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Cogito

AP Calculus AB · Chapter 5 · Lesson 3

Practice Worksheet

Optimisation

The largest, the cheapest, the closest.

12 problems · about 22 minutes · FUN-4.C, FUN-4.D

NameDate

Figure — use these to answer the problems

Flip the parabola downward and find the peak. Optimisation locates that point exactly.
-8-8-6-6-4-4-2-222446688
y = -1x² + 4x + 0
  • Point(2, 4)
  • Slope of the tangent0
a-1
b4
Point of tangency2
Figure 1

Warm Up

Straightforward practice. Get the method working first.

5 problems

  1. Two numbers add to 16. What is their greatest product?

    Answer

  2. What does the exam require beyond finding the critical point?

    • A justification that it is the extremum wanted.
    • A sketch of the graph.

  3. With 24 m of fence, largest rectangular area?

    Answer

  4. Two numbers add to 10. Greatest product?

    Answer

  5. A = 20w − w². What is dA/dw at w = 4?

    Answer

Build It Up

The same ideas with more to keep track of.

3 problems

  1. A = 20w − w². At which w is dA/dw zero?

    Answer

  2. Second derivative is negative at a critical point. Maximum or minimum? 1 max, 2 min.

    Answer

  3. Can the extremum sit at an endpoint of a closed interval? 1 yes, 0 no.

    Answer

Stretch Yourself

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

4 problems

  1. Put the optimisation method in order.

    Write 1 to 4 in the boxes to put these in order.

    • Identify the objective and the constraint.
    • Use the constraint to remove one variable.
    • Differentiate and set the result to zero.
    • Justify the extremum and answer the question asked.

  2. Two numbers add to 30. Greatest product?

    Answer

  3. The Fence

    With 40 m of fence, what is the largest rectangular area, in square metres?

    Answerm²

  4. The Numbers

    Two numbers add to 20. What is their greatest product?

    Answer