AP Calculus AB Increasing, Decreasing and the First Derivative Test Test with Point Values (PDF)

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

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This first derivative test 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.

This test checks increasing and decreasing intervals, sign charts of the derivative, the First Derivative Test and absolute extrema (AP Calculus AB, 5.4, 5.5, 5.6). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.

Time and points

About this test

Grade: AP Calculus AB
Chapter: Increasing, Decreasing and the First Derivative Test
Standards: AP CALC AB 5.4, AP CALC AB 5.5, AP CALC AB 5.6
Time: 45 minutes
Total: 20 points

First derivative test: questions

Question 1 – Intervals of monotonicity [2 pts]

Find the intervals where \(f(x)=xe^{x}\) is increasing and decreasing.

Question 2 – First derivative test [3 pts]

Graph of f(x) = x³ − 6x² + 9x + 2.

Use a sign chart of \(f^{\prime}\) to find the relative extrema of \(f(x)=x^{3}-6x^{2}+9x+2\).

Question 3 – Reading the derivative graph [2 pts]

The graph of f′ (not f).

The graph shows the derivative \(f^{\prime}\) of a function \(f\), with zeros at \(-2\), \(1\) and \(3\). (a) On which interval(s) is \(f\) increasing? (1 point) (b) At which \(x\) does \(f\) have a relative maximum? (1 point)

Question 4 – No sign change [3 pts]

Let \(g^{\prime}(x)=(x-1)^{2}(x+4)\). (a) Find the critical numbers. (1 point) (b) Make a sign chart. (1 point) (c) Classify each critical number. (1 point)

Question 5 – Absolute extrema [3 pts]

Find the absolute maximum and minimum of \(f(x)=x^{4}-4x^{3}\) on \([-1,4]\).

Question 6 – Justification [2 pts]

A function \(h\) has \(h^{\prime}(x)=\frac{x-2}{x^{2}+1}\). Does \(h\) have a relative maximum or minimum at \(x=2\)? Justify.

Question 7 – Concentration over time [3 pts]

Concentration C(t) over time.

The concentration of a chemical in a tank in mg/L is \(C(t)=2+3te^{-t/2}\) for \(t\ge 0\) hours. (a) Find \(C^{\prime}(t)\). (1 point) (b) When is the concentration increasing or decreasing? (1 point) (c) Find the maximum concentration. (1 point)

Question 8 – Finding constants [2 pts]

The function \(f(x)=x^{3}+px^{2}+qx\) has a relative maximum at \(x=-1\) and a relative minimum at \(x=3\). Find \(p\) and \(q\).

Key facts behind this first derivative test

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

Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.

All AP Calculus AB chapters: AP Calculus AB math.

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