1. 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

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)

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

4. 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

5. 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

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

7. 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)

8. 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)

9. 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) 123123 1 1.5 1.416.25.006945 2 3 2.83 -.5.083.002451 1.5 1.416 1.414216

10. 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 =

11. 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

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

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