4.5 Linearization &

Newton’s Method

What you’ll learn about•Linear Approximation•Newton’s Method•Differentials•Estimating Change with Differentials

Linear Approximation

Any differentiable curve is “Locally Linear”

if you zoom in enough times.

Do Exploration 1: Appreciating Local Linearity (p 233)

A fancy name for the equation of the tangent line at a is “ The linearization of f at a

y – f(a) = f’(a)(x – a)

Definition - Linearization

If f is differentiable at x = a, then the equation of the tangent line.

L(x) = f(a) + f’(a) (x - a),

defines the linearization of f at a.

The approximation f(x) =L(x) is the standard linear approximation of f at a.

The point x = a is the center of the approximation.

Just Math TutoringYou Tube

What is Linearization?

Just math tutoringFinding the Linearization at a point

Followed by25) Linear Approximation

10 minutes total time needed – Watch if you miss class this day!

Example 1 Finding a Linearization

Find the linearization of at x = 0 (center of approximation) and use it to approximate without a calculator.

Then use a calculator to determine the accuracy of the approximation.

Point of tangency f ‘(0) = L(x) = Equation of the tangent line:

Evaluate L(.02)

Calculator approximation?

Approximation error:

xxf 1)(02.1

Practice: Find linearization L(x) of f(x) at x = a when and a = 2. How accurate is the approximation L(a + 0.1) ≈ f(a + 0.1)

Point of tangency f(2) = f ’(2)Tangent Line equation: L(x)

Evaluate |L(2.1) – f(2.1)|

Approximation error:

32)( 3 xxxf

Example 2: Find the linearization of f(x) = cos x at x = π/2 and use it to approximate cos 1.75 without a calculator. Then use a calculator to determine the accuracy of the approximation.

Point of tangency f (π/2) f ’(π/2)

Tangent Line equation: L(x)

Evaluate |L(1.75) – cos 1.75 by calculator |

Approximation error:

SummaryEvery function is “locally linear” about a

point x = a. If you evaluate the tangent line at x = a for points close to a, you will have a close approximation to the function’s actual value.

Homework

Page 242

Quick Review 1-10

Exercises 3, 5, 7

Warm Up

a) Find the linearization L(a) of f(a) at x = a for f(x) = ln(x+1), a = 0.

b) How accurate is the approximation

L(0.1) ≈ f(0.1)?

Steps

1) Using f(x), find the equation of a tangent line at some point (a, f(a)).

Find f(a) by plugging a into f(x).

Find the slope from f’(a).

L(x) = f(a) + f’(a) (x - a).

2) Evaluate L(x) for any x near a to get a close approximation of f(x) for points near a.

Example 3: Approximating Binomial Powers using the general formula

Use the formula to find polynomials that will approximate the following functions for values of x close to zero.

a) b) c) d)

How? Rewrite expression as (1 + x) k, Identify coefficients of x and k. Find L(x) = 1 + kx for each expression.

3 1 x

kxx k 1)1(

451 x 21

1

xx11

Example 4: Use linearizations to approximate roots. Find a) and b)

• Identify function: f(x) = • Let a be the perfect square closest to 123. Find L(x) at x = a.• Use L(x) to estimate• Error?

You try b.

3 123

x

123

123

FYI – not testedNewton’s Method for approximating

a zero of a function

Approximate the zero of a function by finding the zeros of linearizations converging to an accurate approximation.

Just Math Tutoring – Newton’s Method (7:29 minutes)

DifferentialsLet y = f(x) be a differentiable function.

Since dy/dx = f ’(x), the “differential dy” is defined as dy = f ’(x) dx,

(With dx as in independent variable and dy a dependent variable that depends on both x and dx.)

Although Liebniz did most of his calculus using dy and dx as separable entities, he never quite settled the issue of what they were. To him, they were “infinitesimals” – nonzero numbers, but infinitesimally small. There was much debate about whether such things could exist in mathematics, but luckily for the early development of calculus it did not matter: thanks to the Chain Rule, dy/dx behaved like a quotient whether it was one or not.

Example 6 Find the differential dy and evaluate dy for the given values of x and dx.

How? Find f ’(x), multiply both sides by dx, evaluate for given values.

a) y = x5 + 37x b) y = sin 3x c) x + y = xy x=1, dx = 0.01 x=π, dx = -0.02 x=2, dx = 0.05

You try:

1.0,2,1

22

dxxx

xy

More Notation…

)(')('

xfdx

dxxf

dx

dy

dxxfdf )('

)(' xfdx

df

Example 7 Finding Differentials of functions. Find dy/dx and multiply both sides by dx.

a) d (tan (2x)) b)

You try: d(e5x + x5)

)1

(xx

d

Estimating Change with Differentials

Suppose we know the value of a differentiable function f(x) at a point a and we want to predict how much this value will change if we move to a nearby point (a + dx).

If dx is small, f and its linearization L at “a” will change by nearly the same amount.

Since the values of L are simple to calculate, calculating the change in L offers a practical way to estimate the change in f.

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

estimate with the true change ∆A, and find the approximation error.

Area formula for a circle: A =

True change: f(10.1) – f(10) =

Estimated change: dA/dr = dA =

Approximation error: |∆A – dA| =

You try: f(x) = x3 - x, a = 1, dx = 0.1

Summary

Linearization: The equation of a tangent line to f at a point a will give a good approximation of the value of a function f at a.

The Linearization of (1 + x)k = 1 + kx

Newton’s Method is used to find the roots of a function by using successive tangent line approximations, moving closer and closer to the roots of f if you start with a reasonable value of a.

Differentials: Differentials simply estimate the change in y as it relates to the change in x for given values of x. We learned how to estimate with linearizations, differentials are simply a more efficient method of finding change.

Homework

Page 242

Exercises 1, 4, 19, 22,

25, 28, 31, 34-40