MargaretandErin

The Fundamental Theorem of Calculus September 6, 2012

//**The Fundamental Theorem of Calculus**// states: If f is continuous on [a,b] and f(x) = F'(x) for some function f, then

//**Definition: **// We say F is an **antiderivative** of f if F'(x) = f(x). This can also be expressed by: (indefinite integral) Solve for the indefinite integral:
 * Example:**
 * answer**

//**Observation:**// if F is an antiderivative of f, so is F+c, where c is any constant. Proof: (using derivative sum rule) because the derivative of a constant is 0 and the derivative of F is f.

||
 * //Chart of Basic Antiderivative Rules //**
 * **f(x) ** || [[image:http://latex.codecogs.com/gif.latex?%5Cint%20f(x)%20dx width="65" height="35"]] ||
 * 0 || c ||
 * A || Ax + c ||
 * [[image:http://latex.codecogs.com/gif.latex?x%5Er]] || (x <span style="font-family: Arial,Helvetica,sans-serif; vertical-align: super;">r+1 <span style="font-family: Arial,Helvetica,sans-serif;">) / (r+1)
 * <span style="font-family: Arial,Helvetica,sans-serif;">x <span style="font-family: Arial,Helvetica,sans-serif; vertical-align: super;">-1 || <span style="font-family: Arial,Helvetica,sans-serif;">ln|x| + c ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">cos(x) || <span style="font-family: Arial,Helvetica,sans-serif;">sin(x) + c ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">sin(x) || <span style="font-family: Arial,Helvetica,sans-serif;">-cos(x)+c ||
 * [[image:http://latex.codecogs.com/gif.latex?e%5Ex]] || [[image:http://latex.codecogs.com/gif.latex?e%5Ex+c]] ||
 * <span style="font-family: Arial,Helvetica,sans-serif;">sec <span style="font-family: Arial,Helvetica,sans-serif; vertical-align: super;">2 <span style="font-family: Arial,Helvetica,sans-serif;">(x) || <span style="font-family: Arial,Helvetica,sans-serif;">tan(x) + c ||
 * [[image:http://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B1+x%5E2%7D width="41" height="34"]] || <span style="font-family: Arial,Helvetica,sans-serif;">arctan(x) +c ||

//**<span style="font-family: Arial,Helvetica,sans-serif;">Properties of Definite Integrals **// <span style="font-family: Arial,Helvetica,sans-serif;">1.

<span style="font-family: Arial,Helvetica,sans-serif;">2.

<span style="font-family: Arial,Helvetica,sans-serif;">typically: a < c < b (will still work if this is not true)

<span style="font-family: Arial,Helvetica,sans-serif;">3.

<span style="font-family: Arial,Helvetica,sans-serif;">4.

<span style="font-family: Arial,Helvetica,sans-serif;">where r is a constant solve for: given f(x)= 8cos(x) - <span style="font-family: Arial,Helvetica,sans-serif;">sec <span style="font-family: Arial,Helvetica,sans-serif; vertical-align: super;">2 <span style="font-family: Arial,Helvetica,sans-serif;">(x)
 * Example:**




 * answer**

//**Definitions:**// 1. We say f is an **even function** if f(-x) = f(x) Examples: , 2. We say f is an **odd function** if f(-x) = - f(x) Examples: , //**Observation:**// if f is an even function:



Find:
 * Real Example:**
 * Answer: 1500**

Class example: goes to 0 because it is odd, so:
 * //Observation://** if f is an odd function

Find:
 * Example:**


 * Answer:** 0

1. if:for
 * //Observation://**

then:

2. if for