MAT 137Y, 20042005, Solutions to Term Test 2
1.
For the following, simplify your answers unless otherwise instructed.
(8%)
(i)
Evaluate lim
x
→
0
cos5
x

cos3
x
x
2
.
Let
L
be the limit above. Then by L’Hˆopital’s Rule,
L
H
=
lim
x
→
0

5sin5
x
+
3sin3
x
2
x
H
=
lim
x
→
0

25cos5
x
+
9cos3
x
2
=

8
.
(8%)
(ii)
Find the derivative of
f
(
x
) =
√

x
, where
x
<
0, using the
definition of derivative
.
f
(
x
) =
lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
=
lim
h
→
0

(
x
+
h
)

√

x
h
·

(
x
+
h
)+
√

x

(
x
+
h
)+
√

x
=
lim
h
→
0

(
x
+
h
)

(

x
)
h
(

(
x
+
h
)+
√

x
)
=
lim
h
→
0

h
h
(

(
x
+
h
)+
√

x
)
=
lim
h
→
0

1

(
x
+
h
)+
√

x
=

1
2
√

x
.
(8%)
(iii)
Find the equation of the tangent line to the curve 2
(
x
2
+
y
2
)
2
=
25
(
x
2

y
2
)
at the point
(
3
,
1
)
.
Implicitly differentiating gives us
4
(
x
2
+
y
2
)
·
(
2
x
+
2
yy
) =
50
x

50
yy
.
Sticking in
x
=
3,
y
=
1 gives 40
(
6
+
2
y
) =
150

50
y
or 130
y
=

90, so
y
=

9
13
. Hence the
equation of the tangent line is
y

1
=

9
13
(
x

3
)
.
(8%)
(iv)
Use Newton’s Method with
x
1
=
2 to find the next approximation
x
2
to the root of
x
4

20
=
0.
Applying Newton’s Method to
f
(
x
) =
x
4

20, we have
x
2
=
x
1

f
(
x
1
)
f
(
x
1
)
=
2

2
4

20
4
·
2
3
=
2
+
4
32
=
17
8
.
(12%)
2.
A car leaves an intersection at noon and travels due south at a speed of 80 km/h. Another car has been
heading due east at 60 km/h and reaches the same intersection at 1:00 p.m. At what time were the cars
closest together? Verify that your answer is correct by showing that at that given time the distance is
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 Spring '08
 UPPAL
 Derivative, Mean Value Theorem, Mathematical analysis, Convex function

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