## Problem Set: Calculus of the Hyperbolic Functions

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

2. From the definitions of $\text{cosh}(x)$ and $\text{sinh}(x),$ find their antiderivatives.

3. Show that $\text{cosh}(x)$ and $\text{sinh}(x)$ satisfy $y\text{″}=y.$

4. Use the quotient rule to verify that $\text{tanh}(x)\prime ={\text{sech}}^{2}(x).$

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

6. Take the derivative of the previous expression to find an expression for $\text{sinh}(2x).$

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

8. Take the derivative of the previous expression to find an expression for $\text{cosh}(x+y).$

For the following exercises (9-18), find the derivatives of the given functions and graph along with the function to ensure your answer is correct.

9. [T] $\text{cosh}(3x+1)$

10. [T] $\text{sinh}({x}^{2})$

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

12. [T] $\text{sinh}(\text{ln}(x))$

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

14. [T] ${\text{cosh}}^{2}(x)-{\text{sinh}}^{2}(x)$

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

16. [T] $\frac{1+\text{tanh}(x)}{1-\text{tanh}(x)}$

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

18. [T] $\text{ln}(\text{sech}(x)+\text{tanh}(x))$

For the following exercises (19-29), find the antiderivatives for the given functions.

19. $\text{cosh}(2x+1)$

20. $\text{tanh}(3x+2)$

21. $x\text{cosh}({x}^{2})$

23. $3{x}^{3}\text{tanh}({x}^{4})$

24. ${\text{cosh}}^{2}(x)\text{sinh}(x)$

25. ${\text{tanh}}^{2}(x){\text{sech}}^{2}(x)$

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

27. $\text{coth}(x)$

28. $\text{cosh}(x)+\text{sinh}(x)$

29. ${(\text{cosh}(x)+\text{sinh}(x))}^{n}$

For the following exercises (30-36), find the derivatives for the functions.

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

31. ${\text{sinh}}^{-1}({x}^{2})$

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

33. ${\text{cosh}}^{-1}({x}^{3})$

34. ${\text{tanh}}^{-1}( \cos (x))$

35. ${e}^{{\text{sinh}}^{-1}(x)}$

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

For the following exercises (37-43), find the antiderivatives for the functions.

37. $\displaystyle\int \frac{dx}{4-{x}^{2}}$

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

39. $\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+1}}$

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

41. $\displaystyle\int -\frac{dx}{x\sqrt{1-{x}^{2}}}$

42. $\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-1}}$

43. $\displaystyle\int -\frac{2x}{{x}^{4}-1}$

For the following exercises (44-46), use the fact that a falling body with friction equal to velocity squared obeys the equation $dv\text{/}dt=g-{v}^{2}.$

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

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

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

For the following exercises (47-49), 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 $c$ is the ratio of cable density to tension.

47. Show that $S=\text{sinh}(cx)$ satisfies this equation.

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

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

For the following exercises (50-53), solve each problem.

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

51. [T] A chain hangs from two posts four meters apart to form a catenary described by the equation $y=4\text{cosh}(x\text{/}4)-3.$ Find the total length of the catenary (arc length).

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

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

54. Prove the formula for the derivative of $y={\text{sinh}}^{-1}(x)$ by differentiating $x=\text{sinh}(y).$ (Hint: Use hyperbolic trigonometric identities.)

55. Prove the formula for the derivative of $y={\text{cosh}}^{-1}(x)$ by differentiating $x=\text{cosh}(y).$

(Hint: Use hyperbolic trigonometric identities.)

56. Prove the formula for the derivative of $y={\text{sech}}^{-1}(x)$ by differentiating $x=\text{sech}(y).$ (Hint: Use hyperbolic trigonometric identities.)

57. Prove that ${(\text{cosh}(x)+\text{sinh}(x))}^{n}=\text{cosh}(nx)+\text{sinh}(nx).$

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

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