Math 1090-004
Midterm 2
April 6, 2015
Please show all of your work as partial credit will be given where
appropriate, and there may be no credit given for problems where there is no
work shown. You may use a scientific calculator and a 4 × 6 inch note card.
Scratch paper will be provided but NOT collected, so please transfer all
finished work onto the proper page in the test. This exam totals 70 points.
Name:
UID:
1. (10 pts) Consider the functions f (x) = log5 (x) and g(x) = log5 (x + 1) − 3.
(a) (3 pts) Fill in the following tables
x
−5
f (x)
1
5
1 5
x − 45 0 4
g(x)
(b) (2 pts) Graph the function f (plot the points you obtained in (a); if f (x) has
asymptotes, make sure your graph reflects that).
(c) (2 pts) Write a sentence describing how to obtain the graph of g from that of f .
(d) (3 pts) Graph the function g according to the description you gave in (c) (if g has
asymptotes, make sure your graph reflects that).
2. (10 pts) Consider the functions f (x) =
2x
√
,
3 2
x −9
g(x) = x3 − 4 and h(x) = ex .
(a) (3 pts) Find the domain of f (x) (namely, find all the values of x for which f (x)
makes sense).
(b) (2 pts) Evaluate f (w2 ).
(c) (3 pts) Find the composition f (g(x))
(d) (2 pts) Evaluate (f − g)(8)
(e) (Extra-credit - 4 pts) Find the composition g(f (h(x))) (there is no need to simplify
your solution).
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3. (10 pts) For the function y = −2x2 + 8x + 10,
(a) (3 pts) Find the coordinates of the vertex. Is it a maximum or a minimum and
why?
(b) (3 pts) Find the x-intercepts.
(c) (4 pts) Sketch the graph of the parabola (you should plot at least 3 points).
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4. (a) (6 pts) Given the function f (x) =
5
√
,
3
3x−2
find its inverse f −1 (x).
(b) (1 pt) What does it mean for 2 functions f (x) and g(x) to be inverses of each other?
(c) (3 pts) Verify that the function you found in (a) is indeed the inverse function of
f.
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5. (10 pts) Solve the following equations:
(a) (10 pts)
1
log(x) − log(x − 1) = 1
2
(b) (Extra-credit 6 pts)
2 log100 (x) − log(x − 1) = 1
Hint: Recall that loga (x) =
log(x)
log(a)
for any number a.
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6. (10 pts) A patient has a painkiller given intravenously with an initial concentration of
. After 2.5 hours, the patient’s blood contains the painkiller in a concentration
500 mg
ml
of only 150 mg
. Assume that the concentration of the painkiller decays exponentially
ml
(namely, the concentration is given by an expression f (t) = aebt , where a, b are real
numbers and where t represents time in hours).
(a) (5 pts) Wite the concentration of painkiller as a function of time.
Hint: You already know that the function looks like f (t) = aebt ; you need to find
a, b.
?
(b) (3 pts) When will the patient’s blood have a concentration of only 20 mg
ml
(c) (2 pts) What is the concentration of painkiller after 5 hours?
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7. (10 pts) Consider the function f (x) =
2+x
1−x
(a) (1 pt) What are (if any) the vertical asymptotes?
(b) (2 pts) For every vertical asymptote x = a, find the limits limx→a f (x) and lima←x f (x).
(c) (2 pts) What are, if any, the horizontal asymptotes?
(d) (3 pts) Find the x-intercepts and the y-intercept.
(e) (3 pts) Sketch the graph of the function (using the information above: draw the
asymptotes and plot the intercepts).
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