1. Taylor Polynomial Approximations
A graphical demonstration

2. Best first order (linear) approximation at x=0.
Approximating
Best first order (linear) approximation at x=0.
OZ calls this straight line function P1(x).
Note: f(0)=P1(0) and f’(0)=P’1(0).

3. Best second order (quadratic) approximation at x=0.
Approximating
Best second order (quadratic) approximation at x=0.
OZ calls this quadratic function P2(x).
Note: f(0)=P2(0), f’(0)=P’2(0), and f’’(0)=P’’2(0).

4. Best third order (cubic) approximation at x=0.
Approximating
Best third order (cubic) approximation at x=0.
OZ calls this cubic function P3(x).
Note: f(0)=P3(0), f’(0)=P’3(0), f’’(0)=P’’3(0), and f’’’(0)=P’’’3(0).

5. Best sixth order approximation at x=0. OZ calls this function P6(x).
Approximating
Best sixth order approximation at x=0.
OZ calls this function P6(x).
P6 “matches” the value of f and its first 6 derivatives at x=0.

6. Best eighth order approximation at x=0. OZ calls this function P8(x).
Approximating
Best eighth order approximation at x=0.
OZ calls this function P8(x).
P8 “matches” the value of f and its first 8 derivatives at x=0.

7. Best tenth order approximation at x=0. This is P10(x).
Approximating
Best tenth order approximation at x=0.
This is P10(x).

8. Best hundedth order approximation at x=0. This is P100(x).
Approximating
Best hundedth order approximation at x=0.
This is P100(x).
Notice that we can’t see any difference between f and P100 on [-3,3].

9. Approximating
What happens on [-6,6]?

10. ---Different “centers”
Approximating
---Different “centers”
Third order approximation at x=0
Third order approximation at x= -1