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Cogito

Calculus · Chapter 6 · Lesson 1

Practice Worksheet

Extrema, Concavity, and Optimisation

Using the derivative to find the best value.

10 problems · about 22 minutes · FUN-4.A, FUN-4.C

NameDate

Figure — use these to answer the problems

Move the point to where the tangent is flat. That is a critical point.
-8-8-6-6-4-4-2-222446688
y = 1x³ − 3x + 0
  • Point(1, -2)
  • Slope of the tangent0
a1
b-3
c0
Point1
Figure 1

Warm-Up Review

From earlier lessons. Loosen up before the new work.

2 problems

  1. From “Linear Approximation”

    L(x) = 4 + 0.5(x − 16). What is L(16.4)?

    Answer

  2. From “Related Rates”

    dV/dt = 3s² · ds/dt. With s = 3 and ds/dt = 2, what is dV/dt?

    Answer

Warm Up

Straightforward practice. Get the method working first.

4 problems

  1. 32 m of fence makes a rectangle. Largest area?

    Answer

  2. What is true of the derivative at a maximum or minimum?

    • It is zero, because the tangent is horizontal.
    • It is at its largest.

  3. f′(x) = 8 − 2x. At what x is it zero?

    Answer

  4. 40 m of fence makes a rectangle. Largest area?

    Answer

Build It Up

The same ideas with more to keep track of.

2 problems

  1. f′(x) = 2x − 6. At what x is it zero?

    Answer

  2. f″ positive at a critical point. Maximum or minimum?

    • Minimum.
    • Maximum.

Stretch Yourself

Mixed problems. Work out what kind of question it is before you start.

2 problems

  1. The Critical Point

    f′(x) = 10 − 2x. At what x is it zero?

    Answer

  2. The Fence

    24 m of fence makes a rectangle. What is the largest area?

    Answersquare m