AP Calculus AB Optimization Worksheet (PDF)
This optimization practice worksheet for AP Calculus AB covers optimization (Common Core AP CALC AB 5.10, AP CALC AB 5.11), with a free PDF to print.
These 20 problems practice optimization problems with fences, boxes, cans, closest points and revenue, including justifying a maximum or minimum (AP Calculus AB, topics 5.10, 5.11). 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 cans, closest points, margins on a poster and pricing situations such as ticket sales. 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 5.10, AP CALC AB 5.11
Optimization practice worksheet: exercises 1 to 10
Exercise 1 – Maximum product
Exercise 2 – Minimum sum of squares
Exercise 3 – Maximum area, fixed perimeter
Exercise 4 – Fence against a wall
A farmer has 120 ft of fence to enclose a rectangular pen against a barn wall (no fence is needed along the wall). Find the dimensions that maximize the area.
Exercise 5 – Minimum fence, fixed area
Exercise 6 – Open box from a square sheet
A square sheet of cardboard 12 in on a side has a square of side \(x\) cut from each corner, and the sides are folded up to make an open box. Find \(x\) that maximizes the volume.
Exercise 7 – Open box from a rectangle
Exercise 8 – Can with least material
A closed can must hold 500 cm³. Find the radius and height that use the least metal.
Exercise 9 – Closest point on a parabola
Find the points on \(y=x^{2}\) closest to \((0,1)\).
Exercise 10 – Closest point on a line
Exercises 11 and up
Exercise 11 – Price for maximum revenue
Exercise 12 – Maximizing profit
Exercise 13 – Rectangle in a semicircle
Exercise 14 – Rectangle under a parabola
Exercise 15 – Fencing with two prices
Exercise 16 – Poster margins
A poster has a printed area of 200 cm² with 4 cm margins at the top and bottom and 2 cm margins on each side. Find the dimensions of the poster that use the least paper.
Exercise 17 – Open-top box
Exercise 18 – Closest point to (4, 0)
Exercise 19 – Two adjacent pens
Exercise 20 – Ticket price
Key facts behind this optimization practice worksheet
- 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 practice 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 optimization practice worksheet as a PDF with the button below and keep it with your notes.
Download the optimization practice worksheet as a PDF with the button below and keep it with your notes.
All AP Calculus AB chapters: AP Calculus AB math.
