AP Calculus BC â€“ 4.5: Linearization and Newtonâ€™s Method - 2

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AP Calculus BC â€“ 4.5: Linearization and Newtonâ€™s Method - 2. Goals : Find linearizations and use Newtonâ€™s method to approximate the zeros of a function. Estimate the change in a function using differentials. Linearization: As long as itâ€™s close, itâ€™s close.

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### 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.

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.

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:

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.

Absolute, Relative, and Percentage Change:

As we move from a to a nearby point a+dx:

True Estimated

Absolute Change

Relative Change

Percentage Change

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.