1 / 7

AP Calculus BC – 4.5: Linearization and Newton’s Method - 2

AP Calculus BC – 4.5: Linearization and Newton’s Method - 2. Goals : Find linearizations and use Newton’s method to approximate the zeros of a function. Estimate the change in a function using differentials. Linearization: As long as it’s close, it’s close.

oleg-beck
Download Presentation

AP Calculus BC – 4.5: Linearization and Newton’s Method - 2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. AP Calculus BC – 4.5: Linearization and Newton’s Method - 2 Goals: Find linearizations and use Newton’s method to approximate the zeros of a function. Estimate the change in a function using differentials.

  2. Linearization: As long as it’s close, it’s close. If f is differentiable at x = a, then the approximating function is 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.

  3. Newton’s Method (Newton-Raphson Method): Newton’s Method is a numerical technique for approximating a zero of a function with zeros of its linearizations. Procedure: 1.Guess a first approximation to a solution of the equation f(X) = 0. A graph of y = f(X) may help. 2. Iterate. Use the first approximation to get a second, and so on, using:

  4. Newton’s Method on the calculator:

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

  6. Absolute, Relative, and Percentage Change: As we move from a to a nearby point a+dx: True Estimated Absolute Change Relative Change Percentage Change

  7. Assignments and Notes: • HW 4.5A: #3, 5-9, 11, 14, 15, 18. • HW 4.5B: #19, 22, 25, 27, 30, 33, 36, 39, 44, 50, 51. Look at #52. • Test next Thursday.

More Related