Department of Mathematics, University of Toronto
Term Test 2 – January 15, 2007
MAT 137Y, Calculus!
Time Alloted: 1 hour 50 minutes
1.
(9%)
(i)
Find the equation of the tangent line to the graph of
y
=
sec
x

2cos
x
at the point
(
π
3
,
1
)
.
(9%)
(ii)
For the equation
x
2
+
4
xy
+
y
3
+
5
=
0, ﬁnd
d
2
y
dx
2
at the point
(
2
,

1
)
.
2.
(10%)
(i)
A balloon is rising at a constant speed of
5
3
meters per second. A boy is cycling along
a straight road at a speed of 5 meters per second. When he passes under the balloon,
it is 15 meters above him. How fast is the distance between the boy and the balloon
changing three seconds later?
(10%)
(ii)
Consider the function
f
(
x
) =
(
x
2
sin
1
x
,
x
6
=
0
,
0
,
x
=
0
.
.
Prove that
f
is differentiable at 0, and ﬁnd
f
0
(
0
)
.
3.
Consider the function
f
(
x
) =
x
/
(
x

1
)
2
.
(5%)
(a)
Find the domain, the
x
 and
y
intercepts, and asymptotes.
(6%)
(b)
Given that
f
0
(
x
) =

x

1
(
x

1
)
3
, locate all critical points of
f
, determine and clearly indicate
the intervals for which
f
is increasing or decreasing, and classify all critical points as
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 Spring '08
 UPPAL
 Calculus, Derivative, Mathematical analysis, Convex function, Toronto Term Test

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