AP Calculus AB Power, Sum and Basic Function Derivative Rules Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Power, Sum and Basic Function Derivative Rules – Maths PDF

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This derivative rules test answers for AP Calculus AB covers power, sum and basic function derivative rules (Common Core AP CALC AB 2.5, AP CALC AB 2.6, AP CALC AB 2.7), with a free PDF to print.

Use this answer key to score the derivative rules 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 2.5, AP CALC AB 2.6, AP CALC AB 2.7

Derivative rules test answers: worked solutions

Question 1 – Polynomial derivative [2 pts]

\(f^{\prime}(x)=20x^{3}-9x^{2}+2\). Final answer: \(20 x^{3} – 9 x^{2} + 2\).

Scoring: 1 point for the first two terms; 1 point for the last two terms.

Question 2 – Roots and reciprocals [2 pts]

Rewrite: \(4x^{1/2}-3x^{-1}+x^{5/2}\). Then \(f^{\prime}(x)=2x^{-1/2}+3x^{-2}+\frac{5}{2}x^{3/2}\).

Final answer: \(\frac{2}{\sqrt{x}}+\frac{3}{x^{2}}+\frac{5}{2}x^{3/2}\).

Scoring: 1 point for rewriting with powers; 1 point for the derivative.

Question 3 – Trig, exponential and log [2 pts]

\(-2\sin x+3e^{x}-\frac{1}{x}\).

Final answer: \(-2\sin x+3e^{x}-\frac{1}{x}\).

Scoring: 1 point for the cosine and exponential terms; 1 point for the logarithm term and signs.

Question 4 – Tangent line [3 pts]

\(f(1)=1-3+2=0\) and \(f^{\prime}(x)=3x^{2}-6x\), so \(f^{\prime}(1)=-3\).

Tangent: \(y-0=-3(x-1)\), so \(y=-3x+3\).

Graph of f(x) = x³ − 3x² + 2.

Scoring: 1 point for f(1); 1 point for f′(1); 1 point for the equation.

Question 5 – Horizontal tangents [3 pts]

\(g^{\prime}(x)=3x^{2}-12x+9=3(x-1)(x-3)\). Zero at \(x=1\) and \(x=3\).

\(g(1)=5\) and \(g(3)=1\).

Final answer: \((1,5)\) and \((3,1)\).

Graph of g(x) = x³ − 6x² + 9x + 1.

Scoring: 1 point for g′; 1 point for the x-values; 1 point for the y-values.

Question 6 – Normal line [2 pts]

\(f(2)=5\) and \(f^{\prime}(2)=4\). The normal slope is \(-\frac{1}{4}\).

\(y-5=-\frac{1}{4}(x-2)\). Final answer: \(y=-\frac{1}{4}x+\frac{11}{2}\).

Scoring: 1 point for the normal slope; 1 point for the equation.

Question 7 – Velocity in context [3 pts]

\(v(t)=3t^{2}-18t+15\).
\(v(2)=12-36+15=-9\) ft/s.
\(3(t-1)(t-5)=0\), so at \(t=1\) and \(t=5\) seconds.

Scoring: 1 point per part.

Question 8 – Unknown constants [3 pts]

Slope: \(f^{\prime}(1)=2+b=3\), so \(b=1\). Point: the tangent line at \(x=1\) gives \(f(1)=3(1)+1=4\).

\(f(1)=1+1+c=4\), so \(c=2\).

Final answer: \(b=1,\ c=2\).

Scoring: 1 point for b; 1 point for the value f(1); 1 point for c.

Total score

Total

20 points

Key facts behind this derivative rules test answers

Key facts

Power rule: d/dx of x^n equals n x^(n-1); the derivative of a constant is 0.
Rewrite roots and reciprocals as powers before differentiating.
The derivatives of sin x, cos x, e^x and ln x are cos x, -sin x, e^x and 1/x.
A normal line has slope equal to the negative reciprocal of the tangent slope.

How to use this derivative rules 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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