business calculus
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calculusTRANSCRIPT
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Section 2.6: The Derivative Business Calculus - p. 1/44
Business CalculusMath 1431
Unit 2.6The DerivativeMathematics Department
Louisiana State University
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 2/44
Introduction
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Introduction
In an earlier lecture we considered the idea offinding the average rate of change of some quan-tity Q(t) over a time interval [t1, t2] and askedwhat would happen when the time interval becamesmaller and smaller. This led to the notion of limits,which we have examined in the previous two lec-tures. In this section we return to our study of av-erage rates of change and apply what we learnedabout limits. This leads directly to the derivative.It turns out that the derivative has an equivalent(purely) mathematical formulation: that of findingthe line tangent to a curve at some point. This iswhere we will start.
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The Tangent Problem
Suppose y = f(x) is some given function and a issome fixed point in the domain. We will explore theidea of finding the equation of the line tangent tothe graph of f at the point (a, f(a)). Look at thepicture below:
a
f(a)
The tangent line at x = a is the unique line thatgoes through (a, f(a)) and just touches the graphthere.
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The Slope?
Recall the an equation of the line is determinedonce we know a point P and the slope m. IfP = (a, f(a)) then the equation of the line tangentto the curve will take the form
y − f(a) = m(x − a)
for some slope m that we have to determine.
How do we determine theslope of the tangent line?
We will do so by a limiting process.
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Secant lines
Given the graph of a function y = f(x) we call asecant line a line that connects two points on agraph.
In the graph to the right a redline joins the points (a, f(a)) and(a+h, f(a+h). ( Think of h as arelatively small number.) a
f(a)
a + h
f(a + h)
Now, what is the slope of the secant line? Thechange in y is ∆y = f(a+h)−f(a) and the changein x is ∆x = a + h − a = h. So the slope of thesecant line is:
∆y
∆x=
f(a + h) − f(a)
h.
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
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Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 9: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/9.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 10: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/10.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 15: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/15.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 16: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/16.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
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IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 18: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/18.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 19: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/19.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 20: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/20.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 21: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/21.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 22: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/22.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 23: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/23.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 24: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/24.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 25: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/25.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 26: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/26.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 27: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/27.jpg)
IntroductionIntroductionThe TangentProblemThe Slope?Secant linesSmall hThe main ideaAn animationSlope of TangentExample
Section 2.6: The Derivative Business Calculus - p. 7/44
Small h
The following graph illustrates what happens whenwe choose smaller values of h. Notice how the se-cant lines get closer to the tangent line.
a
f(a)
![Page 28: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/28.jpg)
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The main idea
The main idea is to let h get smaller and smaller.Remember all we need is the slope of the tangentline so we will compute
limh→0
f(a + h) − f(a)
h.
In the following animation notice how the slopes ofthe secant lines approach the slope of the tangentline.
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Convergence of Secant Lines: An animation
Notice how the secant line and hence its slopes converge to the
tangent line.
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Slope of the Tangent Line
Since the slope of the secant lines are given by theformula
f(a + h) − f(a)
h,
the slope of the tangent line is given by
limh→0
f(a + h) − f(a)
h,
when this exists. We will call this number thederivative of f at a and denote it f ′(a). Thus
The derivative of f at x = a is given by
f ′(a) = limh→0
f(a + h) − f(a)
h.
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Example
To illustrate some of these ideas let’s consider thefollowing example:
Example 1: Let f(x) = x2 and fix a = 1. Computethe slope of the secant line that connects (a, f(a))and (a + h, f(a + h) for■ h = 1
■ h = .5
■ h = .1
■ h = .01
Compute the derivative of f at x = 1, i.e. f ′(1).Finally, find the equation of the line tangent to thegraph of f(x) = x2 and x = 1.
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f(x) = x2
Let mh be the slope of the secant line. Then
mh =f(1 + h) − f(1)
h=
(1 + h)2 − 1
h.
■ For h = 1 m1 = (1+1)2−11
= 4 − 1 = 3.
■ For h = .5 m.5 = (1.5)2−1.5
= 2.5.
■ For h = .1 m.1 = (1.1)2−1.1
= 2.1
■ For h = .01 m.01 = (1.01)2−a
.01= 2.01
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f(x) = x2
Let mh be the slope of the secant line. Then
mh =f(1 + h) − f(1)
h=
(1 + h)2 − 1
h.
■ For h = 1 m1 = (1+1)2−11
= 4 − 1 = 3.
■ For h = .5 m.5 = (1.5)2−1.5
= 2.5.
■ For h = .1 m.1 = (1.1)2−1.1
= 2.1
■ For h = .01 m.01 = (1.01)2−a
.01= 2.01
What do you think the limit willbe as h goes to 0?
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f(x) = x2
The limit is the derivative. We compute
f ′(1) = limh→0
f(a + h) − f(a)
h
= limh→0
(1 + h)2 − 1
h
= limh→0
1 + 2h + h2 − 1
h
= limh→0
2h + h2
h= lim
h→02 + h = 2.
Therefore the slope of the tangent line is m = 2.
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f(x) = x2
The equation of the tangent line is now an easymatter. The slope m is 2 and the point P that theline goes through is (1, f(1)) = (1, 1). Thus, we get
y − 1 = 2(x − 1)
ory = 2x − 1.
On the next slide we give a graphical representa-tion of what we have just computed.
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f(x) = x2
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f(x) = x2
Here h = 1 and the secant line goes through thepoints (1, 1) and (2, 4)
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f(x) = x2
Here h = .5 and the secant line goes through thepoints (1, 1) and (1.5, 2.25)
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f(x) = x2
Here h = .1 and the secant line goes through thepoints (1, 1) and (1.1, 1.21)
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f(x) = x2
Here h = .01 and the secant line goes through thepoints (1, 1) and (1.01, 1.0201)
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Rates of ChangeRates of changeVelocityExample
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Rates of Change
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Rates of ChangeRates of changeVelocityExample
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Rates of change
Given a function y = f(x) the difference quotient
f(x + h) − f(x)
h
measures the average rate of change of y withrespect to x over the interval [x, x + h]. As theinterval becomes smaller, i.e. as h goes to 0, weobtain the instantaneous rate of change
limh→0
f(x + h) − f(x)
h,
which is precisely our definition of the derivativef ′(x). Thus, the derivative measures in an instantthe rate of change of f(x) with respect to x.
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Rates of ChangeRates of changeVelocityExample
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Velocity
If s(t) is the distance travelled by an object (yourcar for instance) as a function of time t then thequantity
s(t + h) − s(t)
h
is the average rate of change of distance over thetime interval [t, t + h]. This is none other than youraverage velocity. The quantity
limh→0
s(t + h) − s(t)
h
is the instantaneous rate of change: This is noneother than your velocity which you would read fromyour speedometer. Thus your speedometer can bethought of as a derivative machine.
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Rates of ChangeRates of changeVelocityExample
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Example
iceExample 2: Suppose the distance travelled by acar (in feet) is given by the function s(t) = 1
2t2 + t
where 0 ≤ t ≤ 20 is measured in seconds.■ Find the average velocity over the time interval
◆ [10, 11]◆ [10, 10.1]◆ [10, 10.01]
■ Find the instantaneous velocity at t = 10.■ Compare the above results.
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Rates of ChangeRates of changeVelocityExample
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s(t) = 1
2t2 + t
■ The average velocity over the given time intervalsare:
◆s(11)−s(10)
11−10= 1
2(11)2 + 11 − (1
2(10)2 + 10) = 11.5
(ft/sec)
◆s(10.1)−s(10)
10.1−10= 1.105
.1= 11.05 (ft/sec)
◆s(10.01)−s(10)
10.01−10= 1.1005
.01= 11.005 (ft/sec)
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Rates of ChangeRates of changeVelocityExample
Section 2.6: The Derivative Business Calculus - p. 21/44
s(t) = 1
2t2 + t
■ We could probably guess that the instantaneousvelocity at t = 10 is 11 (ft/sec). But lets calculatethis using the definition
s′(t) = limh→0
s(t + h) − s(t)
h
= limh→0
12(t + h)2 + (t + h) − (1
2t2 + t)
h
= limh→0
12(t2 + 2th + h2) + t + h − 1
2t2 − t
h
= limh→0
th + 12h2 + h
h
= limh→0
t +1
2h + 1 = t + 1.
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Rates of ChangeRates of changeVelocityExample
Section 2.6: The Derivative Business Calculus - p. 22/44
s(t) = 1
2t2 + t
Notice that we have calculated the derivative at anypoint t:
s′(t) = t + 1.
We now evaluate at t = 10 to get
s′(10) = 11
just as we expected.
The average velocity over the time intervals[10, 10 + h] for h = 1, h = .1 and h = .01 becomecloser to the instantaneous velocity at t = 10. Thisis as we should expect.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 23/44
Finding the derivative of a functionusing the definition
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 24/44
Notation
Differential calculus has various ways of denotingthe derivative, each with their own advantages.We have used the prime notation , f ′(x) (read: "fprime of x"), to denote the derivative of y = f(x).You will also see y′ written when it is clear y = f(x).The prime notation is simple, quick to write, but notvery inspiring.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 25/44
Another notation is
df
dxor
dy
dx.
This notation is much more suggestive. Recall thatthe derivative is the limit of the difference quotient∆y
∆x: the change in y over the change in x. The
notation "dy" or "df " is used to suggest the instan-taneous change in y after the limit is taken and like-wise for dx. One must not read too much into thisnotation. df
dxis not a fraction but the limit of a frac-
tion.
There are other notations that are in use but theseare the two most common.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 26/44
An outline
To compute the derivative df
dx= f ′(x) of a function
y = f(x) using the definition follow the steps:
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 26/44
An outline
To compute the derivative df
dx= f ′(x) of a function
y = f(x) using the definition follow the steps:
1. Find the change in y: f(x + h) − f(x)
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 26/44
An outline
To compute the derivative df
dx= f ′(x) of a function
y = f(x) using the definition follow the steps:
1. Find the change in y: f(x + h) − f(x)
2. Compute f(x+h)−f(x)h
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 26/44
An outline
To compute the derivative df
dx= f ′(x) of a function
y = f(x) using the definition follow the steps:
1. Find the change in y: f(x + h) − f(x)
2. Compute f(x+h)−f(x)h
3. Determine limh→0f(x+h)−f(x)
h.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 27/44
Example
iceExample 3: Find the derivative of y = x3 − x.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 27/44
Example
iceExample 3: Find the derivative of y = x3 − x.
Let f(x) = x3 − x. Then
f(x + h) − f(x) = (x + h)3 − (x + h) − (x3 − x)
= x3 + 3x2h + 3xh2 + h3
−x − h − x3 + x
= 3x2h + 3xh2 + h3 − h
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 27/44
Example
iceExample 3: Find the derivative of y = x3 − x.
Let f(x) = x3 − x. Then
f(x + h) − f(x) = (x + h)3 − (x + h) − (x3 − x)
= x3 + 3x2h + 3xh2 + h3
−x − h − x3 + x
= 3x2h + 3xh2 + h3 − h
Next we get
f(x + h) − f(x)
h=
3x2h + 3xh2 + h3 − h
h
= 3x2 + 3xh + h2 − 1
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 27/44
Example
iceExample 3: Find the derivative of y = x3 − x.
Let f(x) = x3 − x. Then
f(x + h) − f(x) = (x + h)3 − (x + h) − (x3 − x)
= x3 + 3x2h + 3xh2 + h3
−x − h − x3 + x
= 3x2h + 3xh2 + h3 − h
Next we get
f(x + h) − f(x)
h=
3x2h + 3xh2 + h3 − h
h
= 3x2 + 3xh + h2 − 1
Finally, dy
dx= limh→0 3x2 + 3xh + h2 − 1 = 3x2 − 1.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 28/44
Example
iceExample 4: Find the equation of the line tangentto
f(x) =√
x
at the point (4, 2).
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 29/44
f(x) =√
x
We need the slope of the tangent line at this point.This is f ′(4).
f′(4) = lim
h→0
f(4 + h) − f(4)
h
= limh→0
√
4 + h − 2
h
= limh→0
√
4 + h − 2
h
√
4 + h + 2√
4 + h + 2
= limh→0
4 + h − 4
h(√
4 + h + 2)
= limh→0
1√
4 + h + 2=
1
4.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 30/44
f ′(4) = 1
4and P = (4, 2)
Given a point and a slope we compute the line:
y − 2 =1
4(x − 4)
or
y =1
4x + 1.
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DiffereniationNotationAn outlineExampleExampleDo not despair!
Section 2.6: The Derivative Business Calculus - p. 31/44
Do not despair!
Admittedly, the calculation of a derivative using thedefinition can be tedious. However, in the nextchapter we will discuss a set of rules for differenti-ation that will allow us to calculate the derivative ofmany commonly encountered functions very eas-ily. Nevertheless, it is important that you under-stand the definition and the underlying meaning ofthe derivative; at times, it will be necessary to comeback to it.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 32/44
Differentiation and Continuity
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 33/44
Reformulation of Continuity
In the last section we discussed the meaning ofcontinuity. Recall a function y = f(x) is continu-ous at a point a if f(a) is defined and
limx→a
f(x) = f(a).
If we let x = a + h then x approaches a if h ap-proaches 0. This observations allows us the givean equivalent definition for continuity: f(a) is de-fined and
limh→0
f(a + h) − f(a) = 0.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 34/44
Differentiable functions are Continuous
A function is said to be differentiable at a pointx = a if f ′(a) exists. This means that the limitlimh→0
f(a+h)−f(a)h
exists. We say f is differentiableon an interval (a, b) if it is differentiable at everypoint in the interval.
Notice the next theorem:
Theorem: A function that is differentiable at apoint x = a is continuous there.
We have not been proving many theorems but thisone is easy and short enough that we will do so onthe next slide.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 35/44
Proof
Proof: To say f is differentiable at x = a means
limh→0
f(a + h) − f(a)
h
exists and is a finite number, denoted f ′(a). Thus
limh→0
(f(a + h) − f(a)) = limh→0
f(a + h) − f(a)
h· h
= limh→0
f(a + h) − f(a)
h· lim
h→0(h)
= f ′(a) · 0 = 0.
This means that f is continuous at x = a.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
![Page 74: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/74.jpg)
ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 36/44
Continuity does not imply Differentiability
We must not read something that is not in this theo-rem. Though a differentiable function is necessarilycontinuous a continuous function is not necessarilydifferentiable. Consider this classic example:
y = |x| .At x = 0 there are several lines that just touch thegraph at (0, 0); it is not unique.
Remember, tangent lines are unique and sincey = |x| has no unique tangent line it is not differ-entiable at x = 0.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 37/44
y = |x| at x = 0
Consider what happens here in terms of the defini-tion:
y′(0) = limh→0
|0 + h| − 0
h
= limh→0
|h|h
.
Now, to compute this limit we will consider the leftand right-hand limits.
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ContinuityReformulationDifferentiabilityProofContinuiity
Section 2.6: The Derivative Business Calculus - p. 38/44
Left and Right-hand limits of |h|h
If h is positive then |h| = h and
y′(0) = limh→0+
|h|h
= limh→0
h
h= 1.
If h is negative then |h| = −h and
y′(0) = limh→0−
|h|h
= limh→0
−h
h= −1.
The left and right hand limits are not equal there-fore limh→0
|h|h
does not exist.
If y = |x| then y is continuous but notdifferentiable at x = 0.
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SummarySummary
Section 2.6: The Derivative Business Calculus - p. 39/44
Summary
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SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
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SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
![Page 80: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/80.jpg)
SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
■ The meaning: The derivative of a functionrepresents the instantaneous rate of change of f
as a function of x.
![Page 81: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/81.jpg)
SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
■ The meaning: The derivative of a functionrepresents the instantaneous rate of change of f
as a function of x.■ Primary Applications: Tangent lines, velocity
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SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
■ The meaning: The derivative of a functionrepresents the instantaneous rate of change of f
as a function of x.■ Primary Applications: Tangent lines, velocity■ Computation of the derivative.
![Page 83: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/83.jpg)
SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
■ The meaning: The derivative of a functionrepresents the instantaneous rate of change of f
as a function of x.■ Primary Applications: Tangent lines, velocity■ Computation of the derivative.■ The connection between continuity and
differentiation.
![Page 84: Business Calculus](https://reader034.vdocuments.site/reader034/viewer/2022051215/55cf8eac550346703b946f4e/html5/thumbnails/84.jpg)
SummarySummary
Section 2.6: The Derivative Business Calculus - p. 40/44
Summary
This section is very important and likely new tomany students in this course. Here are some keyconcepts to master.
■ The definition of the derivative:f ′(x) = limh→0
f(x+h)−f(x)h
.
■ The meaning: The derivative of a functionrepresents the instantaneous rate of change of f
as a function of x.■ Primary Applications: Tangent lines, velocity■ Computation of the derivative.■ The connection between continuity and
differentiation.■ Notation: y′ or dy
dx.
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In-Class ExercisesICEICEICE
Section 2.6: The Derivative Business Calculus - p. 41/44
In-Class Exercises
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In-Class ExercisesICEICEICE
Section 2.6: The Derivative Business Calculus - p. 42/44
In-Class Exercise
return In-Class Exercise 1: At a fixed temperature thevolume V (in liters) of 1.33 g of a certain gas isrelated to its pressure p (in atmospheres) by theformula
V (p) =1
p.
What is the average rate of change of V with re-spect to p as p increases from 5 to 6 ?
1. 1
2. 16
3. −15
4. − 130
5. −16
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In-Class ExercisesICEICEICE
Section 2.6: The Derivative Business Calculus - p. 43/44
In-Class Exercise
returnIn-Class Exercise 2: Use the definition of thederivative to find y′ if
y = 4x2 − x.
1. 4x2 − 1
2. 8x − 1
3. 8x
4. 4x2 − x
5. None of the above
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In-Class ExercisesICEICEICE
Section 2.6: The Derivative Business Calculus - p. 44/44
In-Class Exercise
returnIn-Class Exercise 3: Find the equation of the linetangent to
y = x2 + x
at the point (1, 2).
1. y = 3x − 1
2. y = 3x − 5
3. y = 2x
4. y = 2x − 3
5. None of the above