AP Calculus AB Product Rule, Quotient Rule and Trig Derivatives Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Product Rule, Quotient Rule and Trig Derivatives – Maths PDF

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This product quotient test answers for AP Calculus AB covers product rule, quotient rule and trig derivatives (Common Core AP CALC AB 2.8, AP CALC AB 2.9, AP CALC AB 2.10), with a free PDF to print.

Use this answer key to score the product rule 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 2.8, AP CALC AB 2.9, AP CALC AB 2.10

Product quotient test answers: worked solutions

Question 1 – Product rule [2 pts]

\(f^{\prime}(x)=2x\,e^{x}+(x^{2}+3)e^{x}=e^{x}(x^{2}+2x+3)\).

Final answer: \(e^{x}(x^{2}+2x+3)\).

Scoring: 1 point for the product rule setup; 1 point for the simplified result.

Question 2 – Quotient rule [3 pts]

\(f^{\prime}(x)=\frac{2(x+4)-(2x-1)(1)}{(x+4)^{2}}=\frac{9}{(x+4)^{2}}\).

Final answer: \(\frac{9}{(x+4)^{2}}\).

Scoring: 1 point for u′ and v′; 1 point for the quotient rule setup; 1 point for simplifying.

Question 3 – Trig and product [2 pts]

\(g^{\prime}(x)=\cos x-x\sin x+3\sec^{2}x\).

Final answer: \(\cos x-x\sin x+3\sec^{2}x\).

Scoring: 1 point for the product term; 1 point for the tangent term.

Question 4 – Tangent line [3 pts]

\(f(e)=e^{2}\ln e=e^{2}\). \(f^{\prime}(x)=2x\ln x+x\), so \(f^{\prime}(e)=2e+e=3e\).

Tangent: \(y-e^{2}=3e(x-e)\). Final answer: \(y=3ex-2e^{2}\).

Graph of f(x) = x² ln x.

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

Question 5 – Table of values [2 pts]

\((fg)^{\prime}(1)=3(-1)+2(4)=5\).

\(\left(\frac{f}{g}\right)^{\prime}(1)=\frac{3(-1)-2(4)}{(-1)^{2}}=-11\).

Final answer: \(5\) and \(-11\).

Scoring: 1 point for each derivative.

Question 6 – Horizontal tangents [3 pts]

\(h^{\prime}(x)=\frac{1(x^{2}+1)-x(2x)}{(x^{2}+1)^{2}}=\frac{1-x^{2}}{(x^{2}+1)^{2}}\). Set \(1-x^{2}=0\): \(x=\pm 1\).

\(h(1)=\frac{1}{2}\) and \(h(-1)=-\frac{1}{2}\).

Final answer: \(\left(1,\frac{1}{2}\right)\) and \(\left(-1,-\frac{1}{2}\right)\).

Graph of h(x) = x/(x² + 1).

Scoring: 1 point for h′; 1 point for the x-values; 1 point for the points.

Question 7 – Rate of sales [3 pts]

\(R(t)=(2+0.1t)(120-2t)\).
\(R^{\prime}(t)=0.1(120-2t)+(2+0.1t)(-2)\), so \(R^{\prime}(10)=0.1(100)+3(-2)=10-6=4\) dollars per day.

Scoring: 1 point for (a); 1 point for the product rule setup; 1 point for the value 4 with units.

Question 8 – Secant and cosecant [2 pts]

\(k^{\prime}(x)=\sec x\tan x+2\csc x\cot x\).

At \(\frac{\pi}{4}\): \(\sqrt{2}\cdot 1+2\cdot\sqrt{2}\cdot 1=3\sqrt{2}\).

Final answer: \(3\sqrt{2}\).

Scoring: 1 point for the derivative; 1 point for the value.

Total score

Total

20 points

Key facts behind this product quotient test answers

Key facts

Product rule: (uv)′ = u′v + uv′.
Quotient rule: (u/v)′ = (u′v – uv′)/v², and the order in the numerator matters.
d/dx tan x = sec²x and d/dx sec x = sec x tan x.
The derivatives of cot x and csc x are negative: -csc²x and -csc x cot x.

How to use this product quotient 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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