**GROUND RULES: 1. You may consult a)your classmates, b)your notes,
c)calculus textbooks, and d) myself.**
**2. You may not consult any other persons (student or faculty)
than those allowed in rule 1.**

1.(10 points) a) Find the Maclaurin polynomial of degree 8 for cos 2x.

b) Find the Maclaurin polynomial of degree 8 for sin

c)Estimate using the first three non-zero terms of the Maclaurin polynomial in b).

Explain how you handled the fact that this is an improper integral in your estimate.

d) Estimate the integral in part c) using Simpson's rule. Explain how you handled the fact that this is an improper integral in your estimate.

e) Discuss the error in your estimate in part c). [Use Taylor's theorem.]

2.(10 points) Suppose that f is a C

a) Find the MacLaurin polynomial of degree 6 for f.

b) Estimate the value of f(1) using this polynomial.

c) Use Euler's method with n = 4 to estimate f(1).

3.(10 points) A fly has been trapped inside a spherical jar of radius 1 foot.

a) What is the probability that at any instant the fly will be at most 1/2 foot from the center of the jar?

b) We observe the fly's movement inside the jar and measure its distance X from the center of the jar at random times. What is the median value of the random variable X measured in this experiment? [Assume the probability that the fly is in any region of space inside the jar is proportional to the volume of the region.]

c) What is the average value of the random variable X that you would expect from repeated observations of the fly's position in the jar?

4. (10 points) [The binomial theorem.] a. Let f(x) = (1 + x)

b. i. Let Find the Maclaurin polynomial of degree 4 for g.

ii. Use your result in part i to estimate g(1). Discuss the error in this estimate.

5. (10 points) Suppose g is a C infinity function with g''(x) = - g(x) for all x and g(0) = 2 and g'(0) = 1.

a. Find the MacLaurin series for g.

b. Use the ratio test to show that the MacLaurin series for g converges for all x.

c. Using the MacLaurin series for sin(x) and cos(x) and your result in part a) find A and B so that g(x) = A sin(x) + B cos(x).