Show all work if full or partial credit is desired. No calculator on this first part, and 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^2-5x)/(x-5)=`
5. `f(x)=7`,   so `f'(x)=`
6. `g(x)=3x-2`,   so `d/(dx)[g(x)]=`
7. `y=5t^3`,   so `dy/(dt)=`
8. `d/(d theta)[cos(theta)]=`
9. `d/dx[3 sqrt(x)]=`
10. `d/dx[x^2*e^x]=`
11. `d/dx[arccos(x)]=`
12. Use the limit definition of the derivative to find `d/dx[2/x]`.

Show all work if full or partial credit is desired. You may use a graphing calculator or Desmos. 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. No derivative calculators allowed.

Part 2 Calculator Allowed

Find the following derivatives.

13. `d/dx[(x^2+3)/(x+3)]=`
14. `d/dx[cos(x) sin(x)]=`
15. `d/dx[sqrt(4-x^2)]=`
16. `d/dx[ln(5x)]=`

Fun with Derivatives

17. 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. `dy/dx=`
    2. `f'(-1)=`
    3. The equation of the tangent line is
    4. Neatly 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.
       
18. 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?
19. Show that the curve defined implicitly by the equation `xy^3 + x^3y = 4` has no horizontal tangent.

Applications

20. 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?
21. 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.
22. 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).
23. 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.)
24. Use L’Hôpital’s rule to find `lim_(x to oo) x^2/e^x`.
25. 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`.