CRMS Calculus May 31, 2010

The Year in Review What the heck is Calculus?

One of the supreme creations of human thought

One of the biggest controversies in history

The Culmination of K-12 mathematics Calculus Trigonometry Elementary Functions Geometry Algebra Arithmetic

Course Goals
To introduce you to the four major concepts of calculus:
Limits
Derivatives
Definite Integral
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:
Numerical (Table)
Visual (Graph)
Algebraic (Function)
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 definition

Limits by Delta - Epsilon definition

Limits by graph

Limits by numerical table

Limits by algebra

Limits involving

Intermediate Value Theorem

Differential Calculus The 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

If y = Displacement Then y’ = Velocity And y’’ = Acceleration

How’s My Deriving?

Power Rule for derivatives Function Notation: Operator Notation:

Related Rates

y = C + AsinB(x – D) y ‘ = ABcosB(x – D)

The Squeeze Theorem

If y = e x , then y’ = e x If y = b x , then y’ = b x · ln b

If y = ln x, then

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 integral And A definite integral?

Antiderivatives of y’ = x 2 Bring the Whole Indefinite Integral Family! Indefinite Integral A Family of Curves With a given Derivative

Definite Integral is a Number

Solving Differential Equations and Slope Fields

Sweetheart, Check out this definition ! Definite Integral;

Rolle’s Theorem The Mean Value Theorem Is just Rolle’s Theorem Rotated

What were those Properties Of Integrals?

Definite Integrals Allow us to find The area between curves

Naked Math Real World

Volumes of Revolution The Disk Method

An Illuminating Volume

This stuff isn’t so hard! In fact …

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CRMS Calculus May 31, 2010

by Colorado Rocky Mountain School on May 31, 2010

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CRMS Calculus May 31, 2010 — Presentation Transcript