4.4 Linearization and Differentials

Linearization• Linearization uses the concept that when

you zoom in to the graph at a point the y-values of the curve are very close to the y-values of the tangent line

3

(3, f(3) 1f x x 1 5

4 4y x

Linearization• We can approximate

without a calculator by using the tangent line to f(x) at (3, f(3)) which is

3

(3, f(3) 1f x x 1 5

4 4y x

3.2 3 .2 3.2f

1 5

4 4y x

Linearization• We can approximate

without a calculator by using the tangent line

3.2 1 3.2 4.2f

1 5

4 4y x

1 53.2 3.2

4 4.25 3.2 1.25

2.05

y

3.2 1 3.2

4.2

2.049

f

Approximate value

Actual value

LinearizationNote that the closer x is to 3 the better the

approximation

x approximation, y(x) actual value, f(x)

3.2 2.05 f(3.2) = 2.0493

3.1 2.025 f(3.1) = 2.0248

3.01 2.0025 f(3.01) = 2.002498

3.001 2.00025 f(3.001) = 2.00024998

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

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.

To find the linearization of f(x) at the point (a, f(a))

Important linearizations for x near zero:

1k

x 1 kx

sin x

cos x

tan x

x

1

x

f x L x

Example

a) Find the linearization L(x) of f(x) at x = a.

b) Compare f(a + 0.1) to L(a + 0.1)

2

6@ 2f x a

x

Example

• Find the linearization of

at x = 0. Use it to approximate:

( ) 1f x x

) 1.2

) 1.05

) 1.005

a

b

c

Example

• Find the linearization of y = cosx at x = 0. Use it to approximate cos(0.5).

Example

• Choose a linearization with center not at a but nearby so that the derivative will be easy to evaluate.

3

2

1) ; 8.5

2) ( ) 1 2 ; 0.9

3) ( ) sin ; 3.2

f x x a

f x x a

f x x a

Example

• Use linearization to find the value of

at x = 0.12 to the nearest hundredth. 4 sinf x x

Hw: p. 229/1-5,11,13,14,37

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.

Let be a differentiable function.

The differential is an independent variable.

The differential is:

y f xdxdy dy f x dx

dy is dependent on x and dx

Examples: find the differential

51) 37

2) sin3

3) tan2

4)1

y x x

y x

d x

xdx

Examples: a) Find dyb) Evaluate dy for the given values of x and

dx

1. y = x2 – 4x ; x = 3, dx = 0.5

2. y = xlnx; x = 1, dx = 0.1

• Differentials can be used to estimate the change in y when x = a changes to

x = a + dx

True Change Estimated Change y f f a dx f a 'dy df f a dx

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

Original Area

Error

Actual Answer.0049751 0.5%

0.01%.0001.01

100

Example

• The earth’s radius is 3959 + 0.1 miles. What effect does a tolerance of + 0.1 miles have on the estimate of the earth’s surface area?

Example

• Estimate the volume of material in a cylindrical shell with a height of 15m, radius 10m and shell thickness 0.2m.

HW: p.229/19-26,31-35,38,39,42