crms calculus may 31, 2010

The Year in Review

Whattheheck isCalculus?

One of the supreme creations of human thought

One of the biggest controversies in history

The Culmination of K-12 mathematics

Calculus

TrigonometryElementary Functions

GeometryAlgebra

Arithmetic

Course Goals

To introduce you to the four major concepts of calculus:

1. Limits

2. Derivatives

3. Definite Integral

4. Indefinite Integral

For each concept you will …

• know the precise definition;• have an intuitive understanding of what the concept

means;• be able to “do” the concept; • be able to apply it in the real world or mathematical

world.

You will develop this knowledge by using four multiple

representations:

1. Numerical (Table)

2. Visual (Graph)

3. Algebraic (Function)

4. Verbally (and written).

Do you Sir

Know the Foundation

For All of Calculus?

The foundation of Calculus

is the concept of a limit.

1/2, 2/3, 3/4, 4/5, 5/6,... 10/11,... 99/100,... 99999/100000,...

Limits by

Delta - Epsilon definition

Limits by graph

Limits by numerical table

Limits by algebra

Limits

involving

Intermediate Value Theorem

Differential CalculusThe mathematics of change

Physical Meaning

Instantaneous rate of change

Slope of the tangent line at a point.

Graphical Meaning

Do you Sir

Know the definition of

The Derivative ?

Derivative

0

( ) ( )'( ) lim

x

f x x f xf x

x

If y = Displacement

Then y’ = Velocity

And y’’ = Acceleration

How’s My Deriving?

Power Rule for derivatives

1If ( ) , then '( )n nf x x f x nx Function Notation:

Operator Notation: 1( )n ndx nx

dx

dV

dt dV

drdr

dt

Related Rates

y = C + AsinB(x – D)

y ‘ = ABcosB(x – D)

The Squeeze Theorem

If y = ex, then y’ = ex

If y = bx, then y’ = bx · ln b

If y = ln x, then1

'yx

I PITY DA FOO’ WHO CAN’T FIND DA DERIVATIVE !!!

Integral Calculus

Areas and Volumes

Counting Squares

Archimedes

Method of Exhaustion

Trapezoidal Rule

Do you sir,Know the difference between

An indefinite integralAnd

A definite integral?

Antiderivatives of y’ = x2

Bring the Whole Indefinite Integral Family!3

3

xy C

Indefinite IntegralA Family of Curves With a given Derivative

Definite Integralis

a Number

Solving Differential Equations andSlope Fields

Definite Integral;

1

lim ( ) ( )n b

i ani

f c x f x dx

Sweetheart,Check out

this definition !

Rolle’s Theorem

The Mean Value Theorem

Is just Rolle’s Theorem

RotatedRotated

What werethose

PropertiesOf Integrals?

Definite Integrals

Allow us to find

The area between curves

Naked Math

Real World

b

aDistance = Velocity time

b

aWork= Force displacement

Volumes of RevolutionThe Disk Method

An Illuminating Volume

This stuff isn’t so hard!

In fact …