Mathematics: Maclaurin and Taylor Series

Chapter 1: Introduction

  • The chapter opens with a casual dialogue indicating some students discussing their transfer status.
    • Student A discusses their application to Purdue and the University of Washington.
    • Student B mentions they are also transferring and waiting for acceptance.
    • They express their opinions about their current institution and share frustrations.

First Lecture

  • The instructor welcomes students back and introduces the course material.
  • Reminds that today’s focus will be on computations, leading to impressive results.
  • Refers to the ebook, section 8.7
  • Topic for today: Maclaurin and Taylor Polynomials.
    • Introduces formulas from Brooke Taylor and Colin Maclaurin aimed to approximate non-polynomial functions.

Maclaurin Series

  • The formula shared for approximating functions:
    • f(0)+f′(0)x+f′′(0)x22!+f′′′(0)x33!+⋯+f(n)(0)xnn!f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots + \frac{f^{(n)}(0)x^n}{n!}
  • Components of the formula are defined as follows:
    • $f(0)$: Function evaluated at zero.
    • $f'(0)$: First derivative of the function at zero.
    • $f''(0)$: Second derivative of the function at zero.
    • Factorials expressed as:
    • n!=n(n−1)(n−2)(…)(3)(2)(1)n! = n(n-1)(n-2)(…)(3)(2)(1)
    • Special note that $0! = 1$ (defined by convention).
    • The initial terms represent the tangent line.

Historical Context

  • The derivation of the formula from Taylor and Maclaurin is noted:
    • Both mathematicians worked independently.
    • Emphasizes the polynomial's property—nonnegative integer powers of x.

Example of Maclaurin Polynomial

  • Function: Let f(x)=sin⁡(x)f(x) = \sin(x)
  • Goal: Derive a third degree Maclaurin polynomial.
  • Calculating derivatives of sine:
    • f′(x)=cos⁡(x),f′′(x)=−sin⁡(x),f′′′(x)=−cos⁡(x)f'(x) = \cos(x), f''(x) = -\sin(x), f'''(x) = -\cos(x)
  • Evaluating at $0$:
    • f(0)=sin⁡(0)=0f(0) = \sin(0) = 0
    • f′(0)=cos⁡(0)=1f'(0) = \cos(0) = 1
    • f′′(0)=−sin⁡(0)=0f''(0) = -\sin(0) = 0
    • f′′′(0)=−cos⁡(0)=−1f'''(0) = -\cos(0) = -1
  • Maclaurin polynomial terms evaluated:
    • The polynomial results in:
    • For f(0)=0,f(0) = 0, this vanishes.
    • For f′(0)=1,f'(0) = 1, include 1x1x.
    • For f′′(0)=0,f''(0) = 0, this vanishes too.
    • For f′′′(0)=−1f'''(0) = -1, include −x36- \frac{x^3}{6}.
  • Resulting expression for third degree Maclaurin polynomial: x−x36x - \frac{x^3}{6}

Significance of Sine Function

  • Domain of sine function: All real numbers (from trigonometric functions).
  • Discusses approximation of values such as sin⁡(14)\sin(\frac{1}{4}):
    • Not a special angle hence gets an approximation using the polynomial derived above.
  • Compares ways to do this manually vs. with calculators.

Generalization to Infinite Series

  • Discusses the need for higher-degree polynomials to refine accuracy:
    • For precise estimation, rely on an infinite degree polynomial or power series.
  • Identifies the formula for a sine function as an infinite series:
    • sin⁡(x)=∑n=0∞(−1)nx2n+1(2n+1)!\sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n + 1}}{(2n + 1)!}
  • The value of knowing the structure of these series extends the understanding of trigonometric functions and calculus.

Properties of Even and Odd Functions

  • Notes that sine is an odd function: sin⁡(−x)=−sin⁡(x)\sin(-x) = -\sin(x).
  • Patterns observed in the powers of the series align with the properties of these functions.

Converting to Sigma Notation

  • Discusses shifting representation of power series into $
    it$sigma$ notation.
  • General sigma notation for Taylor Series:
    • f(x)=∑n=0∞f(n)(0)n!xnf(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n

Chapter 2: Factorial Definitions

  • Factorial Definition: Expressed as n!=n(n−1)(n−2)…3<em>2</em>1n! = n(n-1)(n-2)…3<em>2</em>1.
    • Example:
    • (3!)=3imes2imes1=6(3!) = 3 imes 2 imes 1 = 6
    • Special note: 0!=10! = 1 (by definition).

Calculation of Derivatives for Cosine Function

  • Find the Maclaurin series for cos⁡(x)\cos(x) using derivative approach.
  • Successive derivatives:
    • f(x)=cos⁡(x)f(x) = \cos(x)
    • f′(x)=−sin⁡(x)f'(x) = -\sin(x)
    • f′′(x)=−cos⁡(x)f''(x) = -\cos(x)
    • f′′′(x)=sin⁡(x)f'''(x) = \sin(x)
    • f(4)(x)=cos⁡(x)f^{(4)}(x) = \cos(x)
  • Evaluate at x=0x = 0:
    • f(0)=1,f′(0)=0,f′′(0)=−1,f′′′(0)=0f(0) = 1, f'(0) = 0, f''(0) = -1, f'''(0) = 0.
  • Maclaurin polynomial for cosine obtained:
    • P4(x)=1−x22!+x44!P_4(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!}

Deriving Power Series for Cosine

  • Conjecture based on derivatives leads to the infinite series representation of the cosine function:
    • cos⁡(x)=∑n=0∞(−1)nx2n(2n)!\cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!}
  • Uses these polynomial constructions to highlight their convergence properties.
  • Connection to even functions through calculated powers.

Conclusion: Power Series

  • Introduces exponential function ekxe^{kx} with respective derivatives.

  • Extension leads towards determining polynomials for functions involving natural exponent and rates of change.

  • Ultimately brings attention back to the significance of Taylor's and McClaurin series for calculus and further applications in differential equations, physics, etc.

  • Final notes mention a consistent homework assignment, weekly assessments, and thorough exploration of functions for better understanding of both theoretical and practical applications in mathematics.