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Department of Mathematics, University of Toronto
Term Test 2 – January 16, 2008
MAT 137Y, Calculus!
Time Alloted: 1 hour 50 minutes
Examiners: G. Amir, B. Khesin, R. Stanczak, B. Stephens, J. Sylvestre, S. Uppal
(15%)
1.
A farmer with 1500 feet of fencing wants to enclose a rectangular area and then divide it
into four pens (fenced enclosures) with fencing parallel to one side of the rectangle. Find the
largest possible total area of the four pens; be sure to verify that the area is indeed maximized.
2.
(10%)
(i)
Find the values of
a
and
b
such that lim
x
→
0
sin2
x
+
ax
3
+
bx
x
3
=
0.
(10%)
(ii)
The length of a rectangle is increasing at a rate of 7 cm/sec and the width is decreasing
at 2 cm/sec. When the length is 20 cm and the width is 6 cm, determine whether the
area of the rectangle is increasing or decreasing, and ﬁnd the rate at which the area is
changing.
3.
Let
f
(
x
) =
2
√
x

x
.
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This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto Toronto.
 Spring '08
 UPPAL
 Calculus

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