Assignments


For each set of problems, on or before the "begin by" date you should watch the associated videos, read the appropriate parts of text and related materials.  Work on the assigned problems. Email questions, post them on CANVAS and/or be prepared to ask during the following online session.  Submit your assignment as a pdf in CANVAS on or before the date indicated.  

The first homework file you submit should be named:
hw2yourlastname.pdf
and similarly for  hw3, etc.

An asterisk * beside a listed problem indicates that you do not need to include that in what you submit (i.e. in section 2.1 you should only submit numbers 6, 8, and 10 parts abc.)

 At the bottom of this page there may be a few notes and suggestions relating to each assignment.

With all assigned problems, there should be some indication of your reasoning.  
Problems marked "M&M" in the text are generally accompanied by Mathematica code.  In some cases, these problem ask merely for "observation" regarding what the output suggests.  Otherwise, this sort of computational output is meant to be a resource to suggest a possible conclusion that you would otherwise need to explain. The "prove" part of Problem 5 in section 1.2  is an example of this, while the "what number..." is asking for an observation.

In responding to "prove" or "explain,," merely noting something that you observe in Mathematica or other comutational output is almost never what we are looking for.

Even for numerical answers it should be apparent how you arrived at an answer.  In a problem without an "M&M" designation that asks you to "compute" or "evaluate," always assume that you are being asked to justify an answer based on actual mathematics.  Simple example:  if you are asked to compute
"1 +1/2+1/4+1/8+1/16+..." you could arrive at the answer "2" by citing known facts about geometric sums or providing some (more or less equivalent) ad hoc explanation.  In this case, having Mathematica add up the first 100 terms and then guessing that the sums are getting close to "2" is not going to be an acceptable response.  

Text Section  - Problems	Begin by	Due date:
hw1: 1.2 problems 3, 5, 6, 7 (also...look at the graphs in #4)	5/22	Not necessary to turn in homework 1.  Sketch out what you believe the answers should look like. I will post a sample solution befor 5/26 to compare.
hw2 2.1 - 6,8,9*,10abc (10d*),11* 2.2 - 2,3,5,8*, 4*(only find the target value)	5/24	5/30
hw3 2.3 - 1,4 2.4 - 1,2,4*,8 2.5 - 1*abcd, 2*abcd, 4abcd, 15	5/27	6/2
hw4 3.1 - 1*, 2abc,6a,6b*, 8, 9*, 11 3.2 - 4	5/31	6/6
hw5 3.3 - 1, 3, 5, 8, 17, 18, 19, 20, 21*	6/3	6/9
hw6 3.4 - 1,2,3,6abcdefg (but look at the other parts), 13 (instead of using the result of #12, go ahead and assume the MVT from section 3.5),22* 3.5 2,4,8 (see the note about a typo),11*,16,18ac	6/7	6/13
hw 7 4.1 - 1,3,4,5,13,14abc,15ab, 14d* 4.2 - 1,4,5cd,15,17,20,21, 2*,3*, 5b*	6/10	6/16
hw 8 4.3 - 1abcd, 2abd, 9, 10 14 4.4  - 8, 9, 10	6/14	6/20
hw 9 5.1 - 1, 2, 6, 11, 12, 14 5.2 -  1,4,6	6/17	6/23
hw 10 5.3 - 2 (OK to use section 5.4, Cor 5.12), 6, 7, 8, 5.4 - 1 A-E, 2, 3, 12b	6/21	6/27

Section 1.2  
not much to worry about on 3 and 4 - just do the computation (and anything additonal beyond what the book suggests, if necessary) and give it a go.
6 - yes, you can answer a and b pretty quickly just from the graph
7 - you have to massage the graphs a bit, and they will be deceiving if you don't try to keep the vertical scale on the graphs the same as you take more terms.  Also, "at least 20 terms" is a litte misleading since you might be led to just do 20 and stop there.  "at least 20" think about what happens with 21 or 22 or whatever.  The "what can you prove" is a little open ended, but you may be able to come up with something that is mathematically convincing. 

Section 2.1
#10 For the "harmonic" sums: Up to you ... maybe slightly better to use the "alternate" command from the mathematica harmonic notebook.  On #11, if you use that same mathematica notebook instead of the author's, you would need to change the "alternate" command in some way.
 #6, it is ok to refer to the geometric sum formula in section 2 (equation 2.10)
#10 part d  - this will require some experimenting.  In the harmonic notebook, probably something like Exp[2*Alternate[3, 2, 10000]].  With 10000 that should be large enough to guess the actual value.   But remember, eventually you are looking for a formula for d in terms of r and s.

Section 2.2
#2 and #3 should  be routine using the geometric sum formula equation 2.10. Including the "find a value of n" part.  No mathematica sums needed here.
#5 if you wish, there is a video and solution for similar problem #9, although #5 should be a little less complicated:
https://youtu.be/XoX1mCQ3eSo


Section 2.3
#1 "Justify" - same sort of reasoning as in 2.1.8
#4  "prove" is definitely an overstatement.  Instead - assuming that 2.14 is true, show how we could "infer" the result here in problem #4.  This depends on assuming that the associative property holds for series that converge - we will prove this in chapter 5, although in theory you could explain it now.

Section 2.4
1, 2  - infinite number of possible answers.  A formula is of course OK, but could be a writing out a sum with a pattern that is obvious.
4 - literally,"describe and comment"

Section 2.5
15 - subtle, but you should be able to figure it out

Section 3.1
2, 6 - You smay assume the standard basics about derivative formulas - except at the "transition points" where you should make clear how you arrived at the answer. For example, on 2 part a, recalling that |x| is x when x>0 and -x when x<0, it is ok to write down the formula for the derivative, except at x = 0 where you need a little more... not necessarily the whole E(x,a) business, but at least analysis of the limit of  (f(x)-f(a))/(x-a).  (And ... do not ask mathematica  to calculate the derivatives ... nothing useful is going to come from that)
8, 9 - It would be best if you are able in theory to provide an answer either using algebra or a graph.  OK to turn in either.

Section 3.3
8 - a picture/graph should be helpful.  Eventually, you might want to consider the function g given by g(x) = f(x)-x.
17, 18, 20 - a little justification is required
#17 - one possible approach would use trig identities, and perhaps also the fact that |sin(x)| is less that or equal to |x| (equal only when x=0). You do not need to write down a proof of that inequality, but you SHOULD be able to if anyone asked ... at least a proof with a picture using the unit circle.
#18 - it is often convenient to stipulate up front that you want to have delta<=1. that is, whatever else pops up in a formula for delta, you are going to choose delta to be the SMALLER of that number and 1.
#20 If you cannot think of anything else, it is ok to assume that the MVT is true..  This would mean that e^(x+h)-e^x = e^c*h for some number c between x and x+h.  (Keep in mind that c depends on x, so you will want to replace e^c with a constant that you know is larger).

Section 3.4
1,2,3 - just let your imagination run a bit.
6 - various ways that sets are described.  part d: you are really just explaining why later numbers are between a couple of earlier numbers.
13 - Ok to use MVT

Section 3.5
4b - this is easier if you can change the equation from one with 3 exponentials to 2 exponentials.
8 - typo.. the word "lim" as x->0 should appear before the fraction whose numerator starts "2x sin(1/x)..."  That is NOT the error you are looking for.  Also, even though f not defined at x=0, this is not a problem - extending f by letting f(0)=0 we have a function that is differentiable everywhere, with f'(0)=0.  So THAT is also not the problem
11 This may be a case where the substitution u=1/x and u->infinity helps.  Or maybe not.



Section 4.1
#1 and #3 - use what we know about error terms with geometric series or alternating series, not direct calculation, and certainly not direct calculation when answering about 100 digit accuracy.  
#4,5 "discuss" should also include whether you think this series converges, and why. 
#13 Based on nothing more than what you see in mathematica or wolfram alpha plots, a reasonable guess would be that the series cannot converge.  But as on page 120-121, you are asked to take on faith for now that the series does converge, and make a (wild) guess at the limit.  You are probably not going to get any better information from Mathematica plots than what is visible in figure 4.1 on page 121.
#14c  You probably need the old standby |sin(z)|<=|z|  And in case you need it:  as long as z is not too large (even |z|<Pi/2 is good enough), it is also true that
|sin(z)|>=|z|/2.  The absolute values do not matter in #14 since here we only have sine of positive numbers.  (And both of these inequalities reflect the fact that when z is small, sin(z) approx= z.  

Section 4.2
#21 Try both tests,but not necessary to include both in your solution.

Section 4.3
#1 "Domain of convergence" = "interval of convergence," but also pay particular attention to the endpoints.
#2 b is not a power series, per se.  Suggestion: figure out the domain of convergence if the (2x+1)/x was replaced with some variable z.  Then determine what x values in (2x+1)/x correspond to those z numbers. If you write out the actual terms,  part a in fact is a power series - it is just that the notation with the "2n" exponent makes direct application of the theorem confusing. So I would suggest handling a) in the same way as b).
#11 "radius of convergence" technically is not asking you about the endpoints.
#14 - if you have difficulty with the root test, try the ratio test.

Section 4.4

#8 Here, "bounded" refers to what happens as n gets large, assuming x is fixed.  NOT asking if there is a single upper bound that works for all values of x. So an upper bound here may depend on x, but should not depend on n.
#10 What we referred to as the "tail" of  the series in the notes could also be labeled the "error" if using partial sums to estimate the series.  This also appears at the bottom of page 163.

Section 5.1
#11 - do not be afraid of very simple exammples

Section 5.2
#1 - Up to you, but I would avoid trying to directly apply the epsilon definition and look for some other way to answer.
#4 - Technically, I suppose one should go to the epsilon definition, but you may be able to get away with an explanation that at least partly avoids doing that.

Section 5.3
#6 Be sure to look at the hint in the back of the book.  This is the partial fractions decomposition - Maple has a command to do the same thing - but that is just giving you the equation in the hint. From there, you want to show the algebra to get the G_n(x) formula (not Mathematica).
#7 clearly (I hope!) depends on 6. You probably would want to know the max value for |G_n(x) - G(x)|

Section 5.4 - take a look at the author's hints

#1 - The last  sentence in the instructions should say: If it is not uniformly convergent, determine if it is still convergent.  (I.e. possible that at least one of these does not converge).
#1 a suggestion: symmetry, just worry about [0,infinity)
#1d can you find a way to compare ln(1+z) and just z?  (maybe you do not need to prove that, but the MVT would make it relatively easy to do)
#2 multiple viable approaches