## Journal

Practice Final
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HW 13
will not be graded. Work on it in the help rom and clar leftover questions in  class review.
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HW 13 (due 12/6 at 4:30 pm - the last HW!)

Page 508, A1, 2, 5 (this one was in a past conseling exam), 7, 9,
B 2, 5 (this one was in a past conseling exam), 6.

More on Friday...

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Review for the final (bring your questions!):
We 12/15, 11:50 AM - 1:40 pm, CW 120.

No class on Fri 12/10.
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HW 12 (due 11/29 at 4:30 pm):

Page 166 C 1 (Hint: use a phi such that f  times phi is both continuous and nonnegative, go from there then.Several choices will work.)
C 2. (I f has a positive lower bound you can pull it off in at least two different ways - either with upper and lower sums, or with
Riemann sums.) Can you pull it off without requesting a positive lower bound for f, nor continuity of f?
C 3.

Prove thm 5.7 b. Try to do it without reading the proof in the book. Observe that lemma 5.7 and corollary 5.7 c are immediate consequences of it.
Check that your proof also proves thm. 5.7 f.

Examples and counterexamples:
1) Prove, using  g(x)= sin x in K= integral over [0, 2 pi] of x sin x dx, that thm 5.6 c does *not* hold if g(x) changes sign within the inteval of
integration.

2) Where is the problem in the following integrals computed by substitution?
a) over [0,pi], of t^2 cos t dt , with the substitution x=sin t , equal to the integral form 0 to 0 of  (arc sin x )^2 =0.
b) over [-1,1] of dx =2; with the substitution x^2=t, we get 2=integal from 1 to 1 of (dt / 2 root of t) =0.

Page 172, B 3, 4, and 5. (For #5, you want to "enclose" N in a collection of intervals whose lengths add up to no more than epsilon - you end up
with a series. Borrow ideas from the proof of thm. 5.7 e to come up with a collection of intervals that will do the job.)

HW 11 (due 11/17 at 4:30 pm):

Show that if f is integrable on [a,b]  then so is f^-. (The part we did not do in class of thm. 5.4 g - tweak that proof)

Page 164, A 5 and 6 (very easy);  7 (very short), 8.
B 2: only th. 5.4. f (we did half of it in class).
B 3 (make a picture! That should tell you).
B 5 (hint: combine the fundamental thm of calculus with splitting into positive and negative part).
B 7 (very short).
B 11 (this has been in at least 2 counseling exams).

HW 10 (due 11/10 at 4:30 pm):

Page 150, A 1 d) (this function is continuous, therefore integrable, so you can choose partitions that let you use partial sums of a geometric
series).

C 1 (use the idea in HW 9 #6); C 2 (tricky!!!), C 6 (can you see this a convergence estimate for a numerical method to compute integrals?) What norm of
partition would let you compute the integral over [1,2] of x^2 with error less than 1/10?

Complete the proof  of Lemma 5.3d, page 144 by proving the inequality we did not do in class.

A 2 Page 150.

HW 9 (due 11/3 at 4:30 pm):

1. Use Taylor's thm. to show that for 0<x<pi/2,

x - x^2 /6 < sin x < x.

2. Use Taylor's thm. to find the limit when x tends to 1 of
(x^x -1)/ ln x.

Page 129, B 1, A 1 k), l).

3. How large should the degree n be chosen for the Taylor polynomial p_n (x,0) to have
|e^x - p_n(x,0)|< 10^(-5),  for all x in [-1,1] ?  (this is "epsilon/delta"  work)...

4. Suppose the lower integral of f over [a,b] is positive. Show that there exist a subinterval J of [a,b]
on which f has a strictly positive lower bound. (Somewhat tricky. Use def. of sup like we did today in
class, and lower sums; f does not need to be positive).

5. Consider the function defined on [0,1] as f(x)=1 for x rational, f(x)=0 for x irrational.
Show that the upper integral  of f over [0,1] is 1, and the lower integral 0.

6. Consider the function f(x)=1 if x is a natural number, f(x)=0 otherwise. Given n in N, construct partitions
P_n such that the difference of the upper sum minus the lower sum over the intervals a) [0,5], b) [0,50],
and c) [0,M], where M>0, is smaller than 1/n. (Hint: you want to cut out little intervals around the points where
the function jumps, in such a way that the sum of the little rectangles corresponding to the jump adds up to less
than 1/n. How many reactngles will you have, and how tall are they? Then adjust the width to get what you
want).

HW 8 (due 10/27 at 4:30 pm):

Page 110, B 11, B 12 (compare with p. 73, A 1:  what can you say?) .
B 17 (this one has been in a past counseling exam...).
Page 119, B 1, B 2, B 4 (also give an interpretation of this result).
Page 118, A 1 h), k),  A 2 (this too has been in a past conseling exam...).
Page  124, A 1 d), j), A 2 a) (use uniqueness and a polynomial you know, rather
than compute the derivaties).
A 1 j)  ("expressing a polynomial p(x) in powers of (x-a)...")
B 5 a,), b), d). Give an interpration of this (it tells that a certain sequence converges to a certain limit).

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To the midterm you can bring an index card written both sides with whatever notes you want to make
that you think you might need to use during your exam.
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HW 7 (due 10/20 at 4:30 pm):

1. Show that a function that is continuous on a closed interval achieves its minimum at a point
in that interval.

Page 108: #A.3; A.8; A.10, B.1 (likely tricky); B.10.

Practice test

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The midterm will be on 10/15.
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HW 6 (due 10/6 at 4:30 pm):

Page 84, B.4 (nontrivial!) Steps: 1. Show that f(a,b) is bounded (using uniform continuity).
2. You need a candidate for, say, f(a+0). For this, approximate a with a sequence in (a,b), and
exploit thm 3.5.a.  3. Show that f(x) tends to this candidate as x decreases toward a. You may try by
contradiction, and using that f is uniformly continuous.

Page 83, A.1.a and A.3.a. (These are tame).

Page 84, B. 3 (you have seen this equence before and proved that it converges uisng a diffrent result),
B. 9. This result is very important, this series is used to compare and decide divergence
of many series and improper intergrals.

Read  carefully examples 1 and 2, page 81- 83.

I will skip section 3.7. If we need it later, I will cover it later, but we might just never need it.

HW 5 (due 9/29 at 4:30 pm):

1. Prove that the function f(x)=x^(1/3) is uniformly continuous on [0,1].  (Multiply and divide by some convenient expression
to get rid of the difference of roots, then work along the lines of the x^(1/2) problem).
2. Prove that the function g(x)=x cos(1/x) is uniformly continuous on (0,1]. (Hint: work as we did in class, when you have
to deal with the difference of the 2 cosines, turn it into a product using trig formulas, then you should be in familiar territory).
3. Prove that the function f(x)=x^(-2) is not uniformly continuous on the interval (0,1].
4. Prove that the function f(x)=x^2 is not uniformly continuous on the interval [1, + infinity).
5. Page 73, A.7: find a proof by contradiction using the intermediate value property.

Page 72-73, A.10, B.4.

HW 4 (due 9/20 at 4:30 pm):

Page 59, A. 2 c) and g). B. 4., B. 5., C. 3, C.4.

Page 72-74, A. 6., A. 7., A. 12., B. 2., C.2

HW 3 (due 9/13 at 4:30 pm):

Page 43, C. 1; B. 7.; B. 6.

Page 52-53, A. 1. i),  B. 2, B. 7, B. 4.

Prove Thm 2.5 parts b, e, f, g, and h (page 50-51). (Hint: try to tweak the
proofs of similar properties of converging sequences).

HW 2 (due 9/3):

Page 35, A. 1. f), g); A. 2. c), d), A. 3.

Page 36, B. 2.

Page 37, C. 4. a)

Page 43, B. 1, 2, 3.

HW 1 (due 8/27):

Page 9, # h, i, m, o.
In #o, try starting from what you want to prove (to figure it out), then write down the
way you went "back to front" (the latter would be the proof, if done right).

Page 22, # A 2 e), f), i). Establish that the suprema and infima you guessed satisfy the
difinitions of  suprema and infima.

# A. 1a), B 2, B 9.