Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1 A) -2 C) 1/2 E) 6 B) 1/6 D) 2.

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Presentation transcript:

Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1 A) -2 C) 1/2 E) 6 B) 1/6 D) 2

Problem of the Day No calculator! What is the instantaneous rate of change at x = 2 of f(x) = x2 - 2 ? x - 1 A) -2 C) 1/2 E) 6 B) 1/6 D) 2 (take derivative and then substitute in)

Newton's Method A technique for approximating the real zeroes of a function using tangent lines

Newton's Method A technique for approximating the real zeroes of a function using tangent lines If the function is continuous on [a, b] and differentiable on (a, b) and if f(a) and f(b) differ in sign then by the ___________________________ f must have at least one zero in (a, b) a b y x

Newton's Method A technique for approximating the real zeroes of a function using tangent lines If the function is continuous on [a, b] and differentiable on (a, b) and if f(a) and f(b) differ in sign then by the Intermediate Value Theorem f must have at least one zero in (a, b) a b y x

Newton's Method A technique for approximating the real zeroes of a function using tangent lines Visual Calculus Link

Newton's Method A technique for approximating the real zeroes of a function using tangent lines In summary, the x-intercept will be approximately xn+1 = xn - f(xn) f '(xn)

Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1. Iteration xn f(xn) f '(xn) f(xn) f '(xn) xn - f(xn) f '(xn)

Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1. Iteration xn f(xn) f '(xn) f(xn) f '(xn) xn - f(xn) f '(xn)

Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1. Iteration xn your equation nderiv(Y1,x,x) x - xn - f(xn) f '(xn) Y1 = Ti-84 Y2 = Y3 = Ti-Nspire f1 = your equation f2 = f3 =

Newton's Method will not always produce an answer, such as when 1) the derivative within the interval is zero at any point 2) functions similar to f(x) = x1/3

You can test for convergence to see if it will work with the following formula f(x) f ''(x) [f '(x)]2 < 1

Another precaution Do not round in intermediary steps. Let your calculator carry the numbers.