July 08, 2026 - Calculus 2 - Taylor Series and Polar Coordinates Overview (Concise)

Taylor Polynomials and Taylor Series

• Order of Derivatives: The order of a differential equation corresponds to the degree of the derivative involved (e.g., first-order involves a first derivative).
• Definition of Taylor Series: If a function $f$ has derivatives of all orders, the series is represented by: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n$
• Maclaurin Series: A Taylor series centered at $a = 0$.
• Taylor Polynomials ($P_k$): These are the $k$-th partial sums of the Taylor series:
  • $P_0 = f(a)$
  • $P_1 = f(a) + f'(a)(x-a)$
  • $P_2 = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2$
• Notation Note: Derivatives are denoted with parentheses in the exponent (e.g., $f^{(n)}$) to distinguish them from standard exponents.

Calculus and Convergence of Series

• Example: $f(x) = \ln(x)$ at $x = 1$:
  • $P_1 = x - 1$
  • $P_2 = (x - 1) - \frac{1}{2}(x - 1)^2$
  • $P_3 = (x - 1) - \frac{1}{2}(x - 1)^2 + \frac{1}{3}(x - 1)^3$
  • General Series: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}(x-1)^n$
  • Radius of Convergence: Using the Ratio Test, $R = 1$. The interval of convergence is $(0, 2]$.
• Example: $f(x) = x^{-2}$ at $x = 1$:
  • Derivatives follow the pattern: $f'(x) = -2x^{-3}$, $f''(x) = 6x^{-4}$, $f'''(x) = -24x^{-5}$.
  • Taylor Series: $\sum_{n=0}^{\infty} (-1)^n(n+1)(x-1)^n$
  • Radius of convergence is $R = 1$.
• Convergence Speed: The accuracy of a series depends on how fast its derivatives grow. Functions like $\ln(x)$ converge slowly because they do not curve like polynomials.

Polar Coordinate System

• Definition: Describes space via radius ($r$) and angle ($\theta$) rather than $(x, y)$.
• Conversion Formulas:
  • $x^2 + y^2 = r^2$
  • $x = r \cos(\theta)$
  • $y = r \sin(\theta)$
  • $\tan(\theta) = \frac{y}{x}$
• Coordinate Examples:
  • $(1, 1)$ in Cartesian is $(\sqrt{2}, \frac{\pi}{4})$ in Polar.
  • $(3, 4)$ in Cartesian is $(5, \arctan(\frac{4}{3}))$ in Polar.
  • $(5\sqrt{3}, 5)$ in Cartesian is $(10, \frac{\pi}{6})$ in Polar.
• Polar Curves:
  • Line: Constant $\theta$.
  • Circle: $r = a \cos(\theta) + b \sin(\theta)$.
  • Spiral: $r = a\theta$.
  • Cardioid: $r = a(1 \pm \sin(\theta))$ or $r = a(1 \pm \cos(\theta))$.
  • Limacon: Curves with inner loops.
  • Roses: Multi-petaled curves; number of petals depends on the coefficient of $\theta$.

Area in Polar Coordinates

• Integration Logic: Instead of using rectangles as in Cartesian coordinates, polar integration utilizes sectors of circles.
• Sector Area: The area of a sector with angle $\theta$ is $\frac{1}{2} r^2 \theta$.
• Area Formula: To find the area under a curve from angle $\alpha$ to $\beta$: $\text{Area} = \int_{\alpha}^{\beta} \frac{1}{2} [f(\theta)]^2 \,d\theta$

Questions & Discussion

• Notation Clarification: A student asked about identifying derivatives vs. exponents. The instructor clarified that exponents for derivatives are placed within parentheses between the function symbol and the argument.
• Convergence and Centers: Discussion occurred regarding shifting the center $a$ to calculate values outside the initial interval of convergence. For example, to find values near $x = 3$ for $\ln(x)$, the series must be re-centered close to that value.
• Analogies: The instructor compared the layout of Tallahassee (with roads like Appalachic Parkway, Mkasuki, and Thomasville radiating out) to a polar coordinate system.
• Logistics: Test 4 is scheduled for Wednesday and will cover Taylor series, calculus, and polar coordinates.