For years, there have been various lists of the most common mistakes on the AP Calculus exams. This one was originally drawn up by the late Ben Cornelius. Larry Peterson added some comments and posted it to the AP Calculus Community bulletin board a few days ago. Both Ben and Larry are long time AP Calculus teachers, exam readers, table leaders, and question leaders. I have revised it slightly and added some thoughts of my own starting with the first one.
This is for students; so, be sure to share it with them.
“Common Mistakes that Make Readers Pull Their Hair Out.”
- BAD ALGEBRA. Yes, calculus students should know how to do algebra. The AP exams are calculus exams and the necessity for doing algebra is kept to a minimum. Algebraic simplification is not required. Once you find a derivative using the quotient rule, stop. Don’t take the chance of simplifying a correct answer and making an algebra mistake. Question do not require lines and lines of algebra. If you are doing a lot of algebra, you have made a mistake; start over.
- BAD ARITHMETIC. There is no need to simplify arithmetic. It won’t make the answer any more correct (even a long Riemann sum). If you get 2 + 3 or sin(π/6) stop – that’s good enough.
- Crossed out work. Don’t cross out your work, unless you know you can do better.
- Incorrect or missing units. Units are required only if specifically asked for.
- Go for partial credit.: If you are worried that your result in part a) is incorrect, use it anyway to finish the problem.
- Go for partial credit. When asked to write an integral, start with the limits and any constants of multiplication; that’s worth a point, even if you are not sure of the integrand.
- When using a calculator, show the mathematical set-up (e.g., the definite integral); describe what you are doing clearly in mathematical terms, not in calculator speak. Calculator work is limited to the four required functionalities: graphing, roots, numerical derivative, and numerical integration. You will not be required to do anything else with your calculator and no question will be asked where using an additional feature would give an advantage (e.g., curve fitting). Anything else you do will not earn credit.
- Writing bad math. For example, “slope of the derivative.” or “6.2368 = 6.237″ or (since you know the answer should be positive) “–17.241 = 17.241″)
- Rounding or truncating mistakes. Remember: three or more decimal places, rounded or truncated. Do not round or truncate before the final answer (early rounding may affect the accuracy of the final answer.)
- Say what you mean. Don’t write f(x) = 2(1.5) + 3 if you mean f(1.5) = 2(1.5) + 3. Don’t give a Taylor series when asked for a Taylor polynomial.
- Pronouns needs an antecedent. Name the function you are referring to. Do not say, “It is increasing because it is positive”, say “The function, f, is increasing, because the derivative of f is positive.” Refer to the function by name, especially if there is more than one function.
- Know the difference between increasing and positive: f is increasing when f ’ is positive.
- Know the difference between local and global extrema.
- “Value of a function” means the y-value. Do not give an ordered pair, if only the value is asked for. Know the difference between the extreme value (y-coordinate) and the location of the extreme value (x– and y-coordinates).
- Write justifications and reason in complete sentences (without pronouns). You may use mathematical symbols.
- Make sure the equations flow correctly from one line to the next. Do not use stream of consciousness. Do not connect equations with equal signs, unless you are sure they are equal. (e.g., 3 x+ 12 = 0 = x + 4 = –4 makes no sense). Rather work vertically:
3x + 12 = 0
x + 4 = 0
x = –4
Here is even more advice: “How, not only to Survive, but to Prevail” is the introduction I wrote for to Multiple-choice & Free-response in Preparation for the AP Calculus (AB) Examination and Multiple-choice & Free-response in Preparation for the AP Calculus (BC) Examination published by D&S Marketing Systems, Inc. More advice and information for students about to take the AP Calculus Exams.
And good luck on this year’s exams.