MATH 105 PRACTICE MIDTERM III
Please work out each of the given problems. Credit will be based on the steps that you show towards the final answer. Show your work.
PROBLEM 1
Please answer the following true or false.
If false, explain why or provide a counter example.
If true, explain why.
A) If f(x) is a positive continuous function such that
then
cannot be
equal to 3.
B) If f(x) is a differentiable function such that the equation of the tangent line at
x = 2
is
1 1
y =
x -
2 2
and if x = 2 is the first guess in
Newton’s method, then x = 1
is the second guess.
C)
If f(x) and g(x)
are continuous functions on [a,b],
then
PROBLEM 2 Evaluate the
following integrals:
A.
B.
C.
PROBLEM 3 Use Riemann
Sums to find the area of the region below the curve
y = 9 - x2, above the x-axis,
and between x = 1 and x = 3.
PROBLEM 4
Let
Find F'(x)
PROBLEM 5
You are the
owner of Tahoe Winter Wear and need to determine the best price to sell your
most popular winter jacket. Your
cost for selling the jackets is
C
= 50 + 20x
Where x
is the amount to jackets that you sell.
Your research shows that the relationship between price, p,
and the number of jackets that you can sell, x, is
p = 300 – 10x
How much should you charge for your jacket in order to maximize profit?
PROBLEM 6
You are
manufacturing a square computer chip. Your
machine can construct the square with side length 0.4 
0.0002
cm.
Use differentials to approximate the maximum percent error in the area of
the chip.
PROBLEM
7
Consider the
integral
Use the trapezoidal rule with n = 4 to approximate
this integral and sketch the indicated area and trapezoids.
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