Mathematics

1. (20 pts) Using the deÖnition of Riemann integral Önd the integrals of the following functions

a) f : [0; 1] ! R; f (x) =

1 if x 2 [0; 1] n

1

2

1

if x =

1

2

b) f : [0; 1] ! R; f (x) = 1

n

if x =

1

n

; n 2 N

0 if x 2 [0; 1] n

1

n

j n 2 N

2. (20 pts) Let f : [a; b] ! R be bounded and let M = sup ff (x) j x 2 [a; b]g, m = inf ff (x) j x 2 [a; b]g ;

Mf = sup fjf (x)j j x 2 [a; b]g, me = inf fjf (x)j j x 2 [a; b]g :

a) Show that, for all x; y 2 [a; b] ; jf (x)j

jf (y)j M

m;

b) Show that Mf

me M

m;

c) Show that, if f : [a; b] ! R is Riemann integrable then jfj : [a; b] ! R deÖned by

jfj(x) = jf (x)j

is Riemann integrable.

3. (20 pts) Let f; g : [a; b] ! R be bounded functions and let Mf = sup ff (x) j x 2 [a; b]g, mf =

inf ff (x) j x 2 [a; b]g ; Mg = sup ff (x) j x 2 [a; b]g, mg = inf ff (x) j x 2 [a; b]g :

a) Let M = sup ff (x) + g (x) j x 2 [a; b]g and m = inf ff (x) + g (x) j x 2 [a; b]g. Show that

M Mf + Mg and m mf + mg

b) Show that if f; g : [a; b] ! R are Riemann integrable then f+g : [a; b] ! R; (f + g) (x) = f (x)+g (x)

is Riemann integrable.

4. (20 pts) Let f : [a; b] ! R be an integrable function. DeÖne

f+(x) = (

f(x) if f(x) 0;

0 if f(x) < 0;

and f(x)

= (

0 if f(x) 0;

f(x)

if f(x) < 0:

Prove that f+, f

: [a; b] ! R are integrable and

Z b

a

f(x)dx =

Z b

a

f+(x)dx

Z b

a

f(x)dx:

5. (20 pts) Let f : [a; b] ! R be a Riemann integrable function such that

9 > 0 3 8x 2 [a; b]; f(x) :

Show that the function g(x) = 1

f(x)

is also Riemann integrable on [a; b].

1

Extra Credit Problems

1. (30 pts) Let f : [0; 1] ! R be a di§erentiable function with f(0) = 0 and f

0

(x) 2 (0; 1) for every

x 2 (0; 1). Show that

Z 1

0

f(x)dx

2

>

Z 1

0

(f(x))3dx:

(Hint: Show that the function F(x) = R

x

0

f(t)dt

2

R x

0

(f(t))3dt is increasing.)

2. (30 pts) Compute the following improper integrals

Z 1

0

xe

x cos(x) dx; Z 1

0

xe

x sin(x) dx

PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A GOOD DISCOUNT