chapter 3 limits and the derivative section 4 the derivative (part 1)
TRANSCRIPT
Chapter 3
Limits and the Derivative
Section 4
The Derivative
(Part 1)
2Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 3.4 The Derivative
β Part One
β The student will be able to:
β Calculate slope of the secant line.
β Calculate average rate of change.
β Calculate slope of the tangent line.
β Calculate instantaneous rate of change.
3
Introduction
In Calculus, we study how a change in one variable affects another variable.
In studying this, we will make use of the limit concepts we learned in the previous lessons of this chapter.
Barnett/Ziegler/Byleen Business Calculus 12e
4
Slopes
Slope of a secant Slope of a tangent
Barnett/Ziegler/Byleen Business Calculus 12e
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Example 1Revenue Analysis
The graph below shows the revenue (in dollars) from the sale of x widgets.
When 100 widgets are sold, the revenue is $1800. If we increase production by an additional 300 widgets, the
revenue increases to $4800.
Barnett/Ziegler/Byleen Business Calculus 12e
π (π₯ )=20 π₯β 0.02π₯2
6
Example 1 (continued)
When production increases from 100 to 400 widgets the change in revenue is:
= $3000 The average change in revenue is:
So the average change in revenue is $10 per widget when production increases from 100 to 400 widgets.
Barnett/Ziegler/Byleen Business Calculus 12e
π (400 )βπ (100)400 β100
=$ 3000
300π€πππππ‘π =$ 10ππππ€πππππ‘
7
Rate of Change
This is an example of the βrate of changeβ concept. The average rate of change is the ratio of the change in y
over the change in x. You know this as the βslopeβ between two points.
Barnett/Ziegler/Byleen Business Calculus 12e
8Barnett/Ziegler/Byleen Business Calculus 12e
The Rate of Change
For y = f (x), the average rate of change from x = a to x = a + h is
0,)()(
hh
afhaf
The above expression is also called a difference quotient. It can be interpreted as the slope of a secant.
See the picture on the next slide for illustration.
9Barnett/Ziegler/Byleen Business Calculus 12e
Graphical Interpretation
Average rate of change = slope of the secant line
π¦2 βπ¦ 1
π₯2 βπ₯1
=π (π+h ) β π (π)
(π+h ) βπ
ΒΏπ (π+h ) β π (π)
h
10
What ifβ¦
Suppose the 2nd point (a+h, f(a+h)) gets closer and closer to the first point (a, f(a)). What happens to the value of h?
Barnett/Ziegler/Byleen Business Calculus 12e
Answer: h approaches zero
11Barnett/Ziegler/Byleen Business Calculus 12e
The Instantaneous Rate of Change
If we find the slope of the secant line as h approaches zero, thatβs the same as the limit shown below.
Now, instead of the average rate of change, this limit gives us the instantaneous rate of change of f(x) at x = a.
And instead of the slope of a secant, itβs the slope of a tangent.
h
afhafh
)()(lim
0
12Barnett/Ziegler/Byleen Business Calculus 12e
Visual Interpretation
h
afhaf
h
)()(
0
lim
Slope of tangent at x = a is theinstantaneous rate of change.
Tangent line at x=a
13Barnett/Ziegler/Byleen Business Calculus 12e
Given y = f (x), the instantaneous rate of change at x = a is
provided that the limit exists. It can be interpreted as the slope of the tangent at the point (a, f (a)).
If the slope is positive, then is increasing at x=a.
If the slope is negative, then is decreasing at x=a.
h
afhafh
)()(lim
0
Instantaneous Rate of Change
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Example 3A
Find the avg. rate of change if x changes from 1 to 4.
Answer:
This is equal to the slope of the secant line through (1, 3) and (4, 0).
Barnett/Ziegler/Byleen Business Calculus 12e
π ( 4 )β π (1)4 β1
=0 β 3
3=β 1
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Example 3B
Find the instantaneous rate of change of f(x) at x = 1
Barnett/Ziegler/Byleen Business Calculus 12e
limh β0
π (1+h )β π (1)h ΒΏ
limh β 0
[4 (1+h ) β (1+h )2 ] β(4 β 12)
h
ΒΏlimh β 0
[4+4hβ(1+2h+h2)]β 3
h
ΒΏlimh β 0
[4+4 h β1 β2 hβ h2 ]β 3
h
ΒΏlimh β 0
2 hβ h2
h
Continued on next slideβ¦
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Example 3B - continued
Find the instantaneous rate of change of f(x) at x = 1
This is equal to the slope of the tangent line at x=1.
Barnett/Ziegler/Byleen Business Calculus 12e
ΒΏ limhβ0
(2β h)
ΒΏ2
ΒΏlimh β 0
2 hβ h2
h
ΒΏlimh β 0
h(2β h)
h
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Example 3C
Find the equation of the tangent line at x=1.
When x=1, y=3 and slope = 2
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=ππ₯+π3=2(1)+ππ=1π¦=2 π₯+1
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Application - Velocity
A watermelon that is dropped from the Eiffel Tower will fall a distance of y feet in x seconds.
Find the average velocity from 2 to 5 seconds.β’ Answer:
Barnett/Ziegler/Byleen Business Calculus 12e
π¦=16 π₯2
400 β 645 β 2
π (5 )β π (2)5 β2
=ΒΏ
ΒΏ336 ππ‘3π ππ
=112 ππ‘ /π ππ
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Velocity(continued)
Find the instantaneous velocity at x = 2 seconds.
Barnett/Ziegler/Byleen Business Calculus 12e
πΌππ π‘πππ‘ .π£ππ .=limh β0
π (2+h ) β π (2)h
ΒΏ limhβ0
(64+16 h)
ΒΏ64 ππ‘ /π ππ
ΒΏlimh β 0
16(2+h)2β 16(2)2
h
ΒΏlimh β 0
16(4+4 h+h2)β16 (4)
h
ΒΏlimh β 0
64+64 h+16 h2 β64
h
ΒΏlimh β 0
h(64+16 h)
h
π¦=16 π₯2
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Summary
Slope of a secant Average rate of change Average velocity
Slope of a tangent Instantaneous rate of
change Instantaneous velocity
Barnett/Ziegler/Byleen Business Calculus 12e
π (π+h )β π (π)h
limh β 0
π (π+h )β π (π)
h
21
Homework
#3-4A
Pg 175
(1-4, 27-30)
Barnett/Ziegler/Byleen Business Calculus 12e
Chapter 3
Limits and the Derivative
Section 4
The Derivative
(Part 2)
23Barnett/Ziegler/Byleen Business Calculus 12e
Learning Objectives for Section 3.4 The Derivative
β Part Two
β The student will be able to:
β Calculate the derivative.
β Identify the nonexistence of a derivative.
24
Introduction
In Part 1, we learned that the limit of a difference quotient, can be interpreted as:β’ instantaneous rate of change at x=aβ’ slope of the tangent line at x=aβ’ instantaneous velocity at x=a
In this part of the lesson, we will take a closer look at this limit where we replace a with x.
Barnett/Ziegler/Byleen Business Calculus 12e
limh β0
π (π+h )β π (π)h
25Barnett/Ziegler/Byleen Business Calculus 12e
The Derivative
For y = f (x), we define the derivative of f at x, denoted f (x), to be
if the limit exists.
I refer to as a βslope machineβ. It will allow me to find the slope at any x value.
f (x) lim
h 0
f (x h) f (x)
h
26Barnett/Ziegler/Byleen Business Calculus 12e
Same Meaning as Before
If f is a function, then f has the following interpretations:
β For each x in the domain of f , f (x) is the slope of the line tangent to the graph of f at the point (x, f (x)).
β For each x in the domain of f , f (x) is the instantaneous rate of change of y = f (x) with respect to x.
β If f (x) is the position of a moving object at time x, then v = f (x) is the instantaneous velocity of the object with respect to time.
27Barnett/Ziegler/Byleen Business Calculus 12e
Finding the Derivative
To find f (x), we use a four-step process:
Step 1. Find f (x + h)
Step 2. Find f (x + h) β f (x)
Step 3. Find
Step 4. Find f (x) =
h
xfhxf )()(
h
xfhxfh
)()(lim
0
*Feel free to go directly to Step 4 when youβve got the process down!
28Barnett/Ziegler/Byleen Business Calculus 12e
Find the derivative of f (x) = x 2 β 3x.
Step 1: Find f(x+h)
Step 2: Find f(x+h) β f(x)
Step 3: Find
Example 1
ΒΏ (π₯+h)2β3 (π₯+h)ΒΏ π₯2+2 hπ₯ +h2 β3 π₯β3 h
ΒΏ π₯2+2 hπ₯ +h2 β3 π₯β3 hβ(π₯ΒΏΒΏ2 β3 π₯)ΒΏΒΏ π₯2+2 hπ₯ +h2 β3 π₯β3 hβ π₯2+3 π₯ΒΏ2 hπ₯ +h2β 3 h
ΒΏ 2 hπ₯ +h2 β3 h h
ΒΏ2 π₯+hβ3
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Example 1 (continued)
Step 4: Find f (x) =
Barnett/Ziegler/Byleen Business Calculus 12e
limh β0
π (π₯+h ) β π (π₯)h
π (π₯)=limh β 0
(2 π₯+hβΒΏ3)ΒΏ
π (π₯)=2π₯β3
For x=a, where a is in the domain of f(x),f (a) is the slope of the line tangent to f(x) at x=a.
Find the slope of the line tangent to the graph of f (x) at x = 0, x = 2, and x = 3. f (0) = -3
f (2) = 1
f (3) = 3
30Barnett/Ziegler/Byleen Business Calculus 12e
Find f (x) where f (x) = 2x β 3x2 using the four-step process.
Step 1: Find f(x+h)
Step 2: Find f(x+h) β f(x)
Step 3: Find
Example 2
ΒΏ2 (π₯+h ) β3 (π₯+h)2
ΒΏ2 π₯+2hβ 3(π₯2+2 hπ₯ +h2)ΒΏ2 π₯+2 hβ 3 π₯2 β6 hπ₯ β3 h2
ΒΏ2 π₯+2hβ 3 π₯2 β6 hπ₯ β3 h2β(2 π₯β 3π₯2)ΒΏ2 π₯+2 hβ 3 π₯2 β6 hπ₯ β3 h2β 2π₯+3 π₯2
ΒΏ2 h β6 hπ₯ β 3 h2
ΒΏ 2 hβ6 hπ₯ β3 h2 h
ΒΏ2 β6 π₯β3 h
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Example 2 (continued)
Step 4: Find f (x) =
Barnett/Ziegler/Byleen Business Calculus 12e
limh β0
π (π₯+h ) β π (π₯)h
π (π₯)=limh β 0
(2 β6 π₯β3 hΒΏ)ΒΏ
π (π₯ )=2β 6 π₯
Find the slope of the line tangent to the graph of f (x) at x = -2, x = 0, and x = 1.
f (-2) = 14
f (0) = 2
f (1) = -4
32Barnett/Ziegler/Byleen Business Calculus 12e
Find f (x) where using the four-step process.
Step 1: Find f(x+h)
Step 2: Find f(x+h) β f(x)
Step 3: Find
Example 3
ΒΏβπ₯+h+2ΒΏβπ₯+h+2β(βπ₯+2)ΒΏβπ₯+hββπ₯ΒΏ βπ₯+hββπ₯
h
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Example 3 (continued)
Step 4: Find f (x) =
Barnett/Ziegler/Byleen Business Calculus 12e
limh β0
π (π₯+h ) β π (π₯)h
ΒΏ1
(βπ₯+βπ₯ )
π (π₯ )= 12βπ₯
β βπ₯βπ₯
=βπ₯2π₯
ΒΏ1
2βπ₯
f ( x )=limhβ 0
βπ₯+h ββπ₯h
β βπ₯+h+βπ₯βπ₯+h+βπ₯
f ( x )=limhβ 0
π₯+hβπ₯h (βπ₯+h+βπ₯ )
ΒΏ limhβ0
h
h (βπ₯+h+βπ₯ )f ( x )=lim
hβ 0
1(βπ₯+h+βπ₯ )
34Barnett/Ziegler/Byleen Business Calculus 12e
Nonexistence of the Derivative
The existence of a derivative at x = a depends on the existence of the limit
If the limit does not exist, we say that the function is nondifferentiable at x = a, or f (a) does not exist.
f (a) lim
h 0
f (a h) f (a)
h
35Barnett/Ziegler/Byleen Business Calculus 12e
Nonexistence of the Derivative(continued)
Some of the reasons why the derivative of a function may not exist at x = a are
β The graph of f has a hole or break at x = a, or
β The graph of f has a sharp corner at x = a, or
β The graph of f has a vertical tangent at x = a.
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Examples of Nonexistent Derivatives
Barnett/Ziegler/Byleen Business Calculus 12e
In each graph, f is nondifferentiable at x=a.
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Application β Profit
The profit (in dollars) from the sale of x infant car seats is given by: 0 x 2400
A) Find the average change in profit if production increases from 800 to 850 car seats.
B) Use the 4-step process to find P(x)
C) Find P(800) and P(800) and explain their meaning.
Barnett/Ziegler/Byleen Business Calculus 12e
38
Application β Profit(continued)
The profit (in dollars) from the sale of x infant car seats is given by: 0 x 2400
A) Find the average change in profit if production increases from 800 to 850 car seats.
Barnett/Ziegler/Byleen Business Calculus 12e
π΄π£π hπ ππππππππππππ‘=π (850 ) βπ (800)
850 β800
ΒΏ15187.5 β15000
50ΒΏ3.75
The avg change in profit when production increases from 800 to 850 car seats is $3.75 per seat.
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Application β Profit(continued)
The profit (in dollars) from the sale of x infant car seats is given by: 0 x 2400
B) Use the 4-step process to find P(x)
Barnett/Ziegler/Byleen Business Calculus 12e
πΊππππ :π (π₯+h )=45 (π₯+h ) β0.025 (π₯+h )2 β5000
ΒΏ 45 π₯+45 h β 0.025 (π₯2+2 hπ₯ +h2 ) β5000
ΒΏ 45 π₯+45 h β 0.025π₯2 β0.05 hπ₯ β0.025 h2β 5000πΊππππ :π (π₯+h )β π (π₯)
ΒΏ 45 h β0.05 hπ₯ β 0.025 h2
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Application β Profit(continued)
Barnett/Ziegler/Byleen Business Calculus 12e
πΊππππ :π (π₯+h ) βπ (π₯)
hΒΏ 45 hβ 0.05 hπ₯ β0.025h2
hΒΏ 45 β0.05 π₯β 0.025 h
πΊππππ : limh β 0
π (π₯+h )βπ (π₯ ) h
ΒΏ limhβ0
(45β 0.05 π₯β0.025 h)
ΒΏ 45 β .05 π₯
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Application β Profit(continued)
The profit (in dollars) from the sale of x infant car seats is given by: 0 x 2400
C) Find P(800) and P(800) and explain their meaning.
Barnett/Ziegler/Byleen Business Calculus 12e
π (800 )=45 (800 ) β0.025 (800 )2β 5000π (800 )=$ 15,000π (800 )=45 β 0.05(800)π (800 )=$ 5
At a production level of 800 car seats, the profit is $15,000and it is increasing at a rate of $5 per car seat.
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Homework
Barnett/Ziegler/Byleen Business Calculus 12e
#3-4B Pg 176(7, 21, 25,
31-41, 61, 63)