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Math 105 Practice Exam 3 Solutions
1. (a)
Evaluate
lim
(
x,y
)
→
(1
,

1)
sin(
x
2
+
y
)
x
2
+
y
or show that it doesn’t exist.
We can use the substitution
u
=
x
2
+
y
, so
u
→
1
2

1 = 0:
lim
(
x,y
)
→
(1
,

1)
sin(
x
2
+
y
)
x
2
+
y
= lim
u
→
0
sin(
u
)
u
= lim
u
→
0
sin(
u
)

sin(0)
u

0
=
d
dt
sin(
t
)
±
±
±
t
=0
= cos(0) =
1
.
Here we used the deﬁnition of derivative,
f
0
(
a
) = lim
x
→
a
f
(
x
)

f
(
a
)
x

a
, in reverse.
(b)
Consider the area function
A
(
x
) =
R
x
1
f
(
t
)
dt
, with
A
(2) = 6
and
A
(3) = 5
.
Compute
Z
2
3
f
(
t
)
dt
.
By the Fundamental Theorem of Calculus,
Z
2
3
f
(
t
)
dt
=
A
(2)

A
(3) = 6

5 =
1
.
(c)
A selfemployed software engineer estimates that her annual income over the next
10
years will steadily increase according to the formula
70
,
000
e
0
.
1
t
, where
t
is the
time in years. She decides to save
10%
of her income in an account paying
6%
annual interest, compounded continuously. Treating the savings as a continuous
income stream over a 10year period, ﬁnd the present value.
PV
=
Z
10
0
7000
e
0
.
1
t
e

0
.
06
t
dt
= 7000
Z
10
0
e
0
.
04
t
dt
= 7000
·
1
0
.
04
e
0
.
04
t
±
±
±
10
0
=
175000(
e
0
.
4

1)
(
≈
86
,
000)
(d)
Draw the level curves of the graph of
f
(
x,y
) = 2
x
2
+
y
2
at the heights
0
,
1
,
2
.
For 1 and 2 it’s an ellipse, for 0 it’s just the point (0
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 Fall '10
 MalabikaPramanik
 Math, Calculus

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