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Newton’s Method and other Tangent Line approximations

Newton’s Method and other Tangent Line approximations. Tangent lines provide a useful representation of the curve if we stay close enough to the point of tangency. Newton’s Method is based on the assumption that the graph of f(x) and the tangent line cross the x-axis at about the same place.

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Newton’s Method and other Tangent Line approximations

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  1. Newton’s Method and other Tangent Line approximations

  2. Tangent lines provide a useful representation of the curve if we stay close enough to the point of tangency.

  3. Newton’s Method is based on the assumption that the graph of f(x) and the tangent line cross the x-axis at about the same place.

  4. Example: Calculate 3 iterations of Newton’s Method to approximate a zero of f(x) = x2 – 2. Use x1 = 1 as the initial guess. Step 1: Write the equation of the tangent line to f(x) at x = 1. Step 2: Find the x-intercept of the tangent line by plugging in y = 0. Now use the new x-value and do it again…this is the second iteration. Start over one more time with the new x-value for the third iteration.

  5. Your turn… Approximate the zero of the function f(x)= x3 – x + 1 using 3 iterations of Newton’s Method and x1 = -1

  6. In general, Each iteration…. 0 – f(x1) = f ’(x1)(x2– x1)

  7. To use your calculator… Y1 = f(x) and Y2 = f’(x) Then using recursion on your calculator Type in x1 ENTER Next,

  8. Use Newton’s Method to approximate the zeros of f(x) = 2x3 + x2 – x + 1 . Continue until 2 iterations differ by less than 0.0001

  9. Other tangent line approximations…

  10. Exploration: Graph y = (x2 + 0.0001)1/4 in the “zoom decimal” window. What appears to happen at x = 0? Show algebraically that the derivative of f is defined at x = 0. Write the equation of the tangent line to f at x = 0. What happens when you zoom in around the point (0,0.1)?

  11. Graph y = x2/3 in the standard window What happens when you zoom in around the point where x = 1? Differentiable curves are always locally linear. The “linearization” is the equation of the tangent line… What happens when you zoom in around the point where x = 0?

  12. Write the equation of the tangent line to at x = 0 Use the tangent line to find an approximation of What is the value of f(x)? What is the approximation error?

  13. Use the tangent line to find an approximation of What is the value of f(x)? What is the approximation error?

  14. As we move away from the point of tangency (center of the approximation) we lose accuracy.

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