1. $y=x^2- \sec x+1$

2. $y=3 \csc x+\dfrac{5}{x}$

3. $y=x^2 \cot x$

4. $y=x-x^3 \sin x$

5. $y=\dfrac{\sec x}{x}$

6. $y= \sin x \tan x$

7. $y=(x+ \cos x)(1- \sin x)$

8. $y=\dfrac{\tan x}{1- \sec x}$

9. $y=\dfrac{1- \cot x}{1+ \cot x}$

10. $y= \cos x(1+ \csc x)$

11. [T] $f(x)=−\sin x, \,\,\, x=0$

12. [T] $f(x)= \csc x, \,\,\, x=\frac{\pi}{2}$

13. [T] $f(x)=1+ \cos x, \,\,\, x=\frac{3\pi}{2}$

14. [T] $f(x)= \sec x, \,\,\, x=\frac{\pi}{4}$

15. [T] $f(x)=x^2- \tan x, \,\,\, x=0$

16. [T] $f(x)=5 \cot x, \,\,\, x=\frac{\pi}{4}$

17. $y=x \sin x- \cos x$

18. $y= \sin x \cos x$

19. $y=x-\frac{1}{2} \sin x$

20. $y=\frac{1}{x}+ \tan x$

21. $y=2 \csc x$

22. $y=\sec^2 x$

23. Find all $x$ values on the graph of $f(x)=-3 \sin x \cos x$ where the tangent line is horizontal.

24. Find all $x$ values on the graph of $f(x)=x-2 \cos x$ for [latex]0

25. Let $f(x)= \cot x$. Determine the point(s) on the graph of $f$ for [latex]0

26. [T] A mass on a spring bounces up and down in simple harmonic motion, modeled by the function $s(t)=-6 \cos t$ where $s$ is measured in inches and $t$ is measured in seconds. Find the rate at which the spring is oscillating at $t=5$ s.

27. Let the position of a swinging pendulum in simple harmonic motion be given by $s(t)=a \cos t+b \sin t$ where $a$ and $b$ are constants, $t$ measures time in seconds, and $s$ measures position in centimeters, If the position is $0$cm and the velocity is $3$cm/s when $t=0$, find the values of $a$ and $b$.

28. After a diver jumps off a diving board, the edge of the board oscillates with position given by $s(t)=-5 \cos t$ cm at $t$ seconds after the jump.

1. Sketch one period of the position function for $t\ge 0$.
2. Find the velocity function.
3. Sketch one period of the velocity function for $t\ge 0$.
4. Determine the times when the velocity is 0 over one period.
5. Find the acceleration function.
6. Sketch one period of the acceleration function for $t\ge 0$.

29. The number of hamburgers sold at a fast-food restaurant in Pasadena, California, is given by $y=10+5 \sin x$ where $y$ is the number of hamburgers sold and $x$ represents the number of hours after the restaurant opened at 11 a.m. until 11 p.m., when the store closes. Find $y^{\prime}$ and determine the intervals where the number of burgers being sold is increasing.

30. [T] The amount of rainfall per month in Phoenix, Arizona, can be approximated by $y(t)=0.5+0.3 \cos t$, where $t$ is the number of months since January. Find $y^{\prime}$ and use a calculator to determine the intervals where the amount of rain falling is decreasing.

31. $\frac{d}{dx}(\cot x)=−\csc^2 x$

32. $\frac{d}{dx}(\sec x)= \sec x \tan x$

33. $\frac{d}{dx}(\csc x)=−\csc x \cot x$

34. Use the definition of derivative and the identity

$\cos (x+h)= \cos x \cos h- \sin x \sin h$  to prove that  $\frac{d}{dx}(\cos x)=−\sin x$.

35. $\frac{d^3 y}{dx^3}$ of $y=3 \cos x$

36. $\frac{d^2 y}{dx^2}$ of $y=3 \sin x+x^2 \cos x$

37. $\frac{d^4 y}{dx^4}$ of $y=5 \cos x$

38. $\frac{d^2 y}{dx^2}$ of $y= \sec x+ \cot x$

39. $\frac{d^3 y}{dx^3}$ of $y=x^{10}- \sec x$