Good Question 16

Posted on November 30, 2018 by Lin McMullin

I had an email last week from a teacher asking, how come I can use a substitution to find a power series for $\cos \left( {2x} \right)$ , and for ${{e}^{{\left( {x-1} \right)}}}$ , but not for $\cos \left( {3x+\frac{\pi }{6}} \right)$ ?

The answer is that you can. Substituting (2x) into the cosine’s series give you a Taylor series centered at x = 0, a Maclaurin Series. Substituting (x – 1) into the series for e^x gives you a Taylor series centered at x = 1. And substituting $\left( {3x+\frac{\pi }{6}} \right)$ into the cosine series gives you a Taylor series centered at $x=-\frac{\pi }{{18}}$ . I suspect that she was hoping for or was asked to find a Maclaurin series, not one with such a strange center.

The center of a Taylor series is the value of x that makes its argument zero.

AP Exam Question 2004 BC 6(a)

This brought to mind the AP Exam question 2004 BC 6(a) where students were asked to write the third-degree Taylor polynomial about x = 0 for the function $f\left( x \right)=\sin \left( {5x+\frac{\pi }{4}} \right)$ . The intended method was for students to find the first three derivative and substitute them into the general form for a Taylor series. That’s what students who got this correct did. This is the only time I can remember when students were expected to do that; usually they manipulate a given series or substitute into a known series.

A number of students tried to substitute $\left( {5x+\frac{\pi }{4}} \right)$ into the series for the sine. This gets a very nice Taylor series centered at $x=-\frac{\pi }{{20}}$ . This earned no credit since a Maclaurin series was required.

But there is another way! (I originally wrote, “But there is an easier way!” but it’s only easier if you see how to do it.)

Trigonometry to the Rescue!

$\sin \left( {5x+\frac{\pi }{4}} \right)=\sin (5x)\cos \left( {\frac{\pi }{4}} \right)+\cos \left( {5x} \right)\sin \left( {\frac{\pi }{4}} \right)=\frac{{\sqrt{2}}}{2}\left( {\sin \left( {5x} \right)+\cos \left( {5x} \right)} \right)$

Then using the first two terms each from the series for sine and cosine you get the correct answer:

$\displaystyle \frac{{\sqrt{2}}}{2}\left( {\left( {5x-{{{\frac{{\left( {5x} \right)}}{{3!}}}}^{3}}} \right)+\left( {1-\frac{{{{{\left( {5x} \right)}}^{2}}}}{{2!}}} \right)} \right)=\frac{{\sqrt{2}}}{2}+\frac{{5\sqrt{2}}}{2}x-\frac{{25\sqrt{2}}}{{2\left( {2!} \right)}}{{x}^{2}}-\frac{{125\sqrt{2}}}{{2\left( {3!} \right)}}{{x}^{3}}$

This brings us to $\cos \left( {3x+\frac{\pi }{6}} \right)$ , which can be approached the same way. Here is the entire Maclaurin series.

$\cos \left( {3x+\frac{\pi }{6}} \right)=\cos \left( {3x} \right)\cos \left( {\frac{\pi }{6}} \right)-\sin \left( {3x} \right)\sin \left( {\frac{\pi }{6}} \right)$

$\displaystyle =\frac{{\sqrt{3}}}{2}\cos \left( {3x} \right)-\frac{1}{2}\sin \left( {3x} \right)$

$\displaystyle =\frac{{\sqrt{3}}}{2}\sum\limits_{{n=0}}^{\infty }{{\frac{{{{{\left( {3x} \right)}}^{{2n}}}}}{{\left( {2n} \right)!}}}}-\frac{1}{2}\sum\limits_{{n=0}}^{\infty }{{\frac{{{{{\left( {3x} \right)}}^{{2n+1}}}}}{{\left( {2n+1} \right)!}}}}$

$\displaystyle =\sum\limits_{{n=0}}^{\infty }{{\left( {\frac{{\sqrt{3}\left( {{{3}^{{2n}}}} \right)}}{{2\left( {2n} \right)!}}{{x}^{{2n}}}-\frac{{1\left( {{{3}^{{2n+1}}}} \right)}}{{2\left( {2n+1} \right)!}}{{x}^{{2n+1}}}} \right)}}$

Moral: Trig can be very useful.

Here is a previous post, Geometric Series – Far Out, that shows a “mistake” you may find interesting.

Riemann Sums – the Theory

Posted on November 27, 2018 by Lin McMullin

The series of post leads up to the Fundamental theorem of Calculus (FTC). Obviously, a very important destination.

Working Towards Riemann Sums
Definition of the Definite Integral and the FTC – a more exact demonstration from last Friday’s post and The Fundamental Theorem of Calculus – an older demonstration
More about the FTC The derivative of a function defined by an integral – the other half of the FTC.
Good Question 11 Riemann Reversed – How to find the integral, given the Riemann sum. A problem that appears on the AP Calculus exams and can be confusing for students, at first.
Properties of Integrals
Variation on a Theme – 2 Comparing Riemann sums
Trapezoids – Ancient and Modern – some history.

Revised and updated October 22, 2018

Brushing Up the Blog

Posted on November 20, 2018 by Lin McMullin

Happy Thanksgiving!

I’ve been working this month on making things at the Teaching Calculus blog easier to find. There are about 360 posts and it’s getting difficult for me to find things. Here is the new line up on the black navigation menu at the top of each page. Click each to see more.

HOME

This is the place where the blog post show up in chronological order with the latest on top. Under it are four featured post, which usually are just the four most recent posts. Under that are the remaining posts.

TOPICS a new menu

Under this menu I’ve brought together all the post on each of the main topics in first-year calculus. I hope this helps you find posts more easily. New posts will be added to these lists as they are posted.

OTHER RESOURCES a reorganized menu

Here I’ve listed other posts under these headings:

Before you Start contains links to posts about things you may want to consider before the year begins including some Algebra pre-calculus topics.
Calculators and Technology has information and ideas about calculators, calculator use on the AP Calculus exams, and other technology.
Essays are posts on other topics related to calculus.
Good Questions has links to individual questions I’ve found interesting and/or instructive; questions you can adapt and expand.
The CED has posts related to the AP Calculus Course and Exam Description for AP Calculus.
Presentations contains links to PowerPoint presentations from presentations and talks I have given. You may use them if you like.
Resources from Posts has links to things mentioned in posts that you may be looking for.
Monthly – this monthly list of topics will remain for a while. Since the Topics menu list all these and more, I will be removing this menu soon.

VIDEOS

This menu links to about 57 videos on calculus topics that I made when I was National Director of Mathematics Programs for the National Math and Science Initiative (NMSI). They were made in 2012, but the math hasn’t changed. There are for students and teachers.

WEBSITE

Here are the remnants of a website I abandoned some time ago. Most of this is out of date, but i didn’t want to totally lose it. don’t be surprised if it disappears some time soon

ABOUT & CONTACT

My contact information is here.

As I continue brush things up I will be reorganizing, rearranging, and renaming, and moving some items elsewhere. If you think something has gone lost, if you find a typo, or an incorrect link, please let me know at this email address

If you have any calculus questions or a suggestion for a post, please contact me at the same email address

Please use the LIKE button at the bottom of each post. (assuming you like it); It helps me know what you like and find helpful.

Getting Ready to Integrate

Posted on November 13, 2018 by Lin McMullin

Behind every definite integral is a Riemann sums. Students need to know about Riemann sums so that they can understand definite integrals (a shorthand notation for the limit if a Riemann sum) and the Fundamental theorem of Calculus. Theses posts help prepare students for Riemann sums.

Integration Itinerary Some thoughts on the order of topics in your integration unit.

Some preliminary posts leading up to Riemann sums

The Old Pump Where I start Integration
Flying into Integrationland Continues the investigation in the Old Pump – the airplane problem
Jobs, Jobs, Jobs Integration in real life.
Working Towards Riemann Sums (12-10-2012)

While I prefer to teach antidifferentiation after students have learned about the Fundamental Theorem of Calculus, others prefer to discuss antidifferentiation firsts and the topic often precedes Riemann sums in textbooks. (See Integration Itinerary ) If you are among those, here are posts on antidifferentiation. If you teach this topic later, save this post for then.

ANTIDIFFERENTIATION

Antidifferentiation (11-28-2012)

Why Muss with the “+C”? But still don’t forget it.

Arbitrary Ranges (2-9-2014) Integrating inverse trigonometric functions.

ANTIDIFFERENTIATION BY PARTS This is a BC topic, or you could use it after the exam in an AB course.

Integration by Parts 1 (2-2-2013) Basics

Integration by Parts 2 (2-4-2013) The Tabular Method

Modified Tabular Integration (7-24-2013) A quicker way

Parts and More Parts (8-5-2016) Reduction formulas (Not tested on the AP Calculus exams)

Good Question 12 – Parts with a Constant (12-13-2016) How come you don’t need the “+C”?

Integration

Posted on November 6, 2018 by Lin McMullin

Integration – DON’T PANIC

As I’ve mentioned before, I try to stay a few weeks ahead of where I figure you are in the curriculum. So here. early in November, I start with integration. You probably don’t start integration until after Thanksgiving in early December. That’s about the midpoint of the year. Don’t wait too much longer. True, your kids are not differentiation experts (yet); there will be plenty of differentiation work while your teaching and learning integration. Spending too much time on differentiation will give you less time for integration and there is as much integration on the test as differentiation.

The first thing to decide is when to teach antidifferentiation (finding the function whose derivative you are given). Many books do this at the end of the last differentiation chapter or the first thing in the first integration chapter. Some teachers, myself included, prefer to wait until after presenting the Fundamental Theorem of Calculus (FTC). Still others wait until after teaching all the applications. The reasons for this are discussed in more detail in the first post below, Integration Itinerary.

Integration itinerary – a discussion of when to teach antidifferentiation.

The following posts are on different antidifferentiation techniques.

Antidifferentiation u-substitution

Why Muss with the “+C”?

Good Question 12 – Parts with a Constant?

Arbitrary Ranges Integrating inverse trigonometric functions.

Integration by Parts I (BC only)

Good Question 12 – Parts with a Constant How come you don’t need the “+C”?

The next three posts discuss the tabular method in more detail. This is used when integration by parts must be used more than once. If memory serves, using integration by parts twice on the same function has never shown up on the AP exams. Just sayin’.

Integration by Parts II (BC only) The Tabular method.

Parts and More Parts (BC only) More on the tabular method and on reduction formulas

Modified Tabular Integration (BC only) With this you don’t need to make a table; it’s quicker than the tabular method and just as easy.

Revised and updated November 4, 2018

The Mean Value Theorem

Posted on October 23, 2018 by Lin McMullin

Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year.

See last Fridays post Foreshadowing the MVT for an a series of problems that will get your students ready for the MVT.

Here are some previous post on the MVT:

Fermat’s Penultimate Theorem A lemma for Rolle’s Theorem: Any function extreme value(s) on an open interval must occur where the derivative is zero or undefined.

Rolle’s Theorem A lemma for the MVT: On an interval if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and f(a) = f(b), there must exist a number c in the open interval (a, b) where f ‘(c) = 0.

Mean Value Theorem I Proof

Mean Value Theorem II Graphical Considerations

Darboux’s Theorem The Intermediate Value Theorem for derivatives.

Mean Tables

Revised from a post of October 31, 2017

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Author Archives: Lin McMullin

Good Question 16

Riemann Sums – the Theory

Brushing Up the Blog

Happy Thanksgiving!

Getting Ready to Integrate

Integration

Integration – DON’T PANIC

The Mean Value Theorem

Share this:

Share this:

Happy Thanksgiving!

Share this:

Share this:

Integration – DON’T PANIC

Share this:

Share this:

Share this: