3.7 Optimization

Primary Learning Target:Apply Calculus techniques for finding extrema to solve

real world optimization problems

1. Classic Problem from Pre-calculus:Given 40 ft of fence to enclose a rectangular corral fencing 3 sides, along a river. What dimensions will maximize the area?

2. Optimization strategy• Picture and Variables• Equation of the Model • Must be in terms of a single variable (substitution

is typically used here)• Domain (closed or open?)• Critical points

• Test for Max or Min1. If Domain :

Test endpoints and C.P. 2. If Domain is :

Use FDT or SDT (whichever one is easier)

3. Example

What dimensions of a one liter cylindrical can will use the least amount of material?

4. Example

Maximize the area of the rectangle inscribed between the parabolas and

5. Example

http://www.geogebratube.org/material/show/id/45777Elvis loves to play fetch. We stand at the water's edge at A, and throw the ball into the water to B. Elvis' excitement level suggests that his objective is to retrieve the ball as quickly as possible. That is, he attempts to find a path that minimizes the retrieval time. His running speed is 4 ft/s and swimming speed is 1 ft/s. (You can adjust these using the sliders)

Suppose you decide to race Elvis to the ball. To be fair, you must run and swim at his speeds. To minimize your fetching time, at what point should you jump into the water? That is, what should be the distance from A (starting point) to D (the point where you jump in the water)?

To see Elvis' optimal path, check the box named 'Optimal path'. Can you find this optimal path? Are you as smart as Elvis, the calculus dog?

5. ExamplePoint A: (0,0) Point B: (16,4) Find Point D: the point at which we leave land to swim.Objective: Minimize time to BLand speed is 4 ft/s swimming speed is 1 ft/s.

Homework – collected tomorrowp. 220 #2b,c, 5, 9, 11 13, 19, 21, 25

Elvis the Wonderdog: Suppose he starts at (0,0) .Try a different speed for running and swimming,Between 1 and 6 ft/sec

Where should he get in the water to arrive at the point (16,4) the fastest?

Verify your result with the Geogebra worksheet