AP Calculus AB Mean Value Theorem and Extreme Value Theorem Test with Point Values (PDF)

AP Calculus AB math test: Mean Value Theorem and Extreme Value Theorem – Maths PDF

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This mvt evt test for AP Calculus AB covers mean value theorem and extreme value theorem (Common Core AP CALC AB 5.1, AP CALC AB 5.2, AP CALC AB 5.3), with a free PDF to print.

This test checks the Mean Value Theorem, the Extreme Value Theorem, critical numbers and absolute extrema on closed intervals (AP Calculus AB, 5.1, 5.2, 5.3). 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: Mean Value Theorem and Extreme Value Theorem
Standards: AP CALC AB 5.1, AP CALC AB 5.2, AP CALC AB 5.3
Time: 45 minutes
Total: 20 points

Mvt evt test: questions

Question 1 – Verify the MVT [3 pts]

Graph of f(x) = x² − 2x.

Verify that \(f(x)=x^{2}-2x\) satisfies the hypotheses of the MVT on \([0,4]\) and find the value of \(c\).

Question 2 – Hypotheses fail [2 pts]

Explain why the MVT cannot be applied to \(f(x)=\frac{1}{x-2}\) on \([1,3]\).

Question 3 – MVT conclusion [2 pts]

A differentiable function has \(g(1)=5\) and \(g(5)=-3\). Show that \(g^{\prime}(c)=-2\) for some \(c\) in \((1,5)\).

Question 4 – Critical numbers [3 pts]

Find all critical numbers of \(f(x)=x^{4}-8x^{2}\) and of \(h(x)=(x-1)^{2/3}\).

Question 5 – Absolute extrema [3 pts]

Graph of f(x) = x³ − 12x + 3.

Find the absolute maximum and minimum of \(f(x)=x^{3}-12x+3\) on \([-3,4]\).

Question 6 – Speed and the MVT [2 pts]

A delivery truck travels 150 miles in 2.5 hours on a highway. Use the MVT to explain why its speedometer read exactly 60 mph at some time.

Question 7 – Extrema with an exponential [3 pts]

Graph of f(x) = xe⁻ˣ.

Find the absolute extrema of \(f(x)=xe^{-x}\) on \([0,5]\).

Question 8 – Justifying a maximum [2 pts]

A continuous function \(h\) on \([1,6]\) has exactly one critical number, \(x=3\), with \(h(1)=4\), \(h(3)=9\) and \(h(6)=2\). State the absolute maximum and minimum and name the theorem that guarantees they exist.

Key facts behind this mvt evt test

Key facts

MVT: f′(c) = (f(b) – f(a))/(b – a) for some c in (a, b), if f is continuous and differentiable.
EVT: a continuous function on a closed interval has an absolute maximum and an absolute minimum.
Critical numbers are where f′ = 0 or f′ is undefined (inside the domain).
Candidates test: compare f at endpoints and critical numbers.

How to use this mvt evt 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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