Practice Final
Math 140A, Summer 2014, August 5
Name:
INSTRUCTIONS: This exam is for PRACTICE.
1
Graphing
Chapter 2
1. Write the equation of a circle in standard form with center (2, 1) and radius 3.
Sketch the graph of the circle.
2. The time t that it takes to get to class varies inversely with your average speed
s.
(a) Suppose that it takes you half an hour to get to class when your average speed
is 5 miles per hour. Find the constant of proportionality k.
(b) Write an equation expressing the time t in terms of the average speed s. [3
points]
(c) Suppose that your average speed is 20 miles per hour. How long will it take
you to get to class? [3 points]
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2
Functions
Chapter 3, and 6.1
1.
Let f (x) =
1
and g(x) = x3 − x
x−2
(a) Compute:
i. f (3)
ii. (f + g)(1)
iii. (g ◦ f )(1)
iv. g(−x).
(b) Is g even, odd, or neither?
(c) What is the domain of f ?
(d) What is the domain of f ◦ g?
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(e) Sketch f : (Hint: Use transformations to graph f and g.)
(f) Sketch g:
(g) What is the range of g?
(h) Find and list the coordinates of any local maxima or minima on f and g.
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2.

 x + 2 −5 ≤ x < −2
4
x = −1
h(x) =

x + 5 −1 < x ≤ 3
(a) Sketch the graph of h.
(b) What is the domain of h? [2 points]
(c) What is the range of h? [2 points]
(d) Is h continuous?
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3
Quadratics
Section 4.3 & 4.5
Question 1 Given f (x) = −x2 − 2x + 3
(a) Determine whether the graph opens up or down.
(b) Find the vertex and axis of symmetry.
(c) Find y-intercept and x-intercepts, if any.
(d) Find domain and the range of the function.
(e) Sketch the graph of the function.
(f ) Find all x with f (x) > 0.
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Section 4.4
Question 2 The price p in dollars and the quantity x sold of a certain product obey
the demand equation:
x = 90 − 3p for 0 ≤ p ≤ 30
(a) Express the revenue R = xp as a function of p.
(b) What price should the company charge to maximize revenue? What is the maximum
revenue?
4
Polynomial and Rational Functions
Section 5.1
Question 3 Given f (x) = 9(x − 5)3 (x + 4)2 .
(a) Determine the degree of the polynomial.
(d) Determine the maximum number of turning
points on the graph.
(b) List each real zero and its multiplicity.
(e) Determine the end behavior of f .
(c) Determine whether the graph crosses or
touches the x-axis at each x-intercept.
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Section 5.3
Question 4 Consider the rational function
G(x) =
x2 − 4
x2 − 5x + 6
(a) Find the domain of G.
(b) Put G in lowest terms.
(c) Find the x and y intercepts of G.
(d) Find any vertical asymptotes of G.
(e) Find the horizontal or oblique asymptote of G, if any.
(f ) Find all x such that G(x) ≤ 0.
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Section 5.3
Question 5
f (x) = 2x4 − x3 − 2x2 + x + 2
g(x) = x − 1
Find the remainder R when f (x) is divided by g(x). Is g a factor of f ?
Question 6 List the potential rational zeros of f (x) = 2x3 − x2 − 9x + 4. Check which
of them are actually zeros of f .
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5
Exponents and Logarithms
Section 6.2 & 6.4
Question 7 For each function: (1) determine if the function is one-to-one.
(2) If it is one-to-one, find the inverse. OR If it is not one-to-one, restrict the domain
of f to make it one-to-one and find an inverse on that domain.
(a) f (x) = x3
(b) f (x) = 2x2 − 1
(c) f (x) = ln(x + 2)
Section 6.3, 6.4, 6.5, 6.6
Question 8 Evaluate:
log3 12 − log3 4
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Question 9 Solve. Write exact answers using logarithms or roots as necessary. Be
careful about your domains!
(a) 2x−1 = 8
(b) ex+1 = e3x−4
(c) log2 (x + 1) = 3
(d) 2ex+3 = 3
(e) log3 (x + 2) = 2 − log3 (x − 3)
(f ) 3ex+2 = 2
(g) 2x−1 = 12
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(h) 3x+1 = 2x
Section 6.7
Question 10 For each part, set up the equation. Express your answer in terms of
exponents, logarithms, and square roots.
(a) If $1000 is invested at 4% interest compounded quarterly (4 times a year), how
much is there after a year and a half ?
(b) What if the interest was compounded continuously?
Question 11 How long does it take to double your debt if it accumulates 6% interest
compounded monthly (12 times a year)? Again, show the equation you used.
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Section 6.8
Question 12 The population of a colony of mosquitos obeys the law of uninhibited
growth.
(a) If N is the population of the colony and t is the time in days, express N as a
function of t.
(b) If there are 1000 mosquites initially and there are 1200 after 1 day, how many will
be present in the culture after 5 hours?
(c) How long is it until there are 10,000 mosquitos?
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