Taylor Polynomial Approximations

1
Taylor Polynomial Approximations

• A graphical demonstration

2

• Best first order (linear) approximation at x0.
• OZ calls this straight line function P1(x).
• Note f(0)P1(0) and f(0)P1(0).

3

• Best second order (quadratic) approximation at
  x0.
• OZ calls this quadratic function P2(x).
• Note f(0)P2(0), f(0)P2(0), and
  f(0)P2(0).

4

• Best third order (cubic) approximation at x0.
• OZ calls this cubic function P3(x).
• Note f(0)P3(0), f(0)P3(0), f(0)P3(0),
  and f(0)P3(0).

5

• Best sixth order approximation at x0.
• OZ calls this function P6(x).
• P6 matches the value of f and its first 6
  derivatives at x0.

6

• Best eighth order approximation at x0.
• OZ calls this function P8(x).
• P8 matches the value of f and its first 8
  derivatives at x0.

7

• Best tenth order approximation at x0.
• This is P10(x).

8

• Best hundedth order approximation at x0.
• This is P100(x).
• Notice that we cant see any difference between f
  and P100 on -3,3.

9

• What happens on -6,6?

10
---Different centers
Third order approximation at x0
Third order approximation at x -1