# 6.9E: Exercises for Section 6.9

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1) [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.

$$e^x$$ and $$e^{−x}$$

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

3) Show that $$\cosh(x)$$ and $$\sinh(x)$$ satisfy $$y''=y$$.

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

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

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

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

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

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

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

$$3\sinh(3x+1)$$

10) [T] $$\sinh(x^2)$$

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

$$−\tanh(x)\text{sech}(x)$$

12) [T] $$\sinh(\ln(x))$$

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

$$4\cosh(x)\sinh(x)$$

14) [T] $$\cosh^2(x)−\sinh^2(x)$$

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

$$\dfrac{x\text{sech}^2(\sqrt{x^2+1})}{\sqrt{x^2+1}}$$

16) [T] $$\dfrac{1+\tanh(x)}{1−\tanh(x)}$$

17) [T] $$\sinh^6(x)$$

$$6\sinh^5(x)\cosh(x)$$

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

In exercises 19 - 28, find the antiderivatives for the given functions.

19) $$\cosh(2x+1)$$

$$\frac{1}{2}\sinh(2x+1)+C$$

20) $$\tanh(3x+2)$$

21) $$x\cosh(x^2)$$

$$\frac{1}{2}\sinh^2(x^2)+C$$

22) $$3x^3\tanh(x^4)$$

23) $$\cosh^2(x)\sinh(x)$$

$$\frac{1}{3}\cosh^3(x)+C$$

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

25) $$\dfrac{\sinh(x)}{1+\cosh(x)}$$

$$\ln(1+\cosh(x))+C$$

26) $$\coth(x)$$

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

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

28) $$(\cosh(x)+\sinh(x))^n$$

In exercises 29 - 35, find the derivatives for the functions.

29) $$\tanh^{−1}(4x)$$

$$\dfrac{4}{1−16x^2}$$

30) $$\sinh^{−1}(x^2)$$

31) $$\sinh^{−1}(\cosh(x))$$

$$\dfrac{\sinh(x)}{\sqrt{\cosh^2(x)+1}}$$

32) $$\cosh^{−1}(x^3)$$

33) $$\tanh^{−1}(\cos(x))$$

$$−\csc(x)$$

34) $$e^{\sinh^{−1}(x)}$$

35) $$\ln(\tanh^{−1}(x))$$

$$−\dfrac{1}{(x^2−1)\tanh^{−1}(x)}$$

In exercises 36 - 42, find the antiderivatives for the functions.

36) $$\displaystyle ∫\frac{dx}{4−x^2}$$

37) $$\displaystyle ∫\frac{dx}{a^2−x^2}$$

$$\dfrac{1}{a}\tanh^{−1}\left(\dfrac{x}{a}\right)+C$$

38) $$\displaystyle ∫\frac{dx}{\sqrt{x^2+1}}$$

39) $$\displaystyle ∫\frac{xdx}{\sqrt{x^2+1}}$$

$$\sqrt{x^2+1}+C$$

40) $$\displaystyle ∫−\frac{dx}{x\sqrt{1−x^2}}$$

41) $$\displaystyle ∫\frac{e^x}{\sqrt{e^{2x}−1}}$$

$$\cosh^{−1}(e^x)+C$$

42) $$\displaystyle ∫−\frac{2x}{x^4−1}$$

In exercises 43 - 45, use the fact that a falling body with friction equal to velocity squared obeys the equation $$\dfrac{dv}{dt}=g−v^2$$.

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

44) Derive the previous expression for $$v(t)$$ by integrating $$\dfrac{dv}{g−v^2}=dt$$.

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

$$37.30$$

In exercises 46 - 48, use this scenario: A cable hanging under its own weight has a slope $$S=\dfrac{dy}{dx}$$ that satisfies $$\dfrac{dS}{dx}=c\sqrt{1+S^2}$$. The constant $$c$$ is the ratio of cable density to tension.

46) Show that $$S=\sinh(cx)$$ satisfies this equation.

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

$$y=\frac{1}{c}\cosh(cx)$$

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

In exercises 49 - 52, solve each problem.

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

$$−0.521095$$

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

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

$$10$$

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

53) Prove the formula for the derivative of $$y=\sinh^{−1}(x)$$ by differentiating $$x=\sinh(y).$$

(Hint: Use hyperbolic trigonometric identities.)

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

(Hint: Use hyperbolic trigonometric identities.)

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

(Hint: Use hyperbolic trigonometric identities.)

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

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

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