Increasing and Decreasing Functions

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Presentation transcript:

Increasing and Decreasing Functions Lesson 5.1

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

Definitions Given function f defined on an interval For any two numbers x1 and x2 on the interval Increasing function f(x1) < f(x2) when x1 < x2 Decreasing function f(x1) > f(x2) when x1< x2 X1 X2 X2 X1 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)?

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

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

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

Applications Digitari, the great video game manufacturer determines its cost and revenue functions C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250 R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250 Determine the interval(s) on which the profit function is increasing

Assignment Lesson 5.1 Page 313 Exercises 1 – 57 EOO