Lecture Notes: Taylor Polynomials

Introduction to Taylor Polynomials

• Taylor Form of a Line:
  • Equation: ( L = Y_1 + M(X - X_1) )
  • Represents a first-degree polynomial, part of the Taylor polynomials family.
  • Written in increasing order of degree.

Relationship with Derivatives

• Involves input ( X_1 ), output ( Y_1 ), and the slope at that point.
• Example: If ( f(a) = b ), then ( b = f'(a) + (x-a) ).

Developing Higher Degree Taylor Polynomials

• Possible to develop quadratic, cubic, etc., Taylor polynomials.
• Example: Approximate ( \sin(0.2) ) using Taylor polynomials.
• Sine as a Transcendental function:
  • Like all trig functions, ( e^x ), and ( \ln(x) ).

Calculating Approximations

• Calculators use Taylor polynomials for trigonometric calculations.
  • Example: Tangent line approximation for ( \sin(0.2) ).

Steps to Approximate ( \sin(0.2) ):

1. Find the Point:
   • ( f(0) = \sin(0) = 0 )
   • Point: Origin ((0,0))
2. Find the Slope:
   • Derivative of sine is cosine.
   • ( f'(0) = \cos(0) = 1 )
3. Equation of Tangent Line:
   • ( L(x) = 0 + 1(x-0) = x )
4. Approximate ( \sin(0.2) ):
   • ( f(0.2) \approx L(0.2) = 0.2 )

Accuracy and Concave Down Analysis

• Determine if approximation is over or under the actual function.
• ( \sin(x) ) is concave down from 0 to 0.2, verifying that ( 0.2 ) is an overestimate.

Euler's Method and Higher Order Taylor Polynomials

• Euler's method offers better approximation by breaking steps into smaller increments.
• Higher-order Polynomials:
  • Cubic polynomials approximate better than linear.
  • Higher degree polynomials (cubic, fifth, seventh, etc.) provide even better approximations.

Visual Representation of Polynomial Approximations

• Graphs demonstrate various polynomial degrees (cubic, fifth, seventh, etc.) approximating sine.
• Odd degree polynomials have opposite end behaviors, useful for approximating odd functions like sine.

Taylor Polynomial Formula

• Key Formula for nth Degree Polynomial:
  • ( P_n(x) = f(c) + \frac{f'(c)}{1!}(x-c) + \frac{f''(c)}{2!}(x-c)^2 + ... )
  • nth term: ( \frac{f^{(n)}(c)}{n!}(x-c)^n )
• Brooke Taylor (English Mathematician): Developed the concept of Taylor Polynomials.

Maclaurin Series

• Special case of Taylor series where the center is 0.
• Useful when approximating around the origin.

Conclusion

• Taylor polynomials are essential for approximating functions.
• Important to understand the formula and its application for higher accuracy.