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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!.
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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 y x a b 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)
Newton's Method A technique for approximating the real zeroes of a function using tangent lines y x a b 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 Theoremf must have at least one zero in (a, b)
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. f(xn) f '(xn) f(xn) f '(xn) xn - xn f '(xn) f(xn) Iteration
Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1. f(xn) f '(xn) f(xn) f '(xn) xn - f '(xn) xn f(xn) Iteration -.5 .083 .002451 2 3 2.83 1.5 1.416 1.414216 1 2 3 1 1.5 1.416 -1 .25 .006945
Calculate 3 iterations of Newton's Method to approximate a zero of f(x) = x2 - 2 starting with x = 1. f(xn) f '(xn) xn - Ti-84 xn Iteration Y1 = your equation Y2 = nderiv(Y1,x,x) x - 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.