AP Calculus Slides November 2, 2007

NooNoo studying calculus

a year to remember

Exponential growth rate ...

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?

Now let's look at this numerically ...

remember ... mathematics is the science of patterns

Mandala - the pattern of creation

Base c

Let's flip back and forth from the graph and look for a numerical patternthe search for the "perfect" number ...

Base c

Perfection, Colorado

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)

So ...

But what about exponential functions with other bases ...

If If

then then

and

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 ...

Derivatives of Exponential Functions

The Exponential Function

any exponential function

Let's find these derivatives ...