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ALTERNATE DEFINITION OF A DERIVATIVE
Section 2.1A
Calculus AP/Dual, Revised ©2018
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 1
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−
𝒇 𝒙 −𝒇 𝒄
𝒙−𝒄= 𝐥𝐢𝐦
𝒙→𝒄+
𝒇 𝒙 −𝒇 𝒄
𝒙−𝒄
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 2
REVIEW
Find the slope of the tangent line to the graph of the function 𝒇 𝒙 =𝟓 − 𝒙𝟐, at the given point 𝟐, 𝟏
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 3
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 4
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 5
' 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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 6
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 7
' 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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 8
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 9
' 0 2f
YOUR TURN
Use the alternative form to find the derivative of 𝒇 𝒙 = 𝒙𝟐 + 𝟑, when 𝒄 = −𝟐
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 10
' 2 4f
WHEN I SAY DERIVATIVE, YOU SAY…
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 11
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 12
STEPS
A. Establish any slopes of zero
B. Establish any steep and flat the slopes are
C. Plot the points
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 13
EXAMPLE 4
Sketch the derivative of the function given.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 14
f x
'f x
x axis
Sketch the derivative of the function given.
EXAMPLE 5
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 15
'f x
x axis
f x
EXAMPLE 6
Sketch the derivative of the function given.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 16
1
2Slope
NoSlope
1
2Slope
1Slope
1Slope
EXAMPLE 6
Sketch the derivative of the function given.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 17
0
f x
+ +0
'f x
Increases Increases
YOUR TURN
Sketch the derivative of the function given.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 18
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 19
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.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 20
EXAMPLE 7
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 − 𝟐 . If it does not exist, explain why.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 21
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
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 − 𝟐 . If it does not exist, explain why.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 22
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
Use the graph to determine the derivative of 𝒇 𝒙 = 𝒙 − 𝟐 . If it does not exist, explain why.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 23
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.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 24
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.
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 25
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 26
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?
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 27
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
7/30/2018 12:39 AM §2.1A: Alternate Definition of a Derivative 28