AP Calculus Slides November 2, 2007
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Transcript of AP Calculus Slides November 2, 2007
![Page 1: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/1.jpg)
NooNoo studying calculus
a year to remember
Exponential growth rate ...
![Page 2: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/2.jpg)
What is the derivative rule for exponential functions?
How are we going to remove that multiplier?
Where else can we find this constant? What does it represent?
![Page 3: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/3.jpg)
Now let's look at this numerically ...
remember ... mathematics is the science of patterns
Mandala - the pattern of creation
Base c
![Page 4: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/4.jpg)
Let's flip back and forth from the graph and look for a numerical patternthe search for the "perfect" number ...
Base c
Perfection, Colorado
![Page 5: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/5.jpg)
Symbolically, we're looking for ...
a "magic" number, such that has slope 1 at x = 0.
Numerically ...
we now have 2 different definitions for the number e.
(1) e = the function whose slope of the tangent line to the graph of itself @ x = 0.(2)
![Page 6: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/6.jpg)
So ...
But what about exponential functions with other bases ...
If If
then then
and
![Page 7: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/7.jpg)
remember ... mathematics is the science of patterns
Mandala - the pattern of creation
Base c
Let's explore some more ...
How are all these values related? Let's graph them ...
![Page 8: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/8.jpg)
Derivatives of Exponential Functions
The Exponential Function
any exponential function
Let's find these derivatives ...
![Page 9: AP Calculus Slides November 2, 2007](https://reader033.fdocuments.in/reader033/viewer/2022060200/55982e821a28ab41178b4671/html5/thumbnails/9.jpg)