Problem Set: Calculus of the Hyperbolic Functions

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

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

3. Show that [latex]\text{cosh}(x)[/latex] and [latex]\text{sinh}(x)[/latex] satisfy [latex]y\text{″}=y.[/latex]

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

5. Derive [latex]{\text{cosh}}^{2}(x)+{\text{sinh}}^{2}(x)=\text{cosh}(2x)[/latex] from the definition.

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

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

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

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] [latex]\text{cosh}(3x+1)[/latex]

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

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

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

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

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

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

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

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

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

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

19. [latex]\text{cosh}(2x+1)[/latex]

20. [latex]\text{tanh}(3x+2)[/latex]

21. [latex]x\text{cosh}({x}^{2})[/latex]

23. [latex]3{x}^{3}\text{tanh}({x}^{4})[/latex]

24. [latex]{\text{cosh}}^{2}(x)\text{sinh}(x)[/latex]

25. [latex]{\text{tanh}}^{2}(x){\text{sech}}^{2}(x)[/latex]

26. [latex]\frac{\text{sinh}(x)}{1+\text{cosh}(x)}[/latex]

27. [latex]\text{coth}(x)[/latex]

28. [latex]\text{cosh}(x)+\text{sinh}(x)[/latex]

29. [latex]{(\text{cosh}(x)+\text{sinh}(x))}^{n}[/latex]

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

30. [latex]{\text{tanh}}^{-1}(4x)[/latex]

31. [latex]{\text{sinh}}^{-1}({x}^{2})[/latex]

32. [latex]{\text{sinh}}^{-1}(\text{cosh}(x))[/latex]

33. [latex]{\text{cosh}}^{-1}({x}^{3})[/latex]

34. [latex]{\text{tanh}}^{-1}( \cos (x))[/latex]

35. [latex]{e}^{{\text{sinh}}^{-1}(x)}[/latex]

36. [latex]\text{ln}({\text{tanh}}^{-1}(x))[/latex]

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

37. [latex]\displaystyle\int \frac{dx}{4-{x}^{2}}[/latex]

38. [latex]\displaystyle\int \frac{dx}{{a}^{2}-{x}^{2}}[/latex]

39. [latex]\displaystyle\int \frac{dx}{\sqrt{{x}^{2}+1}}[/latex]

40. [latex]\displaystyle\int \frac{xdx}{\sqrt{{x}^{2}+1}}[/latex]

41. [latex]\displaystyle\int -\frac{dx}{x\sqrt{1-{x}^{2}}}[/latex]

42. [latex]\displaystyle\int \frac{{e}^{x}}{\sqrt{{e}^{2x}-1}}[/latex]

43. [latex]\displaystyle\int -\frac{2x}{{x}^{4}-1}[/latex]

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

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

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

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

For the following exercises (47-49), use this scenario: A cable hanging under its own weight has a slope [latex]S=dy\text{/}dx[/latex] that satisfies [latex]dS\text{/}dx=c\sqrt{1+{S}^{2}}.[/latex] The constant [latex]c[/latex] is the ratio of cable density to tension.

47. Show that [latex]S=\text{sinh}(cx)[/latex] satisfies this equation.

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

49. Sketch the cable and determine how far down it sags at [latex]x=0.[/latex]

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 [latex]y=2\text{cosh}(x\text{/}2)-1.[/latex] 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 [latex]y=4\text{cosh}(x\text{/}4)-3.[/latex] Find the total length of the catenary (arc length).

52. [T] A high-voltage power line is a catenary described by [latex]y=10\text{cosh}(x\text{/}10).[/latex] 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 [latex]y=a\text{cosh}(x\text{/}a).[/latex] 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 [latex]y={\text{sinh}}^{-1}(x)[/latex] by differentiating [latex]x=\text{sinh}(y).[/latex] (Hint: Use hyperbolic trigonometric identities.)

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

(Hint: Use hyperbolic trigonometric identities.)

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

57. Prove that [latex]{(\text{cosh}(x)+\text{sinh}(x))}^{n}=\text{cosh}(nx)+\text{sinh}(nx).[/latex]

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

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