MATHEMATICS 184 (Section 921) - TERM EXAM #2
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STUDENT ID NUMBER:
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INSTRUCTIONS: Notes, books and calculators are not permitted. Answers may
receive no credit if accompanying work is not provided. When indicated, simplify
answers as much as possible, and be sure to clearly indicate the final answer for
each question. Please use the back of the pages if you need extra space.
1. John Dough has now decided to invest his millions in the stock market. He purchases shares
in a new paperclip company, ACME Paperclips Inc., and over the next 100 days John tracks
the price of ACME Paperclips shares.
2
(a) If P(t) is the price, in dollars, of a single share t days after John has purchased his shares,
what are the units of P0 (t)?
2
(b) What does P0 (t) represent?
(c) What is happening to the share price and consequently to John’s investment when:
1
i. P0 (t) > 0?
1
ii. P0 (t) < 0?
1
iii. P0 (t) = 0?
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(d) At the end of the 100 days, John graphs the share price over time and gets the following
graph. Using this graph, fill in the following table with the signs (+,- or 0) of P(t), P0 (t),
and P00 (t) at the indicated points on the graph.
1.8
P(t) - Share Price (dollars per share)
6
C
1.6
1.4
1.2
A
1
B
0.8
0.6
D
0.4
0.2
0
0
20
40
60
t (days since purchase)
80
100
Point Sign of P(t) Sign of P0 (t) Sign of P00 (t)
A
B
C
D
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7
2. (a) Using the limit definition of the derivative (no shortcuts) differentiate the function
(Simplify your answer)
4x − 1
f (x) =
3x + 7
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4
(b) Using the quotient rule, differentiate the same function
(Simplify your answer)
4x − 1
f (x) =
.
3x + 7
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10
3. Find all the points on the curve
y = ex (x2 − 3)
at which the tangent line is horizontal.
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4. Differentiate the following functions: (Do not simplify your answers)
2
(a) f (x) = 7x7 + 5x3 − 2x + 8
2
(b) y = 3ex+7 + xe − π
3
(c) y = (x7 − x6 + x3 − 2x + 3)(2x4 + 6x3 − 3x2 + 1)
4
(d) g(t) =
2
t−5
et (t 2 +2)
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