AP Calculus Unit 4 Day 7 Optimization

Rolle’s Theorem (A special case of MVT) If f is continuous on [a,b] and differentiable on (a,b) AND f(b)=f(a) Then there is at least on c in (a,b) such that f’(c)=0. ***Therefore there is guaranteed to exist a horizontal tangent at some point on f(x).***

Use the mean value theorem to explain why there is guaranteed to exist a horizontal tangent at some location on [-3,1] for the following curve.

Function f is continuous on [0,10] and differentiable on (0,10). Given the following data, what is the minimum number of extrema that must exist on the interval (0,10)? Justify. X f Answer: There are at least two extrema on (0,10). One in the interval (0,4) and one in the interval (4,10). Extrema occur at critical points, where the 1 st derivative equals zero. The average slope on (0,4) is zero. There is guaranteed to exist a point in (0,4) where f’(0) = 0 because of the Mean Value Theorem. The same reasoning guarantees the existence of an extrema in the interval (4,10).

Function f is continuous on [6,10] and differentiable on (6,10). Given the following data, what is the minimum number of extrema that must exist on the interval (6,10)? Justify. Answer: There is at least one extrema on (6,10). Extrema occur at critical points, where the 1 st derivative equals zero. The Intermediate Value Theorem guarantees that f(x) = 3 somewhere in the intervals (6,8) and (8,10). The average slope between those two points is zero. The Mean Value Theorem guarantees another point in that interval where f’(x) = 0. X6810 f425 Note: Picking 3 as the value for f(x) was arbitrary. The key is to find a function value that you can guarantee exists in the two intervals.

WORD PROBLEMS

An open top box is to be made by cutting congruent squares of side length x from the corners of a 20-by- 25 inch sheet of tin and bending up the sides. Draw a sketch: (Together) Write the expression, in terms of x, that would represent the volume of the box. Volume of an open top box

You are designing a rectangular poster to contain 50 square inches of printing with a 4 inch margin at the top and bottom and 2 inch margins on the sides. Let x be the width and y be the length of the printed region Draw a sketch: (Together) Write an equation in terms of x and y that represents the area of the printed region. Write the expression, in terms of x and y, that would represent the area of the entire poster. Now, combine the results from above to write an expression in terms of x ONLY to represent the area of the entire poster Area of a Poster

A 216 m 2 rectangular pea patch is to be enclosed by a fence and divided into 2 equal parts by another fence parallel to one side. Let x be the length of the entire pea patch and y be the length of the fence and two other sides of the patch. Draw a sketch: (TOGETHER) Write an equation in terms of x and y that represents the area of the entire pea patch. Write the expression, in terms of x and y, that would represent the amount of fence. Now, combine the results from above to write an expression in terms of x ONLY to represent the amount of fence. Rectangular Pea Patch

Optimization What is optimization? Maximizing or minimizing something Reminder: Maximums and Minimums occur when the derivative equals zero or does not exist!

1. An open top box is to be made by cutting congruent squares from the corners of a 20-by-25 inch sheet of tin and bending up the sides. What dimensions give the box the LARGEST volume? Steps: 1.Set up an equation to maximize or minimize 2. Take the derivative of this equation 3.Find the c.p.s 4.Create sign line to determine sign of derivative 5.Interpret the sign line to determine x-value of maximum or minimum value 6.Answer the question asked

2. You are designing a rectangular poster to contain 50 square inches of printing with a 4 inch margin at the top and bottom and 2 inch margins on the sides. What OVERALL dimensions will minimize the amount of paper used?

3. A 216 m 2 rectangular pea patch is to be enclosed by a fence and divided into 2 equal parts by another fence parallel to one side. What dimensions for the outer rectangle will require the SMALLEST total length of fence? How much fence will be needed? yyy x

TRAIN If 40 passengers hire a special car on a train, they will be charged $8 each. This fare will be reduced by $.10 per extra person over the minimum number of 40. What number of passengers will produce the maximum income for the railroad?. Income=(#passengers)(fee) One Possible setup: Let x: of extra people Income=(40+x)(8-.10x) Total number of passengers EVERYBODY will get the discounted fare.