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

1)
sin(
x
2
+
y
)
x
2
+
y
or show that it doesn’t exist.
(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
.
(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.
(d) Draw the level curves of the graph of
f
(
x,y
) = 2
x
2
+
y
2
at the heights 0
,
1
,
2.
(e) Evaluate
Z
1
0
cos(
√
x
)
√
x
dx
.
(f) Let
f
(
x,y
) =
x
+
y
x

y
. Use linear approximation to estimate
f
(2
.
95
,
2
.
05).
2. Evaluate
Z
x
+ 2
x
(
x
2

1)
dx
.
3. Find the area of the region in the ﬁrst quadrant bounded by
y
=
1
x
,
y
= 4
x
, and
y
=
1
2
x
.
4. Find
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This note was uploaded on 01/14/2012 for the course MATH 105 taught by Professor Malabikapramanik during the Fall '10 term at The University of British Columbia.
 Fall '10
 MalabikaPramanik
 Math, Calculus

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