Elementary Analysis Math 140B—Winter 2007 Homework answers—Assignment ...

[Pages:1]Elementary Analysis Math 140B--Winter 2007 Homework answers--Assignment 26; March 19, 2007

Exercise 32.6

Let f be a bounded function on [a, b]. Suppose there exist sequences (Un) and (Ln) of upper and

lower Darboux sums for f such that lim(Un - Ln) = 0. Show f is integrable and

b a

f

=

lim Un

=

lim Ln.

Solution: We are given that Un = U (f, Pn) and Ln = L(f, Qn) for certain partitions Pn and Qn of [a, b]. Since

U (f, Pn Qn) - L(f, Pn Qn) Un - Ln 0

it follows that for any > 0, there is a partition P = Pn Qn for some n such that

U (f, P ) - L(f, P ) < .

Hence f is integrable on [a, b].

From Ln L(f ) = U (f ) =

b a

f

Un

we

have

0

b a

f

- Ln

Un

- Ln

and

Ln

- Un

b a

f

0

and therefore limn Ln = limn Un =

b a

f

.

Exercise 33.8 Let f and g be integrable functions on [a, b]

(a) Show that f g is integrable on [a, b].

Solution: Since 4f g = (f + g)2 - (f - g)2 and Exercise 33.7 states that the square of an integrable function is integrable, using linearity (Theorem 33.3) it follows that f g is integrable.

(b) Show that max(f, g) and min(f, g) are integrable on [a, b].

Solution: Since min(f, g) = (f + g)/2 - |f - g|/2 and max(f, g) = - max(-f, -g), using Theorem 33.5 and linearity (Theorem 33.3), it follows that max(f, g) and min(f, g) are integrable.

Exercise 34.2

Calculate

(a)

limx0

1 x

x 0

et2

dt

Solution: Let F (x) = 1.

x 0

et2

dt.

Then

limx0

1 x

x 0

et2

dt

=

limx0

F (x) x

=

F

(0)

=

ex2 |x=0

=

(b)

limh0

1 h

3+h 3

et2

dt

Solution: Let g(x) =

x 3

et2

dt

so

that

g(3)

=

0

and

g

(x)

=

ex2 .

Then

limh0

1 h

limh0

g(3+h)-g(3) h

=

g

(3)

=

e9.

3+h 3

et2

dt

=

Exercise 34.9

Use Example 3 to show

1/2 0

sin-1

x

dx

=

/12

+

3/2

-

1

Solution: Take a = 0, b = /6 and g(x) = sin x in Example 3. Then

/6

sin x dx +

1/2

sin-1 x dx =

1 ?=

0

0

6 2 12

and

/6 0

sin

x

dx

=

-

cos

x

|0 /6

=

3

- 2

+

1.

1

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