2.9 Linear Approximations and Differentials

2.9 Linear Approximations and Differentials - estimate x2 =...

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Linear Approximations and Differentials Section 2.9
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Defn: Linearization If f is differentiable at x = a , then the approximating function Is the linearization of f at a . ( ) ( ) ( )( ) = + - L x f a f a x a
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Finding a Line Linearization l The most important linear approximation of for roots and powers is (1 ) 1 k x kx + ≈ +
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Newton’s Method is a technique for approximating the real zeros of a function. It uses tangent lines to approximate the graph of the function near its x-intercepts.
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Consider a function f that is continuous on [a,b] and differentiable on (a,b). If f(a) and f(b) differ in sign, by the Intermediate Value Thm., f must have at least one zero in (a,b). Estimate where you think the zero should occur: x = x 1 x x 1 2 (x , f(x ) 1 1
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The tangent line passes thru the point (x1 , f(x1) ) with a slope of f ’(x1 ). y – f(x1) = f ’(x1)(x – x1) y = f ’(x1)(x – x1) + f(x1) x-intercept: Let y = 0 and solve for x. 0 = f ’(x1 )x – f ’(x1)x + f(x1) f ’(x )x = f ’(x1 )x - f(x1) x = x - f(x1) f ’(x1)
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From the initial estimate x1, you obtain a new
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Unformatted text preview: estimate: x2 = x1 – f(x1) f ’(x1) Repeated application of this process is called Newton’s Method. Each successive application of this procedure is called an iteration . Newton’s Method for Approximating the Zeros of a Function 1. Make an initial estimate x1 that is “close to” c. (A graph is helpful.) 2. Use the first approximation to get a second, the second to get a third, and so on, using xn+1 = xn – f(xn) f ’(xn) Defn: Differentials Let y = f(x) be differentiable function. The differential dx is an independent variable. The differential dy is ( ) dy f x dx ′ = Differential Estimate of Change ( ) df f a dx ′ = Change in f True Estimated Absolute Change Relative Change Percentage Change ( ) ( ) f f a dx f a ∆ = +-( ) df f a dx ′ = ( ) f f a ∆ ( ) df f a 100 ( ) f f a ∆ • 100 ( ) df f a • Joke Time Who was the dancing Vice-President in 2000? Al-gore rhythum What does a lumberjack do to trees? axi-oms...
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This note was uploaded on 12/01/2011 for the course MATH 112 taught by Professor Jarvis during the Fall '08 term at BYU.

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2.9 Linear Approximations and Differentials - estimate x2 =...

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