Practice Exam 2
Math 140A, Summer 2014, July 17
Name:
INSTRUCTIONS: These problems are for PRACTICE. For the practice
exam, you may use your book, consult your classmates, and use any other
resource available. For the actual exam, calculators will not be allowed.
The actual exam will also be shorter, to allow all work to be done within an
hour. These problems are for practice only.
Question 1 Given f (x) = 9x2 − 6x + 3
(a) Graph the quadratic function by determining whether its graph opens up or down
and by finding its vertex, axis of symmetry, y-intercept and x-intercepts, if any.
(b) Determine the domain and the range of the function.
Question 2 The price p in dollars and the quantity x sold of a certain product obey
the demand equation: x = 1500 − 10p for 0 ≤ p ≤ 150
(a) Express the revenue R as a function of p.
(b) What is the revenue if 100 units are sold?
(c) What price should the company charge to maximize revenue? What is the maximum
revenue?
Question 3 Form a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of a leading coefficient.
Zeros: 3 with multiplicity 3; −1 with multiplicity 3; degree 6
Question 4 Consider the polynomial f (x) = (x − 4)3 (x − 3)2
1. What’s the degree of f (x)?
2. What are the zeros of f (x)? What are their multiplicities?
3. For each zero, does f (x) cross or touch the x axis?
4. Find a function that approximates the behavior of f near each x-intercept.
Question 5 Find the domain of the rational function.
G(x) =
x2
x+2
+ 2x − 15
Question 6 Find the vertical, horizontal, and oblique asymptotes, if any, of the rational function
5x2 − 3x + 1
g(x) =
.
x+2
Question 7 Consider the rational function
G(X) =
x2 − 4
x2 − x − 2
(a) Find the domain of G.
(b) Find the x and y intercepts of G.
(c) Find any vertical asymptotes of G.
(d) Divide the x-axis into intervals separated by the x-intercepts and vertical asymptotes of G. Check a point in each and determine whether the graph in that interval
is located above or below the x-axis.
Question 8
f (x) = 8x3 − 3x2 + x + 4
g(x) = x + 2
Find the remainder R when f (x) is divided by g(x). Is g a factor of f ?
Question 9 List the potential rational zeros of f (x) = x3 − x2 − 10x − 8. Check which
of them are actually zeros of f .