Math
165
Spring 2002
Final Exam
- (20 pt) Find the
following limits:
a) b) c)
d) where is a function such
that for all
- (20 pt) Find the
derivatives of the following functions:
a) b) c)
d)
- (6 pt) Use the definition of the derivative to find the derivative
of the function
- (20 pt) Evaluate the following integrals:
a) b) c) d)
5.
(8
pt) A rock is thrown into a pond resulting in ripples that flow outward in the
shape of a circle. If the ripples are going out at a rate of 2 ft/sec, then how
fast is the area of the circle changing after 5 seconds?
- (8 pt) A circular piece of paper with radius has a circular
sector removed from it. The remaining piece is then folded as to make a
cone shaped paper cup. What is the maximum volume of this cup?
- (5 pt) Use the definition of the definite integral to evaluate Check your
answer.
- (5 pt) Suppose that I use Newton’s method to approximate the
positive square root of a positive number. Explain why all my
approximations (besides my first guess) are overestimates if I start with
a positive initial approximation. What happens if I start with a negative
initial approximation? (Hint: a picture might help you make your
argument).
- (6 pt) Find the tangent line to the curve at the point
- (12 pt) Sketch the graph of the function