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University of Toronto
MAT 137Y1Y
Calculus!
April/May Examinations; April 29, 2003
Time Alloted: 3 hours
Instructors: B. Begun, J. Colliander, K. Consani, A. del Junco, J. Korman, R. Rotman, D. Slepcev, R. Sreekantan, S. Uppal
1.
(a)
Find the derivatives of the following functions.
(4%)
(i)
f
(
x
) = (
sin
x
)
2
x
.
(4%)
(ii)
f
(
x
) =
Z
x
2
0
(
1
+
t
2
)
1
/
3
dt
.
(5%)
(b)
Find the equation for the tangent line to the curve
y
3

x
3
y
+
x

1
=
0 at the point
(
1
,
1
)
.
2.
Evaluate the following integrals.
(7%)
(i)
Z
e
2
x
sin3
x dx
.
(7%)
(ii)
Z
4
0
u
1
/
2
1
+
u
1
/
2
du
.
(7%)
(iii)
Z
2
x
2

2
x
+
1
x
3

2
x
2
+
x
dx
.
(10%)
3.
One end of a rope 20 metres long is attached to a box resting on the ﬂoor. The other end
is passed over a pulley directly above the box, 5 metres above the ﬂoor, and attached to the
back of a truck at a point one metre above the ground. The truck then drives in a straight line
away from the pulley at a speed of 0.5 metres per second. At what speed is the box rising
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 Spring '10
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 Derivative

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