for any function f (x), the tangent is a close approximation of the function for some small distance...

For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.

y

x0 x a

f x f aWe call the equation of the tangent the linearization of the function.

The linearization is the equation of the tangent line, and you can use the old formulas if you like.

Start with the point/slope equation:

1 1y y m x x 1x a 1y f a m f a

y f a f a x a

y f a f a x a

L x f a f a x a linearization of f at a

f x L x is the standard linear approximation of f at a.

Linearization

Example Finding a Linearization

Find the linearization of ( ) cos at / 2 and use it to approximate

cos 1.75 without a calculator.

f x x x

Since ( / 2) cos( / 2) 0, the point of tangency is ( / 2,0). The slope of the

tangent line is '( / 2) sin( / 2) 1. Thus ( ) 0 ( 1) .2 2

To approximate cos 1.75 (1.75) (1.75) 1.75 .2

f

f L x x x

f L

Important linearizations for x near zero:

1k

x 1 kx

sin x

cos x

tan x

x

1

x

1

21

1 1 12

x x x

13 4 4 3

4 4

1 5 1 5

1 51 5 1

3 3

x x

x x

f x L x

This formula also leads to non-linear approximations:

Differentials:

When we first started to talk about derivatives, we said that

becomes when the change in x and change in

y become very small.

y

x

dy

dx

dy can be considered a very small change in y.

dx can be considered a very small change in x.

Estimating Change with Differentials

Let be a differentiable function.

The differential is an independent variable.

The differential is:

y f xdxdy dy f x dx

Example Finding the Differential dy

5

Find the differential and evaluate for the given value of and .

2 , 1, 0.01

dy dy x dx

y x x x dx

45 2

5 2 0.01

0.07

dy x dx

dy

Examples

Find dy if

a. y = x5 + 37x

Ans: dy = (5x4 + 37) dx

b. y = sin 3x

Ans: dy = (3 cos 3x) dx

Differential Estimate of Change

Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is

df = f ‘(a) dx.

Example

The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in circle’s area A.

Compare this to the true change ΔA.

Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change?

2A r

2 dA r dr

2 dA dr

rdx dx

very small change in A

very small change in r

2 10 0.1dA

2dA (approximate change in area)

2dA (approximate change in area)

Compare to actual change:

New area:

Old area:

210.1 102.01

210 100.00

2.01

.01

2.01

Error

Actual Answer.0049751 0.5%