ALTERNATE DEFINITION OF A DERIVATIVE...' 2 4 EXAMPLE 1 Use the alternative form to find slope of the...

ALTERNATE DEFINITION OF A DERIVATIVE

Section 2.1A

Calculus AP/Dual, Revised ©2018

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7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 1

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DEFINITION

A. Alternate form of the Derivative Equation: 𝒇′ 𝒄 = 𝐥𝐢𝐦𝒙→𝒄

𝒇 𝒙 −𝒇 𝒄

𝒙−𝒄

B. This equation will allow the showing of the slope of the tangent line at a particular point

C. When investigating the relationship between differentiability and continuity

D. The existence of this limit requires that the one-sided limits exist and

are equal: 𝐥𝐢𝐦𝒙→𝒄−


𝒙−𝒄= 𝐥𝐢𝐦

𝒙→𝒄+


𝒙−𝒄


REVIEW

Find the slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓 − 𝒙𝟐, at the given point 𝟐, 𝟏


0' lim

h

f x h f xf x

h

2 2

0

5 5' lim

h

x h xf x

h

2 2 2

0

5 2 5' lim

h

x xh h xf x

h

0

2' lim

h

h x hf x

h

2x

REVIEW

Find the slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓 − 𝒙𝟐, at the given point 𝟐, 𝟏


0

2' lim

h

h x hf x

h

2x

' 2f x x

' 2 2 2f

' 2 4f

EXAMPLE 1

Use the alternative form to find slope of the tangent line to the graph of the function 𝒇 𝒙 = 𝟓 − 𝒙𝟐, at the given point 𝟐, 𝟏


' limx c

f x f cf c

x c

2

2

5 1' 2 lim

2x

xf

x

2

2

4' 2 lim

2x

xf

x

2

2

4lim

2x

x

x

EXAMPLE 1

Use the alternative form to find slope of the tangent line to the graph of the function 𝒇 𝒙 = 𝟓 − 𝒙𝟐, at the given point 𝟐, 𝟏


2

2' 2 lim

2x

f x ff

x

2

2 2lim

2x

x x

x

2

lim 2x

x

' 2 4f

2

lim 2 2x

EXAMPLE 2

Use the alternative form to find the derivative of 𝒇 𝒙 =𝟏

𝒙when 𝒄 = 𝟒


' limx c

f x f cf c

x c

4

1 1

4' 4 lim4x

xfx

4

4 1' 4 lim

4 4x

xf

x x

4

4' 4 lim

4 4x

xf

x x

EXAMPLE 2

Use the alternative form to find the derivative of 𝒇 𝒙 =𝟏

𝒙when 𝒄 = 𝟒


4

4 1' 4 lim

4 4x

xf

x x

4

1lim

4x x

1

' 416

f

1

4

4' 4 lim

4 4x

xf

x x

EXAMPLE 3

Use the alternative form to find the derivative of 𝒇 𝒙 = 𝒆𝒙 + 𝟐𝒙when 𝒄 = 𝟎


' 0 2f

YOUR TURN

Use the alternative form to find the derivative of 𝒇 𝒙 = 𝒙𝟐 + 𝟑, when 𝒄 = −𝟐


' 2 4f

WHEN I SAY DERIVATIVE, YOU SAY…


GRAPHING A DERIVATIVE

A. When looking at a graph, identify what type of slope that each point has

1. If 𝒇 has a positive slope, 𝒇′ is increasing

2. If 𝒇 has a negative slope, 𝒇′ is decreasing

B. After establishing all of the slopes, determine the derivative graph by its slope.

C. In other words, plot points of the derivative onto the graph


STEPS

A. Establish any slopes of zero

B. Establish any steep and flat the slopes are

C. Plot the points


EXAMPLE 4

Sketch the derivative of the function given.


f x

'f x

x axis


EXAMPLE 5


'f x

x axis

f x

EXAMPLE 6



1

2Slope

NoSlope

1

2Slope

1Slope

1Slope

EXAMPLE 6



0

f x

+ +0

'f x

Increases Increases

YOUR TURN



Negative Slope

Positive Slope

Negative Slope

Positive Slope

f x

DIRECTION OF A DERIVATIVE FROM A GRAPH

A. A graph which is differentiable is continuous

B. A graph which is continuous, is not always differentiable

C. A graph is neither continuous or differentiable is discontinuous

1. Hole

2. Vertical Asymptotes

3. Jump discontinuities

4. Kinks/Cusps


SPECIAL CASES

A. Differentiable functions are Continuous but continuous functions are not always Differentiable

B. Functions can not be differentiable when:

1. Sharp turns or “cusps” (i.e. 𝒇(𝒙) = |𝒙 − 𝟐|)

2. Vertical Tangents (i.e. 𝒇 𝒙 = 𝒙𝟏/𝟑)

3. Discontinuity

C. A function 𝒇 is said to be differentiable at 𝒙 = 𝒄 if the derivative of 𝒇 exists at 𝒙 = 𝒄. For the derivative to exist at 𝒙 = 𝒄, the graph of 𝒇 must have local linearity.


EXAMPLE 7

Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 − 𝟐 . If it does not exist, explain why.


1

2lim 2x

x

2

2lim

2x

f x f

x

2

2 2 2lim

2x

x

x

2

2lim

2x

x

x

2

1.999 2lim

1.999 2x

EXAMPLE 7



1

2lim 2x

x

2

2lim

2x

f x f

x

2

2 2 2lim

2x

x

x

2

2lim

2x

x

x

2

2.001 2lim

2.001 2x

EXAMPLE 7



2 2lim 2 lim 2x x

x x

2

1m

21m

𝒇 is not differentiable at 𝒙 = 𝟐 because 𝒇has a sharp turn at 𝒙 = 𝟐.

EXAMPLE 8

Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙𝟏/𝟑 at 𝒙 = 𝟎. If it does not exist, explain why.


0

0lim

0x

f x f

x

1/3

0

0lim

0x

x

x

1/3

0limx

x

x

2/30

1limx x

0

1limx

𝒇 is not differentiable at 𝒙 = 𝟐 because

𝒇 has a sharp turn at 𝒙 = 𝟐.1/3

0

;

lim DNEx

m

x

EXAMPLE 9

Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 − 𝟏 𝟐/𝟑. If it does not exist, explain why.


𝒇 is not differentiable at 𝒙 = 𝟏because 𝒇 has a cusp at 𝒙 = 𝟏.

Let 𝒇 be a function such that 𝐥𝐢𝐦𝒉→𝟎

𝒇 𝟐+𝒉 −𝒇 𝟐

𝒉= 𝟓. Which of the following must be

true?I. 𝒇 is continuous at 𝒙 = 𝟐

II. 𝒇 is differentiable at 𝒙 = 𝟐

III. The derivative of 𝒇 is continuous at 𝒙 = 𝟐

(A) I only

(B) II only

(C) I and II only

(D) II and III only


AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR)

Let 𝒇 be a function such that 𝐥𝐢𝐦𝒉→𝟎

𝒇 𝟐+𝒉 −𝒇 𝟐

𝒉= 𝟓. Which of the following must be

true?


AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR)

Vocabulary Connections and Process Answer and Justifications

CContinuity

Differentiable

limx c

f x f c

x c

0

2 2lim

2 2h

f h f

h

0

2 2lim ' 2 5

2 2h

f h ff

h

is Differentiable and therefore,

Continuous at 2

f x

x

Don't Know if ' is continuous

at 2. Not enough info given.

f x

x

Since is differentiable at 2,

therefore we can imply that is

continuous at 2. However, we do

not know the continuity of ' 2 .

f x x

f x

x

f

ASSIGNMENT

Worksheet


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