Math 1100-006
Quiz 3
16 September, 2011
Answer the questions in the spaces provided. If you run out of room for an
answer, you may continue on the back. Show all of your work. Round
answers as appropriate (dollar amounts to the nearest penny, and quantities
to whole numbers where logical). Include units when necessary.
Name:
1. Suppose the total revenue function for a blender is given by
R(x) = 36x − 0.01x2
(a) (6 points) Use the limit definition of the derivative to find the derivative function,
R0 (x).
Solution:
R(x + h) − R(x)
h→0
h
[36(x + h) − 0.01(x + h)2 ] − [36x − 0.01x2 ]
= lim
h→0
h
2
36x + 36h − 0.01x − 0.02xh − 0.01h2 − 36x + 0.01x2
= lim
h→0
h
36h − 0.02xh − 0.01h2
= lim
h→0
h
= lim 36 − 0.02x − 0.01h
R0 (x) = lim
h→0
= 36 − 0.02x
(b) (2 points) Find the marginal revenue when 600 units are sold.
Solution: R0 (600) = 36 − 0.02(600) = 24 dollars per additional unit sold
(c) (2 points) Write a sentence that interprets your answer to part (b).
Solution: The marginal revenue when 600 units are sold is 24 dollars per unit,
which means that if one additional unit is sold (i.e. 601 units instead of 600),
the revenue will increase by approximately 24 dollars.
2. Find f 0 (x) when f (x) =
(x + 1)(x − 2)
.
x2 + 1
Solution: Using the quotient rule:
d 2
d
2
[(x
+
1)(x
−
2)]
−
(x
+
1)(x
−
2)
(x
+
1)
(x
+
1)
dx
dx
f 0 (x) =
(x2 + 1)2
Using the product rule:
d
[(x + 1)(x − 2)] = (x + 1) + (x − 2) = 2x − 1
dx
Substituting in the expression for f 0 (x), we have
(x2 + 1)(2x − 1) − (x + 1)(x − 2)(2x)
(x2 + 1)2
2x3 − x2 + 2x − 1 − 2x3 + 2x2 + 4x
=
x4 + 2x2 + 1
x2 + 6x − 1
= 4
x + 2x2 + 1
f 0 (x) =
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