Increasing & Decreasing Functions & The First Derivative Test (3.3) November 29th, 2012.

Slides:

Advertisements

Similar presentations

Increasing and Decreasing Functions

Advertisements

Maximum ??? Minimum??? How can we tell?

DO NOW: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5

Concavity & the second derivative test (3.4) December 4th, 2012.

Copyright © Cengage Learning. All rights reserved.

Objectives: 1.Be able to determine where a function is concave upward or concave downward with the use of calculus. 2.Be able to apply the second derivative.

Extremum. Finding and Confirming the Points of Extremum.

How Derivatives Affect the Shape of the Graph

Relative Extrema.

Extrema on an interval (3.1) November 15th, 2012.

Relative Extrema Lesson 5.5. Video Profits Revisited Recall our Digitari manufacturer Cost and revenue functions C(x) = 4.8x x 2 0 ≤ x ≤ 2250 R(x)

Increasing and Decreasing Functions and the First Derivative Test.

Section 5.1 – Increasing and Decreasing Functions The First Derivative Test (Max/Min) and its documentation 5.2.

The Shape of the Graph 3.3. Definition: Increasing Functions, Decreasing Functions Let f be a function defined on an interval I. Then, 1.f increases on.

Problem of the Day (Calculator allowed)

Applications of Derivatives

CHAPTER Continuity Derivatives and the Shapes of Curves.

Copyright © Cengage Learning. All rights reserved. Applications of Differentiation.

Using Derivatives to Sketch the Graph of a Function Lesson 4.3.

In the past, one of the important uses of derivatives was as an aid in curve sketching. We usually use a calculator of computer to draw complicated graphs,

Ch. 5 – Applications of Derivatives 5.1 – Extreme Values of Functions.

Copyright © 2016, 2012 Pearson Education, Inc

INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST Section 3.3.

Ch. 5 – Applications of Derivatives

Implicit differentiation (2.5) October 29th, 2012.

Extremum & Inflection. Finding and Confirming the Points of Extremum & Inflection.

Miss Battaglia AP Calculus AB/BC. A function f is increasing on an interval for any two numbers x 1 and x 2 in the interval, x 1

1. Use the graph to determine intervals where the function is increasing, decreasing, and constant.

§3.4 Concavity Concave Up Concave Down Inflection Points Concavity Changes Concave Up Concave Down.

Ch. 5 – Applications of Derivatives 5.1 – Extreme Values of Functions.

Ch. 5 – Applications of Derivatives 5.2 – Mean Value Theorem.

Increasing and Decreasing Functions

Increasing/decreasing and the First Derivative test

Increasing & Decreasing Functions & The First Derivative Test (3.3)

Increasing, Decreasing, Constant

INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST

Ch. 5 – Applications of Derivatives

Calculus Section 4.2 Find relative extrema and graph functions

Learning Target: I will determine if a function is increasing or decreasing and find extrema using the first derivative. Section 3: Increasing & Decreasing.

3.3: Increasing/Decreasing Functions and the First Derivative Test

3.3 Increasing and Decreasing Functions and the First Derivative Test

3.1 Extrema on an Interval Define extrema of a function on an interval. Define relative extrema of a function on an open interval. Find extrema on a closed.

The Derivative and the Tangent Line Problem (2.1)

First Derivative Test So far…

Increasing and Decreasing Functions and the First Derivative Test

Lesson 37 - Second Derivatives, Concavity, Inflection Points

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVE Find relative extrema of a continuous function using the First-Derivative.

The derivative and the tangent line problem (2.1)

Relative Extrema Lesson 5.2.

Applications of the Derivative

Extremas – First Derivative Test & Second Derivative Test

3.1: Increasing and Decreasing Functions

3.2: Extrema and the First Derivative Test

A function f is increasing on an open interval I if, for any choice of x1 and x2 in I, with x1 < x2, we have f(x1) < f(x2). A function f is decreasing.

Application of Derivative in Analyzing the Properties of Functions

Self Assessment 1. Find the absolute extrema of the function

Concave Upward, Concave Downward

3.1 – Increasing and Decreasing Functions; Relative Extrema

5.2 Section 5.1 – Increasing and Decreasing Functions

Increasing and Decreasing Functions and the First Derivative Test

Increasing and Decreasing Functions and the First Derivative Test

Applications of Differentiation

4.2 Critical Points, Local Maxima and Local Minima

The First Derivative Test

Copyright © Cengage Learning. All rights reserved.

Concavity & the second derivative test (3.4)

The Derivative and the Tangent Line Problem (2.1)

Concavity & the 2nd Derivative Test

Unit 4: Applications of Derivatives

Presentation transcript:

Increasing & Decreasing Functions & The First Derivative Test (3.3) November 29th, 2012

I. Increasing & Decreasing Functions Defs. of Increasing & Decreasing Functions: A function f is increasing on an interval if for any two numbers x 1 and x 2 in the interval x 1 < x 2 implies f(x 1 )<f(x 2 ). A function f is decreasing on an interval if for any two numbers x 1 and x 2 in the interval x 1 f(x 2 ).

*Increasing is going up from left to right, decreasing is going down from left to right.

Thm. 3.5: Test for Increasing & Decreasing Functions: 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], (because f has a positive slope on (a, b)). 2. If f’(x)<0 for all x in (a, b), then f is decreasing on [a, b], (because f has a negative slope on (a, b)). 3. If f’(x)=0 for all x in (a, b), then f is constant on [a, b], (because f is a horizontal line & has a slope of 0 on (a, b)).

constant f’(x)=0 decreasing f’(x)<0 increasing f’(x)>0

Ex. 1: Find the open intervals on which is increasing or decreasing.

Guidelines for Finding Intervals on Which a Function is Increasing or Decreasing: Let f be continuous on the interval (a, b). To find the open intervals on which f is increasing or decreasing, use the following steps. 1. Locate the critical numbers of f in (a, b) and use these numbers to determine the test intervals. 2. Determine the sign of f’(x) at one test value in each interval. 3. Use Thm. 3.5 to determine whether f is increasing or decreasing on each interval.

You Try: Identify the open intervals on which the function is increasing or decreasing.

Thm. 3.6: The First Derivative Test: Let c be a critical number of a function f that is continuous on the open interval I containing c. If f is differentiable on the interval, except possibly at c, then f(c) can be classified as follows. II. The First Derivative Test

1. If f’(x) changes from negative to positive at c, then f has a relative minimum at (c, f(c)). f’(x)<0 f’(x)>0 (-)(+) (c, f(c))

2. If f’(x) changes from positive to negative at c, then f has a relative maximum at (c, f(c)). f’(x)>0 f’(x)<0 (+) (-) (c, f(c))

3. If f’(x) is positive on both sides of c or negative on both sides of c, then f(c) is neither a relative minimum nor a relative maximum. f’(x)>0 (+) (c, f(c))

Ex. 2: Find the relative extrema of the function on the interval.

You Try:Find the relative extrema of the function.