Monthly Archives: March 2013

Challenge

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A student was asked to find the volume of the bowl-shaped figure generated when the curve y = x2 from x = 0 to x = 2 is revolved around the y-axis. She used the disk method and found the volume to be \displaystyle \int_{0}^{4}{\pi ydy}=8\pi . To check her work she used the method of cylindrical shells and found the same answer: \displaystyle\int_{0}^{2}{2\pi x\left( 4-{{x}^{2}} \right)dx}=8\pi .

The second part of the question asked for the value of x for which the bowl would be half full. So she first solved the equation \displaystyle \int_{0}^{k}{\pi ydy=\frac{1}{2}\cdot 8\pi }  and found k=2\sqrt{2}. This is a y-value so the corresponding x-value is \sqrt{2\sqrt{2}}\approx 1.682. She again checked her work by shells by solving \displaystyle \int_{0}^{k}{2\pi x\left( 4-{{x}^{2}} \right)dx=\frac{1}{2}\cdot 8\pi } and found that k=\sqrt{-2\left( \sqrt{2}-2 \right)}\approx 1.082. (This is the x-value.)

Both computations are correct. Can you explain to her why her answers are different? Use the comment box below to share your explanations. I will post mine in a week or so.

Looking Back and Ahead

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This blog came about this way. I work with a group of AP mathematics teachers in about 40 schools in Arkansas. Since I could not see them all each week, about a year ago I thought to write the calculus teachers one or two e-mails each week to give them hints and advice on teaching calculus. Then it occurred to me that a blog would be easier to handle, would make past posts easier to find, and be a help (I hoped) to more than just my teachers. 

So last August I started. My plan was to do three posts a week, a schedule I have managed to stick to. Also I hoped to stay a week or two ahead of where they should be, so that what they liked could be incorporated into their lessons.

Now is the time to finish up and start reviewing. And so, having discussed reviewing for the last few weeks, I will consider my first year as completed.

For the next few months, I will probably be posting less often. I will write about whatever I think you may be interested in, or anyway whatever I find interesting.

In August, getting ready for a new school year, I plan to add to the topics I’ve already written about and fill in the gaps with topics I skipped.

I hope you will continue to follow. You can sign up by clicking on “Follow Blog via Email” on the right sidebar. You will get an e-mail when a new post is available and you will not need to come here if there is nothing new.

As always, I would like to hear from you. I would especially like suggestions about what you would like me to expound on. You may use the comment space below or write me privately at lnmcmullin@aol.com.

I hope to hear from you.

Calculator Use on the AP Exams

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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.

Sequences and Series

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AP Type Question 10

Sequences and Series – for BC only

Convergence tests for series appear on both sections of the BC Calculus exam. In the multiple-choice section students may be asked to say if a sequence or series converges or which of several series converge.

The Ratio test is used most often to determine the radius of convergence and the other tests to determine the exact interval of convergence by checking the convergence at the end points. Click here for a Convergence test chart students should be familiar with; this list is also on the resource page.

On the free-response section there is usually one full question devoted to sequences and series. This question usually involves writing a Taylor or Maclaurin polynomial for a series.

Students should be familiar with and able to write several terms and the general term of a series. They may do this by finding the derivatives and constructing the coefficients from them, or they may produce the series by manipulating a known or given series. They may do this by substituting into a series, differentiating it or integrating it.

What Students Should be Able to Do 

  • Use the various convergence tests to determine if a series converges. The test to be used is rarely given so students need to know when to use each of the common tests. For a summary of the tests click: Convergence test chart.
  • Write the terms of a Taylor or Maclaurin series by calculating the derivatives and constructing the coefficients of each term.
  • Distinguish between the Taylor series for a function and the function. Do NOT say that the Taylor polynomial is equal to the function; say it is approximately equal.
  • Determine a specific coefficient without writing all the previous coefficients.
  • Write a series by substituting into a known series, by differentiating or integrating a known series or by some other algebraic manipulation of a series.
  • Know (from memory) the Maclaurin series for sin(x), cos(x), ex and \displaystyle \tfrac{1}{1-x} and be able to find other series by substituting into them.
  • Find the radius and interval of convergence. This is usually done by using the Ratio test and checking the endpoints.
  • Be familiar with geometric series, its radius of convergence, and be able to find the number to which it converges, \displaystyle {{S}_{\infty }}=\frac{{{a}_{1}}}{1-r}. Re-writing a rational expression as the sum of a geometric series and then writing the series has appeared on the exam.
  • Be familiar with the harmonic and alternating harmonic series.
  • Use a few terms of a series to approximate the value of the function at a point in the interval of convergence.
  • Determine the error bound for a convergent series (Alternating Series Error Bound and Lagrange error bound). See my post of  February 22, 2013.
  • Use the coefficients (the derivatives) to determine information about the function (e.g. extreme values).

This list is quite long, but only a few of these items can be asked in any given year. The series question on the exam is usually quite straightforward. As I have suggested before, look at and work as many past exam questions to get an idea of what is asked and the difficulty of the questions. Click on Power Series in the “Posts by Topic” list on the right side of the screen to see previous posts on Power Series.

Polar Curves

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AP Type Questions 9

Polar Curves for BC only.

Ideally, as with parametric and vector functions, polar curves should be introduced and covered thoroughly in a precalculus course. Questions on the BC exams have been concerned with calculus ideas. Students have not been asked to know the names of the various curves (rose, curves, limaçons, etc.). The graphs are usually given in the stem of the problem, but students should know how to graph polar curves on their calculator.

What students should know how to do

  • Find the intersection of two graphs (to use as limits of integration).
  • Find the area enclosed by a graph or graphs using the formula \displaystyle A=\tfrac{1}{2}\int_{{{\theta }_{1}}}^{{{\theta }_{2}}}{(r(}θ\displaystyle ){{)}^{2}}dθ
  • Use the formulas x(θ) = r(θ)cos(θ)  and y(θ) = r(θ)sin(θ) to
    • convert from polar to rectangular form,
    • calculate the coordinates of a point on the graph, and
    • calculate \frac{dy}{d\theta } and \frac{dx}{d\theta } (Hint: use the product rule).
    • Discuss the motion of a particle moving on the graph by discussing the meaning of \frac{dr}{d\theta } (motion towards or away from the pole), \frac{dy}{d\theta } (motion in the vertical direction) or \frac{dx}{d\theta } (motion in the horizontal direction).
    • Find the slope at a point on the graph, \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }.

This topic appears only occasionally on the free-response section of the exam. The most recent were 2007 and 2013. If the topic is not on the free-response then 1, or maybe 2 questions, probably finding area, can be expected on the multiple-choice section.

Parametric and Vector Equations

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AP Type Questions 8

Particle moving on a plane for BC – the parametric/vector question.

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 probably cannot be fully developed in BC calculus. I will try to address here 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 textbooks jump right to vectors in 3-space while the exam only test motion in a plane and 2-dimensional vectors.

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.

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}}}

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 any notation is allowed.
  • 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)}.
  • 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 or down).
  • 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 really just initial value differential equation problems (IVP).
  • Dot product and cross product are not tested on  the BC exam.

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.

Implicit Relations & Related Rates

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AP Type Questions 7

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 the 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.)
  • 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.

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, 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 relation s see November 14, 2012