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Methods of Calculus
MAC3233
Test 1
January 31, 1996
Instructions:
Do not write your answers on this paper, use separate answer sheets. This is a closed
book test. No calculators are allowed to be used. To receive credit, you must show all work. You are
not
required to simplify your answers.
1. (10 pts.) If
f
(
x
) =
1
√
x
, compute
f
(4) and
f
0
(4). Find the equation of the tangent line to the graph
of
y
=
f
(
x
) at the point where
x
= 4.
2. (10 pts.) Find
dy
dx
, if
y
=
x
+ 1
79
.
3. (10 pts. each) Diﬀerentiate:
(a)
f
(
x
) =
5
x
+ 8
x

3
(b)
f
(
x
) =
(
3

2
x
+
x
5
)
2
3
(c)
f
(
x
) =
8
x
2

2
x
+ 161
4. (10 pts.) Suppose a ﬁrm’s proﬁt from producing and selling
x
units of a product is given by
P
(
x
) =
32
x
2

9
x
3
thousands of dollars. Find the marginal proﬁt at a production level of 8.
5. (10 pts.) An object is thrown upward. Suppose
s
(
t
) = 4 + 100
t

16
t
2
is the height of the object, in
feet, after
t
seconds.
(a) Find the velocity function
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