ppt on applications of derivatives

7/30/2019 ppt on applications of derivatives

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Applications ofDerivatives

Curve Sketching


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What the First Derivative Tells

Us: Suppose that a function fhas a derivative at

every pointxof an interval I. Then:

increases on I if ()0 for all in I.ffxx

decreases on I if ()0 for all in I.ffxx


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What This Means: In geometric terms, the first derivative

tells us that differentiable functions

increase on intervals where their graphshave positive slopes and decrease on

intervals where their graphs have

negative slopes.

WHAT HAPPENS IF THE FIRST

DERIVATIVE IS ZERO?


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When The First Derivative is

ZeroA derivative has the intermediate value

property on any interval on which it is

defined. It will take on the value zero when it

changes signs over that interval.

Thus, when the derivative changes signs

on an interval, the graph off(x) musthave a horizontal tangent.


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HOWEVER A derivative need not change sign every time

it is zero. Consider

The derivative is

The derivative is zero at the origin but

positive on both sides of the origin.

3yx

23yx


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Relative Maxima and Minima If the derivative

changes sign, there

may be a local max

or min, as shown

here.

More on this later.


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Concavity Concave downspills water

Concave upholds water

The graph of

is concave down on any interval where

and concave up on any interval where

()yfx

0y

0y


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Points of Inflection A point on the curve where the concavity

changes is called a point of inflection.

If the second derivative is zero for somex, wemay be able to find a point of inflection.

It IS possible for the second derivative to be

zero at a point that is NOT a point of inflection.

A point of inflection may occur where thesecond derivative fails to exist.


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Relative Extrema

Let f(x) be defined on an interval, I, and let x0 be

in I.

1. If f(x) has a relative extremum at x = x0 then eitherf(x)=0

orfis not differentiable at x = x0. 2. Values at which the derivative is zero at x0 or at which fis

not differentiable at x = x0 are called critical numbers.

3. If fis defined on an open interval, its relative extrema

occur at critical numbers.

NOTE: This does NOT mean that a critical

number MUST yield a relative extremum.


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The First Derivative and

Relative Extrema

No relative extremumFirst derivativepositive

x0First derivative

positive

No relative extremumFirst derivativenegativex0First derivative

negative

Relative min at x0First derivativepositive

x0First derivative

negative

Relative max at x0First derivative

negative

x0

First derivative

positive

ResultRight SideLeft Side

This is what happens around the point x0:


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The Second Derivative and

Relative Extrema

Inconclusive. fmay have

a relative max, min, or

neither.

AND

fhas a relative maximumat x0.

AND

fhas a relative minimum

at x0.

AND

ResultSecondDerivative

FirstDerivative

0f

0f

Assume that fis twice differentiable atx0. If:

0f

0f

0f

0f


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An Example:

This first derivative is equal to zero atx=0, x=1 and x= -1.

These are the critical values.

Examine the sign of the derivative around

these values:

42Let ()2fxxx 3()44fxxx


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Sign of the First Derivative:

The change from to +

indicates a relative min.+1-

The change from + to 1

indicates a relative max._0+

The change from to +indicates a relative min.+-1-

ResultRightLeft


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Furthermore

The function is decreasing from ( ) and on(0,1) because the derivative is negative on thoseintervals.

The function is increasing on (-1,0) and on ( 1, )because the derivative is positive on thoseintervals.

We will examine the second derivative for what it

can tell us. The second derivative is:

,1

2()124fxx


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The Second Derivative.

The second derivative is equal to zero at

x =

Examine the sign of the second derivative

around these points:

+++++ ------- +++++

33

3333


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Concavity

The function is concave up in those

areas where the second derivative is

positive and concave down in that areawhere the second derivative is negative.

If you check the sign of the second

derivative at the critical values, you willfind that this reinforces what we said

before about the relative max and min.


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Inflection Points

You can tell where the function changes

concavity by finding the inflection points.

Evaluate thefunction

at those values wherethe second derivative is zero; that is, at x =

Take a look at the graph of the original

function:

33

42()2fxxx


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The Graph


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Does It Check?

Check the intervals on which the function

is increasing and decreasing.

Check the location of relative maximaand/or minima.

Check the concavity of the function.

The graph should match informationdetermined from the derivatives.

ppt on applications of derivatives

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