MATH 184 Quiz 2’
October 30
Grade:
First Name:
Last Name:
Student-No:
Section: 184-104
Short answer questions — you must show your work
1. 6 marks Each part is worth 2 marks.
(a) Differentiate y = (x2 + 3)x .
2x2
)
Answer: (x + 3) (ln(x + 3) + 2
x +3
2
x
2
Solution:
2x2
d
2
2
x
2
x ln(x + 3) = (x + 3) (ln(x + 3) + 2
)
y = (x + 3)
dx
x +3
0
(b) Differentiate y = ln
2
(x2
x
2x
.
+ 1)3
Answer: y 0 =
Solution:
y0 =
1
6x
− 2
x x +1
1
d
6x
ln 2 + ln x − 3 ln(x2 + 1) = − 2
dx
x x +1
(c) You invest $100,000 now at an annual percentage yield of 5%, compounded continuously.
How many years will it take for your investment to become $300,000? (A calculator-ready
form will suffice.)
Answer:
ln 3
ln 1.05
Solution: y(t) = y0 ekt where y0 = 105 . We have
y(1)/y0 = 1 + 0.05 = 1.05.
Thus
ek = 1.05,
k = ln 1.05.
We want y(t) = 300000, thus
3 · 105 = 105 ekt ,
t=
ln 3
ln 3
=
≈ 22.5 (years).
k
ln 1.05
Long answer question — you must show your work
2. 4 marks Currently 1800 people ride a commuter passenger ferry each day and pay $4 for
a ticket. The number of people q willing to ride the ferry at price p is determined by the
relationship
2
q − 3000
, (q < 3000).
p=
600
The company would like to increase its revenue. Use the price elasticity of demand to give
advice to management on whether it should increase or decrease its price from $4 per passenger.
p dq
Recall that = E =
.
q dp
Answer: increase the price
Solution: We have
−p1/2 =
Thus
q − 3000
,
600
q = 3000 − 600p1/2 .
dq
= −300p−1/2 . When p = 4, we have q = 1800 and
dp
=
4
1
p dq
=
(−300/2) = − .
q dp
1800
3
Since || < 1, it is inelastic. The company can increase the price to increase its revenue.