The following strategies for answering the free-response questions will help you on exam day.
Keep an eye on your time.
Monitor your time carefully. Make sure not to spend too much time on any one question so you’ll have enough time to answer all of them. You may want to look over all the questions as you begin each part of the free-response section before starting work.
During the second timed portion of the free-response section (Part B), you are permitted to continue work on problems in Part A, but you are not permitted to use a calculator during this time.
If you do work that you think is incorrect, simply put an “X” through it instead of spending time erasing it completely: crossed-out work won’t be graded.
Show your work, even when you’re using a calculator.
Show all the steps you took to reach your solution on questions involving calculations, even if a question may not explicitly remind you to do so. The exam reader wants to see if you know how to solve the problem. Answers without supporting work will usually not receive credit.
Clearly label any functions, graphs, tables, or other objects that you use. Justifications require that you give mathematical reasons, and that you verify the needed conditions under which relevant theorems, properties, definitions, or tests are applied.
If you use your calculator to solve an equation, compute a numerical derivative, or find a definite integral, then be sure to write the equation, derivative, or integral first: an answer without this information might not get full credit, even if the answer is correct. Remember to write your work in standard notation (e.g. ∫𝑥2𝑑𝑥51) rather than calculator syntax (e.g. fnInt(X2,X,1,5)), as calculator syntax is not acceptable.
Other important notes:
- Unless otherwise specified, answers (numeric or algebraic) need not be simplified. If you use decimal approximations in calculations, your work will be scored on accuracy. Unless otherwise specified, your final answers should be accurate to 3 places after the decimal point.
- Unless otherwise specified, the domain of a function f is assumed to be the set of all real numbers x for which f(x) is a real number.
Try to solve each part of the question.
Many free-response questions are divided into parts such as (a), (b), (c), and (d), with each part calling for a different response. Credit for each part is awarded independently, so you should attempt to solve each part. For example, you may receive no credit for your answer to part (a), but still receive full credit for part (b), (c), or (d). If the answer to a later part of a question depends on the answer to an earlier part, you may still be able to receive full credit for the later part, even if that earlier answer is wrong.
Be sure to fully answer the question being asked.
For example, if a question asks for the maximum value of a function, do not stop after finding the x-value at which the maximum value occurs. Be sure to express your answer in correct units if units are given and always provide a justification when it is asked for.
When asked to justify or to explain an answer, think about how that can be done.
For example, if you are asked to justify a point of inflection, you need to show that the sign of the second derivative changes. Simply saying that the second derivative equals zero or is undefined is not a justification.
Do not round partial answers.
Store partial answers in your calculator so that you can use them unrounded in further calculations.
Answering free-response questions from previous AP Exams is a great way to practice: it allows you to compare your own responses with those that have already been evaluated and scored. Go to the Exam Questions and Scoring Information section of the AP Calculus AB page on AP Central to review the latest released free-response questions and scoring guidelines. Older questions and scoring information are available on the Past Exam Questions page.
Pay close attention to the task verbs used in the free-response questions. Each one directs you to complete a specific type of response. Here are the task verbs you’ll see on the exam:
- Approximate: Use rounded decimal values or other estimates in calculations, which require writing an expression to show work.
- Calculate/Write an expression: Write an appropriate expression or equation to answer a question. Unless otherwise directed, calculations also require evaluating an expression or solving an equation, but the expression or equation must also be presented to show work. “Calculate” tasks might also be formulated as “How many?” or “What is the value?”
- Determine: Apply an appropriate definition, theorem, or test to identify values, intervals, or solutions whose existence or uniqueness can be established. “Determine” tasks may also be phrased as “Find.”
- Estimate: Use models or representations to find approximate values for functions.
- Evaluate: Apply mathematical processes, including the use of appropriate rounding procedures, to find the value of an expression at a given point or over a given interval.
- Explain: Use appropriate definitions or theorems to provide reasons or rationales for solutions and conclusions. “Explain” tasks may also be phrased as “Give a reason for...”
- Identify/Indicate: Indicate or provide information about a specified topic, without elaboration or explanation.
- Interpret: Describe the connection between a mathematical expression or solution and its meaning within the realistic context of a problem, often including consideration of units.
- Interpret (when given a representation): Identify mathematical information represented graphically, symbolically, verbally, and/or numerically, with and without technology.
- Justify: Identify a logical sequence of mathematical definitions, theorems, or tests to support an argument or conclusion, explain why these apply, and then apply them.
- Represent: Use appropriate graphs, symbols, words, and/or tables of numerical values to describe mathematical concepts, characteristics, and/ or relationships.
- Verify: Confirm that the conditions of a mathematical definition, theorem, or test are met in order to explain why it applies in a given situation. Alternately, confirm that solutions are accurate and appropriate.