Author Archives: Lin McMullin

Ted Gott’s Exam Index

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My friend Ted Gott has just prepared a new index to the AP Calculus exam free-response questions from 1998 to the present. Expanding on an earlier spread sheet by Mark Howell, Ted has referenced all the questions to the Learning Outcomes (LO) and Essential Knowledge (EK) statements of the new Course and Exam Description. AND has live links to the individual questions and their scoring standards.

AND HERE IT IS: FRQ Index by topic 1998 to 2017 (Updated to include 2017 FRQs. August 2, 2017)

And for Multiple-choice questions MC unsecure Index by topic 1998 to 2018

Clicking on the arrows at the top of each column allows you to search by LOs and EKs, and to find other questions on the same topic.The right most column provides a direct link to the scoring standard for that question.

What a great resource. THANK YOU TED !

Parametric/Vector Question (Type 8 for BC only)

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I have always had the impression that the AP exam assumed that parametric equations and vectors were first studied and developed in a pre-calculus course. In fact, many schools do just that. It would be nice if students knew all about these topics when they started BC calculus. Because of time considerations, this very rich topic is not fully developed in BC calculus.

That said, the parametric/vector equation questions only concern motion in a plane. I will try to address the minimum that students need to know to be successful on the BC exam. Certainly, if you can do more and include a unit in a pre-calculus course do so.

Another concern is that most calculus textbooks jump right to vectors in 3-space while the exam only tests motion in a plane and 2-dimensional vectors. (Actually, the equations and ideas are the same with an extra variable for the z-direction)

In the plane, the position of a moving object as a function of time, t, can be specified by a pair of parametric equations x=x\left( t \right)\text{ and }y=y\left( t \right) or the equivalent vector \left\langle x\left( t \right),y\left( t \right) \right\rangle . The path is the curve traced by the parametric equations or the tips of the position vector. .

The velocity of the movement in the x- and y-direction is given by the vector \left\langle {x}'\left( t \right),{y}'\left( t \right) \right\rangle . The vector sum of the components gives the direction of motion. Attached to the tip of the position vector this vector is tangent to the path pointing in the direction of motion.

The length of this vector is the speed of the moving object. \text{Speed }=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}. (Notice that this is the same as the speed of a particle moving on the number line with one less parameter: On the number line \text{Speed}=\left| v \right|=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}}.)

The acceleration is given by the vector \left\langle {{x}'}'\left( t \right),{{y}'}'\left( t \right) \right\rangle .

What students should know how to do:

  • Vectors may be written using parentheses, ( ), or pointed brackets, \left\langle {} \right\rangle , or even \vec{i},\vec{j} form. The pointed brackets seem to be the most popular right now, but all common notations are allowed and will be recognized by readers.
  • Find the speed at time t: \text{Speed }=\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}
  • Use the definite integral for arc length to find the distance traveled \displaystyle \int_{a}^{b}{\sqrt{{{\left( {x}'\left( t \right) \right)}^{2}}+{{\left( {y}'\left( t \right) \right)}^{2}}}}dt. Notice that this is the integral of the speed (rate times time = distance).
  • The slope of the path is \displaystyle \frac{dy}{dx}=\frac{{y}'\left( t \right)}{{x}'\left( t \right)}. See this post for more on finding the first and second derivatives with respect to x.
  • Determine when the particle is moving left or right,
  • Determine when the particle is moving up or down,
  • Find the extreme position (farthest left, right, up, down, or distance from the origin).
  • Given the position find the velocity by differentiating; given the velocity find the acceleration by differentiating.
  • Given the acceleration and the velocity at some point find the velocity by integrating; given the velocity and the position at some point find the position by integrating. These are just initial value differential equation problems (IVP).
  • Dot product and cross product are not tested on the BC exam, nor are other aspects.

Shorter questions on these ideas appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

Next Posts:

Friday March 31: For BC Polar Equations (Type 9)

Tuesday April 4: For BC Sequences and Series.(Type 10)

 

 

 

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Other Topics (Type 7)

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Any topic in the Course and Exam Description may be the subject of a free-response question. The two topics listed here have been the subject of full free-response questions or major parts of them.

Implicitly defined relations and implicit differentiation

These questions may ask students to find the first or second derivative of an implicitly defined relation. Often the derivative is given and students are required to show that it is correct. (This is because without the correct derivative the rest of the question cannot be done.) The follow-up is to answer questions about the function such as finding an extreme value, second derivative test, or find where the tangent is horizontal or vertical.

What students should know how to do

  • Know how to find the first derivative of an implicit relation using the product rule, quotient rule, the chain rule, etc.
  • Know how to find the second derivative, including substituting for the first derivative.
  • Know how to evaluate the first and second derivative by substituting both coordinates of a given point. (Note: If all that is needed is the numerical value of the derivative then the substitution is often easier if done before solving for dy/dx or d2y/dx2 and as usual the arithmetic need not be done.)
  • Analyze the derivative to determine where the relation has horizontal and/or vertical tangents.
  • Write and work with lines tangent to the relation.
  • Find extreme values. It may also be necessary to show that the point where the derivative is zero is actually on the graph and to justify the answer.

Simpler questions about implicit differentiation my appear on the multiple-choice sections of the exam.

Related Rates

Derivatives are rates and when more than one variable is changing over time the relationships among the rates can be found by differentiating with respect to time. The time variable may not appear in the equations. These questions appear occasionally on the free-response sections; if not there, then a simpler version may appear in the multiple-choice sections. In the free-response sections they may be an entire problem, but more often appear as one or two parts of a longer question.

What students should know how to do

  • Set up and solve related rate problems.
  • Be familiar with the standard type of related rate situations, but also be able to adapt to different contexts.
  • Know how to differentiate with respect to time, that is find dy/dt even if there is no time variable in the given equations. using any of the differentiation techniques.
  • Interpret the answer in the context of the problem.
  • Unit analysis.

Shorter questions on both these concepts appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on related rate see October 8, and 10, 2012 and for implicit relations see November 14, 2012

Next Posts:

Friday March 31: For BC Polar Equations (Type 9)

Tuesday April 4: For BC Sequences and Series.

Friday April 7, 2017 The Domain of the solution of a differential equation.

Differential Equations (Type 6)

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Differential equations are tested every year. The actual solving of the differential equation is usually the main part of the problem, but it is accompanied by a related question such as a slope field or a tangent line approximation. BC students may also be asked to approximate using Euler’s Method. Large parts of the BC questions are often suitable for AB students and contribute to the AB sub-score of the BC exam.

What students should be able to do

  • Find the general solution of a differential equation using the method of separation of variables (this is the only method tested).
  • Find a particular solution using the initial condition to evaluate the constant of integration – initial value problem (IVP).
  • NEW Determine the domain restrictions on the solution of a differential equation. See this post for more on this. 
  • Understand that proposed solution of a differential equation is a function (not a number) and if it and its derivative are substituted into the given differential equation the resulting equation is true. This may be part of doing the problem even if solving the differential equation is not required (see 2002 BC 5 – parts a, b and d are suitable for AB)
  • Growth-decay problems.
  • Draw a slope field by hand.
  • Sketch a particular solution on a (given) slope field.
  • Interpret a slope field.
  • Use the given derivative to analyze a function such as finding extreme values
  • For BC only: Use Euler’s Method to approximate a solution.
  • For BC only: use the method of partial fractions to find the antiderivative after separating the variables.
  • For BC only: understand the logistic growth model, its asymptotes, meaning, etc. The exams so far, have never asked students to actually solve a logistic equation IVP

Look at the scoring standards to learn how the solution of the differential equation is scored, and therefore, how students should present their answer. This is usually the one free-response answer with the most points riding on it. Starting in 2016 the scoring has changed slightly. The five points are now distributed this way:

  • one point for separating the variables
  • one point each for finding the antiderivatives
  • one point for including the constant of integration and using the initial condition – that is, for writing “+ C” on the paper with one of the antiderivatives and substituting the initial condition; finding is value is included in the “answer point.” and
  • one point, the “answer point”, for the correct answer. This point includes all the algebra and arithmetic in the problem.

A future post suggested by a reader and tentatively schooled for April 4, 2017, will discuss the domain of the solution of a differential equation. The domain is often included on the scoring standard, but unless it is specifically asked for in the question students do not need to include it. This is a new topic not previously on the Course Description.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on differential equations see January 5, 2015 and for post on related subjects see November 26, 2012,  January 21, 2013 February 1, 6, 2013

Revised 3-21-17, 08:21 to reflect new point distribution.

Next Posts:

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

Tuesday March 28: for BC Parametric Equation (Type 8)

Friday March 31: For BC Polar Equations (Type 9)

Tuesday April 4: For BC Sequences and Series.

Friday April 7, 2017 The Domain of the solution of a differential equation.

Table & Riemann Sum Questions (Type 5)

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Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, position-velocity-acceleration, theory and theorems among others. Numbers and working with numbers is part of the Rule of Four and table problems are one way this is tested.

 What students should be able to do:

  • Find the average rate of change over an interval
  • Approximate the derivative using a difference quotient. Use the two values closest to the number at which you are approximating.  This amounts to finding the slope. Show the quotient even if you can do the arithmetic in your head.
  • Use Riemann sums (left, right, midpoint), or a trapezoidal approximation to approximate the value of a definite integral using values in the table (typically with uneven subintervals). The Trapezoidal Rule, per se, is not required; it is expected that students will add the areas of a small number of trapezoids without reference to a formula.
  • Average value, average rate of change, Rolle’s theorem, the Mean Value Theorem and the Intermediate Value Theorem. (See 2007 AB 3 – four simple parts that could be multiple-choice questions; the mean on this question was 0.96 out of a possible 9.)
  • These questions are usually presented in some context and answers should be in that context.
  • Unit analysis.

 Do’s and Don’ts

Do: Remember that you do not know what happens between the values in the table unless some other information is given. For example, don’t assume that the largest number in the table is the maximum value of the function.

Do: Show what you are doing even if you can do it in your head. If you’re finding a slope, show the quotient.

Do Not do arithmetic: A long expression consisting entire of numbers such as you get when doing a Riemann sum, does not need to be simplified in any way. If you simplify correct answer incorrectly, you will lose credit. However, do not leave expression such as R(3) – pull its numerical value from the table.

Do Not: Find a regression equation and then use that to answer parts of the question. While regression is perfectly good mathematics, regression equations are not one of the four things students may do with their calculator. Regression gives only an approximation of our function. The exam is testing whether students can work with numbers.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

Next Posts:

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

Tuesday March 28: for BC Parametric Equation (Type 8)

 

 

 

Area & Volume (Type 4)

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Given equations that define a region in the plane students are asked to find its area and the volume of the solid formed when the region is revolved around a line or used as a base of a solid with regular cross-sections. This standard application of the integral has appeared every year since 1969 on the AB exam and all but one year on the BC exam.

What students should be able to do:

  • Find the intersection(s) of the graphs and use them as limits of integration (calculator equation solving). Write the equation followed by the solution; showing work is not required. Usually no credit is earned until the solution is used in context (as a limit of integration). Students should know how to store and recall these values to save time and avoid copy errors.
  • Find the area of the region between the graph and the x-axis or between two graphs.
  • Find the volume when the region is revolved around a line, not necessarily an axis or an edge of the region, by the disk/washer method.
  • The cylindrical shell method will never be necessary for a question on the AP exams, but is eligible for full credit if properly used.
  • Find the volume of a solid with regular cross-sections whose base is the region between the curves. For an interesting variation on this idea see 2009 AB 4(b)
  • Find the equation of a vertical line that divides the region in half (area or volume). This involves setting up and solving an integral equation where the limit is the variable for which the equation is solved.
  • For BC only – find the area of a region bounded by polar curves: A=\tfrac{1}{2}\int\limits_{{{\theta }_{1}}}^{{{\theta }_{2}}}{{{\left( r\left( \theta  \right) \right)}^{2}}}d\theta

If this question appears on the calculator active section, it is expected that the definite integrals will be evaluated on a calculator. Students should write the definite integral with limits on their paper and put its value after it.  It is not required to give the antiderivative and if a student gives an incorrect antiderivative they will lose credit even if the final answer is (somehow) correct.

There is a calculator program available that will give the set-up and not just the answer so recently this question has been on the no calculator allowed section. (The good news is that in this case the integrals will be easy or they will be set-up-but-do-not-integrate questions.)

Occasionally, other type questions have been included as a part of this question. See 2016 AB5/BC5 which included an average value question and a related rate question along with finding the volume.

Shorter questions on this concept appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on this subject see January 9, 11, 2013

Next Posts:

Friday March 17: Table and Riemann sums (Type 5)

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

Tuesday March 28: for BC Parametric Equation (Type 8)

Graph Analysis (Type 3)

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The long name is “Here’s the graph of the derivative, tell me things about the function.”

Most often students are given the graph identified as the derivative of a function. There is no equation given and it is not expected that students will write the equation (although this may be possible); rather, students are expected to determine important features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points.

The graph may be given in context and student will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion and position. (Type 2)

Less often the function’s graph may be given and students will be asked about its derivatives.

What students should be able to do:

  • Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
  • Find and justify where the function is increasing or decreasing.
  • Find and justify extreme values (1st and 2nd derivative tests, Closed interval test aka.  Candidates’ test).
  • Find and justify points of inflection.
  • Find slopes (second derivatives, acceleration) from the graph.
  • Write an equation of a tangent line.
  • Evaluate Riemann sums from geometry of the graph only.
  • FTC: Evaluate integral from the area of regions on the graph.
  • FTC: The function, g(x), maybe defined by an integral where the given graph is the graph of  the integrand, f(t), so students should know that if,  \displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{t}{f\left( t \right)dt} then  {g}'\left( x \right)=f\left( x \right)  and  {{g}'}'\left( x \right)={f}'\left( x \right). In this case students should write {g}'(t)=f\left( t \right) on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

The ideas and concepts that can be tested with this type question are numerous. The type appears on the multiple-choice exams as well as the free-response. They have accounted for almost 20% of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with these topics.

Study past exams; look them over and see the different things that can be asked.

For some previous posts on this subject see October 15, 17, 19, 24, 26 (my most read post), 2012 and  January 25, 28, 2013

An activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Added 4-1-17

Next Posts:

Tuesday March 14: Area and Volume (Type 4)

Friday March 17: Table and Riemann sums (Type 5)

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)