Calculus Volume 1 · 6.9 Calculus of the Hyperbolic Functions

Calculus of the Hyperbolic Functions: practice problems

27 problems from the book, with its answers. Check your working, or have any of them explained to you on the whiteboard.

1. Exercises 377

   [T] Find expressions for $\cosh x+\sinh x$ and $\cosh x-\sinh x.$ Use a calculator to graph these functions and ensure your expression is correct.

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2. Exercises 379

   Show that $\cosh (x)$ and $\sinh (x)$ satisfy $y^{\prime \prime }=y.$

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3. Exercises 381

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

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4. Exercises 383

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

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5. Exercises 385

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

   [T] $\cosh (3x+1)$

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6. Exercises 387

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

   [T] $\frac{1}{\cosh (x)}$

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7. Exercises 389

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

   [T] ${\cosh }^2(x)+{\sinh }^2(x)$

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8. Exercises 391

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

   [T] $\tanh (\sqrt{x^2+1})$

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9. Exercises 393

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

   [T] ${\sinh }^6(x)$

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10. Exercises 395

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

    $\cosh (2x+1)$

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11. Exercises 397

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

    $x\cosh (x^2)$

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12. Exercises 399

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

    ${\cosh }^2(x)\sinh (x)$

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13. Exercises 401

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

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

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14. Exercises 403

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

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

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15. Exercises 405

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

    ${\tanh }^{-1}(4x)$

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16. Exercises 407

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

    ${\sinh }^{-1}(\cosh (x))$

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17. Exercises 409

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

    ${\tanh }^{-1}(\cos (x))$

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18. Exercises 411

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

    $\ln ({\tanh }^{-1}(x))$

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19. Exercises 413

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

    $\int \frac{dx}{a^2-x^2}$

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20. Exercises 415

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

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

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21. Exercises 417

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

    $\int \frac{e^x}{\sqrt{e^{2x}-1}}dx$

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22. Exercises 419

    For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation $dv\text{/}dt=g-v^2.$

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

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23. Exercises 421

    For the following exercises, use the fact that a falling body with friction equal to velocity squared obeys the equation $dv\text{/}dt=g-v^2.$

    [T] Estimate how far a body has fallen in $12$ seconds by finding the area underneath the curve of $v(t).$

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24. Exercises 423

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

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25. Exercises 425

    For the following exercises, solve each problem.

    [T] A chain hangs from two posts $2$ m apart to form a catenary described by the equation $y=2\cosh (x\text{/}2)-1.$ Find the slope of the catenary at the left fence post.

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26. Exercises 427

    For the following exercises, solve each problem.

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

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27. Exercises 428

    For the following exercises, solve each problem.

    A telephone line is a catenary described by $y=a\cosh (x\text{/}a).$ 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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