Department of Mathematics, University of Toronto
Term Test 2 – January 18, 2006
MAT 137Y, Calculus!
Time Alloted: 1 hour 50 minutes
Examiners: I. Alexandrova, M. Harada, V. Ivrii, G. Lynch, M. Saprykina, A. Savage
1.
(10%)
(a)
Find a constant
k
for which
lim
x
→
0
sin
x

x

kx
3
x
5
exists and the limit is also nonzero, and determine the value of the limit.
(10%)
(b)
Find the equation of all tangent lines to the curve
y
=
x
2
that intersect the point
(
1
4
,

3
2
)
.
(12%)
2.
A poster is to have an area of 180 square inches with 1 inch margins at the bottom and sides
and a 2 inch margin at the top. What dimensions will give the largest printed area? Make
sure to verify that your answer yields a maximum.
3.
Consider the function
f
(
x
) =
x
x
2
+
9
.
(4%)
(a)
Show that
f
00
(
x
) =
2
x
(
x
2

27
)
(
x
2
+
9
)
3
.
(4%)
(b)
Find the domain, all intercepts, and asymptotes.
(6%)
(c)
Locate all critical points of
f
, determine and clearly indicate the intervals for which
f
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 Spring '08
 UPPAL
 Calculus, Derivative, Mathematical analysis, Inch, Toronto Term Test, camera elevation angle

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