Linear Approximation and Differentials

Lesson 4.8

Tangent Line Approximation

• Consider a tangent to a function at a point 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)

Tangent Line Approximation

• We claim that

• This is called linearization of the function at the point a.

• Recall that when we zoom in on an interval of a 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

dydx

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

dydx

'( )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 dyf 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

Try It Out

• Note the rules for differentialsPage 274

• Find the differential of

3 – 5x2

x e-2x

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 23% 3.5 0.105

V x x x xdx

Propagated Error

• The propagated error is the dy sometimes called the df

• The relative error is

• The percentage of error relative error * 100%

7.7175 0.09( ) 85.75dyf x

Assignment

• Lesson 4.8• Page 276• Exercises 1 – 45 odd