AP Calculus AB The Derivative as a Limit Test Solutions and Scoring (PDF)
This derivative definition test answers for AP Calculus AB covers the derivative as a limit (Common Core AP CALC AB 2.1, AP CALC AB 2.2, AP CALC AB 2.3, AP CALC AB 2.4), with a free PDF to print.
Use this answer key to score the derivative definition test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.
AP Calculus AB · Common Core State Standards: AP CALC AB 2.1, AP CALC AB 2.2, AP CALC AB 2.3, AP CALC AB 2.4
Derivative definition test answers: worked solutions
Question 1 – Average and instantaneous rate [2 pts]
- \(\frac{25-4}{3}=7\).
- \(\frac{4.0401-4}{0.01}=4.01\), so \(f^{\prime}(2)\approx 4\).
Scoring: 1 point per part.
Question 2 – Limit definition, polynomial [3 pts]
\(f(2)=-2\). \(f(2+h)=4+4h+h^{2}-6-3h=-2+h+h^{2}\).
\(\frac{h+h^{2}}{h}=1+h\to 1\). Final answer: \(f^{\prime}(2)=1\).
Scoring: 1 point for the difference quotient; 1 point for simplifying; 1 point for the limit.
Question 3 – Limit definition, reciprocal [3 pts]
\(\frac{\frac{2}{3+h}-\frac{2}{3}}{h}=\frac{6-2(3+h)}{3(3+h)h}=\frac{-2h}{3(3+h)h}=\frac{-2}{3(3+h)}\).
As \(h\to 0\): \(f^{\prime}(3)=-\frac{2}{9}\).
Scoring: 1 point for the difference quotient; 1 point for simplifying; 1 point for the limit.
Question 4 – Tangent line [2 pts]
\(g(3)=9-6=3\). Tangent: \(y-3=4(x-3)\), so \(y=4x-9\).
Scoring: 1 point for g(3); 1 point for the equation.
Question 5 – Differentiability of a piecewise function [3 pts]
- Left value: \(\frac{4}{2}=2\); right limit: \(2(2)-2=2\); \(k(2)=2\). Continuous.
- Left slope: \(k^{\prime}(x)=x\) gives \(2\). Right slope: \(2\). The slopes match, so k is differentiable at x = 2 with \(k^{\prime}(2)=2\).
Scoring: 1 point for (a); 1 point for each one-sided slope in (b) with the conclusion.
Question 6 – Recognizing a derivative [2 pts]
Take \(f(x)=x^{4}\) and \(a=3\), since \(f(3)=81\). Then the limit is \(f^{\prime}(3)=4\cdot 27=108\).
Final answer: \(108\).
Scoring: 1 point for identifying f and a; 1 point for the value.
Question 7 – Cost of a pizza oven [3 pts]
- \(P(10)=200\), \(P(12)=216\), so \(\frac{216-200}{2}=8\) dollars per pizza.
- \(P(10+h)=-(100+20h+h^{2})+300+30h=200+10h-h^{2}\), so the quotient is \(10-h\to 10\). \(P^{\prime}(10)=10\)
Scoring: 1 point for (a); 1 point for the quotient in (b); 1 point for the limit.
Question 8 – Slopes on a graph [2 pts]
At A the curve falls, so the slope is negative. At B the curve rises, so positive. At C the curve has a peak, so the tangent is horizontal: zero.
Scoring: 1 point for A and C correct; 1 point for B correct and a reason for each.
Total score
20 points
Key facts behind this derivative definition test answers
- The derivative is the limit of secant slopes, which are average rates of change.
- f′(a) is the slope of the tangent line at x = a; the tangent is y – f(a) = f′(a)(x – a).
- Both forms of the definition give the same value when the limit exists.
- Differentiable implies continuous, but continuous does not imply differentiable.
How to use this derivative definition test answers
Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.
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