NCTM 2006 St.Louis AP Calc: Lessons from 2005 Free-response Problems Speaker: Craig Wright, Education Testing Service, New Jersey Typed by Sean Bird Also included are “Global Tips” by Dan Kennedy, “Be Careful” by Dave Slomer, Instructions for the AP Calc Exam, AP Calc course description, and more. Good Advice for Students taking the AP Calculus Exam “Get plenty of rest and have a good breakfast with quality protein.” Study “Stuff You MUST Know Cold” (printable)“Stuff You MUST Know Cold” (printable) “Take practice tests & grade them yourself”

General Comments Show work. Answers w/o supporting work may not receive credit. Communicate reasoning clearly in a concise way using proper notation. “Precise & Concise” Justify conclusions using mathematical (CALCULUS) arguments There will be application problems (like we’ve been doing all along). Real life math (no 2.34 people) TRY EACH part of EACH free-response problem. It doesn’t go in increasing difficulty. Graphical, Numerical (tabular), & Analytical

10 Reminders about calculator 1. Set the calculator to RADIAN mode 2. Report decimal approximation to [at least] three decimal places after the decimal point. E.g [truncate or round] 3. Be proficient with the 4 expected capabilities. Graph, zeros, numerically differentiate & integrate 4. SHOW set up but don’t try to do things by hand on the calculator portion.

10 Reminders about calculator 5. Watch parenthesis 6. Store function in y1(x) (or something like that) 7. Be able to store important values (e.g. zero or point of intersection) for a short cut! [“xc” on 89] (trace won’t cut it for precision. Avoid TRACE) Use the variable in subsequent calculations.

2005#1 Look at the graph provided. Consider the period of the sin curve. It looks like ½ a cycle. Pi is ½ a cycle. Perhaps x is around 1. Assign xmin and xmax, then ZoomFit. I picked xmin= -0.1 and xmax a bit more than 1. In fact, clearly, algebraically you can see that x = 1 is an intersection. On 89,

2005#1 On 89,

On the TI 83/84

10 Reminders about calculator 5. Watch parenthesis 6. Store function in y1(x) (or something like that) 7. Be able to store important values (e.g. zero or point of intersection) for a short cut! “xc” on 89 (trace won’t cut it for precision. Avoid TRACE) Use the variable in subsequent calculators 8. If you round too much then your solution will be wrong & unacceptable. E.g. Definite integral 9. “Because my calculator said so” will never get you the justification point. 10. Use standard mathematical notion, not calculator syntax, on the exam. Never us “it”! Tell them what equation are you solving, or what are you differentiating. Now you in the back can see #10, but you can’t see this.

Wisdom from 2005 FR BC2 candidates test AB/BC3 – never use a regression. Do the problem they give; don’t make up your own. It will be hard for you to get any points. #3 has will likely be a problem that you can reasonably come back to without the use of your calculator. BC4 “most common error was not doing something that it told you to do” and then not making a numerical answer even when you can’t use your calculator.

Wisdom from 2005 FR BC6 check endpoints on interval of convergence. Be careful of arithmetic errors Most common error 2n! Instead of (2n)! …NO CREDIT GIVEN. You are scored on what you show on paper and NOT ON WHAT YOU WERE THINKING DON’T USE DECIMAL APPROXIMATION OF pi!!! (Unless you use the decimal approximation out to 10 decimal places.)

Global Tips for Students Show all work. Remember that the grader is not really interested in finding out the answer to the problem. The grader is interested in seeing if you know how to solve the problem. Do not round partial answers. Store them in your calculator so that you can use them unrounded in further calculations. Do not let the points at the beginning keep you from getting the points at the end. If you can do part (c) without doing (a) and (b), do it. If you need to import an answer from part (a), make a credible attempt at part (a) so that you can import the (possibly wrong) answer and get your part (c) points. By Dan Kennedy, Chattanooga, TN – from apcentral.com If it seems like this is repetitive, that probablymeans it is REALLY important. We need reminded again and again of some things (see 2 Peter 1).

Global Tips for Students continued If you use your calculator to find a definite integral, write the integral first. An answer without an integral will not get full credit, even if it is correct. [Always at least write the limits of integration and constant] Do not waste time erasing bad solutions. If you change your mind, simply cross out the bad solution after you have written the good one. Crossed-out work will not be graded. If you have no better solution, leave the old one there. It might be worth a point or two. Do not use your calculator for anything except: (a) graph functions, (b) compute numerical derivatives, (c) compute definite integrals, and (d) solve equations. In particular, do not use it to determine max/min points, concavity, inflection points, increasing/decreasing, domain, and range. (You can explore all these with your calculator, but your solution must stand alone.) By Dan Kennedy, Chattanooga, TN – from apcentral.com If it seems like this is repetitive, that probablymeans it is REALLY important. We need reminded again and again of some things (see 2 Peter 1).

Global Tips for Students continued Be sure you have answered the problem. For example, if it asks for the maximum value of a function, do not stop after finding the x at which the maximum value occurs. Be sure to express your answer in correct units if units are given. If you can eliminate some incorrect answers in the multiple-choice section, it is advantageous to guess. Otherwise it is not. Wrong answers can often be eliminated by estimation, or by thinking graphically. [Don’t be fooled by distractors] If they ask you to justify your answer, think about what needs justification. They are asking you to say more. If you can figure out why, your chances are better of telling them what they want to hear. For example, if they ask you to justify a point of inflection, they are looking to see if you realize that a sign change of the second derivative must occur. By Dan Kennedy, Chattanooga, TN – from apcentral.com If it seems like this is repetitive, that probablymeans it is REALLY important. We need reminded again and again of some things (see 2 Peter 1).

Top Ten Student Errors Not unless f ’’ changes from + to –, or – to + Not unless f ’ changes from + to –, or – to + Avoid “it” Show set up

“Be Careful” by Dave Slomer posted Saturday 4/29/2006 FWIW, here're my booboos. 1. Find the min value of x ln x. At x = 1/e, the function has a min. Since 1/e was not an alternative and since "none" was, I selected 'none' because of the ln approaching -inf.. While explaining to the class why this was correct [and while Julie was frowning], I took the limit as x -> 0 and got... AWK! ZERO instead of -inf. D'OH!! THEN I realized that I hadn't even FOUND the FUNCTION VALUE, which was -1/e. Dumb. Dumb. 2. Find the derivative of y = cuberoot(x^2+8) DIVIDED BY fourthroot(2x+1). Since it was calculator legal, I did Nderiv but omitted the division sign, essentially omitting the negative exponent. CARELESS!!! [It's a wonder I got one of the alternatives.] 3. Particle's position is -4 cos t - (t^2/2) Find velocity when acceleration is first zero. The acceleration was first zero when t = 1.32, alternative C. End of problem. D'OH!! We want the VELOCITY. CARELESS!! DUMB. 4. The top of a 25-foot ladder is sliding down a wall at 3 feet per minute yadda yadda yadda. Needless to say, I didn't make this rate NEGATIVE. How dumb can ya get? Of course, if I had only thought about NEGATIVE 7/8 ft/min not being logical since the distance was increasing... GAH! … But my point is maybe to share these common easy to make errors with your kids Mon or Tue. It's never too late to emphasize being careful.

Final IMPORTANT advice Read the instructions before test day 05_calc_rev_comment_22817.pdf And the course description, especially pg “5”ff (pdf page 11ff) 836apcoursdesccalc0_4313.pdf