Exercises

1. 1. Calculate the derivative of `sec theta = 1/(cos theta) .`
   2. Express the answer to part (a) in terms of the functions `sec theta` and `tan theta.`
   3. Find the second derivative of `sec theta.`
   4. Use your graphing tool to draw the graph of `sec theta` on the interval from `-2 pi` to `2 pi,` and match the features of the graph with properties of the function as defined in part (a), as well as properties of the first and second derivatives.
   5. What is the period of `sec theta`? Explain.

2. Calculate each of the following derivatives.

   `d/(dt) tan 5t`	`d/(dx) sin 5x tan 5x`
   `d/(dx) tan^(-1) 5x`	`d/(dt) cos 5t tan 5t`
   `d/(dt) e^(-t) tan 5t`	`d/(dx) sin^(-1) 5x tan 5x`
   `d/(dx) e^(-2x) tan^(-1) 5x`	`d/(dt) sin 5t tan^(-1) 5t`

3. Calculate each of the following second derivatives. You may need the result of Exercise 1, parts (a) and (b).

   `d^2/(dt^2) tan 5t`	`d^2/(d t^2) sin^(-1) 5t`
   `d^2/(d t^2) tan^(-1) 5t`	`d^2/(d theta^2) cos 5theta tan 5theta`

4. 1. Calculate the derivative of the cotangent function, `cot theta = (cos theta)/(sin theta) .`
   2. Express the answer to part (a) in terms of the cosecant function `csc theta=1/(sin theta) .`
   3. Use your graphing tool to draw the graph of `cot theta` on the interval from `-2 pi` to `2 pi,` and match the features of the graph with properties of the function as defined in part (a), as well as properties of the derivative.
   4. What is the period of `cot theta`? Explain.
5. 1. Calculate the derivative of `csc theta = 1/(sin theta) .`
   2. Express the answer to part (a) in terms of the functions `csc theta` and `cot theta.`
   3. Use your graphing tool to draw the graph of `csc theta` on the interval from `-2 pi` to `2 pi,` and match the features of the graph with properties of the function as defined in part (a), as well as properties of the derivative.
   4. What is the period of `csc theta`? Explain.
6. In Example 3, we determined the optimal distance for placing the museum bench to maximize the vertical viewing angle for a picture on the wall. If Figure 12 were redrawn with that optimal distance, it would look more like Figure E1.
1. If you were the designer, would you place the bench this close to the wall? Why or why not?
2. Figure E2 repeats Figure 14 showing the vertical viewing angle as a function of distance from the wall. What do you see in this figure to suggest that a different placement might be a better design? Explain.

Figure E1   Optimal viewing angle	Figure E2   Viewing angle vs. distance

7. Figure E3 shows a right triangle in which `y` is the angle whose cosine is `x.` That is, `y=cos^(-1) x.`

   
   Figure E3   `y=cos^(-1) x`

   1. How is the angle `z` related to `x`?
   2. Explain why `cos^(-1) x =pi/2-sin^(-1) x.`
   3. What is the derivative of `cos^(-1) x`? (You should be able to answer this with only mental calculation.)
   4. What is the domain of the inverse cosine function? What is its range?
   5. Use your graphing tool to graph both arcsine and arccosine.
   6. What do you think we meant in the Section Summary (preceding page) when we said the inverse cosine "is less interesting and less useful than the inverse sine"?
8. 1. What is the domain of the inverse tangent function? What is its range?
   2. Use your graphing tool to graph the arctangent function.
   3. Use your graphing tool to graph the derivative of the arctangent function.
   4. What is the steepest slope on the graph of the arctangent function, and where does it occur?
9. 1. Explain why `cot^(-1) x =pi/2-tan^(-1) x.`
   2. What is the derivative of `cot^(-1) x`? (You should be able to answer this with only mental calculation.)
   3. What is the domain of the inverse cotangent function? What is its range?
   4. Use your graphing tool to graph both arctangent and arccotangent.
10. Figure E4 shows a right triangle in which `y` is the angle whose secant is `x.` That is, `y=sec^(-1) x.`

    
    Figure E4   `y=sec^(-1) x`

    1. What is the tangent of the angle `y`?
    2. Use implicit differentiation and the result of Exercise 1(a) to find `dytext[/]dx`. You will need the answer to part (a) to write the result as a function of `x.`
    3. What is the domain of the secant function? What is its range?
    4. Use your graphing tool to graph both secant and its derivative. Match features of the two graphs.
11. Figure E5 shows a right triangle in which `y` is the angle whose cosecant is `x.` That is, `y=csc^(-1) x.`

    
    Figure E5   `y=csc^(-1) x`

    1. How is the angle `z` related to `x`?
    2. Explain why `csc^(-1) x =pi/2-sec^(-1) x.`
    3. What is the derivative of `csc^(-1) x`? (After doing Exercise 10, you should be able to answer this with only mental calculation.)
    4. What is the domain of the inverse cosecant function? What is its range?
    5. Use your graphing tool to graph both arcsecant and arccosecant.