AP Calculus AB Mean Value Theorem and Extreme Value Theorem Worksheet (PDF)

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

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This mvt evt worksheet 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.

These 22 problems practice the Mean Value Theorem, the Extreme Value Theorem, critical numbers and absolute extrema on closed intervals (AP Calculus AB, topics 5.1, 5.2, 5.3). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems cover hypotheses that fail, the closed interval method, and situations such as a speeding ticket, greenhouse temperature and shop revenue. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 5.1, AP CALC AB 5.2, AP CALC AB 5.3

Mvt evt worksheet: exercises 1 to 10

Exercise 1 – MVT with a parabola

The secant from x = 1 to x = 4.

Verify the Mean Value Theorem for \(f(x)=x^{2}\) on \([1,4]\) and find \(c\).

Exercise 2 – MVT with a cubic

Find all \(c\) guaranteed by the MVT for \(f(x)=x^{3}\) on \([0,2]\).

Exercise 3 – MVT with a logarithm

The tangent at c is parallel to the secant.

Find the value \(c\) guaranteed by the MVT for \(f(x)=\ln x\) on \([1,e^{2}]\).

Exercise 4 – MVT with a reciprocal

Find \(c\) for \(f(x)=\frac{1}{x}\) on \([1,4]\).

Exercise 5 – Two values of c

Find all \(c\) guaranteed by the MVT for \(f(x)=x^{3}-3x\) on \([-2,2]\).

Exercise 6 – When the MVT fails

The graph of |x| on [−1, 2].

Does the MVT apply to \(f(x)=|x|\) on \([-1,2]\)? Is there a \(c\) with \(f^{\prime}(c)=\frac{1}{3}\)?

Exercise 7 – Discontinuity

Explain why the MVT does not apply to \(f(x)=\frac{1}{x}\) on \([-1,1]\).

Exercise 8 – Speeding ticket

A car passes a mile marker 25 at 2:00 p.m. and mile marker 130 at 3:30 p.m. The speed limit is 65 mph. Use the MVT to show that the driver was speeding at some moment.

Exercise 9 – MVT from a table

A differentiable function has \(f(0)=3\) and \(f(4)=11\). What does the MVT guarantee? Could \(f^{\prime}(x)\) be less than 2 for all \(x\) in \((0,4)\)?

Exercise 10 – Rolle's theorem

Find the \(c\) guaranteed for \(f(x)=x^{2}-4x+3\) on \([1,3]\), where \(f(1)=f(3)\).

Exercises 11 and up

Exercise 11 – EVT with a parabola

The graph of x² on [−1, 3].

Find the absolute maximum and minimum of \(f(x)=x^{2}\) on \([-1,3]\).

Exercise 12 – Critical numbers with a Gaussian factor

Find the critical numbers of \(f(x)=xe^{-x^{2}/2}\).

Exercise 13 – Undefined derivative

Find the critical numbers of \(f(x)=x^{2/3}\).

Exercise 14 – Product with an exponential

Find the critical numbers of \(f(x)=xe^{-x}\).

Exercise 15 – Trig critical numbers

Find the critical numbers of \(f(x)=\sin x+\cos x\) on \((0,2\pi)\).

Exercise 16 – Closed interval method, cubic

Candidates for extrema on [−2, 3].

Find the absolute extrema of \(f(x)=x^{3}-3x\) on \([-2,3]\).

Exercise 17 – Closed interval method, second cubic

Find the absolute extrema of \(f(x)=x^{3}-6x^{2}+9x+1\) on \([0,5]\).

Exercise 18 – Closed interval method, rational

Find the absolute extrema of \(f(x)=\frac{x}{x^{2}+1}\) on \([0,4]\).

Exercise 19 – Endpoint critical number

Find the absolute extrema of \(f(x)=\ln x-x\) on \([1,4]\).

Exercise 20 – Greenhouse temperature

Greenhouse temperature in °F.

The temperature in a greenhouse in °F is \(T(t)=-t^{2}+12t+40\) for \(0\le t\le 10\) hours. Find the highest and lowest temperatures.

Exercise 21 – Maximum revenue

A skate shop’s revenue in dollars from selling \(x\) boards a week is \(R(x)=120x-x^{2}\) for \(0\le x\le 100\). Find the number of boards that gives the greatest revenue and the revenue.

Exercise 22 – Parallel tangent

For \(f(x)=x^{3}-x\) on \([0,2]\), find the value \(c\) where the tangent is parallel to the secant line, and write the equation of that tangent line.

Key facts behind this mvt evt worksheet

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 worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the mvt evt worksheet as a PDF with the button below and keep it with your notes.

All AP Calculus AB chapters: AP Calculus AB math.

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