Review Exam 3: Math 203 Calculus I
Sample Problems
1) Find the absolute minimum and absolute maximum of
23)(
23
xxxxf
over the interval [-2,2]
2) Let f(x) =
2
32
2
x
xx
.
a. Determine if Rolle’s Theorem can be applied to f over [-1,3]. If so, find
)3,1(c
that satisfies Rolle’s theorem.
b. Determine if the Mean Value Theorem can be applied to f over [-1,3]. If so,
find
)3,1(c
that satisfies the mean Value Theorem.
3) Let
4
)5(
)(
2
xx
xf
. Sketch f by doing all off the following:
a. Determine the Domain
b. Compute the intercepts
c. Determine any symmetry of the function
d. Find asymptotes
e. Use first derivative test to find:
i. Intervals of increase & decrease
ii. Local Max and Min
f. Use 2nd derivative test to determine
i. Intervals and concavity
ii. Max and Min values
4) Show the indeterminate form and evaluate the limits using L’Hospitals rule:
a.
1
)4/(arctan
lim
1
x
x
x
b.
x
x
x
1
1lim
5) What point of the graph of
2
4xy
is closest to (0,2)?
6) A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must
have an area of 180,000m2. Find the dimensions that will minimize the perimeter if
no fencing is necessary along the river.
7) Let the cost function of a product be defined as
2
001.064000)( xxxC
and the
demand function be defined as
xxp 15.0200)(
a. Find the cost, average cost, and marginal cost of producing 50, 100, and 200
items.
b. At what production level will the average cost be lowest and what is the
minimum average cost?
c. Find the revenue function, marginal revenue function, and profit function.
d. Find the production level that will maximize the profit.
8) Use Newton’s Method to solve
13
23
xxx
accurate to 5 decimal places.
