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CALCULUS I FALL 2013

Applications of the Derivitive

Newton's Method

approximating zeros

Types of Motion

Simple Harmonic

Rectilinear

Optimization

Graphing

THEOREMS

ROLLES THEOREM

MEAN VALUE THEOREM

Second Derivitive Test

determines if the rate of change of a function is increasing or decreasing

determines concavity

First Derivitive Test

used to find critical values

determines if a function is increasing or decreasing

Extrema of Functions

Critical Values

Local and Extreme minimun and maximum values

Antiderivatives and Integrals

THEOREMS

Average (mean) Value of Integrals

Mean Value Theorem of Definate Integrals

Fundamental Theorem of Calculus

Numerical Integration

Simpson's Rule

Trapezoidal Rule

Summation Notation and Area

Finding the area under a curve

Upper Darboux Sum

Lower Darboux Sum

Definate Integral

Properties of the Definate Integral

Reimann Sum

Indefinite Intergral

Change of Variables (limits of integration)

Substitution

Applications of the Definate Integral

Moments and Centers of Mss

Theorem of Pappus

Work/Force

Variable Force

Constant Force

Force exerted by a liquid

Area between functions

Solids of Revolution

finding the volume of a solid revolved around an axis

Cylindrical Shells Method

Washer/Disk method

Definate Intergal

Volume by Cross Sections

finding the volume of a solid using a cross section with the base of that cross section perpendicular to the x axis

Arc Length and Surfaces of Revolution

surface area

uses the definate integral

revolving a line segment or multiple connected line segments (graph of a function) around an axis

determining the length of the graph of a function by using the limit of sums of lengths of line segments

The Derivative

Related Rates

Implicit Differentiation

How to find the derivative of a function whose equation has two variables.

Increments and Differentials

Linear Approximation

aprroximating the value of a function by using the tangent line

DIFFERENTIATION-the slope of the secant is tending towards the slope of the tangent line

Trig Functions

Chain Rule: f'(g(x))g'(x)

Power Rule

Rates of Change

Instantaneous Velocity

Difference Quotient

rate of change at a point

the slope of the tangent line to a graph at a given point

Average Velocity

Rate of change over an interval

The Limit

Epsilon/Delta Proofs & Problems

REGULAR DEFINITION

ALTERNATIVE DEFINITION

f'(x)=

f(x) - f(a)

_______

x - a

Limits

Computations

Quotient Rule

Product Rule

Subtraction

Addition

Continuity

3 things make a function continuous at a number "c":

the limit of the function must equal the value of the function at that point, i.e. limit f(x) as x approaches c = f(c).

the limit of f(x) as x approaches that number "c" must exist

f(c) is defined

Discontinuity types

infinite

removable

jump

Theorems

INTERMEDIATE VALUE

if there is a continuous function on a closed interval [a,b] and w is any number between f(a) and f(b) then there is a point c in [a,b] where f(c) = w.

SANDWICH

Example. Find lim x→0 x 2 cos(1 x ) . Hide Solution

squeeze theoremSince −1≤cos(1 x )≤1 for all x (actually we are interested only in x near 0) then −x 2 ≤x 2 cos(1 x )≤x 2 . Since lim x→0 x 2 =lim x→0 −x 2 =0 then by Squeeze theorem lim x→0 x 2 cos(1 x )=0 .

On the figure you can see that x 2 cos(1 x ) is squeezed between x 2 and −x 2 .

CHAPTER ONE...The Beginning (of the end)

Pre-Cal Review

Binomial Theorem

Used to prove the Power Rule for Differentiation

expanding expressions with large exponents

Absolute Value

lal = a, a> or = 0 and lal = -a, a<0

Definition of a funtion

for any given x value in a domain there is a unique y value in the range such that f(x) = y.