UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
MAT 235 Y - CALCULUS II
TEST #1. OCTOBER 30, 2008.
PROBLEMS AND SOLUTIONS.
1. Let
C
be the curve defined by the parametric equations
x
= 4
t
–
t
2
and
y
= 2
t
3
.
a) (10 marks) Find the coordinates of each of the points on the curve
C
where the tangent line has slope
3 .
Solution:
We just have to find the points on
C
where
3
dy
dx
=
. Computing the derivates, we obtain
2
/
6
/
4
2
dy
dy dt
t
dx
dx dt
t
=
=
−
.
So, we need
6
t
2
= 12 – 6
t
or
t
2
+
t
– 2 = 0 . That is:
t
= – 2
or
t
= 1 .
The points on the curve
C
where the tangent has slope 3 are ( – 12 , – 16 )
and
( 3 , 2 ) .
b) (10 marks) Find the values of
t
for which the curve
C
is concave upward.
Solution:
We just have to find the values of
t
for which
2
2
0
d
y
d x
>
. Again, computing the corresponding derivatives, we
obtain
2
2
3
3 (4
)
2(2
)
d
dy
d
y
t
t
dt
dx
dx
d x
t
dt
⎛
⎞
⎜
⎟
−
⎝
⎠
=
=
−
. So, we need
3
3 (4
)
0
2(2
)
t
t
t
−
>
−
and the values of
t
for which this condition
holds are
0 <
t
< 2
or
t
> 4 .

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