Cogito
AP Calculus AB · Chapter 5 · Lesson 3
Optimisation
The largest, the cheapest, the closest.
12 problems · about 22 minutes · FUN-4.C, FUN-4.D
What this lesson teaches
The student solves optimisation problems and justifies the extremum found.
- Pair an objective with a constraint and reduce to one variable.
- Differentiate, set to zero, then justify the extremum.
- Check endpoints and answer the question actually asked.
Warm Up
Straightforward practice. Get the method working first.
5 problemsTwo numbers add to 16. What is their greatest product?
Answer 64
Why 64.
What does the exam require beyond finding the critical point?
Answer A justification that it is the extremum wanted.
Why An explicit justification.
With 24 m of fence, largest rectangular area?
Answer 36
Why A 6 by 6 square.
Two numbers add to 10. Greatest product?
Answer 25
Why 5 × 5.
A = 20w − w². What is dA/dw at w = 4?
Answer 12
Why 20 − 8.
Build It Up
The same ideas with more to keep track of.
3 problemsA = 20w − w². At which w is dA/dw zero?
Answer 10
Why 20 − 2w = 0.
Second derivative is negative at a critical point. Maximum or minimum? 1 max, 2 min.
Answer 1
Why Concave down.
Can the extremum sit at an endpoint of a closed interval? 1 yes, 0 no.
Answer 1
Why The derivative need not vanish there.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the optimisation method in order.
Answer 1. Identify the objective and the constraint. 2. Use the constraint to remove one variable. 3. Differentiate and set the result to zero. 4. Justify the extremum and answer the question asked.
Why Nothing can be differentiated until one variable remains.
Two numbers add to 30. Greatest product?
Answer 225
Why 15 × 15.
The Fence: With 40 m of fence, what is the largest rectangular area, in square metres?
Answer 100 m²
Why 100 m².
The Numbers: Two numbers add to 20. What is their greatest product?
Answer 100
Why 100.