Download presentation

Presentation is loading. Please wait.

Published byVivian Pelter Modified over 2 years ago

1
Increasing & Decreasing Functions and 1 st Derivative Test Lesson 4.3

2
Increasing/Decreasing Functions Consider the following function For all x < a we note that x 1

3
Increasing/Decreasing Functions Similarly -- For all x > a we note that x 1 f(x 2 ) If a function is either strictly decreasing or strictly increasing on an interval, it is said to be monotonic f(x) a The function is said to be strictly decreasing

4
Test for Increasing and Decreasing Functions If a function is differentiable and f ’(x) > 0 for all x on an interval, then it is strictly increasing If a function is differentiable and f ’(x) < 0 for all x on an interval, then it is strictly decreasing Consider how to find the intervals where the derivative is either negative or positive

5
Test for Increasing and Decreasing Functions Finding intervals where the derivative is negative or positive Find f ’(x) Determine where Try for Where is f(x) strictly increasing / decreasing f ‘(x) = 0 f ‘(x) > 0 f ‘(x) < 0 f ‘(x) does not exist Critical numbers

6
Test for Increasing and Decreasing Functions Determine f ‘(x) Note graph of f’(x) Where is it pos, neg What does this tell us about f(x) f ‘(x) > 0 => f(x) increasing f ‘(x) f(x) decreasing

7
First Derivative Test Given that f ‘(x) = 0 at x = 3, x = -2, and x = 5.25 How could we find whether these points are relative max or min? Check f ‘(x) close to (left and right) the point in question Thus, relative min f ‘(x) < 0 on left f ‘(x) > 0 on right

8
First Derivative Test Similarly, if f ‘(x) > 0 on left, f ‘(x) < 0 on right, We have a relative maximum

9
First Derivative Test What if they are positive on both sides of the point in question? This is called an inflection point

10
Examples Consider the following function Determine f ‘(x) Set f ‘(x) = 0, solve Find intervals

11
Assignment A Lesson 4.3A Page 226 Exercises 1 – 57 EOO

12
Application Problems Consider the concentration of a medication in the bloodstream t hours after ingesting Use different methods to determine when the concentration is greatest Table Graph Calculus

13
Application Problems A particle is moving along a line and its position is given by What is the velocity of the particle at t = 1.5? When is the particle moving in positive/negative direction? When does the particle change direction?

14
Application Problems Consider bankruptcies (in 1000's) since 1988 Use calculator regression for a 4 th degree polynomial Plot the data, plot the model Compare the maximum of the model, the maximum of the data 1988198919901991199219931994 594.6643.0725.5880.4845.31042.1835.2

15
Assignment B Lesson 4.3 B Page 227 Exercises 95 – 101 all

Similar presentations

OK

Section 4.1 Using First and Second Derivatives. Let’s see what we remember about derivatives of a function and its graph –If f’ > 0 on an interval than.

Section 4.1 Using First and Second Derivatives. Let’s see what we remember about derivatives of a function and its graph –If f’ > 0 on an interval than.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on rural retailing in india Ppt on tunnel diode application Mis ppt on nokia phone Ppt on mathematical games Ppt on nuclear waste in india Ppt on extinct species of plants and animals Converting pdf to ppt online Ppt on power quality monitoring Ppt on brand marketing definition Ppt on micro operations