# Monday 23 October and Wednesday 25 October classes

(Wednesday after Test 2)

Topics:

• Taylor’s Theorem and the error term

• The Mean Value Theorem and Rolle’s Theorem (Ch. 3 section 3.2)

Here are my notes on the examples I used to illustrate these theorems:

MAT1575MeanValueThmRollesThmillustrations

You can also view this video on the MVT from PatrickJMT

We will use the Mean Value Theorem to prove the error term in Taylor’s Theorem, and then use the error term bound to figure out how many terms of the Taylor polynomial we would need to make the error be less than a specified amount.

Homework:

• Review the examples discussed in class

• Do the WeBWorK:

• There are no problems in the textbook assignment on the Mean Value Theorem, so make sure that you do that assignment on WeBWorK!

• For extra practice on Taylor polynomials you can do the problems p. 475 #5-20

• No quiz next time.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

# Test 2 review (updated, now with answers and some hints)

Test 2 is scheduled for the first hour or so of class on Wednesday 25 October.

Test 2 review self-tests here: MAT1575Test2Review

You probably should allow 90 minutes per self-test: the test itself will not have quite so many problems in it!

Important note: there are some errors in a couple of the problems, which I did not catch before publishing these:

In Self-Test 1, problem 4, the integral should go from 7 to 9 (not 4 to 9)

In Self-Test 2, problem 2, there is a typesetting error: the integral should read $\displaystyle \int \frac{x}{\sqrt{9x^{2}-1}}\textrm{d}x$

Answers and hints are at the end of this post, after the “more”

Topics:
• Trigonometric integrals of the form $\int \tan^{m}(x) \sec^{n}(x)\textrm{d}x$, where m and n are positive integers.
• Trigonometric substitution, including completing the square to use it
• Partial Fractions, including repeated factors and irreducible quadratic factors
• Improper integrals and the two comparison theorems

Also make sure that you are familiar with the additional antiderivatives we found and added to our basic list:

MAT1575Integration-Formulas-Theorems-OurMethods-slideshow

It may be useful to fill out the Outline Summary of Integration Methods:

MAT1575OutlineSummaryOfIntegrationMethods

Here is a nice resource: an online, free Integral calculator. Don’t let it do everything for you, but it’s great for checking your work. This can be very helpful in locating errors since it shows step-by-step how it arrived at the integrals.

# Wednesday 18 October class

Topics:

• More on improper integrals: the Limit Comparison Theorem (and why it is more powerful than the direct Comparison Theorem); improper integrals when the interval of integration contains a vertical asymptote

• Introduction to Taylor polynomials

Notes to follow!

Homework:

• Review the examples we discussed in class. Especially make sure that you understand why direct comparison cannot be used in the example where I used limit comparison.

• Do the WeBWorK: there are two assignments due by Sunday evening, but do not wait to the last minute! And don’t forget to use Piazza to discuss! I have also extended the extra credit assignment on partial fractions.

• There will be a quiz next time: the topic will be improper integrals and the comparison theorems.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

# Monday 16 October class (updated)

I’ve started typing this up (as you can see, it’s got a lot of notes) but I’m getting error messages in some of the new things I’m trying to put in, so it’s incomplete. I’ll keep working on it, but wanted to give you these notes for now.

UPDATE: Here are the notes I have managed to get to work on this post. Some details remain for you to fill in! There is also a link to the Paul’s Notes page on L’Hôpital’s Rule which I highly recommend if you feel the urge to brush up on your skills! (Or even if you’ve totally forgotten what it is!)

Many of you have already started the WeBWorK, and I also extended the extra credit assignment: both are due Tuesday night!

There will be a quiz on partial fractions with either long division or irreducible quadratic factors (or both), plus one improper integral, next time.

Topics:

• Some homework problems from Partial Fractions 3

• Improper integrals (continued): the type where the interval of integration is infinite

Note: for working with improper integrals of this type, where we are taking limits as $x\rightarrow \pm \infty$, it is very useful to keep in mind that any time we take a limit of a rational expression whose numerator is a constant and the denominator goes to infinity, that limit will be 0. So for example

$\displaystyle \lim_{x\rightarrow \infty}\left(\frac{1}{x^{p}}\right) = 0$ for any $p>0$

$\displaystyle \lim_{x\rightarrow \infty}\left(\frac{1}{e^{x}}\right) = 0$

which means that $\displaystyle \lim_{x\rightarrow \infty}\left(e^{-x}\right) = 0$

and even

$\displaystyle \lim_{x\rightarrow \infty}\left(\frac{1}{\ln(x)}\right) = 0$

Also, we may have to use L’Hôpital’s Rule when we encounter the situation where the limit has one of the indefinite forms:

$\frac{\infty}{\infty}$, $\frac{0}{0}$, $\infty - \infty$, etc.

Here are examples I discussed: (some details omitted, you should be able to fill in the missing steps though!)

$\displaystyle \int_{1}^{\infty}\frac{1}{x}\textrm{d}x = \lim_{b\rightarrow\infty}\left[\ln(x)\right]_{1}^{b}$ diverges to infinity because $\lim_{b\rightarrow\infty}\left(\ln(x)\right) = \infty$

$\displaystyle \int_{-\infty}^{0}e^{x}\textrm{d}x = \lim_{a\rightarrow-\infty}\left[e^{x}\right]_{a}^{0} = 1$ because $\displaystyle \lim_{a\rightarrow -\infty}e^{a} = \lim_{x\rightarrow \infty}\left(e^{-x}\right) = 0$

$\displaystyle\int_{-\infty}^{\infty}\frac{1}{1+x^{2}}\textrm{d}x=\lim_{a\rightarrow-\infty}\left[\tan^{-1}(x)\right]_{a}^{0}+\lim_{b\rightarrow \infty}\left[\tan^{-1}(x)\right]_{0}^{b}$

$=0 - \left(-\frac{\pi}{2}\right) + \frac{\pi}{2} - 0 =\pi$

Comment: I find the limits for the inverse tangent by considering the graph of tangent in the period between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$; as $\tan(x) \rightarrow -\infty$ in this part of the graph, the angle x is approaching $-\frac{\pi}{2}$, and as $\tan(x) \rightarrow \infty$ in this part of the graph, the angle x is approaching $\frac{\pi}{2}$.

You can find the indefinite integral for $\int\frac{1}{1+x^{2}}\textrm{d}x$ either by remembering that $\frac{1}{1+x^{2}}$ is the derivative of $\tan^{-1}(x)$, or else by using a trig substitution $x=\tan^{2}\theta$.

Example using L’Hôpital’s Rule:

$\displaystyle \int_{1}^{\infty}\frac{\ln(x)}{x^{2}}\textrm{d}x$

Note: this is Example 194 on p. 336 in the textbook, done there in considerable detail. We end up using L’Hôpital’s Rule to find $\displaystyle \lim_{b\rightarrow\infty}\frac{\ln(b)}{b}$, which is an indefinite form of the type $\frac{\infty}{\infty}$.

A general rule for integrals involving powers of x in the denominator: this will be useful for comparison to prove whether or not other indefinite integrals converge.

Consider the improper integral $\displaystyle \int_{1}^{\infty}\frac{1}{x^{p}}\textrm{d}x$ with $p>0, p\neq 1$.

What values of p will make it converge? When we integrate, we end up having to compute

$\displaystyle \lim_{b\rightarrow\infty}\frac{b^{-p+1}}{-p+1}$

This will go to 0 if the exponent is negative: otherwise, it will diverge to infinity.

So in order to have the integral converge, we need $-p+1<0$, which means that $p>1$.

We already saw that this type of integral diverges when p =1, so now we know the following:

$\displaystyle \int_{1}^{\infty}\frac{1}{x^{p}}\textrm{d}x$ converges when p>1, and it diverges when $0 < p \le1$.

Comparison Theorem:

Let f(x) and g(x) be functions such that both f(x) and g(x) are continuous on the interval $[a, \infty)$

and $0 \le f(x) \le g(x)$ for all x in $[a, \infty)$

Then:

• If $\displaystyle \int_{a}^{\infty}g(x)\textrm{d}x$ converges, then also $\displaystyle \int_{a}^{\infty}f(x)\textrm{d}x$ converges.

• If $\displaystyle \int_{a}^{\infty}f(x)\textrm{d}x$ diverges, then also $\displaystyle \int_{a}^{\infty}g(x)\textrm{d}x$ diverges.

Interpretation:

f(x) and g(x) have to be continuous on the given interval. Don’t forget to check this condition.

The condition $0 \le f(x) \le g(x)$ for all x in $[a, \infty)$ means that both f(x) and g(x) are non-negative functions on the interval, and the graph of f(x) is squeezed in between the x-axis and the graph of g(x).

Now remember that the question of whether or not an improper integral of this type will converge, depends on how quickly the integrand goes to 0 as $x\rightarrow\infty$.

The first part of this theorem,

• If $\displaystyle \int_{a}^{\infty}g(x)\textrm{d}x$ converges, then also $\displaystyle \int_{a}^{\infty}f(x)\textrm{d}x$ converges,

is saying that g(x) goes to 0 “fast enough” that its integral converges. Since the graph of f(x) is squeezed between the graph of g(x) and the x-axis, that means that f(x) goes to 0 at least as fast as g(x) does, maybe faster. So the integral of f(x) will also converge.

The second part of this theorem,

• If $\displaystyle \int_{a}^{\infty}f(x)\textrm{d}x$ diverges, then also $\displaystyle \int_{a}^{\infty}g(x)\textrm{d}x$ diverges,

is saying that f(x) does not go to 0 “fast enough” for its integral to converge. Since the graph of f(x) is squeezed between the graph of g(x) and the x-axis, that means that g(x) goes to 0 no faster than f(x) does, maybe slower. So the integral of g(x) will also diverge.

(A picture may help. See the graphs on p. 340 of the textbook.)

Example: see Ex. 197 part 2 on p. 340 (to the end of the page).

I also discussed the convergence of $\displaystyle \int_{1}^{\infty}\frac{1-e^{-x}}{x^{2}}\textrm{d}x$.

Since $0\le e^{-x} <1$ for x in the interval $[1,\infty)$, the numerator $1-e^{-x}$ will always be a positive number less than 1 in that interval. So we can use comparison:

$0\le \frac{1-e^{-x}}{x^{2}} \le \frac{1}{x^{2}}$ on $[1,\infty)$

We know from before that $\displaystyle \int_{1}^{\infty}\frac{1}{x^{2}}\textrm{d}x$ converges, so it follows that $\displaystyle \int_{1}^{\infty}\frac{1-e^{-x}}{x^{2}}\textrm{d}x$ converges.

General comment on this comparison theorem: The theorem can only tell you that a certain integral converges: it does not tell you what the value of that integral is. In the example I just worked, we know that $\displaystyle \int_{1}^{\infty}\frac{1-e^{-x}}{x^{2}}\textrm{d}x$ converges, but the most we can say about its value is that $\displaystyle \int_{1}^{\infty}\frac{1-e^{-x}}{x^{2}}\textrm{d}x\le\int_{1}^{\infty}\frac{1}{x^{2}}\textrm{d}x$.

You may wonder why it is useful to know that something converges if we don’t know its actual value. One reason that it is useful to know this is that once we know that the integral converges, we can use numerical methods to estimate its value if all else fails. It is very dangerous to try to use numerical methods on an integral if we don’t know whether or not it converges!

Numerical methods are very important in applications (physics, etc) where many of the integrals we want to compute cannot be found explicitly even by the many methods you learn in this course!

# Delayed posts (UPDATED)

UPDATE: I accidentally posted topics here which are actually for a different class! Sorry, I’m working on this today (Sunday) and will post a final version when all the editing is finished. I am also trying to re-post the resources which needed corrections.

The resources are still linked in the last post [corrected link] and here is the  Homework:

• Review the methods for integration by partial fractions when there is an irreducible quadratic factor, and the definitions of improper integrals when the integration interval is infinite

• Do the problems assigned in the Course Outline

for the partial fractions (second part)

• Do the WeBWork

• Here is some review on L’Hopital’s Rule (which we will need to use next time), and there are also practice problems with solutions included. [From Paul’s Notes]

• There will be a Quiz on Monday, the topics will be integrating by partial fractions, and there will be the possibility of) denominators which are not linear so you have to use trig substitution.

# National Voter Registration Day (bumped up and updated)

UPDATE: In NY State, the deadline to vote in the November elections is this Friday 13 October 2017.

If you want to vote in the 2018 primaries, you must register in that party NOW by Friday 13 November. The 2018 elections will include every member of Congress, among much else.

Today (26 September 2017) is National Voter Registration Day.

If you are a resident of NY, you can find information about registering to vote here. [NY State Board of Elections site]

If you are a resident of NJ, you can find information about registering to vote here. [NJ Department of State site]

# resources for Wednesday 11 October class

Here are some resources for today’s class:

Integration using partial fractions when there is an irreducible quadratic factor:

(corrected typos)

Table of Derivatives and Antiderivatives (this is what we started with):

Slideshow of new indefinite integrals to add to this table: (using methods we have so far learned)

MAT1575Integration-Formulas-Theorems-OurMethods-slideshow

(corrected typos)

Outline notes for summarizing methods so far:

MAT1575OutlineSummaryOfIntegrationMethods

# Request for homework problems on the board next time

By request, I am posting here some homework problems which I would like to have students put on the board on Wednesday. To help save time, if you get to the classroom before class starts, you could put one or more of these on the board at that time, before the start of class – you don’t have to wait! (Reminder: You get extra credit quiz points for putting problems on the board, as well as for answering student questions on Piazza.)

From the course outline:

Any of the indefinite integrals which are mentioned in the problem section from Session 6 (they are not in the textbook). Here they are:

$\displaystyle \int \frac{x^{2}}{\sqrt{4-x^{2}}}\textrm{d}x$

$\displaystyle \int \frac{x^{3}}{\sqrt{x^{2}+9}}\textrm{d}x$

$\displaystyle \int \frac{\sqrt{x^{2}=9}}{x^{4}}\textrm{d}x$

$\displaystyle \int \frac{1}{x^{2}\sqrt{4-x^{2}}}\textrm{d}x$

$\displaystyle \int \frac{1}{x^{2}\sqrt{x^{2}+9}}\textrm{d}x$

Also: Textbook p. 304 #18 and #32

# Wednesday 4 October class

Topics:

• Completing the square in order to use trig substitution. See my notes on Piazza from my WeBWorK problem which I showed on the board in class. (WeBWorK “Trigonometric Substitution 3” problems 9, 18, 21, and 3 use this. 3 is a little more challenging.)

• Back to partial fractions expansion when the denominator has only linear factors: Examples 183 and 184. I showed the details of the long division for Ex. 184, and I will try to type that up when I have time. I have not yet found any nice videos that show long division used in partial fractions except for this one from Khan Academy. If you need to review long division of polynomials, Khan academy has a number of videos on that topic (in Algebra 2, not specific to partial fractions).

We have to use long division to reduce the degree of the numerator whenever the degree of the numerator is the same as or greater than the degree of the denominator. Partial fractions only works for rational expressions where the numerator has smaller degree than the denominator.

Homework:

• Review the examples discussed in class, and review long division of polynomials as well if you feel the need!

• Do the WeBWorK: there is an additional short assignment “Partial Fractions 2” which is also due next Tuesday. PLEASE do not wait to the last minute!

Please also remember that one of the purposes of assigning WeBWorK is that, since you get immediate feedback if your answer is wrong, the intention is that you will seek help (read the book, look at the videos I linked, post a question to Piazza, talk to a classmate or tutor… etc…) AS SOON AS POSSIBLE, not waiting for the next class meeting! Not to mention that if you post a question to Piazza, you give your fellow classmates an opportunity to help (which is always a good thing) and to earn a few quiz points (which is also a good thing).

• There will be a quiz next time: the topic will be completing the square in trig substitution (see problems 18, 21 in the WeBWorK), and partial fractions used to integrate (no long division).

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!

# Monday 2 October class

Topics:

• More on trig substitutions: substituting back at the end, for indefinite integrals

(Example from last time and also the examples below)

• Trig substitutions when there are no radical signs over the sum or difference of squares: I discussed Examples 178-180 in the textbook.

• Completing a square to use a trig substitution (Example 179)

The videos on Trig Substitution from Patrick’s Just Math Tutorials are linked in the post from last time: Example 3 uses completing the square to get to the trig substitution.

• Partial fractions: introduction, and one integral.

Patrick has a video on finding the partial fractions expansion, which unfortunately contains an error (wrong number). See if you can find the error and correct it! (There will be a blue annotation square when the error occurs.)

He also gives a nice trick for finding the coefficients when you only have linear factors in your denominator.

Homework:

• Review the examples discussed in class. I hope you find these videos helpful also!

• Do the WeBWorK: Note there is a new assignment I linked after class: it is just on finding the partial fractions expansions, no integration yet. These are not due until next week, but please do not wait to the last minute! Try to do at least a few by next time.

• There will be a quiz next time: the topic will be Trigonometric integrals of the type that includes powers of tangent and/or secant, and also a definite integral which uses trigonometric substitution.

Don’t forget, if you get stuck on a problem, you can post a question on Piazza. Make sure to give your question a good subject line and tell us the problem itself – we need this information in order to answer your question. And please only put one problem per posted question!