Transcendentals


 * Unit 5: Transcendentals**

Pre-requisite Information (Kendalyn Moulder)
~Understand basic derivation and integration techniques~ ~Know derivatives and integrals of trig functions by heart <3~ ~Understand basic logarithmic properties and rules~ (even though we were not thoroughly taught this) :/ ~ DON'T STRESS AND TAKE THIS ONE STEP AT A TIME!!!! :) ~

Notes (Kendalyn)
What is a log? Logarithms are how we determine exponents and denote exponential functions. They are denoted as such a^x = log (base a) of b= x So log (base 2) of 8= 3 because 2^3 = 8
 * If a base is not stated the it is understood that the base is 10***

Natural Logaritms
 * denoted ln
 * have a base of //e//
 * //e// is an irrational constant similar to pi and is approximately 2.71.....

Integration and Differentiation of e and ln
Remember how we couldn't integrate 1/x because we were left with a 0 in the denominator which made the function undefined.... Well now wwe will learn how to do that

__**Natural logs**__ This is the rule for integrating natural logs
 * It's basically something you need to memorize
 * That the integral of some constant over a variable x is equal to that constant times the ln of the variable x

__**Derivative of ln**__

The derivative of a natural log is as follws
 * one over the function with the ln
 * times the derivative of the function
 * This works for all basic functions as well as polynomials and trig functions as well

Helpful Websites and Videos:

 * Katie Miklausich **

Okay so we know a lot of us still struggle with dealing with basic logarithms. So I've found a video and a website that will hopefully refresh us on what we SHOULD have known about logarithms..

Help with basics of logarithms video

Website for help with basic properties of logs -provides you with some notes, example problems, a quiz and answers to the quiz

Now for the harder stuff..

__Videos:__ Derivatives of Natural Logs (Katie) -starts from the beginning of the unit talking about natural logs and examples of deriving a problem involving natural logs

Deriving Logarithms (Katie) -Shows a couple quick examples of deriving with logarithms.

Deriving One over x (Katie) -explains how to derive one over x. Which up until now, we thought was impossible. Of course we learn in calculus that nothing is impossible.

Deriving Exponential Functions (Katie) -Okay the guy who is speaking is very quiet so turn your volume up!!! It's worth it I promise! Very helpful video, I actually used this to help me make it through figuring out these types of problems!

[|Integration of Natural Exponential Functions](Emily) This video is about 16 minutes long, but the guy who does it is nerdy and adorable. He starts with the basics of using U-substitution in integrating exponential functions and works his way through more complicated examples.

__Websites:__ This website I personally found VERY cool! Not only does it give you examples of practice problems, it also gives you a place to where you can type a problem in and it will show you how to  ones similar to the problem you are stuck on. (Which I find extremely cool)
 * @http://www.freemathhelp.com/derivative-log-exponent.html** (Katie)

[] (Katie) Goes over anything and everything dealing with deriving logarithmic functions.

@http://www.themathpage.com/acalc/exponential.htm (Katie) Words everything just a bit differently in case you had problems figuring out the others.

[|Intro to Integration of Eponential and Logarithms Functions] (Emily) This website provides an introduction into integration with logarithms and exponential functions. There are also practice problems with solutions at the bottom of the page.

[|Integrals of Exponential and Logarithmic Functions] (Emily) Includes a chart which has the rules of taking the integral of exponential and logarithmic function, has a step-by-step u substitution practice problem, and a short video.

__Careers: (Emily) __

 * Different types of engineers, computer programmers, and financiers use logarithms and their derivatives to calculate the quantity of items that increase and decrease at an exponential rate. Specifically, aerospace engineers, computer hardware engineers, and statisticians use logarithms.

__Annotated Bibliography: (Emily) __
Hockett, Shirley, and David Bock. // AP Calculus //. 10th edition. New York: Barron's Educational Series, 2008. 378-390. #|Print.

This AP test prep book provided an overview of the rules of differentiation involving logarithms, of using logarithms in exponential growth and decay, and properties of exponential and logarithmic functions. This source was helpful because it gave brief, accurate information about properties and theorems dealing with differentiation and logarithms. Overall it is a very reliable source.