1. (25 pt) Compute the following derivatives:

a) b) c)

c) d)

2. (10 pt) Find the tangent line to at the origin.

3. (10 pt ) Find the polynomial of degree 3 that satisfies

.

4. (13 pt) A parchment is found to contain 80% of its original ¹⁴C (a naturally occurring radioactive isotope of carbon with a half-life of 5730 years). Assuming exponential decay, how old is the parchment?

5. (10 pt) Water is put into a cone-shaped cup with height twice the radius of the top. The water is heated. How fast is the water evaporating if the water level is decreasing at 2 inches per minute when the water is 6 inches deep?

6. (10 pt) A cube is measured and found to be 4 feet long per side with an error of ± 1/10 in. Use differentials to estimate the maximum error in using this measurement to compute the volume. What is the relative error?

7. (12 pt) Consider the following functions

What initial values (if any) for x (marked by the vertical slashes on the x-axis) will make Newton’s method find the root r₁. Justify your answer (using a picture to justify on your test is OK).

8. (10 pt) Let f(x) be a one to one function. If the tangent line to f(x) is given as y=2x+4 at the point (2,8), find the tangent line to f^-1(x) when x=8.

9. (Extra Credit 10 pt). Use the piece of secant shown to define the inverse secant function. That is y=sec^-1(x) (| x| ³ 1) Û sec(y)=x and y is in [0,p /2)È (p /2,p ]. Find the derivative of y=sec^-1(x). (Hint, what can you say about the slopes of the tangent lines?)