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AP Calculus AB/BC No-Calculator FRQ Practice

59 no-calculator FRQs written in the AP format for Calculus AB and BC. Pick one, read the full question, and write your response.

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Your first scored response is included. A plan unlocks unlimited scoring.

New to the No-Calculator FRQ? Read the exam guide

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Piecewise function continuity and function rates of change

Unit 1: Limits and Continuity

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The function ff is defined by

f(x)={x2−5x+6x−3,x<32x−5,x≥3f(x) = \begin{cases} \dfrac{x^2 - 5x + 6}{x - 3}, & x < 3 \\ 2x - 5, & x ≥ 3 \end{cases}

The function gg is continuous on the closed interval 0≤x≤80 ≤ x ≤ 8. Selected values of g(x)g(x) are given in the table shown.

xx

00

22

44

66

88

g(x)g(x)

−3-3

−1-1

22

55

44

A.

Find lim⁡x→3−f(x)\lim_{x \to 3^-} f(x) and lim⁡x→3+f(x)\lim_{x \to 3^+} f(x). Is ff continuous at x=3x = 3? Justify your answer using the definition of continuity.

B.

Use the average rate of change of gg over the interval 2≤x≤42 ≤ x ≤ 4 to approximate the instantaneous rate of change of gg at x=3x = 3. Show the work that leads to your answer.

C.

Must there be a value cc, for 0<c<20 < c < 2, such that g(c)=−2g(c) = -2? Must there be a value kk, for 6<k<86 < k < 8, such that g(k)=4.5g(k) = 4.5? Justify your answers.

D.

A new function hh is defined by

h(x)={x2−5x+6x−3,x<3ax2−4,x≥3h(x) = \begin{cases} \dfrac{x^2 - 5x + 6}{x - 3}, & x < 3 \\ a x^2 - 4, & x ≥ 3 \end{cases}

where aa is a constant. Find the value of aa for which hh is continuous at x=3x = 3. Justify your answer using the definition of continuity.

What the No-Calculator FRQ asks

No-calculator FRQs are questions 3 to 6 on the AP Calculus AB and BC exams, in Part B of the FRQ section. Each has four parts with numbers built to be worked by hand, and you justify answers with the right theorem or test.

60 min
for the 4 questions in Part B
9 points
across parts A to D
50%
of your score, for all six FRQs
read the exam guide

  1. Part A: Evaluate

    A value, rate, or estimate worked by hand, often with units

    2 pts

  2. Part B: Apply

    A derivative or integral applied to the function or context

    2 pts

  3. Part C: Extend

    Another calculus task on the same setup, such as an area or an approximation

    2 pts

  4. Part D: Conclude

    A final result, often backed by a reason or a named theorem

    3 pts

How No-Calculator FRQ practice works

Each question follows the exam’s format, from the full prompt to a score on every part.

  1. Read the full question

    The setup, any graph or table, and parts A to D, laid out the way the exam shows them.

  2. Write on a 15-minute timer

    About the time each question gets on exam day. Pause it or turn it off, and your response saves as you go.

  3. Submit for a score out of 9

    Your response is scored against the scoring guidelines written for that question. Your first score is included.

Feedback on every part

A summary at the top tells you what to work on next. Below it, each part shows whether you earned the point and what the scoring guidelines were looking for.

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Detailed Feedback

Part A: 1/1 point

Earned the point. You correctly set up the difference quotient (f(2.1) - f(1.9))/(2.1 - 1.9) = 0.2/0.2 to approximate the derivative.

Part A: 1/1 point

Earned the point. You obtained the correct numerical answer of 1 with supporting work shown.

Part C: 0/1 point

Did not earn the point. While you mentioned that g(x) approaches a value of 3, you described it as 'approaching a maximum value of 3' rather than identifying it as a horizontal asymptote. To earn this point, you needed to explicitly state that the graph has a horizontal asymptote at y = 3, or describe the end behavior as the graph approaching the horizontal line y = 3 as x increases without bound.

Detailed feedback on a practice No-Calculator FRQ

Questions about the No-Calculator FRQ

How many responses can I get scored?

You can read every question without a plan. Your first scored response is included. A plan unlocks unlimited scoring.

How is the no-calculator FRQ different from the calculator FRQ?

Calculator questions lean on integrals you can’t do by hand and on solving messy equations. No-calculator questions use friendly numbers and reward exact algebra and clear justifications.

How do I type math in my answer?

Type it in plain text, use the superscript button for exponents, and pick symbols like ∫ and √ from the special characters menu. You can also switch to handwrite and add a photo of your work.

Are these for Calculus AB or BC?

Both. The AB and BC exams use the same FRQ format, and some of these questions cover BC-only topics like series and parametric equations.

How are the 9 points split?

Across parts A to D, most often 2, 2, 2, and 3. The split changes by question, and on the exam each part is scored on its own.

More AP Calculus AB/BC practice

Calculator FRQ practice

Calculator-required FRQs with the full setup and all four parts.

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AP Calculus AB/BC FRQs

Every FRQ type for AP Calculus AB/BC, with practice for each.

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AP Calculus AB/BC study guides

Unit-by-unit review for the whole course.

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Write your first No-Calculator FRQ

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Teaching AP Calculus AB/BC? Assign these No-Calculator FRQs to your class