Applied Optimization

Turning a real situation into a surface to climb.

12 problems · about 24 minutes · F-IF.C.7, A-CED.A.2

NameDate

Figure — use these to answer the problems

A cost surface seen from above. The cheapest point is where the rings shrink to nothing.

a, the cost of overshooting in x1

b, the cost of overshooting in y2

Figure 1

Straightforward practice. Get the method working first.

5 problems

Two numbers sum to 24 with the largest product. What is that product?
Answer
2
Why must the boundary be checked on a closed region?
- The extreme may sit on the edge, where the gradient need not be zero.
- The boundary never matters.
Two numbers sum to 20 with the largest product. What is that product?
Answer
On a closed bounded region, is an absolute maximum guaranteed? 1 yes, 0 no.
Answer
A rectangular region has how many corners to check separately?
Answer

The same ideas with more to keep track of.

3 problems

An open box with square base side 2 and volume 12. What is its height?
Answer
Two numbers sum to 14 with the largest product. What is that product?
Answer
A model gives a side length of −3 metres. Is that a valid answer? 1 yes, 0 no.
Answer

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

4 problems

Sort each location by whether it must be checked when hunting absolute extrema on a closed region.
Write each item under the heading it belongs to: Interior critical points · The boundary curve · The corners · Points outside the region
Must be checked
Not a candidate
A square base of side 5 with height 4. What is the volume?
Answer
The Box
A box has a square base of side 4 and volume 32. What is its height?
Answer
The Two Numbers
Two positive numbers sum to 30 and their product is largest. What is that product?
Answer