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Antiderivatives

Connections with differential equations

Lack of uniqueness of antiderivatives

Some examples

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A basic differential equation

Consider

dy/dx = f(x) with y = y0 when x = x0

Need to find a function y satisfying y=f(x)

and whose graph goes through the point

(x0, y0)

As well see, the latter condition imposes

uniqueness

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Example

10,000is0at timepopulationtheofsizetheand

,0for)(2 ttNdtdN

solution.uniquetheis

000,10)(

:Claim

2tetN

ating.differentiofprocessthe

reversemustweequations,oftypesthesesolveTo

ating.differentibysolutionait wasthatVerified

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Definition of antiderivative

Definition: A functionFis called an antiderivativeof

fon an intervalIif

F(x) = f (x)forxinI.

Example: Find antiderivatives of 4x3

One given byx4

Another byx4+347 Any of the formx4+C, where Cis a constant work

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Consequences of Mean Value Theorem

],[on

constantisthen,),(allfor0)('with

),(intervalopentheonabledifferentiand],[

intervalclosedtheoncontinuousisIf:1Fact

ba

fbaxxf

baba

f

IxCxFxG

C

Ixf

xGxF

allfor)()(

thatsoconstantaexists

therethen,intervalanon)(functioncontinuous

theoftivesantiderivaare)(and)(If:2Fact

constantabydiffermust

functioncommonaoftivesantiderivaTwo:Conclusion

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Examples

Find the general antiderivatives of the

following functions

23)( xxf xxf sin)(

xexf 5)( 103)( xxf

134)( 2 xxxf )3sec()3tan()( xxxf

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An antiderivative table

Function tiveAntiderivaParticular

)(xkf )(xkF

)()( xgxf )()( xGxF

1, nxn1

1

1

nxn

x

1 ||ln x

ax

e

axea

1

)sin(ax )cos(1

axa

)cos(ax )sin(1

axa

)(sec2 ax )tan(1

axa

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Differential equations

0,2

equationaldifferentitheofsolutiongeneraltheFind

3 xx

xdx

dy

10),sin(

equationaldifferentitheofsolutiongeneraltheFind

ssds

dy

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Initial value problems

1when2with0for,2

problemvalueinitialtheSolve

xyxxdx

dy

0when0with0for,2

problemvalueinitialtheSolve

x-

xyxee

dx

dy x

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Gravity

An object is dropped from a height of 100

ft. The acceleration due to gravity is

32 ft/sec2.

When will the object hit the ground?

What will be its speed at impact?