AP Calculus AB Optimization Test with Point Values (PDF)

AP Calculus AB math test: Optimization – Maths PDF

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This optimization test for AP Calculus AB covers optimization (Common Core AP CALC AB 5.10, AP CALC AB 5.11), with a free PDF to print.

This test checks optimization problems with fences, boxes, cans, closest points and revenue, including justifying a maximum or minimum (AP Calculus AB, 5.10, 5.11). 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: Optimization
Standards: AP CALC AB 5.10, AP CALC AB 5.11
Time: 45 minutes
Total: 20 points

Optimization test: questions

Question 1 – Maximum product [2 pts]

Two numbers add to 30. Find the numbers whose product is largest.

Question 2 – Pen against a wall [3 pts]

Pen against a wall; 80 ft of fence.

A rancher has 80 ft of fence for a rectangular pen against a wall (three sides need fence). Find the dimensions that maximize the area and state the maximum area.

Question 3 – Open box [3 pts]

An 18-inch square sheet.

An open box is made from an 18-in square sheet by cutting squares of side $x$ from the corners. Find $x$ that gives the greatest volume and that volume.

Question 4 – Closest point [2 pts]

Find the point on the line $y=x+3$ that is closest to the origin.

Question 5 – Revenue [3 pts]

A hat stand sells $q=300-6p$ hats per week at price $p$ dollars. (a) Write the revenue function. (1 point) (b) Find the price that maximizes revenue. (1 point) (c) Justify and give the maximum revenue. (1 point)

Question 6 – Can design [3 pts]

A can holding 1000 cm³.

A closed can must hold 1000 cm³. Find the radius that minimizes the surface area, and show that it is a minimum. (Round to two decimals.)

Question 7 – Justifying an extremum [2 pts]

For the function $A(x)=x(60-2x)$ on $0 < x < 30$, show that the critical number gives an absolute maximum using the First Derivative Test.

Question 8 – Ticket price [2 pts]

A concert sells 400 tickets at \$20. Each \$2 increase sells 20 fewer tickets. What price gives the greatest revenue?

Key facts behind this optimization test

Key facts

Name the variables, write an objective function, and use a constraint to reduce it to one variable.
Find critical numbers by solving f′ = 0 on the allowed domain.
Justify the extreme with a sign change of f′, the sign of f″, or by comparing endpoint values.
Check the domain and give the final answer in context with units.

How to use this optimization 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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