4.5: Linear Approximations, Differentials and Newton’s Method

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

Start with the point/slope equation: linearization of f at a is the standard linear approximation of f at a. The linearization is the equation of the tangent line, and you can use the old formulas if you like.

Important linearizations for x near zero: This formula also leads to non-linear approximations:

dy can be considered a very small change in y. 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. 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:

Example: Consider a circle of radius 10. If the radius increases by 0 Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change? very small change in r very small change in A (approximate change in area)

(approximate change in area) Compare to actual change: New area: Old area:

Newton’s Method Finding a root for: We will use Newton’s Method to find the root between 2 and 3.

Guess: (not drawn to scale) (new guess)

Guess: (new guess)

Guess: (new guess)

Guess: Amazingly close to zero! This is Newton’s Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) This is a recursive algorithm because a set of steps are repeated with the previous answer put in the next repetition. Each repetition is called an iteration.

Guess: Newton’s Method: Amazingly close to zero! This is Newton’s Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) This is a recursive algorithm because a set of steps are repeated with the previous answer put in the next repetition. Each repetition is called an iteration.

This technique is sometimes called the Newton-Raphson method. Actually, the techniques that we will see in the next slides were developed by an English mathematician named Joseph Raphson, and published in 1690. Isaac Newton developed a similar formula in 1671, but it was not published until 1736 and is not as easy to use as Raphson’s method. Very little is known about Joseph Raphson, who lived approximately 1648-1715. He was acquainted with Newton, so he may have gotten the idea from Newton. Anyway, Newton tends to get the credit.

Find where crosses .

There are some limitations to Newton’s method: Looking for this root. Bad guess. Wrong root found Failure to converge

There are much easier methods of finding roots using our calculator. We learn Newton’s method of finding roots for historical interest and to deepen our appreciation of calculus. Newton’s method is sometimes tested on the AP exam, and will therefore sometimes appear on a test in this class. There are much easier methods of finding roots using our calculator. Statue of Isaac Newton as a college student in Trinity College Chapel at Cambridge University, Cambridge, England

Approximate the negative root of: Example: Approximate the negative root of: If you have the function graphed, you can find the roots by using: F5 Math 2 Zero Use the arrow keys to select the lower and upper bounds, and press ENTER each time. You would need to find each root separately.

p Example: Approximate the negative root of: An even quicker way to find roots is to use the following when in the home screen: F2 Algebra 4 Zeros Press ENTER p

Homework: 4.5a 4.5 p242 5,7,11 4.4 p228 45 3.6 p153 11,25,41,55   4.5b 4.5 p242 5,8,14,18,25,33 3.6 p153 17,33,47,63 4.5c 4.5 p242 19,27,36,50 3.6 p153 29; p156 1,2,3,4