Using Derivatives to Sketch the Graph of a Function

Lesson 4.3

How It Was Done BC(Before Calculators)

• How can knowledge of a function and it's derivative help graph the function?

• How much can you tell about the graph of a function without using your calculator's graphing?

Regis might be calling for this information!

Regis might be calling for this information!

Increasing/Decreasing Functions

• Consider the following function

• For all x < a we note that x1<x2 guarantees that f(x1) < f(x2)

f(x)

a

The function is said to be strictly increasing

The function is said to be strictly increasing

Increasing/Decreasing Functions

• Similarly -- For all x > a we note that x1<x2 guarantees that f(x1) > f(x2)

• 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

The function is said to be strictly

decreasing

Monotone Function Theorem

• 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

Monotone Function Theorem

• 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

31( ) 9 2

3f x x x

Monotone Function Theorem

• Determine f ‘(x)

• Note graphof f’(x)

• Where is it pos, neg

• What does this tell us about f(x)f ‘(x) > 0 => f(x) increasing f ‘(x) > 0 => f(x) increasingf ‘(x) < 0 => f(x) decreasing

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) < 0on left

f ‘(x) > 0on right

First Derivative Test

• Similarly, if f ‘(x) > 0 on left, f ‘(x) < 0 on right,

• We have a relative maximum

First Derivative Test

• What if they are positive on both sides of the point in question?

• This is called aninflection point

Examples

• Consider the following function

• Determine f ‘(x)

• Set f ‘(x) = 0, solve

• Find intervals

2 2( ) (2 1) ( 9)f x x x

Concavity

• Concave UP

• Concave DOWN

• Inflection point:Where concavitychanges

Inflection Point

• Consider the slope as curve changes through concave up to concave down

Slope starts

negative

Becomes less negative

Slope becomes (horizontal) zero

Slope becomes positive, then more positive

At inflection point slope reaches

maximum positive value

After inflection point, slope

becomes less positive

Graph of the slope

Inflection Point

• What could you say about the slope function when the original function has an inflection point

Graph of the slope

• Slope function has a maximum (or minimum

• Thus second derivative = 0

• Slope function has a maximum (or minimum

• Thus second derivative = 0

Second Derivative

• This is really the rate of change of the slope

• When the original function has a relative minimum– Slope is increasing (left to right) and goes

through zero– Second derivative is positive– Original function is concave up

Second Derivative

• When the original function has a relative maximum– The slope is decreasing (left to right) and

goes through zero– The second derivative is negative– The original function is

concave down

Second Derivative

• If the second derivative f ’’(x) = 0– The slope is neither increasing nor decreasing

• If f ’’(x) = 0 at the same place f ’(x) = 0– The 2nd derivative test fails– You cannot tell what the function is doing

4( )f x x2''( ) 12f x x

Not an inflection point

Not an inflection point

Example

• Consider

• Determine f ‘(x) and f ’’(x) and when they are zero

3( ) 3 4f x x x

2'( ) 3 3 0 when 1

''( ) 6 0 when 0

f x x x

f x x x

Example

f(x)

f ‘(x)

f ‘’(x)

f ‘’(x) = 0 this is an inflection point

f ‘(x) = 0, f ‘’(x) < 0this is concave

down, a maximum

f ’(x) = 0, f’’(x) > 0, this is concave up, a

relative minimum

3( ) 3 4f x x x

Example

• Try

• f ’(x) = ?

• f ’’(x) = ?

• Where are relative max, min, inflection point?

2( ) 1f x x

Algorithm for Curve Sketching

• Determine critical points – Places where f ‘(x) = 0

• Plot these points on f(x)

• Use second derivative f’’(x) = 0– Determine concavity, inflection points

• Use x = 0 (y intercept)

• Find f(x) = 0 (x intercepts)

• Sketch

Assignment

• Lesson 4.3

• Page 214

• Exercises 9 – 43 odd, 47, 49, 53