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 = (2x3 + x)x .
Solution:
d
x(6x2 + 1)
3
3
3
x
y = (2x + x)
x ln(2x + x) = (2x + x) ln(2x + x) +
.
dx
2x3 + x
0
3
x
√
(x − 1)2 x + 2
√
(b) Differentiate y = ln
.
x2 + 1
Answer: y 0 =
2
1
x
+
− 2
x − 1 2(x + 2) x + 1
Solution:
d
y =
dx
0
1
1
2
1
x
2
2 ln(x − 1) + ln(x + 2) − ln(x + 1) =
+
− 2
2
2
x − 1 2(x + 2) x + 1
(c) I invest $5,000 in an RESP account with an annual percentage yield of 4%, compounded
continuously. How much will the balance be after 18 years when my son is ready for
college? (A calculator-ready form will suffice.)
Answer: 5000e18 ln 1.04 = 5000(1.04)18
Solution: y(t) = y0 ekt where y0 = 5000. We have
y(1)/y0 = 1 + 0.04 = 1.04.
Thus
ek = 1.04,
k = ln 1.04.
We want y(18) which is
y(18) = 5000e18 ln 1.04 = 5000(1.04)18 ≈ 10129.1 (dollors).
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
= 2200.
20
If the current price per unit is $100, 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
q=
p
44000 − 40p,
−20
dq
=√
.
dp
44000 − 40p
When p = 100, we have q = 200 and
=
100 −20
p dq
=
= −0.05.
q dp
200 200
Since || < 1, the revenue will increase if the price is raised.
p dq
to decide
q dp