Linearization, Newton’s Method Section 4.5a

Starting with a Calculator Exploration… Let 1. Find the line tangent to the graph of at . 2. Set and . Zoom in on the two graphs at . The line tangent to a curve lies close to the curve near the point of tangency…

y x Slope = Point-slope equation for the tangent line: x Point-slope equation for the tangent line: Thus, the tangent is the graph of the linear function  As long as the line remains close to the graph of f, L(x) gives a good approximation of f (x)…

Definition: Linearization If is differentiable at x = a, then the approximating function is the linearization of f at a. The approximation is the standard linear approximation of at a. The point x = a is the center of the approximation.

Practice Problems Let’s take a look at the graph!!! For each of the following, find the linearization of the function at the given point. Let’s take a look at the graph!!!

Practice Problems Again, consider the graph… For each of the following, find the linearization of the function at the given point. Again, consider the graph…

Newton’s Method Newton’s method is a numerical technique for approximating a zero of a function with zeros of its linearizations… (note: this is a technique utilized by many calculators…)

y x The process of Newton’s method: To find a solution of an equation f(x) = 0, we begin with an initial estimate x , found either by looking at the graph or by guessing. 1 y Root sought x 1st Approximations

y x The process of Newton’s method: Then we use the tangent curve at that point to approximate the curve  the point where the tangent hits the x-axis gives the next approximation, x . y 2 Root sought x 2nd 1st Approximations

y x The process of Newton’s method: We continue this process, using each approximation to generate the next  usually, we will eventually get close enough to the actual zero to stop. y Root sought x 4th 3rd 2nd 1st Approximations

y Root sought x Point-slope equation for the tangent to the curve at

y Root sought x We can find where it crosses the x-axis by setting y = 0:

y x This value of x is the next approximation!!! Root sought Solve for x:

Procedure for newton’s method 1. Guess a first approximation to a solution of the equation . A graph may help. 2. Use the first approximation to get a second, the second to get a third, and so on, using the formula:

Practice Problems Use Newton’s method to estimate the solution. Make your answer accurate to 6 decimal places. The graph of the function suggests that x = – 0.3 is a good first approximation: 1

Practice Problems Final Answer: Use Newton’s method to estimate the solution. Make your answer accurate to 6 decimal places. Let and Store into Then store into and press the “Enter” key over and over. Watch as the numbers converge to the zero of f. Use different values for your initial approximation, and repeat steps 2 and 3… Final Answer: