MATH 147
Quiz Key #5
3/3/2016
(1) Suppose that water is stored in a circular cylindrical tank of radius 5 m. If
the water is drained at a rate of 250 cubic meters per minute, what is the rate
at which the water level in the tank decreases?
Solution: First, we need to write an equation for the relationship between the
water volume V, the water radius r, and the water level h. This is the equation
for the volume of a cylinder:
V = πr2 h.
However, note that the radius is constant (5 m). Hence,
V = 25πh.
Now differentiating with respect to time, we see that
dh
dV
= 25π .
dt
dt
dh
.
dt
dh
10
.
∴− =
π
dt
∴ −250 = 25π
Therefore, the water level decreases at a rate of
1
10
π
m/min.
(2) Find the first and second derivatives of g ( x ) =
answer using positive exponents.
√
4x3 + 2x. Express your
Solution:
1/2
g ( x ) = 4x3 + 2x
.
−1/2 d h
i
1 3
4x + 2x
·
4x3 + 2x .
2
dx
−
1/2
1
6x2 + 1
∴ g0 ( x ) =
4x3 + 2x
12x2 + 2 =
.
1/2
2
(4x3 + 2x )
∴ g0 ( x ) =
Now, differentiating again using the quotient and chain rules:
h
2
i
d
3 + 2x 1/2 − 6x2 + 1 · d
3 + 2x 1/2
6x
+
1
·
4x
4x
dx
dx
g00 ( x ) =
.
3
4x + 2x
00
12x 4x3 + 2x
1/2
∴ g (x) =
− 6x2 + 1 ·
6x2 +1
1/2
(4x3 +2x )
4x3 + 2x
00
∴ g (x) =
·
2
12x 4x3 + 2x − 6x2 + 1
(4x3 + 2x )
3/2
4x3 + 2x
1/2
(4x3 + 2x )
1/2
.
.
2