4.5 linearization and newton’s method newtons... · 4.5 linearization and newton’s method. ......

4.5 Newtons method complete.notebook September 07, 2018

Locally Linearderivative – tangent line – useful

representations of a curve. If we keep really close to the point of tangency, we say differentiable curves are locally linear.

4.5 Linearization and Newton’s Method


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

L(x)=f(a)+f’(a)(xa)

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.


Finding a Linearization Ex.1 Find the linearization of f(x)=sinx at x=π/3, and use it to approximate sin(1.25) without a calculator.

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


Approximating Roots Ex. 2 Use linearizations to approximate√26 and ∛26


Approximating Binomial Powers General linearization or binomials

(1+x)k≈1+kx

This is for very small values of x.

Ex. 3 Using the formula above, find a linear approximation for ∛(1x)

Try Using the formula above, find a linear approximation for √(1+5x 4)


Newton’s MethodNewton’s Method is a numerical technique for approximating a zero of a function using linearizations.


Procedure for Newton’s Method

• Guess a first approximation to the solution of the equation f(x)=0. Draw a sketch to help get you started.• Use the first approx to get the second, use the 2nd to get the 3rd• Use Newton’s formula:


Ex. 4 Use Newton’s method to solve

x3 + 7x + 3=0

A word of caution – if your initial guess is way off, you may end up with the wrong root, or indeed no root at all :(


Differentials – how small changes in x affect the overall function.

Let y=f(x) be a differentiable function. The differential dx is an independent variable. The differential dy is dy=f’(x)dx.


Ex. 5

Find the differential dy and evaluate dy for the given value of x and dx when x2y=3xy, x=4 and dx=0.03.

TRY Repeat the above for y=x3+29x, x=2 and dx=0.02.


Finding Differentials of FunctionsEx. 6

Estimating ChangeEx. 7 The length of a side of a cube increases from 5in to 5.01in. Use dV to estimate the change in the cube’s volume. Compare this estimate with the true change.


Absolute, relative and %age ChangeAs we move away from a point a to a nearby point a+dx, the function can be described in three ways.

True Estimated

Absolute

Relative

Percentage

Change

Δf=f(a+dx)f(a) df=f ' (a)dx


Ex. 8

Inflating a bicycle tire changes its radius from 12” to 13”. Use differentials to estimate the absolute, relative and percentage change in the perimeter of the tire.


Ex. 9 About how accurately must you measure the radius of a sphere to calculate the surface area S=4πr2 within 1% of its true value?

Ex. 10

Using the formula V=kr4 (which is used to determine volume of fluid flow through tubes) determine how a 10% increase in r affects V.

4.5 linearization and newton’s method newtons... · 4.5 linearization and newton’s method. ......

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