1. Warm Up Write an equation of the tangent line to the curve at the given point. 1)f(x)= x 3 – x + 1 where x = -1 2)g(x) = 3sin(x/2) where x = π/2 3)h(x) = where x = 0

2. Newton’s Method and other Tangent Line approximations

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

4. 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.

5. Example: Calculate 3 iterations of Newton’s Method to approximate a zero of f(x) = x 2 – 2 (the value of ). Use x 1 = 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.

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

7. In general, Each iteration…. 0 – f(x 1 ) = f ’(x 1 )(x 2 – x 1 )

8. To use your calculator… Y 1 = f(x) and Y 2 = f’(x) Then using recursion on your calculator Type in x 1 ENTER Next,

9. Use Newton’s Method to approximate the zeros of f(x) = 2x 3 + x 2 – x + 1. Continue until 2 iterations differ by less than 0.0001

10. Other tangent line approximations…

11. Exploration: Graph y = (x 2 + 0.0001) 1/4 in the “zoom decimal” window. What appears to happen at x = 0? What happens when you zoom in around the point (0,0.1)? Show algebraically that the derivative of f is defined at x = 0. Write the equation of the tangent line to f at x = 0.

12. Graph y = x 2/3 in the standard window What happens when you zoom in around the point where x = 1? What happens when you zoom in around the point where x = 0? Differentiable curves are always locally linear. The “linearization” is the equation of the tangent line…

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

14. What is the value of f(x)? What is the approximation error?

15. Let f(x) = x 4 – 1. 1) Write the equation of the tangent line T to the graph of f at the point where x = -1. 2) Use this linear approximation to complete the table. x-1.1-1.01-.99-.9 f(x) T(x)

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