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Q1 (a)

1
2
, (b)
1
3
, (c)

π
2
, (d)
1
3
, (e)

5
2
.
Q2 (a)
y
=
π

1
,
y
=

π

1
, (b)
x
=

2
,
x
= 2
, (c)
5
288
π
cm/s.
Q3 (a)
e
x
, (b)

u
√
1

u
2
, (c)

2 + cosh(
x
)
, (d)
14
25
.
Q4 (a) When
x
=

1
,
y
=

6
. When
x
= 0
,
y
= 1
. Since
y
depends continuously on
x
, by
the Intermediate Value Theorem,
y
must vanish at least once in

1
< x <
0
. On the other hand
y
0
= 2 + 3
x
2
+ 20
x
4
≥
2
. If
y
were to vanish at two distinct points
x
=
a
and
x
=
b
, then by
Rolle's Theorem,
y
0
would vanish at some point between
a
and
b
. It doesn't. Hence there is at most
one solution in
x
to the equation
y
= 0
.
(b)
K
= 2
.
Q5 At
x
= 1
,y
= 2
,
y
0
=

1
and
y
00
=

32
9
. The linear approximation is 2.03, the quadratic
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This note was uploaded on 01/11/2011 for the course MATH MATH 140 taught by Professor Drury during the Fall '10 term at McGill.
 Fall '10
 Drury

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