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Haiman
Math 1A—Calculus
Fall, 2006
Second Midterm Exam Solutions
Name
Student ID
Discussion Section (Time and GSI)
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are 8 questions, on front and back. Write answers on the exam and turn in only this paper.
Show enough work so that we can see how you arrived at your answers.
1. Find the equation of the tangent line to the hyperbola
x
2
+ 2
xy

y
2
+
x
= 9 at the
point (2
,
1).
Diﬀerentiate implicitly with respect to
x
, to get
2
x
+ 2
y
+ 2
xy
0

2
yy
0
+ 1 = 0
.
Set
x
= 2,
y
= 1, and solve for
y
0
=

7
/
2. The equation of the tangent line is then
y
= (

7
/
2)(
x

2) + 1 = 8

7
x/
2
.
2. Diﬀerentiate the function (ln
x
)
cos
x
.
Using logarithmic diﬀerentiation,
f
0
(
x
) =
f
(
x
)
d
dx
ln
f
(
x
) = (ln
x
)
cos
x
(
cos
x
x
ln
x

(sin
x
)(ln ln
x
)
)
3. The surface area of a melting spherical ball of ice decreases at a rate of 1 cm
2
/
min.
How fast is the volume decreasing when the radius of the ball is 10 cm? For your information,
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This note was uploaded on 10/31/2010 for the course MATH 53603 taught by Professor Christ during the Fall '08 term at University of California, Berkeley.
 Fall '08
 CHRIST

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