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 = (x3 + 4)x+2 .
Solution:
0
3
y = (x + 4)
x+2
d
(x + 2)(3x2 )
3
3
3
x+2
ln(x + 4) +
(x + 2) ln(x + 4) = (x + 4)
.
dx
x3 + 4
√
(x + 1)3 x + 2
(b) Differentiate y = ln
.
(x2 + 1)2
Answer: y 0 =
3
1
4x
+
− 2
x + 1 2(x + 2) x + 1
Solution:
d
y =
dx
0
1
3 ln(x + 1) + ln(x + 2) − 2 ln(x2 + 1)
2
=
3
1
4x
+
− 2
x + 1 2(x + 2) x + 1
(c) You invest $2,000 in a certificate of deposit with an annual percentage yield of 2%, compounded continuously. How many years will it take for your investment to become $3,000?
(A calculator-ready form will suffice.)
Answer:
ln 1.5
ln 1.02
Solution: y(t) = y0 ekt where y0 = 2000. We have
y(1)/y0 = 1 + 0.02 = 1.02.
Thus
ek = 1.02,
k = ln 1.02.
We want y(t) = 3000, thus
3000 = 2000ekt ,
t=
ln 1.5
ln 1.5
=
≈ 20.5 (years).
k
ln 1.02
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
2p2 +
q2
= 2200.
100
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: decrease
Solution: We have
p
q = 10 2200 − 2p2 ,
dq
5(−4p)
.
=p
dp
2200 − 2p2
When p = 30, we have q = 200 and
=
p dq
30 −20 · 30
9
=
=− .
q dp
200 20
2
Since || > 1, the revenue will decrease if the price is raised.
p dq
to decide
q dp