3.7 OPTIMIZATION. Slide 1. The McGraw- Hill Companies, Inc. Permission required for reproducfon or display.

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1 Preliminaries In this secfon, we bring the power of the calculus to bear on a number of applied problems involving finding a maximum or a minimum. We start by giving a few general guidelines. If there s a picture to draw, draw it! Don t try to visualize how things look in your head. Put a picture down on paper and label it. Determine what the variables are and how they are related. Decide what quanfty needs to be maximized or minimized. Slide 1

2 Preliminaries Write an expression for the quanfty to be maximized or minimized in terms of only one variable. To do this, you may need to solve for any other variables in terms of this one variable. Determine the minimum and maximum allowable values (if any) of the variable you re using. Solve the problem and be sure to answer the quesfon that is asked. Slide 2

3 EXAMPLE 3.7 OPTIMIZATION 7.1 ConstrucFng a Rectangular Garden of Maximum Area You have 40 (linear) feet of fencing with which to enclose a rectangular space for a garden. Find the largest area that can be enclosed with this much fencing and the dimensions of the corresponding garden. Slide 3

4 EXAMPLE 7.1 ConstrucFng a Rectangular Garden of Maximum Area We see the maximum value of the funcfon Slide 4

5 EXAMPLE 3.7 OPTIMIZATION 7.1 ConstrucFng a Rectangular Garden of Maximum Area Since x is a distance, we must have 0 x. Further, since the perimeter is 40, we must have x 20. The only crifcal number is x = 10 and this is in the interval under considerafon. Slide 5

6 EXAMPLE 3.7 OPTIMIZATION 7.1 ConstrucFng a Rectangular Garden of Maximum Area The maximum and minimum values of a confnuous funcfon on a closed and bounded interval must occur at either the endpoints or a crifcal number. maximum The rectangle of perimeter 40 with maximum area is a square 10 c on a side. Slide 6

7 EXAMPLE 3.7 OPTIMIZATION 7.3 Finding the Closest Point on a Parabola Find the point on the parabola y = 9 x 2 closest to the point (3, 9). Slide 7

8 EXAMPLE 7.3 Finding the Closest Point on a Parabola Slide 8

9 EXAMPLE 7.3 Finding the Closest Point on a Parabola Instead of minimizing d(x) directly, we minimize the square of d(x): Slide 9

10 EXAMPLE 7.3 Finding the Closest Point on a Parabola Instead of minimizing d(x) directly, we minimize the square of d(x): Slide 10

11 EXAMPLE 7.3 Finding the Closest Point on a Parabola Recognize that x = 1 is a zero of f (x), which makes (x 1) a factor. So, x = 1 is a crifcal number. In fact, it s the only crifcal number, since (2x 2 + 2x + 3) has no zeros. Now compare the value of f at the endpoints and the crifcal number. minimum Slide 11

12 EXAMPLE 7.3 Finding the Closest Point on a Parabola Thus, the minimum value of f (x) is 5. This says that the minimum distance from the point (3, 9) to the parabola is and the closest point on the parabola is (1, 8). Slide 12

13 EXAMPLE 7.6 Minimizing the Cost of Highway ConstrucFon The state is building a new highway to link an exisfng bridge with a turnpike interchange, located 8 miles to the east and 8 miles to the south of the bridge. There is a 5- mile- wide stretch of marshland adjacent to the bridge that must be crossed. The highway costs $10 million per mile to build over the marsh and $7 million per mile to build over dry land. Slide 13

14 EXAMPLE 7.6 Minimizing the Cost of Highway ConstrucFon How far to the east of the bridge should the highway be when it crosses out of the marsh? Slide 14

15 EXAMPLE 7.6 Minimizing the Cost of Highway ConstrucFon Slide 15

16 EXAMPLE 7.6 Minimizing the Cost of Highway ConstrucFon Slide 16

17 EXAMPLE 7.6 Minimizing the Cost of Highway ConstrucFon Note that the only crifcal numbers are where C (x) = 0. Slide 17

18 EXAMPLE 3.7 OPTIMIZATION 7.6 Minimizing the Cost of Highway ConstrucFon Compare the value of C(x) at the endpoints and at this one crifcal number: The highway should be about 3.56 miles to the east of the bridge when it crosses over the marsh. Slide 18