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Cogito

Calculus · Chapter 6 · Lesson 3

Practice Worksheet

Optimisation

The largest, the cheapest, the fastest.

12 problems · about 23 minutes · FUN-4.C

NameDate

Figure — use these to answer the problems

Flip the parabola downward and find the peak. Optimisation is locating that point exactly.
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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. Why must the constraint be used before differentiating?

    • The objective must be a function of a single variable first.
    • It makes the arithmetic faster.

  3. With 24 m of fence, what is the 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. Two numbers add to 30. Greatest product?

    Answer

  3. Can the extremum sit at an endpoint of a closed interval? 1 for yes, 0 for 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.
    • Confirm it is the extremum you wanted, and check the endpoints.

  2. With 100 m of fence, what is the largest rectangular area?

    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 possible product?

    Answer