Taylor Polynomial Approximations A graphical demonstration

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).

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).

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).

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.

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.

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

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].

Approximating What happens on [-6,6]?

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