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 = (2x2 + 1)−x .
Solution:
0
2
y = (2x + 1)
−x
(b) Differentiate y = ln
d
4x2
2
2
2
−x
− ln(2x + 1) − 2
(−x) ln(2x + 1) = (2x + 1)
.
dx
2x + 1
(x + 2)2 (x − 1)1/3
.
(x2 + 2)2
Answer: y 0 =
2
1
4x
+
− 2
x + 2 3(x − 1) x + 2
Solution:
d
y =
dx
0
2
1
4x
1
2
+
− 2
2 ln(x + 2) + ln(x − 1) − 2 ln(x + 2) =
3
x + 2 3(x − 1) x + 2
(c) You invest $10,000 in a mutual fund with an annual percentage yield of 8%, compounded
continuously. How many years will it take for your investment to become $25,000? (A
calculator-ready form will suffice.)
Answer:
ln 2.5
ln 1.08
Solution: y(t) = y0 ekt where y0 = 104 . We have
y(1)/y0 = 1 + 0.08 = 1.08.
Thus
ek = 1.08,
k = ln 1.08.
We want y(t) = 25000, thus
25000 = 104 ekt ,
t=
ln 2.5
ln 2.5
=
≈ 11.9 (years).
k
ln 1.08
Long answer question — you must show your work
2. 4 marks The price p (in dollars) and the demand q for a product are related by
2p +
q2
= 160.
400
If the current price per unit is $30, use the price elasticity of demand = E =
whether the revenue will increase or decrease if the price is raised slightly.
Answer: increase
Solution: We have
p
q = 20 160 − 2p,
dq
−20
.
=√
dp
160 − 2p
When p = 30, we have q = 200 and
=
p dq
30 −20
=
= −0.3.
q dp
200 10
Since || < 1, the revenue will increase if the price is raised.
p dq
to decide
q dp