Warm Up

Page 214

Quick Review

#1 as a class

#2 individually

Correct Homework

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

What you’ll learn•First Derivative Test for Local Extrema•Concavity•Points of Inflection•Second Derivative Test for Local Extrema•Learning about Functions from Derivatives

Why?Differential calculus is a powerful problem-solving tool precisely because of its usefulness for analyzing functions.

Example 1:For each of the following functions, use the First Derivative Test to find the local extreme

values. Identify any absolute extrema.

a)

Find f’(x) and critical pts where f’(x) = 0.

Divide graph into intervals Use critical points and find the

sign of f’ on each interval.Use 1st derivative test to identify

the behavior of y.Check GraphIdentify extrema

b) xexxg )3()( 2 512)( 3 xxxf

Concavity

The graph of a differentiable function y = f(x) isa) Concave up on an open interval I if

y” is increasing on Ib) Concave down on an open interval I

if y’ is decreasing on I.

If a function has a 2nd derivative we can conclude that

y’ increases when y”> 0, and y’ decreases if y” < 0.

http://justmathtutoring.com/Use 2nd derivative test to find concavity, max, min

Homework

Page 214

Quick Review 3-10

Page 215

Exercises 1-15 odds

4.3 Continued

Warm Up

Page 211 Example 6

Put data into your calculator and verify logistics equation

Graph equation over points using window [12, 108] by [0, 730,000]

Example 6 Population Growth in Alaska(using a calculator to find derivatives,

analyze data)

Table 4.2 shows the population of Alaska in each 10-year census between 1920 and 2000.

a) Find the logistic regression for the datab) Use the regression equation to predict the Alaskan

population in the 2020 census.c) Find the inflection point of the regression equation. What

significance does the inflection point have in terms of population growth in Alaska?

d) What does the regression equation indicate about the population of Alaska in the long run?

Theorem 4 (review) First Derivative Test for Local Extrema

P205 / 206 for graphs

The following test applies to a continuous function f(x).

At a critical point c:

1. If f’ changes sign from positive to negative at c, then f has a local maximum value at c.

2. If f’ changes sign from negative to positive at c, then f has a local minimum value at c.

3. If f’ does not change sign at c, then f has no local extreme value at c.

1st Derivative Test Continued…

At a Left Endpoint a:

If f ’ < 0, for x near a, then f has a local maximum at a.

If f ’ > 0, for x near a, then f has a local minimum at a.

At a Right Endpoint b:

If f ’ < 0, for x near b, then f has a local minimum at b.

If f ’ > 0, for x near b, then f has a local maximum at b.

Concavity Test

The graph of a twice-differentiable function y = f(x) is

a) Concave up on any interval where y” > 0.

b) Concave down on any interval where y” < 0.

Example 2: Use the Concavity Test to determine the concavity of the given functions on the given intervals.

)10,3(,2xy )2,0(,sin3 xy

Point of Inflection (reminder)

A point where the graph of a function has a tangent line

(derivative exists & function is continuous)

and where the concavity changes is a Point of Inflection.

f” = 0

Example 3 Finding Points of Inflection

Find all points of inflection of the graph of

HOW?Find the 2nd derivative – recalling product and chain rules.

Find critical points of 2nd derivative, set y” = 0 and solve for x(these are the x values of your points of inflection)

Evaluate f(critical points) to find the y values of your points of inflection. (use original function!)

2xey

Example 4 Reading the Graph of the DerivativePositive Negative Zero

f ’(x) f is increasing f is decreasing x is an extrema

f ”(x) f ’ increasing, f is concave up

f ’ decreasing, f is concave down

x might be a point of inflection – check it

Caution: It is tempting to oversimplify a point of inflection as a point where the second derivative is zero, but that can be wrong for 2 reasons.

1. The 2nd derivative can be zero at a non-inflection point. Consider f(x) = x4 and figure 4.27 on p 209

2. The 2nd derivative need not be zero at an inflection point. Consider f(x) = cube root of x. At x = 0 there is a vertical tangent and both f’ and f” fail to exist although there is a change in concavity.

3. The safest way to test for a point of inflection is to confirm algebraically that there was a sign change in the 2nd derivative!

Given the graph of the derivative function on p209, figure 4.25, answer the following questions about f, justifying

each answer with information obtained from the graph of f’.

1. On what intervals is f increasing? (f’ +)

2. On what intervals is the graph of f concave up? (f” +)

3. At which x-coordinates does f have local extrema? (f’ = 0)

4. What are the x-coordinates of all inflection points of the graph of f? (f” = 0)

5. Use info from 1-4 to sketch a possible graph of f on the interval [-4,4].

Example 5 Studying Motion along a Line

A particle is moving along the x-axis with position function , t ≥ 0.

Find the velocity and acceleration – take first and 2nd derivative

Describe the motion of the particle – use derivatives f ’ positive: increasing, right / f ’ negative: decreasing, left

f ” positive: concave up, accelerating f ” negative: concave down, decelerating

522142)( 23 ttttx

Theorem 5 Second Derivative Test for Local Extrema

1. If f’(c) = 0 and f”(c) < 0, then f has a local maximum at x = c.

2. If f’(c) = 0 and f”(c) > 0, then f has a local minimum at x = c.

Good news: Only need to know f’ and f” at c itself so easy to apply.

Bad news: Test fails if f”(c) = 0 or f”(c) fails to exist. When that happens we revert to the 1st Derivative test.

Example 7: Using the 2nd Derivative Test

Find the local extreme values of .

Find f’Find f”Find the critical points (f’(x) = 0) and test them with f”.

Give your conclusions in sentences.

512)( 3 xxxf

Example 8 Using f’ and f” to Graph f

Let

a) Identify where the extrema of f occur.

b) Find the intervals on which f is increasing and the intervals on which f is decreasing.

c) Find where the graph of f is concave up and where it is concave down.

d) Sketch a possible graph for f.

23 124)(' xxxf

Learning about Functions from Derivatives

We are able to recover almost everything we need to know about a differentiable function y = f(x) by examining y’ and y”. The only info we can’t get from the derivative is where to place the graph on the coordinate plane, we need one point from f(x) to position the graph.

Study the graphs on p213 to further develop an eye for finding f(x) from its derivative.

Also, interactive derivatives athttp://www.slu.edu/classes/maymk/GeoGebra/Derivatives.html

Example 9: Analyzing a Discontinuous Derivative

A function f is continuous on the interval [-4,4]. The discontinuous function f ’, with domain [-4,0) U (0,2) U (2,4], is shown in figure 4.33 on page 214.

a) Find the x-coordinates of all local extrema ( where f ’ changes sign) and points of inflection (where f ” changes sign so f ’ changes

direction) of f.

b) Sketch a possible graph of f.

Homework

Summary

AP Test Practice P121 thru 124 (due Friday)

Homework

Page 216

Exercises 18-48 (3n, nЄI)

Study for quiz on 4.1 thru 4.3