Show all work if full or partial credit is desired. You may use a graphing calculator or Desmos on the second part. If Desmos is used, then the device must be in airplane-mode (i.e., no wifi, cellular, or Bluetooth connections). No notes, books or websites allowed.

Part 1 No Calculator

1. `lim_{x to -3} (x^2-2x+5)=`
2. `lim_{x to 0+} 1/x=`
3. `lim_{x to oo} 1/x=`
4. `lim_{x to 5} (x^2-25)/(x-5)=`
5. `f(x)=7`,   so `f'(x)=`
6. `g(x)=x`,   so `d/(dx)[g(x)]=`
7. `q(t)=t^3`,   so `d/(dt)[q(t)]=`
8. `d/(d theta)[cos(theta)]=`
9. Use the limit definition of the derivative to find the slope of the tangent line to the curve `r(x)=2/x` at `x=-3`.

Part 2 Calculator Allowed

Find the following derivatives.

10. `d/dx[2x^2-3x+5]=`
11. `d/dx[(x-3)^2]=`
12. `d/dx[3 sqrt(x)]=`
13. `d/dx[(x^2+3x+6)(x^2-3x+-5)]=`
14. `d/dx[(x^2+3)/(x+3)]=`
15. `d/dx[cos(x) sin(x)]=`
16. `d/dx[sqrt(4-x^2)]=`
17. `d/dx[{sin(pi x+1)}^3]=`
18. `d/dx[(e^(x)-e^(-x))/2]=`
19. `d/dx[arccos(x)]=`
20. `d/dx[ln(5x)]=`
21. `d/dx[ln(cos(x))]=`
22. `d/dx[2*3^x]=`

Fun with Derivatives

23. Calculate the slope of the tangent line (rate of change) to `f(x)=3/(x-1)`   at   `x=-1` by finding the derivative function.
    1. `d/dx[f(x)]=`
    2. `f'(-1)=`
    3. The equation of the tangent line is
    4. Sketch a graph of `f(x)` on the interval `[-3,1]` along with the tangent line when `x=-1`.
       Be sure to label and scale the axes, and use a ruler if necessary.
       
24. A ball is dropped from a height of 25 meters. Its height above ground (in meters) t seconds later is given by `h(t)=−4.9t^2+25`.
    1. At what time does the ball hit the ground?
    2. What is the instantaneous velocity of the ball when it hits the ground?
    3. What is the average velocity during its fall?
25. Find the slope of the tangent line to the curve `x^3-9xy+y^3=0` at the point `(2,4)`.
26. Show that the curve defined implicitly by the equation `xy^3 + x^3y = 4` has no horizontal tangent.
27. Use logarithmic differentiation to find `dy/dx`   if   `y=(1+x)^(1/x)`.
28. If `P(t)=1000e^(0.3t)` describes a mosquito population after `t` days, what is the rate of change of the population after 4 days?

Part 3: Chapter 4

29. A highway patrol plane is flying 1 mile above a long, straight road, with constant ground speed of 120 m.p.h. Using radar, the pilot detects a car whose distance from the plane is 1.5 miles and decreasing at a rate of 136 m.p.h. How fast is the car traveling along the highway?
30. Consider the volume of a sphere, `V=4/3 pi r^3`. Suppose that you fill the balloon with air at a constant rate, `100" cm"^3/"s"`.
    1. At what rate does the radius increase when the radius is 2cm?
    2. At what rate does the radius increase when the radius is 4cm?
31. The diameter of a spherical ball bearing was measured to be 5mm with a possible error of 0.05mm. Use linear approximation (aka differentials) to estimate the maximum error in the volume of the ball bearing.
32. Sketch the graph of the function `f(x)=x^3-7x^2+8x+1`
    Be sure to identify in writing all local maxs and mins, regions where the function is increasing/decreasing, concavity, points of inflection, symmetries, and any asymptotes (if any of these behaviors occur).
33. Sketch the graph of the function `f(x)=(x^2+7x+10)/(x+1)`
    Be sure to identify in writing all local maxs and mins, regions where the function is increasing/decreasing, concavity, points of inflection, symmetries, and any asymptotes (if any of these behaviors occur).
34. Consider an open-top box with a square base and a volume of `216" in."^3`. Suppose the cost of the material for the base is 20¢/`"in."^2` and the cost of the material for the sides is 30¢/`"in."^2` and we are trying to minimize the cost of this box. Write the cost as a function of the side lengths of the base. (Let `x` be the side length of the base and `y` be the height of the box.)
35. A poster is to be designed with `50" in"^2` of printed type, 4 inch margins on both the top and the bottom, and 2 inch margins on each side. Find the dimensions of the poster which minimize the amount of paper used. (Be sure to indicate why the answer you found is a minimum.)
36. Use L’Hôpital’s rule to find `lim_(x to oo) x^2/e^x`.
37. Use Newton's method to approximate a root of `f(x)=x^3-3x+1` on the interval `[1,2]`. Let `x_0=2` and find `x_1`, `x_2`, `x_3`, `x_4`, and `x_5`.