Calc 2 - 11/3

Taylor and Maclaurin Polynomials

  • Taylor Polynomial represents a function as a finite series around a center point a.
  • Maclaurin Polynomial is a specific case of the Taylor Polynomial where a = 0.
  • Nth degree Taylor/Maclaurin Polynomial approximates a function; useful due to approximation ability.
  • Error in approximation can be bounded using the (n+1)th derivative.

Example: Sine Function

  • For f(x)=sin⁡(x)f(x) = \sin(x):
    • Maclaurin series includes only odd powers: sin⁡(x)=x−x33!+x55!−⋯\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots.
    • Polynomials:
    • T1=xT_1 = x (degree 1)
    • T3=x−x33!T_3 = x - \frac{x^3}{3!} (degree 3)
    • T5=x−x33!+x55!T_5 = x - \frac{x^3}{3!} + \frac{x^5}{5!} (degree 5)
  • As nn increases, polynomial approximations become more accurate.

Intro to Differential Equations

  • Ordinary Differential Equation (ODE): relates a function y to its derivatives with one independent variable.
  • Solutions to ODEs produce a function, not just a number.
  • Autonomous vs Non-Autonomous ODEs:
    • Autonomous: Independent variable does not appear in the equation.
    • Non-Autonomous: Independent variable appears in the equation.
  • Example of ODE: dydx=y\frac{dy}{dx} = y (first order).

General vs Specific Solutions

  • General Solution demonstrates a family of solutions; can be narrowed down with initial conditions (initial value problems).
  • Example: y=Cex−x−1y = C e^{x} - x - 1 is a general solution; finding C gives a unique solution.
  • Equilibrium Solutions: solutions that remain constant; e.g., zero solution for certain ODEs.

Parameters in Differential Equations

  • Parameters provide information; in growth/decay equations like dydt=ky\frac{dy}{dt} = k y, k influences growth (positive) or decay (negative).
  • Understanding these metrics is critical for real-world applications.