Increasing and Decreasing Functions Lesson 5 1 The

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Increasing and Decreasing Functions Lesson 5. 1

Increasing and Decreasing Functions Lesson 5. 1

The Ups and Downs Think of a function as a roller coaster going from

The Ups and Downs Think of a function as a roller coaster going from left to right Uphill • Slope > 0 • Increasing function Downhill • Slope < 0 • Decreasing function 2

Definitions Given function f defined on an interval • For any two numbers x

Definitions Given function f defined on an interval • For any two numbers x 1 and x 2 on the interval Increasing function • f(x 1) < f(x 2) when x 1 < x 2 Decreasing function X 1 X 2 f(x) • f(x 1) > f(x 2) when x 1< x 2 3

Increasing/Decreasing and the Derivative Assuming existence of derivative on interval If f '(x) >

Increasing/Decreasing and the Derivative Assuming existence of derivative on interval If f '(x) > 0 for each x • f(x) increasing on interval If f '(x) < 0 for each x • f(x) decreasing on interval What if f '(x) = 0 on the interval? What could you say about f(x)? 4

Check These Functions By graphing on calculator, determine the intervals where these functions are

Check These Functions By graphing on calculator, determine the intervals where these functions are • Increasing • Decreasing 5

Critical Numbers Definition Numbers c in the domain of f where • f '(c)

Critical Numbers Definition Numbers c in the domain of f where • f '(c) = 0 • f '(c) does not exist Critical Points 6

Applying Derivative Test Given a function f(x) Determine the derivative f '(x) Find critical

Applying Derivative Test Given a function f(x) Determine the derivative f '(x) Find critical points … • Where f '(x) = 0 or f '(x) does not exist Evaluate derivative between or on either side of the critical points Try it with this function 7

Applications Digitari, the great video game manufacturer determines its cost and revenue functions •

Applications Digitari, the great video game manufacturer determines its cost and revenue functions • C(x) = 4. 8 x -. 0004 x 2 • R(x) = 8. 4 x -. 002 x 2 0 ≤ x ≤ 2250 Determine the interval(s) on which the profit function is increasing 8

Assignment Lesson 5. 1 Page 313 Exercises 1 – 57 EOO 9

Assignment Lesson 5. 1 Page 313 Exercises 1 – 57 EOO 9