AP Calculus Slides December 12, 2007
-
Upload
darren-kuropatwa -
Category
Technology
-
view
742 -
download
2
description
Transcript of AP Calculus Slides December 12, 2007
![Page 1: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/1.jpg)
Acme adding machine
Properties of Integrals
![Page 2: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/2.jpg)
What does The Fundamental Theorem of Calculus say?
Let the function ƒ be continuous on [a, b] with derivative ƒ'. Then ...
Total change in ƒ
Also recall:
In words: Integrating a rate of change function ƒ' over an interval [a, b] gives the total change in ƒ, ƒ(b) - ƒ(a), over the same interval.
In other words, the integral of a derivative is the same thing as the total change in it's parent function over the same interval.
and ...
![Page 3: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/3.jpg)
Use The Fundamental Theorem of Calculus to evaluate:
![Page 4: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/4.jpg)
Properties of Integrals ...
Constant Multiple Rule
Sum and Difference Rule
Additive Interval Rule
Inequality Rule
Let's see some examples ...
If ƒ(x) ≤ g(x) for all x
![Page 5: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/5.jpg)
Constant Multiple Rule Consider each of these integrals and the related graphs
![Page 6: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/6.jpg)
Sum and Difference Rule Combining this rule with the constant multiple rules allows us to integrate expressions like this:
![Page 7: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/7.jpg)
Additive Interval Rule This rule is illustrated in the graph below.It allows us to solve problems like this one:Given: and
Find:
Ye Olde Buh-cetO Integrals +
–=
Build your own equations from the bucket:
= +
= –
-3
4 = -3 – 4
![Page 8: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/8.jpg)
Inequality Rule Consider each of these integrals and the related graphsIf ƒ(x) ≤ g(x) for all x
![Page 9: AP Calculus Slides December 12, 2007](https://reader034.fdocuments.in/reader034/viewer/2022042815/557e7797d8b42a7b5c8b53d0/html5/thumbnails/9.jpg)
The Definite Integral of a Rate Give Total ChangeLuke! Remember the FTC!
obi-wan kenobi
A company estimates that its rate of sales of an item is given by:
Where t is measured in years. Find the total sales during the years 2 ≤ t ≤ 4.