Linear Approximation and Differentials Lesson 3.9.

Linear Approximation and Differentials

Lesson 3.9

Tangent Line Approximation

• Consider a tangent to a function at a point where x = a

• Close to the point, the tangent line is an approximation for f(x)

•

a

f(a)

y=f(x)

•The equation of the tangent line:y = f(a) + f ‘(a)(x – a)

•The equation of the tangent line:y = f(a) + f ‘(a)(x – a)

Tangent Line Approximation

• We claim that

• This is called linearization of the function at x=a.

• Recall that when we zoom in on an interval of a differentiable function far enough, it looks like a line.

1 1( ) ( ) '( )( )f x f a f a x a

New Look at

• dy = rise of tangent relative to x = dx y = change in y that occurs relative to

x = dx

dy

dx

x

x = dx

• dy y

•• x + x

New Look at

• We know that

then

• Recall that dy/dx is NOT a quotient it is the notation for the derivative

• However … sometimes it is useful to use dy and dx as actual quantities

dy

dx

'( )y

f xx

'( )y f x x

The Differential of y

• Consider

• Then we can say

this is called the differential of y the notation is d(f(x)) = f ’(x) * dx it is an approximation of the actual change of y

for a small change of x

'( )y dy

f xx dx

'( )dy f x dx y

Animated Graphical View

• Note how the "del y" and the dy in the figure get closer and closer

Example: Find the differential of the function:

Remember a differential is dy = f ‘(x)· dx (assuming y=f(x))

DifferentialsDifferentials

2)35(

)43(5)35(3

x

xx

dx

dy2)35(

2015915

x

xx

2)35(

11

xdx

dy

35

43

x

xy

dxx

dy2)35(

11

Try It Out

• Note the examples and rules for differentialson page 238.

• Find the differential of:

1) y = 3 – 5x2

2) f(x) = xe-2x

DifferentialsLet ( ) be a differentiable function. The is an

independent variable. The is '( ) .

y f x

dy f x dx

differential

differential

dx

dy

Example Finding the Differential dyFind the differential dy and evaluate dy for the given value of x and dx.

01.0 ,1 ,25 dxxxxy

dxxdy 25 4

01.0215 4 dy

07.0

Differential Estimate of Change (needed for #34 and #37 on the homework)

Three types of changes that can be found. Absolute, Relative and Percent Change

Absolute

True

Estimated

afxafxf

dxxfdf

Relative and Percent Change

True Estimate

Relative

Percent .

afxf

afdf

100

af

xf 100af

df% %

Example: A company makes ball-bearings with a radius of 2 inches. The measurements are considered to be correct to within 0.01 in. Use differentials to determine the approximate error in volume, approximate relative error in volume, AND approximate relative error percentage in volume. To solve this problem, we will use the equation:

∆V ≈ dV = V ‘(r)· dr

DifferentialsDifferentials

3

3

4 sphere a of volume)( rrV

24 )(

rdr

rdV drrrdV 24)(

01.0)2(4 2 dV35027.016.0 indV <- Approximate error in

volume

Example: A company makes ball-bearings with a radius of 2 inches. The measurements are considered to be correct to within 0.01 in. Use differentials to determine the approximate error in volume, approximate relative error in volume, AND approximate relative error percentage in volume.The estimate of relative error is given by:

DifferentialDifferential

afdf

The estimate percentage of relative error is given by: %100

af

df

015. )2(

34

16.

3

%5.1100%* )2(

34

16.

3

<- Approximate relative error in volume

<- Approximate relative error percentage in volume

Differentials for Approximations

• Consider

• Use

• Then with x = 25, dx = .3 obtain approximation

25.3

( )

1( ) '( )

2

f x x

f x x f x f x dx x dxx

Propagated Error

• Consider a rectangular box with a square base Height is 2 times length

of sides of base Given that x = 3.5 You are able to measure with 3% accuracy

• What is the error propagated for the volume?

xx

2x

Propagated Error

• We know that

• Then dy = 6x2 dx = 6 * 3.52 * 0.105 = 7.7175This is the approximate propagated error for the volume

32 2

3% 3.5 0.105

V x x x x

dx

Propagated Error

• The propagated error is the dy sometimes called the df

• The relative error is

• The percentage of error relative error * 100% (in this case, it would be

9%)

7.71750.09

( ) 85.75

dy

f x