AP Calculus AB Increasing, Decreasing and the First Derivative Test Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Increasing, Decreasing and the First Derivative Test – Maths PDF

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This first derivative test answers for AP Calculus AB covers increasing, decreasing and the first derivative test (Common Core AP CALC AB 5.4, AP CALC AB 5.5, AP CALC AB 5.6), with a free PDF to print.

Use this answer key to score the first derivative test 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 5.4, AP CALC AB 5.5, AP CALC AB 5.6

First derivative test answers: worked solutions

Question 1 – Intervals of monotonicity [2 pts]

\(f^{\prime}=e^{x}+xe^{x}=e^{x}(x+1)\). Since \(e^{x} > 0\), the sign of \(f^{\prime}\) is the sign of \(x+1\).

Final answer: decreasing on (-∞, -1); increasing on (-1, ∞).

Scoring: 1 point for the derivative and critical number; 1 point for the intervals.

Question 2 – First derivative test [3 pts]

\(f^{\prime}=3x^{2}-12x+9=3(x-1)(x-3)\). Signs: \(+\) on \((-\infty,1)\), \(-\) on \((1,3)\), \(+\) on \((3,\infty)\).

Final answer: relative maximum f(1) = 6; relative minimum f(3) = 2.

Graph of f(x) = x³ − 6x² + 9x + 2.

Scoring: 1 point for f′ and the critical numbers; 1 point for the sign chart; 1 point for the classification with values.

Question 3 – Reading the derivative graph [2 pts]

For \(x < -2\), \(f^{\prime} < 0\); on \((-2,1)\), \(f^{\prime} > 0\); on \((1,3)\), \(f^{\prime} < 0\); for \(x > 3\), \(f^{\prime} > 0\).

\((-2,1)\) and \((3,\infty)\).
\(x=1\), where \(f^{\prime}\) changes from positive to negative.

The graph of f′ (not f).

Scoring: 1 point per part.

Question 4 – No sign change [3 pts]

\(x=1\) and \(x=-4\).
\(g^{\prime} < 0\) on \((-\infty,-4)\), \(g^{\prime} > 0\) on \((-4,1)\) and \((1,\infty)\).
At \(-4\): \(-\) to \(+\), relative minimum. At \(1\): no sign change, no extremum.

Scoring: 1 point per part.

Question 5 – Absolute extrema [3 pts]

\(f^{\prime}=4x^{3}-12x^{2}=4x^{2}(x-3)\). Critical numbers: \(0\) and \(3\).

Candidates: \(f(-1)=5\), \(f(0)=0\), \(f(3)=-27\), \(f(4)=0\).

Final answer: absolute maximum 5 at x = -1; absolute minimum -27 at x = 3.

Scoring: 1 point for the critical numbers; 1 point for the four values; 1 point for the conclusions.

Question 6 – Justification [2 pts]

The denominator is always positive, so the sign of \(h^{\prime}\) is the sign of \(x-2\): negative for \(x < 2\) and positive for \(x > 2\).

Since \(h^{\prime}\) changes from negative to positive at \(x=2\), h has a relative minimum at x = 2.

Scoring: 1 point for the sign analysis; 1 point for the conclusion with justification.

Question 7 – Concentration over time [3 pts]

\(C^{\prime}(t)=3e^{-t/2}-\frac{3}{2}te^{-t/2}=\frac{3}{2}e^{-t/2}(2-t)\).
Increasing on \((0,2)\) (positive), decreasing on \((2,\infty)\).
\(C(2)=2+6e^{-1}\approx 4.21\text{ mg/L}\).

Concentration C(t) over time.

Scoring: 1 point per part.

Question 8 – Finding constants [2 pts]

\(f^{\prime}=3x^{2}+2px+q\) has zeros at \(-1\) and \(3\), so \(f^{\prime}=3(x+1)(x-3)=3x^{2}-6x-9\).

Therefore \(p=-3\) and \(q=-9\).

Scoring: 1 point for the derivative with both zeros; 1 point for p and q.

Total score

Total

20 points

Key facts behind this first derivative test answers

Key facts

A positive f′ means f is increasing; a negative f′ means f is decreasing.
Sign changes of f′ can occur only at critical numbers or where f′ is undefined.
First Derivative Test: + to − gives a relative maximum; − to + gives a relative minimum.
A critical number without a sign change, such as x = 0 for x³, is not an extremum.

How to use this first derivative test answers

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