
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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

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    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

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    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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