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UNIVERSITY OF TORONTO
DEPARTMENT OF MATHEMATICS
MAT 235 Y – CALCULUS II
FALLWINTER 200809
ASSIGNMENT #1. DUE ON OCTOBER 3.
PROBLEMS AND SOLUTIONS
1.
A brief review of single variable calculus.
a) Evaluate
( )
2
0
lim
9 17ln
9ln
3 1 ln
ln
x
2
x
xx
+
→
++
−
+
+
x
, if the limit exists.
Solution:
Let
2
91
7
l
n 9
l
n
A
=+
+
and
2
31l
n l
n
B
+
.
Now,
22
AB
−
−=
+
, where
.
2
2
7
l
l
n
9
(
1l
n ) 8
l
n
x
x
x
x
+
+
−+
+
=
x
Notice that
7
7
9 17ln
ln
(
9)
ln
9
ln
ln
ln
ln
Ax
x
x
x
+
=
+
+=
+
+
and similarly,
2
11
3ln
1
ln
ln
Bx
x
x
+
. So,
7
1
1
ln
(
9
3
1 )
ln
ln
ln
ln
x
+ ++
+ +
.
Notice also that
and
0
lim ln
x
x
+
→
=−∞
ln
ln
x
x
=−
when
0 <
x
< 1 .
Therefore,
()
00
8ln
8
4
lim
lim
33
3
7
1
1
ln
(
9
3
1 )
ln
ln
ln
ln
x
x
→→
−
=
+
+
.
b) Let
P
(
t
,
f
(
t
) )
be any point on the curve
(1
)
x
fx
x
=
+
and let
Q
be the point at which the
y
axis
intersects the line that is tangent to the given curve at the point
P
. Suppose that
A
(
t
)
denotes the area of the
triangle
OPQ
, where
O
= ( 0 , 0 ) . Is there an absolute maximum value for the area
A
(
t
) ? If so, find this
absolute maximum value and determine the coordinates of all the points
P
for which that maximum is reached.
Solution:
An equation of the line that is tangent to the given curve at the given point is
y
–
f
(
t
) =
f
′
(
t
) (
x
–
t
) .
If
x
= 0
then the value of
y
in this equation is
y
=
f
(
t
) –
t
f
′
(
t
) . So,
Q
is the point with coordinates
( 0 ,
f
(
t
) –
t
f
′
(
t
) )
and
OPQ
is a triangle whose base has a length of
f
tt
f
t
′
−
and whose height has
a length of
t
. Therefore, the area of the triangle
OPQ
is
1
( ()
)
2
At
t ft tf t
′
.
Now,
2
23
13
)
x
x
−
′
=
+
, then
24
1(
1
3
)
(
)
2(
1)(
1)
(
t
t
t
−
=
+
2
t
.
Finally,
32
4(
2 )
)
t
−
′
=
+
which shows that
A
(
t
)
reaches its absolute maximum at
2
t
=±
.
The absolute maximum value for the area of the triangle
OPQ
is
8
(2
)
27
A
±=
and this maximum area is
reached only when the tangency point
P
has coordinates
2
,
9
±
±
)
.
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2. More review on single variable calculus.
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This note was uploaded on 10/16/2009 for the course MATH MAT235 taught by Professor Recio during the Fall '08 term at University of Toronto Toronto.
 Fall '08
 Recio
 Calculus, Multivariable Calculus

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