Student Tips for Free Response Questions These are suggestions from someone who has graded AP the free-response portion of the calculus exams. It is hoped they will help you to do your best. First, understand that there are no tricks to doing well – the more calculus you know and have practiced, the better you will do. Still, there are ways to avoid losing points which you can earn. 1. Read, understand, and follow the "General Instructions for Section II" (the free response section). In particular, "Numerical answers must be correct to three places after the decimal point."
This is enforced 100% and you can lose point after point on it. Graphs must be completely labeled. If it is a graph or a sign chart for the derivative, it must be labeled as such. ( f '(x), h'(x), etc., as appropriate.) An unlabeled graph will not earn the points, even if it is otherwise correct. If the problem refers to a function as g(x), you must not label its graph f(x), or call its derivative f '(x). If the independent variable is "t", you must not change it to "x". Don't simplify algebraic or numerical expressions unless you are asked to. Practice the first three of these things all year long in all your work so they will be automatic when you take the exam. As for simplifying, you should practice simplifying – just remember not to do it on the AP exam unless asked to – you risk making a minor error and losing a major point! 2. If calculators are permitted and helpful, use them. Recent examinations have included questions where numerical solutions, derivatives and definite integrals are clearly anticipated, and yet some students waste time attempting "exact" solutions – even when exact solutions may be impossible to get. But be sure to show what you are calculating – don't just give the calculator's result.
3. Many questions have "multiple entry points". Even though part (b) may depend on part (a), part (c) may not depend on either of them. Do not give up if you cannot get part (a). Read the entire question and look for a "second entry point" – a place to start earning points! 4. Do not do the work for part (a) in the space allotted to part (b) unless you clearly draw an arrow and label it "work for part (a)". 5. Pay attention to units. Some questions will ask you to give the units such as m/sec2, degrees F/ day, etc. You must do this correctly. Practice it whenever possible. Other comments: • Details can be very important. The requirement that you give numerical answers accurate to three figures to the right of the decimal point is an example. Another example is the correct notation for open versus closed intervals. If ∞ is an endpoint, that end of the interval is open. [0, ∞) is correct but [0, ∞] is not. • A general AP grading principle is not to give credit "just for a formula". You must apply the formula to the specific problem. For example, if the problem gives velocity v(t)=t2 and requests the change in position from t = 3 to t = 5, no credit would be given for "formulas" b
like s(t) = ! v(t)dt or s = ! v(t )dt , but part credit would surely be given 5
a
for s = ! t 2 dt . 3
• Students must convince the reader they know how to do the mathematics guaranteeing their solution. For example, in a problem asking for a local or relative maximum, it is not enough to just to find it, it is not enough to show you found it by setting the derivative to zero - you must convince the reader you have correctly applied either the first or second derivative test to be sure it is a maximum, minimum or neither. And don't forget to check endpoints and other critical points if a global or absolute maximum is asked for.
• Some sloppiness may be forgiven - but don't try to guess which. In 14 2000 AB 4, a student who showed ! t + 1dt = , the correct 3
numerical answer, lost a point for omitting the limits of integration, even though he or she must have used them. On the other hand, in AB 6 students were not penalized for omitting the dy and dx in 2y 2 ! e = ! 3x . At any rate, thank your teacher for grading hard – it helps you learn better! • Use standard mathematical notation. The instructions make it clear that expressions such as fnInt(x^2,x,1,3) may not be used in place of 3
! x dx . 2
1
• When you use your calculator, make sure to tell the reader what you calculated. For example, a "calculator permitted" question asks for the area between the graph of y = x , the x-axis and the line x = 3. A "bald answer" of 3.464 may earn one point at best. To get full credit 3
you must show you calculated
!
xdx .
0
Remember, the grading standards will reflect the calculus your book and teacher have taught you – make sure you learn and practice how to do things correctly, and avoid losing points unnecessarily.
