=========================preview======================
(math202)[2007](f)final~PPSpider^_10472.pdf
Back to MATH202 Login to download
======================================================
MATH202 Introduction to Analysis (2007 Fall-2008 Spring)
Extra Practice Problems for Final

You may try to work on the following exercises to get familiar with the materials.

Integral Criterion and Lebesgue Theorem
.Exercise 1
Show that the function f x =|cosx|
is Riemann Integrable on [0,2] using integral criterion. (Hint: Draw graph first)

.Exercise 2
Determine whether the function f x = 2x if x 0,12 x.2 if x 12,1
is Riemann Integrable or not on [0,1] by using a) Lebesgue Theorem, b) Integral Criterion
(Hint: You may draw the figure to help you and here note that f(x) is not continuous at x=12. Use Lebesgue Theorem first to see whether the function is Riemann Integrable or not (It is an important technique!!). Next, when using integral criterion, when you draw your partition, note that the function is discontinuous at x=12, so you need to first draw a .interval around x=12 and for the remaining part, apply uniform cutting (since the function is continuous)

.Exercise 3
Determine whether the function defined by f x = x if x.. 0 if x..\..
is Riemann Integrable or not on [0,1] by using a) Lebesgue Theorem, b) Integral Criterion .



.Exercise 4
Let f1 x ,f2 x ,f3(x) be bounded Riemann Integrable function on [a,b] and define h x = f1 x if x a,c1 f2 x if x c1,c2 f3 x if x(c2,b]
where a<c1<c2<b. (Draw the simple graph).
Determine (using BOTH a) Lebesgue Theorem, b) Integral Criterion ) whether h(x) is Riemann Integrable on not

Hint: In part b), when you draw the partition, since h(x) may be discontinuous at x=c1 or x=c2. So you first draw .intervals around x=c1 and x=c2 respectively. Then for other parts, since f1, f2 and f3 are Riemann Integrable, there exists P1, P2,P3 on [0,1] such that U Pi,fi .L Pi,fi <5 for i=1,2,3
Draw you partition by fill in the blank below
---------------[-----------------(--x--)--------------------(--x--)---------------------]


a c1 c2




h x =f1(x) h x =f2(x) h x =f3(x)
(Pick P1) (Pick P2) (Pick P3)
.interval .interval

.Exercise 5
Let f(x) be Riemann Integrable on [0,2] and g(x) is monotone function on [0,2], define h x = f x +g(x) if x[0,,1) f x if x[1,2]
Determine with proof whether the f(x) is Riemann Integrable on [0,2]

Improper Integral and Cauchy principal Value
.Exercise 6
Determine the convergence of the following integrals
a) dx x x+1 0

b) dx 4.x22.2

c) x+1 (x12.3 x+230dx


d) 1sinxdx20 (Hint: limx0+..........=1)

e) xsin ex dx0 (Hint: Let u=ex)



.Exercise 7
Determine all possible such that the improper integral 1x sinx ....0
Converges.
(Hint: Since there are two trouble points (at x=0,x=), so we split the integral into 2 parts: 1x sinx ....