Monthly Archives: February 2017

The Writing Questions on the AP Exams

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The goals of the AP Calculus program state that, “Students should be able to communicate mathematics and explain solutions to problems both verbally and in well written sentences.” For obvious reasons the verbal part cannot be tested on the exams; it is expected that you will do that in your class. The exams do require written answers to parts of several questions. The number of points riding on written explanations on recent exams is summarized in the table below.

 Year AB BC
2007 9 9
2008 7 8
2009 7 3
2010 7 7
2011 7 6
2012 9 7
2013 9 7
2014 6 3
2015 8 6
2016 6 6

The average is between 6 and 8 points each year with some years having 9. That’s the equivalent of a full question. So, this is something that should not be overlooked in teaching the course and in preparing for the exams. Start long before calculus; make writing part of the school’s math program.

That a written answer is expected is indicated by phrases such as:

  • Justify you answer
  • Explain your reasoning
  • Why?
  • Why not?
  • Give a reason for your answer
  • Explain the meaning of a definite integral in the context of the problem.
  • Explain the meaning of a derivative in the context of the problem.
  • Explain why an approximation overestimates or underestimates the actual value

How do you answer such a question? The short answer is to determine which theorem or definition applies and then show that the given situation specifically meets (or fails to meet) the hypotheses of the theorem or definition.

Explanations should be based on what is given in the problem or what the student has computed or derived from the given, and be based on a theorem or definition. Some more specific suggestions:

  • To show that a function is continuous show that the limit (or perhaps two one-sided limits) equals the value at the point. (See 2007 AB 6)
  • Increasing, decreasing, local extreme values, and concavity are all justified by reference to the function’s derivative. The table below shows what is required for the justifications. The items in the second column must be given (perhaps on a graph of the derivative) or must have been established by the student’s work.
Conclusion Establish that
y is increasing y’ > 0  (above the x-axis)
y is decreasing y’ < 0   (below the x-axis)
y has a local minimum y’ changes  – to + (crosses x-axis below to above) or {y}'=0\text{ and }{{y}'}'>0
y has a local maximum y’ changes + to –  (crosses x-axis above to below) or {y}'=0\text{ and }{{y}'}'<0
y is concave up y’ increasing  (going up to the right) or {{y}'}'>0
y is concave down y’ decreasing  (going down to the right) or {{y}'}'<0
y has point of inflection y’ extreme value  (high or low points) or {{y}'}' changes sign.
  •  Local extreme values may be justified by the First Derivative Test, the Second Derivative Test, or the Candidates’ Test. In each case the hypotheses must be shown to be true either in the given or by the student’s work.
  • Absolute Extreme Values may be justified by the same three tests (often the Candidates’ Test is the easiest), but here the student must consider the entire domain. This may be done (for a continuous function) by saying specifically that this is the only place where the derivative changes sign in the proper direction. (See the “quiz” below.)
  • Speed is increasing on intervals where the velocity and acceleration have the same sign; decreasing where they have different signs. (2013 AB 2 d)
  • To use the Mean Value Theorem state that the function is continuous and differentiable on the interval and show the computation of the slope between the endpoints of the interval. (2007 AB 3 b, 2103 AB3/BC3)
  • To use the Intermediate Value Theorem state that the function is continuous and show that the values at the endpoints bracket the value in question (2007 AB 3 a)
  • For L’Hôpital’s Rule state that the limit of the numerator and denominator are either both zero or both infinite. (2013 BC 5 a)
  • The meaning of a derivative should include the value and (1) what it is (the rate of change of …, velocity of …, slope of …), (2) the time it obtains this value, and (3) the units. (2012 AB1/BC1)
  • The meaning of a definite integral should include the value and (1) what the integral gives (amount, average value, change of position), (2) the units, and (3) what the limits of integration mean. One way of determining this is to remember the Fundamental Theorem of Calculus \displaystyle \int_{a}^{b}{{f}'\left( x \right)dx}=f\left( b \right)-f\left( a \right). The integral is the difference between whatever f represents at b and what it represents at a. (2009 AB 2 c, AB 3c, 2013 AB3/BC3 c)
  • To show that a theorem applies state and show that all its hypotheses are met. To show that a theorem does not apply show that at least one of the hypotheses is not true (be specific as to which one).
  • Overestimates or underestimates usually depend on the concavity between the two points used in the estimates.

A few other things to keep on mind:

  • Avoid pronouns. Pronouns need antecedents. “It’s increasing because it is positive on the interval” is not going to earn any points.
  • Avoid ambiguous references. Phrases such as “the graph”, “the derivative”, or “the slope” are unclear. When they see “the graph” readers are taught to ask “the graph of what?” Do not make them guess. Instead say “the graph of the derivative”, “the derivative of f”, or “the slope of the derivative.”
  • Answer the question. If the question is a yes or no then say “yes” or “no.” Every year students write great explanations but never clearly say whether they are justifying a “yes” or a “no.”
  • Don’t write too much. Usually a sentence or two is enough. If something extra is in the explanation and it is wrong, then the credit is not earned even though the rest of the explanation is great.

As always, look at the scoring standards from past exam and see how the justifications and explanations are worded. These make good templates for common justifications. Keep in mind that there are other correct ways to write the justifications.

QUIZ

Here is a quiz that can help your students learn how to write good explanations.

Let f\left( x \right)={{e}^{x}}\left( x-3 \right) for 0\le x\le 5.

Find the location of the minimum value of f(x). Justify your answer three different ways (without reference to each other).

Don’t tell your students the three ways – they should know that!

The minimum value occurs at x = 2. The three ways to justify this are the First Derivative Test, the Second Derivative Test and the Candidates’ Test (aka: the Closed Interval Test). Let them discuss and constructively criticize each other’s answers. As a class, compare and contrast the students’ answers.

Next Posts:

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

Friday March 10: Type 3: Graph Analysis

Revised from a post of March 9, 2015.

Using Practice Exams

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bubble-sheetThe multiple-choice exams from 2003, 2008 and 2012 and all the free-response questions and solutions from past years are available online. The students can easily find them. Starting in 2012 the College Board provided full actual AP Calculus exams, AB and BC, for teachers who had an audit on file to use with their students as practice exams. These included multiple-choice and free-response questions. However, the rules about using the exams are quite restrictive. I quote:

AP Practice Exams are provided by the College Board for AP Exam preparation. Teachers are permitted to download the materials and make copies to use with their students in a classroom setting only. To maintain the security of the exams, teachers should collect all materials after their administration and keep them in a secure location. Exams may not be posted on school or personal websites, nor electronically redistributed for any reason. Further distribution of these materials outside of the secure College Board site disadvantages teachers who rely on uncirculated questions for classroom testing. Any additional distribution is in violation of the College Board’s copyright policies and may result in the termination of Practice Exam access for your school as well as the removal of access to other online services such as the AP Teacher Community and Online Score Reports.(Emphasis in original)

Practice exams are a good thing to use to help get your students ready for the real exam. They

  • Help students understand the style and format of the questions and the exam,
  • Give students practice in working under time pressure
  • Help students identify their calculus weaknesses, to pinpoint the concepts and topics they need to brush up on before the real exam.
  • Give students an idea of their score 5, 4, 3, 2, or 1.

Teachers sometimes assign a grade on the exam and count it as part of the students’ averages. The problem is that some of the exams in whole or part have found their way onto the internet. (Imagine.) The College Board does act to remove the exams when they learn of such a situation. Nevertheless, students have often able to, shall we say, “research” the questions ahead of their practice exams or homework assignments. Teachers are, quite rightly, upset about this and considered the “research” cheating.To deal with this situation I offer …

A Modest Proposal

If you give a practice exam, DON’T GRADE IT or count it as part of the students’ average. Don’t grade their homework if you assign the released questions.

Athletes are not graded on their practices; only the game counts. Athletes practice to maintain their skills and improve on their weakness. Make it that way with your practice tests.

Calculus students are intelligent. Explain to them why you are asking them to take a practice exam; how they will use to it maintain their skills, identify their weaknesses, and improve on them, and how this will help them on the real exam. By taking the pressure of a grade away, students can focus on improvement.

Make an incentive of this, by not making students concerned about a grade.

This post is a revision of my post of June 6, 2015. There are some good comment and suggestions from readers of the blog. Check them out here

Next posts:

 

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

 

(Confession: When I was teaching I often had nothing to base a fourth quarter grade on. The school started after Labor Day and the fourth quarter began about two weeks before the AP exam (and ran another 6 or 7 week after it). Students were required to take a final exam given the week after the AP exam and then they were done. The fourth quarter grade was usually the average of the first three quarters.)

 

 

 

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Resources for Reviewing

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Here are several resources that will help you get started with your review

  • Released free-response questions from the College Board. AB and BC.
  • Released multiple-choice questions.
    • 2012 from the College Board are here for AB and here for BC FREE (.PDF)
    • 2008 AB and BC  College Board store cost $30.00 Paper (Search on-line and you should be able to find a copy, but so can your students.)
    • 2003 AB and BC College Board store cost $42.00 Paper (Search on-line and you should be able to find a copy, but so can your students.)
    • 1998  AB Exam Free (.PDF)
    • The 2013 – 2016 Secure Exams are available at your audit website.
  • Type Analysis 2018 by the 10 type questions that will be discussed in later posts. (by. Lin McMullin)
  • The  AB Directions and BC Directions. Yes, this is boiler plate stuff, but take a few minutes to go over it with your students. They should not have to see the directions for the first time on the day of the exam.The free-response instruction have changed slightly from previous years. The change is not a policy change, but rather made to emphasize certain things that students should be doing. For more on the changes see NCTM Calculus Panel Notes.
  • Calculator Skills needed on the AP Exams – share this information with your students, if you have not already done so. There are only about 12 -15 points on the entire exam which require a calculator. A calculator alone will not get anyone a 5 (or even a 2). Nevertheless, the points are there and usually pretty easy to earn. The real reason calculators and other technology are so important is that when used throughout the year, they help students better understand the calculus.

The next posts:

Friday February 24: Using Practice Exams

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

Revised 4-17-17

 

 

AP Exam Review

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Don’t panic! It is not time to start reviewing.

I try to keep these posts ahead of the typical AP Calculus timeline so you can have time to think them over and include what you want to use from them (if anything).

Over the next 6 weeks I will post several times each week. The post will be previous posts on reviewing slightly revised and updated. Today’s post is “Ideas for reviewing for the AP Exam” originally posted on February 25, 2013.

Ideas for reviewing for the AP Exam

Part of the purpose of reviewing for the AP calculus exams is to refresh your students’ memory on all the great things you’ve taught them during the rear. The other purpose is to inform them about the format of the exam, the style of the questions, the way they should present their answer, and how the exam is graded and scored.

Using AP questions all year is a good way to accomplish some of this. Look through the released multiple-choice exams and pick questions related to whatever you are doing at the moment. Free-response questions are a little trickier since the parts of the questions come from different units. These may be adapted or used in part.

At the end of the year I suggest you review the free-response questions by type – table questions, differential equations, area/volume, rate/accumulation, graph, etc. That is, plan to spend a few days doing a selection of questions of one type so that student can see how that type question can be used to test a variety of topics. Then go onto the next type. Many teachers keep a collection of past free-response questions filed by type rather than year. This makes it easy to study them by type.

In the next few posts I will discuss each type (there are 10) in turn and give suggestions about what to look for and how to approach the question.

Simulated Exam

Plan to give a simulated (mock) exam. Each year the College Board makes a full exam available. The exams for 1998, 2003, 2008, and 2012 are available at AP Central  and the secure 2013 – 2016 exams are available through your audit website. If possible, find a time when your students can take the exam in 3.25 hours. Teachers often do this on a weekend day or in the evening. This will give your students a feel for what it is like to work calculus problems under test conditions. If you cannot get 3.25 hours to do this give the sections in class using the prescribed time. Some teachers schedule several simulated exams. Of course, you need to correct them and go over the most common mistakes.

Explain the scoring

There are 108 points available on the exam; each half is worth the same – 54 points. The number of points required for each score is set after the exams are graded.

For the AB exam, the points required for each score out of 108 point are, very approximately:

  • for a 5 – 69 points,
  • for a 4 – 52 points,
  • for a 3 – 40 points,
  • for a 2 – 28 points.

The numbers are similar for the BC exams are again very approximately:

  • for a 5 – 68 points,
  • for a 4 – 58 points,
  • for a 3 – 42 points,
  • for a 2 – 34 points.

The actual numbers are not what is important. What is important is that students to know is that they can omit or get wrong many questions and still earn a good score. Students may not be used to this (since they skip or get so few questions wrong on your tests!). They should not panic or feel they are doing poorly if they miss a number of questions. If they understand and accept this in advance they will calm down and do better on the exams. Help them understand they should gather as many points as they can, and not be too concerned if they cannot get them all. Doing only the first 2 parts of a free-response question will probably put them at the mean for that question. Remind them not to spend time on something that’s not working out, or that they don’t feel they know how to do.

Directions

Print a copy of the directions for both parts of the exam and go over them with your students. Especially, for the free-response questions explain the need to show their work, explain that they do not have to simplify arithmetic or algebraic expressions, and explain the three-decimal place consideration. Be sure they know what is expected of them.The directions are here: AB Directions and BC Directions. Yes, this is boiler plate stuff, but take a few minutes to go over it with your students. They should not have to see the directions for the first time on the day of the exam.

Next Posts:

Thursday February 23, 2017: A list of resources for you and your students in preparation for the exams.

Friday February 24: Using Practice Exams

Tuesday February 28: The Writing Questions on the AP Exams

Friday March 3: Type 1 of the 10 type questions: Rate and Accumulation

Tuesday March 7: Type 2 Linear Motion

 

 

 

Geometric Series – Far Out

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One of the great things – at least I like it – about the Taylor series of a function is that it is unique. There is only one Taylor series for any function centered at a given point, what that means is that any way you get it, it’s right.

Faced with writing the power series for, say, \displaystyle f\left( x \right)=\frac{3x}{1-2x}, instead of cranking out a bunch of derivatives, we can say this looks a lot like the formula for the sum of a geometric series,

\displaystyle \sum\limits_{k=1}^{\infty }{a{{r}^{k-1}}}=\frac{a}{1-r}. Taking  a = 3x and r = 2x, the series is

\displaystyle \frac{3x}{1-2x}=3x+6{{x}^{2}}+12{{x}^{3}}+24{{x}^{4}}+\cdots =3\cdot \sum\limits_{k=1}^{\infty }{{{2}^{k-1}}{{x}^{k}}}.

Furthermore, since a geometric series converges only when \left| r \right|<1, the interval of convergence for this series is \left| 2x \right|<1 or -\tfrac{1}{2}<x<\tfrac{1}{2} and we don’t even have to check the endpoints.

There are other choices as well.  We could write \displaystyle f\left( x \right)=3x{{\left( 1-2x \right)}^{-1}} and then expand the binomial using the binomial theorem. Or we could use the technique of long division of polynomials to divide 3x by (1 – 2x) – leaving the divisor as written here.

This works even in more complicated situations. Let \displaystyle g\left( x \right)=\frac{3x}{{{x}^{2}}-4}. Begin by dividing each term by –4. This gives \displaystyle g\left( x \right)=\frac{-\tfrac{3}{4}x}{1-\tfrac{1}{4}{{x}^{2}}}. Then treating this as a geometric series

\displaystyle g\left( x \right)=\sum\limits_{k=1}^{\infty }{-\tfrac{3}{4}x{{\left( \tfrac{1}{4}{{x}^{2}} \right)}^{k-1}}=-\frac{3}{4}x-\frac{3}{16}{{x}^{3}}-\frac{3}{64}{{x}^{5}}-\frac{3}{256}{{x}^{7}}-\cdots }

The interval of convergence is \displaystyle \left| \tfrac{1}{4}{{x}^{2}} \right|<1, or –2 < x < 2

Now the fun part

I once heard of a student making one of those great “mistakes.” For the series above, she divided by (–x2) and found that \displaystyle g\left( x \right)=\frac{-\frac{3}{x}}{1-\frac{4}{{{x}^{2}}}} and then wrote:

\displaystyle g\left( x \right)=\sum\limits_{k=1}^{\infty }{\left( -\frac{3}{x} \right){{\left( \frac{4}{{{x}^{2}}} \right)}^{k-1}}=-\frac{3}{x}-\frac{12}{{{x}^{3}}}-\frac{48}{{{x}^{5}}}-\frac{192}{{{x}^{7}}}-\cdots }

So, what’s wrong with that?

Nothing actually.

Okay, it’s not a Taylor Series since a Taylor series is allowed only non-negative exponents, but it’s still a geometric series. Let’s take a look at its interval of convergence: \displaystyle \left| \frac{4}{{{x}^{2}}} \right|<1, or \displaystyle \left| \frac{{{x}^{2}}}{4} \right|>1,  or the union of x>2 and x<-2, Whoa, that’s different and not even an interval.

The graph will make things clear (as usual):

geom-series

The original function graphed as a rational expression is shown in black. The Taylor polynomial (4 terms) is shown in blue; it approximates the function well between –2 and 2 as we should expect. The red graph is the student’s series (4 terms) and it is a good approximation of the series outside of the interval (–2, 2), far outside! Way Cool!

Of course, this kind of series is not studied in beginning calculus. It may make a good topic for a report or project for someone in your class.

Graphing Taylor Polynomials

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The eighth in the Graphing Calculator / Technology series

Here are some hints for graphing Taylor polynomials using technology. (The illustrations are made using a TI-8x calculator. The ideas are the same on other graphing calculators; the syntax may be slightly different.)

Each successive term of a Taylor polynomial consists of all the previous terms plus one new term. To show students how Taylor polynomials closely approximate a function (in the interval of convergence, of course), enter the function as Y1. Then enter the first term of the polynomial as Y2. Enter the next polynomial as Y3 = Y2 + the second term; enter the next as y4 = Y3 + the next term, and so on.

The example is the McLaurin series for sin(x) centered at the origin:

\displaystyle \sin \left( x \right)=x-\frac{{{x}^{3}}}{3!}+\frac{{{x}^{5}}}{5!}+\cdots +\frac{{{(-1)}^{2n-1}}{{x}^{2n-1}}}{\left( 2n-1 \right)!}=\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n+1}}\frac{{{x}^{2n-1}}}{\left( 2n-1 \right)!}}

Each will graph one at a time. Watching them graph, one at a time, is instructive as well; each curve approximates the sine curve (in black) further and further away from the origin.

series-1

series-2

Another way to graph the polynomials is to enter them as a sequence of sums. The example this time is ln(x) centered at x = 2:

\displaystyle \ln \left( x \right)=\ln \left( 2 \right)+\frac{x-2}{2}-\frac{{{\left( x-2 \right)}^{2}}}{8}+\frac{{{\left( x-2 \right)}^{3}}}{24}+...=\ln \left( 2 \right)+\sum\limits_{n=1}^{\infty }{{{\left( -1 \right)}^{n+1}}\frac{{{\left( x-2 \right)}^{n}}}{{{2}^{n}}n}}

The syntax is seq( series in sigma notation, indexing variable, start value, end value [,step]). Notice from the figure that the indexing variable, K, is above the sigma.

series-4

The individual polynomials graph in the same color (blue); the function is shown in red.

series-3Comparing the two graphs (sin(x) and ln(x)) is a good way to start a discussion about the interval of convergence – ask what is different about the graphs as more polynomials are graphed on each. Notice that unlike the sin(x) series the ln(x) polynomials only come close to the function in a limited interval (0, 4) centered at x = 2.

Desmos is also a good program to use to illustrate Taylor and McLaurin polynomials (as are Geogebra and Winplot). The use of the sliders makes it possible to see the successive polynomials quickly. They also help students see the interval of convergence as higher degree polynomials hug the graph on wider intervals (sin(x)), or stay within the same interval (ln(x)). The Desmos illustration with slider for the sin(x) centered at the origin is here and for ln(x)  centered at x = 2 is here. Study the input on the left side and enter Taylor polynomials for other functions.

The fifth degree Taylor polynomial for sin(x) centered at the origin.The interval of convergence is all real numbers. Each polynomial “hugs” the graph on wider intervals.

The fifth degree Taylor polynomial for ln(x) centered at x = 2. The interval of convergence is 0 < x < 4. The polynomials all leave the graph outside of this interval.

Coming soon

Feb 14th, Geometric Series – Far Out