Sunday, October 15, 2006

3.1: Derivatives of Polynomials and Exponential Functions

HI EVERYONE!! I'm here to impart on you my knowledge of derivatives. So...sit back, relax and enjoy the ride!!

3.1 is basically about finding derivatives of polynomial and exponential functions in ways other than the equations we learned in Chapter 2. 3.1 teaches us to use tools such as the power rule, the derivative of a constant rule and the definition of e to find a functions derivative.

First of all, let's start with the equation we already know about the derivative of a function:

Photobucket - Video and Image Hosting

using this equation, let's find these derivatives:


1. f(x)=x

Photobucket - Video and Image Hosting

2. f(x)=x²

Photobucket - Video and Image Hosting

3. f(x)=x³

Photobucket - Video and Image Hosting

DO YOU SEE A PATTERN?!?!?! I do!!

if Photobucket - Video and Image Hosting then Photobucket - Video and Image Hosting

THAT, my friends, is the good ol' Power Rule.

Now, let's talk about the derivative of a constant rule:
Photobucket - Video and Image Hosting

More simply put, let's think about the meaning of the equation
Photobucket - Video and Image Hosting
What does this mean? This equation is telling us that the function is a horizontal line and we all know that the slope of a horizontal line is 0. Therefore, the derivative of any constant will always be 0.

So, using our knowledge of the power rule and derivative of a constant function let's find the derivatives of these problems:

Photobucket - Video and Image Hosting
Answer:
Photobucket - Video and Image Hosting

Photobucket - Video and Image Hosting
Answer:
Photobucket - Video and Image Hosting

Exponential Functions

Definition of the number e:

Photobucket - Video and Image Hosting

e is the number such that
Photobucket - Video and Image Hosting

Therefore, the derivative of the natural exponential function is, in fact, the natural exponential function itself.

Photobucket - Video and Image Hosting

Use this information to find the derivative of the following problem:

Photobucket - Video and Image Hosting
Answer:
Photobucket - Video and Image Hosting

So now, you're probably all going to ask, "What does this all come to Gianna?" Well, my answer is this:


Photobucket - Video and Image Hosting

...haha good luck KRISTIN, you're next!!

hope this helped with 3.1 guys!!

0 Comments:

Post a Comment

Links to this post:

Create a Link

<< Home