Linearization and Differentials - PowerPoint Presentation

Slide1

Linearization and Differentials

Chapter 5.5Slide2

Linear Approximation

A useful characteristic of the tangent line to a curve at a point is that, for

-values near the point, the curve is approximately linear

In fact, the function values of the curve are approximated by the derivative values near the point of tangencyWe say that differentiable curves are always locally linearIn this section we will explore local linearity

2Slide3

Linear Approximation

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Linear Approximation

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Linear Approximation

5Slide6

Linearization

DEFINITION:

If

is differentiable at

, then the equation of the tangent line,

defines the

linearization of

at

. The approximation

is the

standard linear approximation of

at

. The point

is the

center of approximation

.

6Slide7

Example 1: Finding a Linearization

Find the linearization of

at

, and use it to approximate

without a calculator. Then use a calculator to determine the accuracy of the approximation.

7Slide8

Example 1: Finding a Linearization

Find the linearization of

at

, and use it to approximate

without a calculator. Then use a calculator to determine the accuracy of the approximation.

Note that we are approximating

. The center of the approximation is at

; that is,

in

Find the derivative:

; find

and

. So our linear approximation is

Therefore,

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Example 1: Finding a Linearization

Find the linearization of

at

, and use it to approximate

without a calculator. Then use a calculator to determine the accuracy of the approximation.

The calculator value is

The difference is

Note that as we choose values farther from zero, the approximation becomes less accurate. For example, the approximation for

would be

, which differs from the true value by

or about

. We

could

get a better approximation by choosing a center closer to

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Example 1: Finding a Linearization

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Example 1: Finding a Linearization

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Example 2: Finding a Linearization

Find the linearization of

at

and use it to approximate

without a calculator. Then use a calculator to determine the accuracy of the approximation.

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Example 2: Finding a Linearization

Find the linearization of

at

and use it to approximate

without a calculator. Then use a calculator to determine the accuracy of the approximation.

Here,

, so

The approximation is

The calculator value to 9 decimal places is

. The difference is

This amounts to a percent difference of about

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Example 3: Approximating Binomial Powers

Example 1 introduces a special case of a general linearization formula that applies to powers of

for small values of

:

If

is a positive integer this follows from the Binomial Theorem, but the formula actually holds for

all

real values of

. Use this formula to find polynomials that will approximate the following functions for values of

close to zero:

a)

b)

c)

d)

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Example 3: Approximating Binomial Powers

a)

b)

c)

d)

The first example does not seem to follow the pattern. One technique to use when this happens is to substitute a variable so that the pattern is the same. Let

. The function is therefore

. Now back-substitute to get

This example shows that we need merely use the term in place of

in the formula. So the rest of the examples are:

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Example 4: Approximating Roots

Use linearization to approximate. State how accurate your approximation is.

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Example 4: Approximating Roots

Use the function

and center it around

. Then

and

,

. So,

. Linearization means that

at values of

near the center. Therefore,

.

The difference between

and our approximation is:

This number is less than

, so

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Example 4: Approximating Roots

The nearest perfect cube to 123 is

, so we will center around

. Our function is

and we want an approximation to

. Linearization requires

.

Now,

and

. Our linearization formula is

. Use this to approximate

:

The difference between

and

is

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Differentials

The notation

was Leibniz means of representing the derivative of a function

It is written as a ratio because Leibniz believed that it was a ratio (so did Newton)

But this was their thinking prior to the development of the limit concept almost 200 years laterRecall that we define this notation as

It is not a ratio, then, but a

limit

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Differentials

This matters because it is a mistake to think that, since it is written as a ratio, then all the laws relating to ratios also apply

In mathematics, we can often decide what we want something to mean (within reason) by defining what it means

Our definition cannot be such that other theorems are contradicted by results obtained from our definitionWe will define the “numerator” and “denominator” of

separately in a way that allows us to use them separately

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Differentials

DEFINITION:

Let

be a differentiable function so that

. The

differential

is an independent variable (that is, it’s a number) and the

differential

is

Note that

is a dependent variable and also a function dependent on both

and

.

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Example 5: Finding the Differential

Find the differential

for the given values of

and

.

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Example 5: Finding the Differential

Find the differential

for the given values of

and

.

In differential form, we can write

Now,

We could also take the derivative as we have in the past

Then rewrite this is

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Example 5: Finding the Differential

Find the differential

for the given values of

and

.

In differential notation,

Then

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Example 5: Finding the Differential

Find the differential

for the given values of

and

.

We can begin by solving the equation for

, but we can also use implicit differentiation

Now gather like terms together on each side and solve for

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Example 5: Finding the Differential

Find the differential

for the given values of

and

.

If

, then using the original equation we have

. Therefore,

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Differential Notation

Using differential notation can make our work more concise

, which corresponds to

, which corresponds to

, which corresponds to

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Example 6: Finding Differentials of Functions

Find

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Example 6: Finding Differentials of Functions

Find

We need the Chain Rule here to deal with the inner function,

:

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Example 6: Finding Differentials of Functions

Find

Use the Quotient Rule:

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Estimating Change with Differentials

We can use local linearity to approximate the change in some differentiable function

; that is, to approximate

The equation of the tangent line at some point

is

If

is some small number, then

is a number near

and

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Estimating Change with Differentials

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Differential Estimate of Change

DEFINITION

Let

be differentiable at

. The approximate change in the value of

when

changes from

to

is

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Example 7: Estimating Change with Differentials

The radius

of a circle increases from

m to 10.1 m. Use

to estimate the change in the circle’s area . Compare this estimate with the true change

, and find the approximation error (i.e., the difference between the true value and the approximate value).

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Example 7: Estimating Change with Differentials

The actual change in area is

The approximate change in area (given

) is

Note that

and

. So the approximate change in area is

The actual change is

The error of approximation is

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Absolute, Relative, and Percentage Change

We can describe the change in

as

changes from to

in three ways

Absolute change:

actual

;

approximate

Relative change:

actual

;

approximate

Percentage change:

actual

;

approximate

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Example 8: Changing Tires

Inflating a bicycle tire changes its radius from 12 inches to 13 inches. Use differentials to estimate the absolute change, the relative change, and the percentage change in the perimeter of the tire.

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Example 8: Changing Tires

We have

and

, so

. Since

, then the approximate change (absolute) is

The relative change is

The percentage change is

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Example 9: Estimating the Earth’s Surface Area

Suppose the earth were a perfect sphere and we determined its radius to be

miles. What effect would the tolerance of

miles have on our estimate of the earth’s surface area?

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Example 9: Estimating the Earth’s Surface Area

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Example 9: Estimating the Earth’s Surface Area

We can use algebra to determine the upper and lower values of the interval where the “true” surface area lies.

From this we see that the “true” surface area is within about

square miles of the measured value.

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Example 9: Estimating the Earth’s Surface Area

Now let’s compare this with the estimate using differentials.

We are looking for the approximation of

, where

and

Your textbook claims that this is an area about the same size as the state of Maryland.

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Example 10: Determining Tolerance

About how accurately should we measure the radius

of a sphere to calculate the surface area

within 1% of its true value?

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Example 10: Determining Tolerance

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Example 10: Determining Tolerance

We see from the previous slide that

Since

, then

The radius should be measured to within 0.5% of its true value.

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Example 11: Unclogging Arteries

In the late 1830s, the French physiologist Jean

Poiseuille

, discovered the formula we use today to predict how much the radius of a partially clogged artery has to be expanded to restore normal flow. His formula,

, says that the volume

of fluid flowing through a small pipe or tube in a unit of time at a fixed pressure is a constant time the fourth power of the tube’s radius

. How will a 10% increase in

affect

?

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Example 11: Unclogging Arteries

First recognize that the percentage change is

, and that we seek

.

So a 10% increase in the radius results in a 40% increase in blood flow.

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