AP Calculus AB: Optimization — Drill 1

About This Drill

AP Calculus AB: Optimization — Drill 1 is a Math practice drill covering Optimization. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice setting up and solving optimization problems using the Candidates Test on closed intervals and applied problems involving area, cost, and distance. These AP Calculus AB skills from Unit 5 appear regularly on the free-response section.

Questions in This Drill

1. Find the absolute maximum value of \( f(x) = x^3 - 3x^2 + 1 \) on the interval \( [0, 3] \).
2. Which of the following lists all critical points of \( f(x) = x^4 - 8x^2 + 3 \)?
3. A farmer has 200 feet of fencing and wants to enclose a rectangular area using a straight barn wall as one side (so fencing is needed for only three sides). What dimensions maximize the enclosed area?
4. A cylindrical can is to be manufactured with a fixed volume \( V \). The material for the top and bottom costs twice as much per square unit as the material for the lateral (side) surface. Using \( h = \dfrac{V}{\pi r^2} \) to eliminate \( h \), which of the following correctly represents the total cost \( C \) as a function of radius \( r \) (taking the cost per unit area of the lateral surface as \( 1 \))?
5. A 10-foot wire is cut into two pieces. One piece is bent into a square and the other into a circle. If \( x \) represents the length of wire used for the square (so \( 10 - x \) is used for the circle), where \( 0 \leq x \leq 10 \), which value of \( x \) minimizes the combined area of the two shapes?