Practice a College Board-style free response question on rates of change, the tangent line, and differentiability. Write your response, then reveal the model answer to see exactly what earns each point.
Free Response Question · Unit 2 · Definition of the Derivative
Let f be the function defined by f(x) = x³ − 4x + 1.
Let g be the function defined by g(x) = |x − 1|.
A
Find the average rate of change of f on the interval [1, 3].
✓ Model answer
The average rate of change is [f(3) − f(1)]/(3 − 1). Since f(3) = 27 − 12 + 1 = 16 and f(1) = 1 − 4 + 1 = −2, this equals (16 − (−2))/2 = 18/2 = 9.
Why it scores: Uses the average-rate-of-change (secant slope) formula with correct values of f(3) and f(1), and reports 9. Confusing this with the derivative would lose credit.
B
Find f′(x), and use it to write an equation of the line tangent to the graph of f at x = 2.
✓ Model answer
By the power and sum/difference rules, f′(x) = 3x² − 4. At x = 2, the slope is f′(2) = 3(4) − 4 = 8, and f(2) = 8 − 8 + 1 = 1. The tangent line through (2, 1) with slope 8 is y − 1 = 8(x − 2) (equivalently y = 8x − 15).
Why it scores: Differentiates correctly, evaluates f′(2) = 8 and f(2) = 1, and writes a correct tangent-line equation in point-slope form. Forgetting to compute f(2) for the point would lose a point.
C
Is g(x) = |x − 1| differentiable at x = 1? Justify your answer.
✓ Model answer
g is not differentiable at x = 1. The graph of g has a corner at x = 1: for x < 1 the slope is −1, and for x > 1 the slope is +1. Because the slopes from the left and right do not match, the derivative (the limit of the difference quotient) does not exist there, even though g is continuous at x = 1.
Why it scores: States "not differentiable," identifies the corner, and justifies it with the mismatched one-sided slopes (−1 vs +1) — noting that continuity alone is not enough. A bare "no" without the slope justification would lose credit.
How to score points on AP Calculus FRQs
Distinguish average from instantaneous rate. Average = secant slope over an interval; instantaneous = the derivative at a point.
Show the derivative and the point. A tangent line needs both f′(a) (slope) and f(a) (the point).
Justify differentiability. Point to corners, cusps, or vertical tangents, and compare one-sided slopes.
Use correct notation. Write f′(x), evaluate at the requested point, and keep exact values.
Answer the verb. "Find" needs the computation; "justify" needs supporting reasoning.