3.3 Increasing and Decreasing and the First Derivative TestObjective: Determine intervalues in which a function is increasing or

decreasing and apply the First Derivative Test.

Miss BattagliaAP Calculus AB/BC

A function f is increasing on an interval for any two numbers x1 and x2 in the interval, x1<x2 implies f(x1)<f(x2)A function f is decreasing on an interval for any two numbers x1 and x2 in the interval, x1<x2 implies f(x1)>f(x2)

Increasing and Decreasing Functions

Increasing! Pierre the Mountain Climbing Ant is climbing the hill from left

to right.

Decreasing! Pierre is walking downhill.

Let f be a function that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b).1. If f’(x)>0 for all x in (a,b), then f is

increasing on [a,b]2. If f’(x)<0 for all x in (a,b), then f is

decreasing on [a,b]3. If f’(x)=0 for all x in (a,b), then f is contant

on [a,b]

Test for Increasing and Decreasing Functions

Find the open intervals on which is increasing or decreasing.

Intervals on Which f is Increasing or Decreasing

Find the first derivative. Set the derivative equal to zero and solve for

x. Put the critical numbers you found on a

number line (dividing it into regions). Pick a value from each region, plug it into the

first derivative and note whether your result is positive or negative.

Indicate where the function is increasing or decreasing.

The First Derivative Test

Find the relative extrema of the function in the interval (0,2π)

Applying the First Derivative Test

Find the relative extrema of

Applying the First Derivative Test

Find the relative extrema of

Applying the First Derivative Test

Read 3.3 Page 179 #1, 8, 12, 21, 27, 29, 35, 43, 45, 63, 67, 79, 99-103

Classwork/Homework