MAC 2311
Dec 13, 2006
Final Exam and Key
Prof. S. Hudson
1) (20 pts) Compute and simplify. Include an arbitrary constant, if appropriate;
3
x
2
1+
x
3
dx
=
3
1+
x
2
dx
=
sin
2
t dt
=
e
3
x
+2
dx
=
2) (20 pts) Compute;
d
dx
x
x
d
dx
(cos 2
x
tan 2
x
)
d
dx
log
2
x
3
Find
dy/dx
, given that
xy
2
+
x
2
y
=
y
.
3) (10 pts) Solve this initialvalue problem; find a formula for
y
(
t
) so that
y
(
t
) = 6(
t
+ 1)
2
and
y
(0) = 7.
4) (10 pts) Compute;
lim
x
→
0
+
(1 + 3
x
)
1
/x
lim
x
→
1
ln(
x
)
sin(
πx
)
5) (10 pts) Answer TRUE or FALSE:
The slope of the graph of tan

1
(
x
) is positive for all
x
.
The domain of sin

1
(
x
) is [

1
,
1].
Every polynomial has exactly one antiderivative whose graph contains the point (3,4).
The function
f
(
x
) = sec(2
x
) has an inverse on [0
, π/
6].
If
F
is an antiderivative of an antiderivative of
f
, then
F
(
x
) =
f
(
x
).
6) (10 pts) Given parametric equations,
x
(
t
) =
t
2
+ 1 and
y
(
t
) =
t
3
+ 1, find the equation
of the tangent line at the point where
t
= 2.
7a) (5pts) State the Intermediate Value Theorem.
1
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7b) (5pts) State the Extreme Value Theorem.
8) (10 pts) CHOOSE ONE;
A) Prove the Product Rule.
B) Prove (as in Ch2) that lim
x
→
0
sin
x
x
= 1
C) Prove (using
) that lim
x
→
0
4
x
+ 7 = 7.
Bonus (5pts): Use Newton’s method to approximate
√
3 (a positive root of
x
2

3 = 0),
starting from
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 Calculus, Derivative, pts, Continuous function, 5pts

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