4.3 How Derivatives Affect the Shape of a Graph. Facts If f ’( x ) > 0 on an interval ( a,b ), then f (x) is increasing on ( a,b ). If f ’( x ) < 0 on.

Slides:

Advertisements

Similar presentations

First Derivative Test, Concavity, Points of Inflection Section 4.3a.

Advertisements

AP Calculus Section 3.4 Concavity and the Second Derivative Test

Section 3.4 – Concavity and the Second Derivative Test

Concavity and the Second Derivative Test

4.3 Derivatives and the shapes of graphs 4.4 Curve Sketching

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

Copyright © Cengage Learning. All rights reserved.

Aim: Concavity & 2 nd Derivative Course: Calculus Do Now: Aim: The Scoop, the lump and the Second Derivative. Find the critical points for f(x) = sinxcosx;

Section 3.3 How Derivatives Affect the Shape of a Graph.

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

What does say about f ? Increasing/decreasing test

Sec 3.4: Concavity and the Second Derivative Test

How Derivatives Affect the Shape of the Graph

Relative Extrema.

 Recall MARGINAL Costs, Revenue, Profit & Sales are ALL first derivatives of C(x), R(x), P(x), S(x)  For our purposes, marginal functions represent.

3.2 The Second-Derivative Test 1 What does the sign of the second derivative tell us about the graph? Suppose the second derivative is positive. But, is.

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.

1 f ’’(x) > 0 for all x in I f(x) concave Up Concavity Test Sec 4.3: Concavity and the Second Derivative Test the curve lies above the tangentsthe curve.

Concavity f is concave up if f’ is increasing on an open interval. f is concave down if f’ is decreasing on an open interval.

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.

5.3:Higher Order Derivatives, Concavity and the 2 nd Derivative Test Objectives: To find Higher Order Derivatives To use the second derivative to test.

Problem of the Day (Calculator allowed)

In this section, we will investigate some graphical relationships between a function and its second derivative.

2.1.  Remember that the first derivative test will tell where a function is increasing or decreasing.

Applications of Derivatives

Increasing/ Decreasing

Section 13.1 – 13.2 Increasing/Decreasing Functions and Relative Extrema.

CONCAVITY AND SECOND DERIVATIVE RIZZI – CALC BC. WARM UP Given derivative graph below, find a. intervals where the original function is increasing b.

Chapter Four Applications of Differentiation. Copyright © Houghton Mifflin Company. All rights reserved. 4 | 2 Definition of Extrema.

5.3:Higher Order Derivatives, Concavity and the 2 nd Derivative Test Objectives: To find Higher Order Derivatives To use the second derivative to test.

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

CHAPTER 3 SECTION 3.4 CONCAVITY AND THE SECOND DERIVATIVE TEST.

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

How derivatives affect the shape of a graph ( Section 4.3) Alex Karassev.

Increasing/decreasing and the First Derivative test

Ch. 5 – Applications of Derivatives

Section 4.4 The Shape of a Graph

Relative Extrema and More Analysis of Functions

4.3 Using Derivatives for Curve Sketching.

Review Problems Sections 3-1 to 3-4

4.3 Derivatives and the shapes of graphs 4.5 Curve Sketching

Lesson 37 - Second Derivatives, Concavity, Inflection Points

Let’s get our brains back on track …

Concavity of Functions

First and Second Derivatives Tests

Applications of the Derivative

3.2 – Concavity and Points of Inflection

Section 3.6 Calculus AP/Dual, Revised ©2017

Concavity and the Second Derivative Test

Second Derivative Test

1 2 Sec 4.3: Concavity and the Second Derivative Test

Application of Derivative in Analyzing the Properties of Functions

Sec 3.4: Concavity and the Second Derivative Test

Introduction to Graph Theory

3.1 – Increasing and Decreasing Functions; Relative Extrema

4.3 Connecting f’ and f’’ with the graph of f

Critical Numbers – Relative Maximum and Minimum Points

Derivatives and Graphing

Packet #14 First Derivatives and Graphs

1 2 Sec4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH

5-3 Day 1 connecting f graphs with f' and f" graphs

Section 3.4 – Concavity and the Second Derivative Test

4.2 Critical Points, Local Maxima and Local Minima

Copyright © Cengage Learning. All rights reserved.

Concavity & the second derivative test (3.4)

Analyzing f(x) and f’(x) /

Math 1304 Calculus I 4.03 – Curve Shape.

- Derivatives and the shapes of graphs - Curve Sketching

Presentation transcript:

4.3 How Derivatives Affect the Shape of a Graph

Facts If f ’( x ) > 0 on an interval ( a,b ), then f (x) is increasing on ( a,b ). If f ’( x ) < 0 on an interval ( a,b ), then f (x) is decreasing on ( a,b ). Example Find the intervals where the function is increasing or decreasing.

The First Derivative Test Let c be a critical number of f (x).  If f ’(x) changes from + to – at c, then f has a local maximum at c.  If f ’(x) changes from – to + at c, then f has a local minimum at c.  No sign change at c means no local extremum (maximum or minimum)

How to find local max/min and interval of increasing/decreasing 1)Find all critical values by solving f ’( x ) = 0 and f ’( x ) = undefined 2)Put all critical values on the number line and pick some test values to determine the sign of the derivative for each interval. 3)Determine the interval of increasing/decreasing based on the sign of the first derivative.

Examples Find the intervals of increase/decrease and all local extrema.

Examples

Definition  A function f ( x ) is concave up on an interval ( a,b ) if the graph of f (x) lies above its tangent line at each point on ( a,b ).  A function f ( x ) is concave down on an interval ( a,b ) if the graph of f (x) lies below its tangent line at each point on ( a,b ).  A point where a graph changes its concavity is called an inflection point. o If the slopes of the tangent lines are increasing, then the function is concave up. o If the slopes of the tangent lines are decreasing, then the function is concave down. Facts

Test For Concavity  If f ’’( x ) > 0 on an interval ( a,b ), then f (x) is concave up on ( a,b ).  If f ’’( x ) < 0 on an interval ( a,b ), then f (x) is concave down on ( a,b ).  If f ’’( x ) changes signs at c, then f (x) has an inflection point at c. Example : Find the intervals where the function is concave up or concave down, and find all inflection points.

How to find inflection points and interval of concavity 1)Find all numbers c such that f ’’(c) = 0 or f ’’(c) is undefined. 2)Put all values found in step 1 on the number line and use test values to determine the sign of the second derivative for each interval. 3)Determine the interval of concavity based on the sign of the second derivative.

Examples Find the intervals of concavity and all inflection points.

Second Derivative test Let c be a critical number of a function f (x). If f ’’( c ) > 0, then f (x) has a local minimum at c. If f ’’( c ) < 0, then f (x) has a local maximum at c. If f ’’( c ) = 0 or dne, then this test fails  must use the first derivative test! Example : Find all local extrema

Examples a) Find the intervals where the function is increasing /decreasing, and find all local extrema. b) Find the intervals where the function is concave up or concave down, and find all inflection points.