MATH 115 Test 2 (Sec: 1.5 - 2.1) Version 1
Sec Number:
Name: KEY
Answer the following questions to the best of your ability showing all appropriate work to obtain full
credit. Questions that involve word problems should be answered using a sentence, and appropriate
units provided, in order to receive full credit.
1. (15 pts) For each of the following functions find an expression for the functions instantaneous
rate of change.
• H(x) = (x8 + 9x)16
H 0 (x) = 16 x8 + 9x
• G(x) =
√
4
15
8x7 + 9
10x + 6
1
G(x) = (10x + 6) 4
3
1
G0 (x) =
(10x + 6)− 4 10
4
3
5
0
(10x + 6)− 4
=⇒ G (x) =
2
• f (x) =
8x−14
17x7 +1
18
f 0 (x) = 18
8x − 14
17x7 + 1
17
!
17x7 + 1 (8) − (8x − 14) 119x6
(17x7 + 1)2
• g(x) = (15x − 12)13 (7 − 9x)12
g 0 (x) = 13 (15x − 12)12 (15) (7 − 9x)12 + 12 (7 − 9x)11 (−9) (15x − 12)13
= 195 (15x − 12)12 (7 − 9x)12 + −108 (7 − 9x)11 (15x − 12)13
• h(x) =
q
3
(x+8)
9x18
1
(x + 8) 3
9x18
2
1 (x + 8) − 3
3
9x18
h(x) =
h0 (x) =
!
9x18 (1) − (x + 8) 162x17
(9x18 )2
2. (6 pts) Given the function f (x) = 2x8 + 6x4 + 3x2 compute:
(a)
df (x)
dx :
f 0 (x) = 16 x7 + 24 x3 + 6 x
(b)
d2 f (x)
:
dx2
f 00 (x) = 112 x6 + 72 x2 + 6
(c) f 000 (x):
f 000 (x) = 672 x5 + 144 x
3. (10 pts) Given the following graph of the function f (x) sketch the derivative f 0 (x) in the space
provided.
3
2
1
-4
-2
2
-1
-2
-3
4
6
4. (5pts) Differentiate the function: s(t) =
s0 (t) = −
3t4
(7−3t)5
45 t4
12 t3
+
(3 t − 7)5 (3 t − 7)6
Or with out simplification:
(7 − 3t)5
ds(t)
=
dt
12t3 − 3t4 5(7 − 3t)4 (−3)
((7 − 3t)5 )2
5. (9pts) Given that
F (2) = 6; F 0 (2) = 7; H(2) = 4; H 0 (2) = 5
find:
(a) G0 (2) if G(z) = F (z)H(z)
G0 (2) = F 0 (2)H(2) + F (2)H 0 (2)
=⇒ G0 (2) = 58
(b) G0 (2) if G(w) = F (w)/H(w)
H(w)F 0 (w) − F (w)H 0 (w)
(H(w))2
1
= −
8
G0 (2) =
(c) G0 (2) if G(w) = 4 (F (w))3
G0 (2) = 12 (F (2))2 F 0 (2)
= 3024
= 3024.0
6. (8 pts) Given the function f (x) =
6x7
(7−6x)4
find the equation of the line tangent to the graph
of f at x = 2.
Here we need the derivative of f to give the proper slope for the tangent line. We also need
the value of the function at the given point of interest. This gives a point and a slope to use
with point slope form of a line to find the requested equation.
f 0 (x) =
144 x7
42 x6
−
(6 x − 7)4 (6 x − 7)5
4992
= −1.597
3125
768
Known Point (x∗ , y∗ ) = (2,
) = (2, 1.229)
625
m = f 0 (2) = −
y = m(x − x∗ ) + y∗
=⇒ y = −
4992
13824
x+
3125
3125
7. (10 pts) A software company estimates that it will sell N units of a program after spending a
dollars on advertising, where
N (a) = −a2 + 300a + 6, 0 ≤ a ≤ 300,
and a is in thousands of dollars. Find the maximum number of units that can be sold and the
amount that must be spent on advertising in order to achieve that maximum. Be sure to use
your knowledge of Calculus to justify that the value you found is a maximum.
Find the critical points:
N 0 (a) = 0 =⇒ −2a + 300 = 0
=⇒ a = 150
Using the second derivative N 00 (a) = −2 we know that N (a) is always concave down. Thus,
a = 150 is a local maximum.
We can conclude that the maximum number of units that can be sold is 22,506 when $150,000
is spent on advertising.