MAT135 Written Quiz 1 Practice Problems
1. Let y = f (x) be a function. A log plot of f (x) is obtained by transforming the expression to plot
either ln y against x, y against ln x, or ln y against ln x. For the following expressions, determine the
appropriate transformation that makes the log plot linear.
(a)
y = 2x+2
(b)
y = log10 3x
(c)
2y = e4x+1
2. Consider the piecewise function


ax + b
f (x) = − cos(x/2) + 1


cx + d
x < −2π
−2π ≤ x ≤ π .
(a) Find all possible values of a, b, c, d which makes f continuous everywhere.
(b) Find all possible values of a, b, c, d which makes the limits lim
x→−2π
f (−2π) − f (x)
f (π) − f (x)
and lim
x→π
−2π − x
π−x
exist, and numerically estimate the limits.
3. Let d(t) be the distance in feet that a roller coaster moves along the track t seconds after the ride
begins. The entire ride lasts exactly 60 seconds. Several values of d(t) are shown in the following table.
t
d(t)
0
0
10
401
25
1056
30
1366
40
1817
45
2189
55
2816
60
3102
(a) Find the average velocity of the roller coaster over the last 15 seconds of the ride.
(b) Estimate the instantaneous velocity at 55 seconds into the ride.
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4. The graph of a function f (x) are given below:
Compute the following limits if they exist and explain your solution. If the limit does not exist, explain
why.
(a) limx→−1− f (x)
(b) limx→−1 f (x)
(c) limx→0 f (f (x))
(d) limx→1 f −1 (x)
5. Lui recorded his movement on a morning walk. The data collected is shown in the following table.
time
total distance walked today (metres)
8:00 AM
500
8:10 AM
1202
8:20 AM
1999
8:30 AM
2604
(a) Did Lui walk at a constant velocity from 8:00 AM to 8:30 AM? Circle your answer below. (No
explanation is required)
yes
no
it cannot be determined
(b) Find the average velocity between 8:00 AM and 8:30 AM. You must include the units in your
answer.
(c) Let D(t) be the total distance Lui walks in metres, where t is the time in minutes after 8:00 AM
and 0 ≤ t ≤ 30. Write a limit involving D that gives Lui’s instantaneous velocity at 8:10 AM.
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6. As part of a test of the acoustics of a concert hall, consultants asked a trombone player to play a single
note at maximum volume. Once the sound had reached its maximum intensity, the player stopped
and the sound intensity was measured for the next 0.2 seconds at regular intervals. The consultants
scaled the intensity measurements so that the maximum intensity at time 0 was 1, and then plotted
their data on graph (a) below.
The consultants suspect that this data is best modelled by an exponential model of the form I = I0 10kt
where I is the relative intensity and t is time in seconds after the maximum intensity is reached. To
check their suspicion, they plotted the graph (b) above, where the logarithm is a base 10 logarithm.
(a) Use the properties of logarithms to show that if I = I0 10kt , the function in the second plot must
be linear.
(b) Use your work in part (a) to calculate values for I0 and k, and use these to write an algebraic
model for the relationship between I and t.
7. Consider the rational function
a(x) =
(x − 2)2 (x + 8)
(x2 − 4)(2x + 7)
(a) What are the vertical asymptotes of the function a(x)? (Hint: There is at least one vertical
asymptote of this function.)
(b) Compute limx→∞ a(x). Justify your answer in full.
(c) Let x = V be one of the vertical asymptotes (so V is one of the numbers that you wrote in
part (a)). Indicate whether limx→V + a(x) and limx→V − a(x) are limits approaching ∞ or −∞ by
circling two options below.
Choice of vertical asymptote: V =
lim a(x) =
−∞
or
∞
lim a(x) =
−∞
or
∞
x→V +
x→V −
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(d) Which of the following has a shape closest to the graph of a(x)? Circle a letter.
a.
b.
c.
d.
8. On the axes provided, sketch the graph of a single function `(x) that satisfies all of the following
properties:
• The domain of the function `(x) includes −8 < x < 8
• limx→−5− `(x) = 2
• limx→−5+ `(x) = 2
• `(x) is not continuous at x = −5
• the slope of the secant line between −4 and −2 is equal to 1
• `(x) = `(−x) for −2 < x < 2
• `(x) has a vertical asymptote at x = 6
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9. A small Toronto coffee shop has reached out to the Math Department to help them model the sales of
their signature hot chocolates throughout the year. They tell us that:
• total daily sales of signature hot chocolate reach a high of $500 per day around February 1
• total daily sales of signature hot chocolate reach a low of only $50 per day around August 1
Let S(t) be a function that models the total daily sales of signature hot chocolate, in dollars per day,
where t is the time in months since February 1. February 1 is the 31st day of the year and August 1
is the 213th day of the year.
(a) Give a 1-sentence reason why a sinusoidal function is a good choice for S(t).
(b) If S(t) is a sinusoidal function, what is the period of S(t)?
(c) Write a formula for the function S(t), assuming that S(t) is sinusoidal.
10. Professor Mayes-Tang has been reading some of your reading recommendations. She was happy to
discover that in two of the books the word “proportional” came up! Choose ONE of the passage below
and write an equation that provides the same information. Remember that you will need to define
your own variables and constants.
Passage 1: “Artificially-induced intelligence deteriorates at a rate of time directly proportional to the
quantity of that increase.” - from Flowers for Algernon by Daniel Keyes
Passage 2: “Personal density is directly proportional to temporal bandwidth.” - Kurt Moudaugen,
quoted in Breaking Bread with the Dead by Alan Jacobs (Aside: “Temporal bandwidth is the width
of your present, your “now.”)
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