Introducing Power Series

The posts for the next several weeks will be on topics tested only on the BC Calculus exams. Continuing with some posts on introducing power series (the Taylor and Maclaurin series)

Introducing Power Series 1 Two examples to lead off with.

Introducing Power Series 2 Looking at the graph of a power series foreshadows the idea of the interval of convergence.

Introducing Power Series 3 The Taylor Approximating Polynomial with examples of using a series to approximate.

Graphing Taylor Polynomials Graphing calculator hints.







Getting Ready

Today we continue to look at some previous posts that I hope will help you and your students throughout the year. We begin with some posts on graphing calculator use and then a few general things in three posts on beginning the year, followed by some mathematics I hope students know before they start studying the calculus.

Graphing Calculators

There are four things that students may, and are required to know how to do, for the AP Exams. But graphing calculators are not required just to answer a few questions on the exams. They are to encourage investigations and experimentation in all math classes. And not just graphing calculator use but all kinds of appropriate technology. So, don’t restrict yourself and your students to only those operations required on the exam. That said, here are previous posts on exam calculator use; as the year goes on there will be other posts on the use of graphing calculators and other technology in your class.

Starting School

A little late perhaps …

These posts discuss basic ideas that I always hoped students knew about mathematics before starting calculus





My Favorite Function

My favorite function is f\left( x \right)=\sin \left( \ln \left( x \right) \right). I like to ask folks how many zeros this function has on the interval 0<x\le 1.

Most folks will get their calculator out and graph the function on the interval

Fav 1

Two zeros: one at 1 and the other about 0.05 more or less.

So then I suggest they look at 0<x\le 0.1. This is the left 10% of the first window.

Fav 2

Sure enough there is the zero near 0.05 but there is another near 0

So another window 0<x\le 0.01

Fav 3

Pretty soon they get the idea. Every time we stretch out the graph, there are more roots.

What is going on? The first thing is that this is not a question to be answered on a graphing calculator, the nice graphs notwithstanding.

So try to solve it by hand. Since \sin \left( x \right)=0 for x=k\pi  where k is an integer, we need to see when \ln \left( x \right)=k\pi . That will be when \displaystyle x={{e}^{k\pi }}. And since our domain is proper fractions it must be that k\le 0. So the zeros are infinite in number, namely \displaystyle x={{e}^{0}},{{e}^{-\pi }},{{e}^{-2\pi }},{{e}^{-3\pi }},\cdots . Which answers the original question but raises others.

Why can’t we see the zeros on the graph?

This is not a calculator glitch; in fact computers can do no better. Each root is the next largest root divided by \displaystyle {{e}^{\pi }}\approx 23. So each root is about \displaystyle \tfrac{1}{23}of the larger next root.

The calculator screen is made up of pixels. The number you choose for xmin is the center of the column of pixels; the number you choose for xmax is the center of the right-most column of pixels. The distance between the two ends is divided and assigned evenly to the centers of the other columns of pixels. The y-coordinates of the pixels are calculated the same way. The calculator evaluates the function at each pixel value and turns on the pixel in that column closest to (rarely at) the function’s value. A lot can go on between the pixels and the graphing calculator and its operator will not see what is happening there.

In this example, all the missing roots are between the first and second pixels on the left! When you change xmax to see the left 10% of the screen you see one and every now and then two roots, but the rest are still between the two pixels on the left.

Would a wider screen help? Perhaps a little, but not much.

Here’s a good exercise for a class: Suppose you could print the graph on a paper 1 mile (5280 feet) wide with the root at x = 1 on the right edge. Where would the next several roots be?

  •  \displaystyle {{e}^{-\pi }} is 228.169 feet from the left edge
  •  \displaystyle {{e}^{-2\pi }} is 9.860 feet from the left edge
  •  \displaystyle {{e}^{-3\pi }} is 0.426 feet or 5.113 inches from the left edge
  •  \displaystyle {{e}^{-4\pi }} is 0.221 inches from the left edge (less than ¼ inch)
  • And all the remaining roots are within 0.00955 inches from the left edge.

If the paper stretched from the earth to the sun you could see a few more. At 93,000,000 miles, the zero at \displaystyle {{e}^{-10\pi }} is about 0.134 inches from the edge.

So why do I like this problem?

Look at all the math we did.

  • We learned that graphing is not always the path to the answer.
  • We learned how calculators choose the points they graph, and which they miss.
  • We practiced how to solve a trig equation.
  • We practiced how to solve a natural logarithm equation.
  • We consider the actual size of the negative powers of e and saw how they got exponentially smaller.
  • We did a practical problem in scaling to illustrate how fast the numbers diminish.

Why do I like this function?

What’s not to like?

Update (February 7, 2015) Chip Rollinson made this cool Geogebra applet to illustrate My Favorite Function. Use the slider on the screen and notice the x-axis scale as it changes. Thank you, Chip. 

Calculator Use on the AP Exams

In my final post on reviewing for the AP Calculus exams I return to calculator use.

I hope everyone has been using calculators all year long. (My opinion is that students should be given a CAS calculator, or CAS computer program, or now even an iPad CAS app on the first day of Algebra 1 before they get their books – or earlier. But we’ll save that for another post.)

The exam will not instruct students when or if to use a calculator on a particular question. Sometimes the multiple-choice answers will provide a big hint (if the choices are decimals), other times not. On the free-response questions if the problem can be done by calculator it is expected that students will use their calculator – they don’t have to, but there is no credit for finding an antiderivative or a derivative by hand. It is the numerical answer that counts.

Here again is what students are allowed to do on the exams with their calculator without showing any additional work.

  • Graph a function in a window. They may have to determine the window themselves.
  • Solve an equation. This is usually best done by graphing both sides and finding the point(s) of intersection, but any equation solving routine in the calculator may be used.
  • Find the numerical value of a derivative at a point.
  • Evaluate a definite integral.

For the last three items students should write what they are doing on their paper. That is, write the equation, the definite integral or {v}'\left( 3.5 \right)= followed by the number from their calculator. Use standard notation not calculator syntax.

For anything else, the work must be shown. Calculators can find extreme values, but readers will look for the appropriate “calculus” and not just the answer. In fact, a correct answer with no supporting work may not receive any credit.

Another handy skill is to be able to store and recall the numbers found without retyping them on their calculator. This saves time, students are less likely to make a typing mistake, and it avoids round off error.

Answers from calculators should be correct to three places after the decimal point. This does not mean they must be rounded or truncated; more places may be given as long as the first three are correct.

Remember that arithmetic simplification is not required. Only answers from calculators need to be given as decimals. If the computation gives 1 + 1, or 6\pi  or ½ + cos(3) or \sin \left( \tfrac{\pi }{6} \right) then the answer may be left that way. This also applies to a long expression resulting from a Riemann sum. Once the answer consists of all numbers, stop! You cannot make the answer any better and if you make a mistake or type the wrong key, then the final answer will be wrong and the point will be lost.

Be careful not to round too soon. If, for example, a student find the limits of integration and rounds them to three places to write on their paper, they will have earned the point given for limits of integration. BUT if these rounded values are then used to compute the integral and the rounding causes the final answer to not be correct to three places then they will lose the answer point.

Finally, get a copy of the directions for the two parts of the exam and go over them with your students before the test. Be sure students understand them.

Good luck to your students on the exam. Nah, luck has nothing to do with it. You prepared them well, they will do well.