MAT 16A
(A002)
NAME:
March 20, 2003
1
2
3
4
total
Problem 1.
(estimated time: 25mn ) (50 points)
(a) Which of the following lines are parallel? Which of them are perpendicular?
L
1
:
1
3
x

1
2
y
+ 1 = 0
L
2
:
y
=

3
2
x
+ 5
L
3
:
x

3
y
+ 3 = 0
L
4
passing by the points of coordinates (1
,
1) and (4
,
3)
.
(b) Find the standard form of the circle of equation
1
4
x
2
+
1
4
y
2
=
y

x
+ 2
,
and determine its center and its radius.
(c) Find the standard form of the translation of the circle in question (b)
two units left
and
one unit up
.
(d) Find the following limit
lim
x
7→
1
√
x
2
+
x
+ 3

√
5
x

1
.
Problem 2.
(estimated time: 30mn ) (60 points)
(i) Find the second derivatives of the following functions and simplify your results as much
as possible.
(a)
f
(
x
) =
x
2

2
x
2
+ 3
.
(b)
f
(
x
) =
x
2
cos(3
x

1)

sin(
x
2
+ 1)
.
(ii) Using the implicit differentiation find
dy/dx
for the following relations.
(a)

y
3
+
x
3
=
x
2
y
2

1
(b)
y
2
cos
x

x
sin
y
= 0
.
(iii)
The radius
r
of a sphere is increasing at a rate of 3 inches per second. Find the
rate of change of the volume when
r
= 5 inches.
Problem 3.
(estimated time: 40mn ) (70 points) Consider the function defined by
f
(
x
) =
2
x
x
2
+ 1
+ 1
.
(a) Find the domain of
f
.
(b) Find the
x
intercepts and
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 Spring '09
 Zhu
 Calculus, Derivative, Standard form, Implicit Differentiation Find

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