Cogito
Calculus · Chapter 6 · Lesson 1
Extrema, Concavity, and Optimisation
Using the derivative to find the best value.
10 problems · about 22 minutes · FUN-4.A, FUN-4.C
What this lesson teaches
The student uses derivatives to find extrema, determine concavity, and solve optimisation problems.
- Extrema occur where the derivative is zero.
- The sign change of f′, or the sign of f″, tells you which kind.
- Optimisation writes the quantity as one function, then differentiates.
Warm-Up Review
From earlier lessons. Loosen up before the new work.
2 problemsReview — Linear Approximation: L(x) = 4 + 0.5(x − 16). What is L(16.4)?
Answer 4.2
Why 4.2.
Review — Related Rates: dV/dt = 3s² · ds/dt. With s = 3 and ds/dt = 2, what is dV/dt?
Answer 54
Why 54.
Warm Up
Straightforward practice. Get the method working first.
4 problems32 m of fence makes a rectangle. Largest area?
Answer 64
Why 64 m².
What is true of the derivative at a maximum or minimum?
Answer It is zero, because the tangent is horizontal.
Why The derivative is zero.
f′(x) = 8 − 2x. At what x is it zero?
Answer 4
Why 8 = 2x.
40 m of fence makes a rectangle. Largest area?
Answer 100
Why A 10 by 10 square.
Build It Up
The same ideas with more to keep track of.
2 problemsf′(x) = 2x − 6. At what x is it zero?
Answer 3
Why 2x = 6.
f″ positive at a critical point. Maximum or minimum?
Answer Minimum.
Why Concave up is a bowl.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
2 problemsThe Critical Point: f′(x) = 10 − 2x. At what x is it zero?
Answer 5
Why x = 5.
The Fence: 24 m of fence makes a rectangle. What is the largest area?
Answer 36 square m
Why 36 m².