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Linear Approx, Differentials, Newton S Method
1. 4.5: Linear Approximations, Differentials and Newtonâs Method Greg Kelly, Hanford High School, Richland, Washington
2. 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.
3. The linearization is the equation of the tangent line, and you can use the old formulas if you like. Start with the point/slope equation: linearization of f at a is the standard linear approximation of f at a.
5. 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 .
6. Let be a differentiable function. The differential is an independent variable. The differential is:
7. 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 A very small change in r (approximate change in area)
13. Guess: Amazingly close to zero! This is Newtonâs Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) It is sometimes called the Newton-Raphson method 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 .
14. This is Newtonâs Method of finding roots. It is an example of an algorithm (a specific set of computational steps.) It is sometimes called the Newton-Raphson method Guess: Amazingly close to zero! 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 . Newtonâs Method:
16. There are some limitations to Newtonâs method: Wrong root found Looking for this root. Bad guess. Failure to converge
17. Newtonâs method is built in to the Calculus Tools application on the TI-89. Of course if you have a TI-89, you could just use the root finder to answer the problem. The only reason to use the calculator for Newtonâs Method is to help your understanding or to check your work. It would not be allowed in a college course, on the AP exam or on one of my tests.
18. Now letâs do one on the TI-89: APPS Select and press . Calculus Tools ENTER If you see this screen, press , change the mode settings as necessary, and press again. ENTER APPS Approximate the positive root of:
19. Now letâs do one on the TI-89: Select and press . Calculus Tools Press (Deriv) Enter the equation. (You will have to unlock the alpha mode.) Set the initial guess to 1. Approximate the positive root of: Set the iterations to 3. APPS ENTER F2 Press (Newtonâs Method) 3 Press . ENTER
20. Press to see the summary screen. ESC Press to see each iteration. ENTER
21. ï° Press and then to return your calculator to normal. ESC HOME