CRMS Calculus May 31, 2010
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Transcript of 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 …