Monday 18 September class (Updated)

 

 

Topics:

• Some more on integration by parts: Integrating by parts after a substitution; definite integrals that use integration by parts. (Examples 165 and 166)

• Trigonometric integrals of the form \int \sin^{m}(x)\cos^{n}(x)\textrm{d}x with m and n positive integers.

These integrals can become rather long as we see, so keeping track of terms and signs is very important.

Here are some nice videos on trigonometric integrals from PatrickJMT: Highly recommended to view even before we discuss in class!

Trignonometric integrals part 1 of 6 

(what we did today, for \int \sin^{m}(x)\cos^{n}(x)\textrm{d}x when at least one of the powers is odd)

 

Trignonometric integrals part 2 of 6 

(what we did today, for \int \sin^{m}(x)\cos^{n}(x)\textrm{d}x when both powers are even)

 

Trignonometric integrals part 3 of 6 

(Similar to what we did today, but for \int \tan^{m}(x)\sec^{n}(x)\textrm{d}x which works a similar way)

Trignonometric integrals part 4 of 6 

(More examples of \int \tan^{m}(x)\sec^{n}(x)\textrm{d}x)

 

Trignonometric integrals part 5 of 6 

(Integrating \int \sin(mx)\cos(nx)\textrm{d}x)

Trignonometric integrals part 6 of 6 

(More examples of different kinds of trig integrals that don’t fit neatly into the above categories)

Homework:

• Review the examples discussed in class, and at least the first two videos above.

• Do the WeBWorK:  please do not wait to the last minute! There are two assignments: Integration by Parts 2, and Trigonometric Integrals. You should try at least to do the problems in Trigonometric Integrals which are like the examples we worked in class.

• The quiz on Monday will be on Substitution and Integration by Parts.

• Try to watch the other four videos also, before next time. It will help make the lesson go more smoothly! (I hope)

• Don’t forget that Test 1 is scheduled for the first hour or so of class on Wednesday 27 September. The review self-test and answers will be in a separate post. Please check the course policies on tests.

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!

 

Wednesday 13 September class

Topics:

• Substitution method WeBWorK problem #7

• Integration by parts: polynomial times trig function, polynomial times exponential, exponential times trig function (Examples 159-162 in the textbook)

• Integrating by parts to find antiderivative of \ln(x) (Example 163 in the textbook)

 

Homework:

• Review and study the examples we discussed in class (listed above).

• Do the WeBWorK:  please do not wait to the last minute!

• I have posted the solutions to the first two quizzes here.

I believe that I have corrected or fixed all of the links in the previous posts on this blog, but if you find a link in any post that is broken or leads to the wrong thing, please let me know!

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!

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow

Monday 11 September class

Topics:

• More on the FTC and problems 16, 17, and 18 of the WeBWorK “FTC part 2” – see also the student questions on Piazza on these problems.

• Integration by substitution.

Here are the examples I discussed in class: I may post slideshows or complete solutions later if I have time

 

\int\cos^{3}(x)\sin(x)\textrm{d}x: here we take u(x) = \cos(x), so \textrm{d}u = -\sin(x)\textrm{d}x

The result is -\frac{\cos^{4}(x)}{4} +C

 

\int x^{2}\sqrt{x^{3}+5)\textrm{d}x: here we take u(x) = x^{3} + 5, so \textrm{d}u = 3x^{2}\textrm{d}x

The result is \frac{2}{9}\left(x^{3}+5\right)^{\frac{3}{2}} +C

 

\int\frac{2x+3}{x^{2}+3x}\textrm{d}x: here we take u(x) = x^{2}+3x, so \textrm{d}u = (2x+3)\textrm{d}x

The result is \ln\left|x^{2}+3x\right| +C

 

\displaystyle \int_{0}^{1} e^{-3x}\textrm{d}x: here we take u(x) = -3x, so \textrm{d}u = -3\textrm{d}x: also, don’t forget to change the limits of integration when you substitute!

The result is \frac{1}{3} - \frac{e^{-3}}{3}

 

Homework:

• Last reminder: make sure that you have done everything that is listed on the first day post! Most especially, make sure that your City Tech email address is in the User Information in WeBWorK. There are still a number of student who have not done this, and it means that if you send me an email from inside of WeBWorK I will not be able to reply to your email.

• Review and study the examples we discussed in class (listed above).

• Do the WeBWorK: some is due by 11 PM tomorrow, Tuesday, and please do not wait to the last minute! The homework on the Substitution method is not due until Sunday, but try to do at least 5 problems by tomorrow.

• Tomorrow is primary election day in NY, so if you are a registered voter check out the NYC Campaign Finance Board’s Voter Guide and don’t forget to vote!

I believe that I have corrected or fixed all of the links in the previous posts on this blog, but if you find a link in any post that is broken or leads to the wrong thing, please let me know!

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!

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow

Wednesday 6 September class

Apologies for the late post!

 

Topics:

• The Fundamental Theorem of Calculus part 1 (more)

• Using the FTC together with the Chain Rule

There are three videos from Khan Academy on the FTC linked in the previous post.

 

Homework:

• Review the examples we discussed in class, and you may also want to view the videos

• Finish the WeBWorK “FTC part 2” – I have delayed the due date until Tuesday because of the lateness of this post.

• There is also a WeBWorK assignment “Substitution Method” but we have not discussed this method yet. You may try some of it if you are interested. We will discuss the Substitution Method next time.

• Make sure that you have done everything from the first day post.

• There will be a quiz at the start of class next time. Be on time! It will be on finding definite integrals.

 

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.

 

Here is my slideshow on how to use Piazza to best effect: MAT1375:1575-UsingPiazza-slideshow

Important note about the WeBWorK “FTC part 2”

You don’t need any formulas for antiderivatives for any of the problems in FTC part 2. They only use the Fundamental Theorem of Calculus part 2 (you may want to view the video I posted on the previous post), and fundamental properties of definite integrals: see here (the blue box in the middle of the page).

 

In particular, you certainly do not need any techniques of integration which are to be covered later in this course! If anyone gives you advice like that, they are leading you in the wrong direction.

 

The point in this set of problems is to understand the relationship between definite integrals and areas, and to the functions which definite integrals define when the upper limit is a variable.

See the answers which I have posted over on Piazza as well.

 

Wednesday 30 August class

Topics: (may be updated when I have time)

• Antiderivatives problem #18 on WeBWorK

• Definite integrals and the Fundamental Theorem of Calculus

Note: there are two different things which have very similar notations, even though they are not the same type of object at all. They are:

The indefinite integral: this represents the set of all of the antiderivatives of the function. (Recall that an antiderivative of f(x) is a function whose derivative is f(x)). The indefinite integral is a function containing an arbitrary, undetermined constant (the constant of integration).

Notation for the indefinite integral: \int f(x)\textrm{d}x

 

A definite integral: this represents a signed area or a sum of signed areas, the areas between the graph of &latex f(x)$ and the x-axis, between two bounds. A definite integral is generally a number.

Notation for a definite integral: \int_{a}^{b} f(x)\textrm{d}x

 

Even though these two things are unrelated to begin with, the Fundamental Theorem of Calculus (part II) comes in to tell us that we can evaluate a definite integral by using antiderivatives, so the same type notation is used for both things!

[This is like the fact that \frac{2}{5} represents two different things: it is the quotient when 2 is divided by 5, and it is also the amount we get when we cut a unit length into 5 pieces and take 2 of them. It happens that the second thing is the answer to the first division problem, so we can use the same notation and usually don’t worry about it!]

 

Here are some videos from Khan Academy:

The Fundamental Theorem of Calculus (part I)

The Fundamental Theorem of Calculus (part II)

A worked example

Homework:

• Make sure that you have done everything from the first day post.

• Review finding definite integrals and the Fundamental Theorem of Calculus.

• Finish the WeBWorK on Antiderivatives and do the new assignments on the Fundamental Theorem of Calculus – due by Tuesday 11 PM, but don’t wait to the last minute!

• There will be a quiz at the start of class next time. Be on time! It will be on finding antiderivatives (and solving differential equations)

• Monday is a holiday. Happy Labor Day! Next class is next Wednesday, the 6th of September.

 

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.

WeBWorK assignment “Antiderivatives” problem 18

The problem gives you two values for the function f(x) at two different x-values. (Different students will get different numbers.)

This is OK, you will have to find f(x) by integrating twice and it will have two constants of integration in it. You can then use the two values of f , one at a time, to find the two constants. Hint: Use f(0) first!

 

If that’s not enough of a hint you can post a question over on Piazza. Don’t forget to tell us exactly what your problem said, as everyone gets a different version of the problem!

First day post

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This website is your “one-stop shopping” for all matters related to our class. Please check for the post of the day every time we meet!

 

Here is the news from the first day:

You should have received a copy of my course policies with the instructions for WeBWorK on the other side.  If you were absent or loose them ever, they are available as pages here on this blog. (Follow the links.)

 

Our course textbook is available at this link (which is also in the updated course policies page): there are several formats available, including interactive graphics, which might be useful!

Here are the topics of the day: to be updated when I have time. Quick run-through now.

• Antiderivatives and indefinite integrals

This should be review from MAT 1475, but also take the opportunity to think more deeply about the concepts here.

An antiderivative of a function f(x) is basically another function whose derivative is f(x). We often call the antiderivative F(x), which requires distinguishing upper-case from lower-case. It’s annoying. A better notation is to subsume all the antiderivatives into one, called the indefinite integral.

A theorem of calculus tells us that once we have one antiderivative of f(x), all the others are found by adding an arbitrary constant, called the constant of integration.

So all we need to do is to find one antiderivative F(x), and the we know that the indefinite integral of f(x) is

\int f(x) \textrm{d}x = F(x) + C

where C is any constant.

 

In practice, antiderivatives are used to solve differential equations, which are equations where we do not know a function but we know one of its derivatives. In order to find out the constant or constants of integration, we need some other information, which is usually called initial values or boundary values.

 

Homework:

• Find and deal with your City Tech email: you must use this email address in WeBWorK and to join the Piazza discussion board. Also, City Tech is already sending you emails here!

• WeBWorK has been set up for this course. The WeBWorK is here.

Please follow these instructions: MAT1575 WeBWorK information,  in order to log in. After changing your password and entering your citytech email, you may start on the Orientation assignment to get used to how WeBWorK works.

A common problem that occurs when people try to log in using iPhone is that there is automatic capitalization for the first letter when you type in your username or password. Check to make sure the capitalization is turned off (or turn it off) while you type!

AFTER you’ve done at least 7-8 of the Orientation problems, you should start on the assignment which is due by Sunday evening 11 PM. Don’t wait to the last minute!

• Join the Piazza discussion board for this course by following this link. You will need to enter your City Tech email in order to join. I will post extra credit questions here from time to time!

• Also don’t forget the extra practice problems from the textbook. See the Course Outline for section 5.1: course outline here:MAT1575

See you on Wednesday!