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Antiderivatives - PowerPoint PPT Presentation


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Antiderivatives. Antiderivatives . Mr. Baird knows the velocity of particle and wants to know its position at a given time. Ms. Bertsos knows the rate a population of bacteria is increasing and she wants to know what the size of the population will be at a future time.

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Presentation Transcript
antiderivatives
Antiderivatives
  • Mr. Baird knows the velocity of particle and wants to know its position at a given time
  • Ms. Bertsos knows the rate a population of bacteria is increasing and she wants to know what the size of the population will be at a future time.
  • In each case the rate of change (the derivative) is known….but what is the original function?
  • The original function is called the

ANTIDERIVATIVE of the rate of change.

slide3

A function is called an antiderivative of on an interval if for all x in .

DEFINITION

suppose

Suppose

We can make some guesses

What is its antiderivative?

They all fit!

theorem

If is an antiderivative of on an interval , then the most general antiderivative of on

is

whereis an arbitrary constant.

Theorem
slide6

Finding an antiderivative is also known as Indefinite Integration and the Antiderivative is the Indefinite Integral

(Especially for us old guys!)

And the symbol for integration is an elongated S

More on why it’s an S later!

slide7

Constant of Integration

Integrand

Variable of Integration

This is read: The antiderivative of f with respect to x or the indefinite integral of f with respect to x is equal to…..

what is the antiderivative of

We know what to differentiate

to get

What is the Antiderivative of

Derivative

We “kinda” multiply

Take the integral of

both sides

some general rules
Some General Rules

They are just the derivative rules in reverse

Differentiation Formula Integration Formula

“Pulling out a konstant”

some general rules10
Some General Rules

Differentiation Formula Integration Formula

Sum / Difference Rule for Integrals

Power Rule for Integrals

some general rules11
Some General Rules

Differentiation Formula Integration Formula

All the other trig functions follow