## Getting Ready for the AP Exams

Another month and it will be time to start reviewing for the AP exams. The exams this year are on Wednesday morning May  7, 2014.

To help you plan ahead, below are links to previous posts specifically on reviewing for the exam and on the type questions that appear on the free-response sections of the exams. I try to review by topic spending 1-2 days on each so that students can see  the things that are asked for each general type. Many of the same ideas are tested in smaller “chunks” on the multiple-choice sections, so looking at the type should help with not only free-response questions but many of the multiple-choice questions as well. Of course, I will also spend some time on just multiple-choice questions as well.

February 25, 2013: Ideas for Reviewing for the AP Calculus Exams

February 25, 2013: The AP Calculus Exams

February 27, 2013: Interpreting Graphs AP Type Questions 1

March 2, 2013: The Rate/Accumulation Question AP Type Question 2

March 4, 2013: Area and Volume Questions AP Type Question 3

March 6, 2013: Motion on a Line AP Type Question 4

March 8, 2013: The Table Question AP Type Question 5

March 10, 2013: Differential Equations AP Type Question 6

March 15, 2013: Implicit Relations and Related Rates AP Type Question 7

March 15, 2013: Parametric and Vector Equations AP Type Question 8 (BC)

March 18, 2013: Polar Curves AP Type Question 9 (BC)

March 20, 2013: Sequences and Series AP Type Question 10 (BC)

March 22, 2013: Calculator Use on the AP Exams (AB & BC)

Filed under AP Calculus Exams, Reviewing

At the school where I am teaching this year, all of the students, K – 12, are issued iPads. Whether this is the coming thing in education or not, I cannot say. I like the idea, but then I like technology in teaching and learning. My school issued iPad is my fourth. I offer today a few observations, anecdotal to be sure, for those who are curious about this growing trend.

First, the school owns the iPads. Therefore, the school restricts what apps students can use on them. The school can see what is on each iPad. Students are able to download apps only from the school’s approved list. The school pays for some of the recommended apps. The iPads do not have Apps Store access. The school owns and uses software to make this possible. Students who manage to get around the system are called in and the problem is corrected.

Websites that are not approved are blocked on the school’s server. Students can still access the entire web away from the school.

Yes, the students have games on their iPads, and yes, they try to play them in class. There is also instant messaging and e-mail. The teachers have to keep an eye on what the kids are doing – nothing new about that.

Many of the teachers require students to do their reports and essays using one of the apps available. Students are getting very good at note talking on their machines. Notability (about \$3) seems to be the most popular app for this. Even in math classes students can take their notes and do their homework without benefit of paper. Some students e-mail me their homework on days when I collect it.

There are a variety of graphing apps available all of which produce far better graphs than graphing calculators. Good Grapher Pro is my favorite and very easy to use for bot 2D and 3D graphs.

Graphing by hand is a problem. Note-taking apps have grid backgrounds, but it is difficult to plot points, and draw lines or curves as neatly as you can on paper.

My calculus classes have access to an electronic copy of their textbook online. It is available anywhere there is internet access. They have a full copy of everything in the text and it looks just like the text. Most of the drawings are animated in the online version – this is a big plus. Also, it is easy to copy an individual problem, say a definite integral, and paste it into Notability or another app and work on it.

My Algebra 1 students do not have an online copy available. They do the next best thing. They photograph the homework page and do their problems from the picture.

It turns out that I am not 100% technology: I still give most of my notes and work the homework problems on a whiteboard. Some students photograph what I write. Then they take the picture home and use it to study from – at least that’s what they tell me. I hope this is a help. I can talk and write on the board much faster than students can write. It seems to me that sometimes note taking can be a distraction. That is, kids are so busy writing down everything that they are not following the flow of ideas.  So if listening and then taking a picture helps them learn better, I’m all for it.

I also post the assignments, worksheets, and so forth online. Students download them to their iPads and always have them handy.

In a previous post I discussed how I use an app called Socrative in my classes.

Filed under Uncategorized

## Arbitrary Ranges

In my last post I discussed the idea that the ranges of the inverse trigonometric functions are chosen somewhat arbitrarily. For good reasons, the ranges always include the first quadrant and the adjoining quadrant (II or IV) where the function is negative. If possible, the range is also chosen to be continuous. Still the choices are arbitrary.

I discussed the range of the inverse tangent function in relation to the value of the improper integral $\int_{0}^{\infty }{\frac{1}{1+{{x}^{2}}}dx}$. I noted that if we used some other continuous range for the inverse tangent that the result of this or any other definite integral of this function gives the same value. Thus a range for the inverse tangent of $\left( -\tfrac{\pi }{2},\tfrac{\pi }{2} \right),\left( \tfrac{\pi }{2},\tfrac{3\pi }{2} \right),\left( \tfrac{3\pi }{2},\tfrac{5\pi }{2} \right)$, etc. will give the same result.

For antiderivatives involving the inverse tangent or inverse cotangent this is true, but what about the other inverse trigonometric functions?

When evaluating the difference between two values as one does when evaluating a definite integral any range which results in a graph “parallel” to the graph over the commonly accepted range gives the same value.

However, for an integral requiring the inverse sine, if we use the range $\left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right]$,

$\displaystyle \int_{0}^{1/2}{\frac{1}{\sqrt{1-{{x}^{2}}}}dx}=\left. {{\sin }^{-1}}\left( x \right) \right|_{0}^{1/2}$

$={{\sin }^{-1}}\left( \tfrac{1}{2} \right)-{{\sin }^{-1}}\left( 0 \right)=\tfrac{5\pi }{6}-\pi =-\tfrac{\pi }{6}$

Indicating that a region above the x-axis has a negative area!

So $\left[ \tfrac{\pi }{2},\tfrac{3\pi }{2} \right]$ is not a good choice. We could use other ranges for the inverse sine function but they would have to be such that they result in inverse sine graphs “parallel” to the usual graph. So we could use $\left( \tfrac{3\pi }{2},\tfrac{5\pi }{2} \right)$ or $\left( \tfrac{7\pi }{2},\tfrac{9\pi }{2} \right)$, but not $\left( \tfrac{5\pi }{2},\tfrac{7\pi }{2} \right)$.

The same problem arises with the inverse cosine, the inverse secant, and the inverse cosecant.

It is best to stick with the commonly accepted ranges. Still, going off on tangents often helps sharpen a student’s understanding.

Filed under Integration Theory, Uncategorized

## Improper Integrals and Proper Areas

A few years ago on the old AP Calculus discussion group a teachers asked a question about this improper integral:

$\displaystyle \int_{0}^{\infty }{\frac{1}{1+{{x}^{2}}}dx}=\underset{b\to \infty }{\mathop{\lim }}\,\int_{0}^{b}{\frac{1}{1+{{x}^{2}}}dx}$

$=\underset{b\to \infty }{\mathop{\lim }}\,\left. \left( {{\tan }^{-1}}\left( x \right) \right) \right|_{0}^{b}$

$=\underset{b\to \infty }{\mathop{\lim }}\,\left( {{\tan }^{-1}}\left( b \right)-{{\tan }^{-1}}\left( 0 \right) \right)=\frac{\pi }{2}$

His (quite perceptive) student pointed out that the range of the inverse tangent function is arbitrarily restricted to the open interval $\left( -\tfrac{\pi }{2},\tfrac{\pi }{2} \right)$. The student asked if some other range would affect the answer to this problem. The short answer is no, the result is the same. For example, if range were restricted to say $\left( \tfrac{5\pi }{2},\tfrac{7\pi }{2} \right)$, then in the computation above:

$\underset{b\to \infty }{\mathop{\lim }}\,\left( {{\tan }^{-1}}\left( b \right)-{{\tan }^{-1}}\left( 0 \right) \right)=\tfrac{7\pi }{2}-3\pi =\tfrac{\pi }{2}$

The value is the same. While that is pretty straightforward, there are other things going on here which may be enlightening. The original indefinite integral represents the area in the first quadrant between the graph of $y=\frac{1}{1+{{x}^{2}}}$ and the x-axis. Let’s consider the function that gives the area between the y-axis and the vertical line at various values of x.

$A \displaystyle\left( x \right)=\int_{0}^{x}{\frac{1}{1+{{t}^{2}}}}\ dt$

Pretending for the moment that we don’t know the antiderivative, we can use a calculator to graph the area function. Of course we recognize this as the inverse tangent function, but what is more interesting is that whatever this function is, it seems to have a horizontal asymptote at $y=\tfrac{\pi }{2}$. The area is approaching a finite limit as x increases without bound.  The unbounded region has a finite area. The connection with improper integrals is obvious.

$\displaystyle \underset{b\to \infty }{\mathop{\lim }}\,A\left( b \right)=\underset{b\to \infty }{\mathop{\lim }}\,\int_{0}^{b}{\frac{1}{1+{{x}^{2}}}dx=}\int_{0}^{\infty }{\frac{1}{1+{{x}^{2}}}dx}$

Also, the improper integral is defined as the limit of the area function. This may give some insight as to why improper integrals are defined as they are.

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Filed under Integration, Integration Theory