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

Derivative Orders: The order of a differential equation refers to the degree of the derivatives involved. For Taylor series, it is assumed that the function $f$ has derivatives for all orders.
Taylor Series Definition: A power series representation starting at the zero derivative ( $n = 0$ ), which is the function itself ( $f$ ).
Maclaurin Series: A specific case of the Taylor series where the center $a = 0$ .
Taylor Polynomials ( $P_k$ ): These are the $k^{th}$ partial sums of a Taylor series.
- $P_0 = f(a)$
- $P_1 = f(a) + f'(a)(x - a)$
- $P_2 = P_1 + \frac{f''(a)}{2!} (x-a)^2$
- $P_k = \sum_{n=0}^{k} \frac{f^{(n)}(a)}{n!} (x - a)^n$

Natural Logarithm ( $f(x) = \ln(x)$ ) at $a = 1$ :
- $P_0 = \ln(1) = 0$
- $P_1 = x - 1$
- $P_2 = (x - 1) - \frac{1}{2} (x - 1)^2$
- $P_3 = (x - 1) - \frac{1}{2} (x - 1)^2 + \frac{1}{3} (x - 1)^3$
- General Series: $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} (x - 1)^n$
- Convergence: radius of convergence $R = 1$ ; interval of convergence is $(0, 2]$ . It diverges at $0$ (harmonic series) and converges at $2$ (alternating harmonic series).
Reciprocal Square ( $f(x) = \frac{1}{x^2}$ ) at $a = 1$ :
- Derivatives: $f'(x) = -2x^{-3}$ , $f''(x) = 6x^{-4}$ , $f'''(x) = -24x^{-5}$ .
- Series Pattern: $\sum_{n=0}^{\infty} (-1)^n (n + 1) (x - 1)^n$
- Convergence: Radius of convergence $R = 1$ with an interval of $(0, 2)$ .

System Definition: Instead of horizontal ( $x$ ) and vertical ( $y$ ) distances, points are defined by a radius ( $r$ ) and an angle ( $\theta$ ).
Conversion Formulas:
- $x^2 + y^2 = r^2$
- $\tan(\theta) = \frac{y}{x}$
- $x = r \cos(\theta)$
- $y = r \sin(\theta)$
Coordinate Points: Usually written as $(r, \theta)$ .
- Example: The Cartesian point $(1, 1)$ converts to polar as $(\sqrt{2}, \frac{\pi}{4})$ .
- Angle Orientation: Negative angles sweep in the opposite direction (clockwise).

Common Shapes:
- Spiral: $r = a\theta$ , where the radius grows linearly with the angle.
- Cardioid: $r = a(1 \pm \cos(\theta))$ or $r = a(1 \pm \sin(\theta))$ .
- Limacon: French term for curves that often feature an internal loop.
- Roses: Curves such as $r = \cos(b\theta)$ where the number of petals depends on the coefficient $b$ .
Symmetry: Many polar functions are highly symmetric, allowing calculus operations (like area) to be performed on one section and multiplied by the total number of segments.

Area of a Sector: The geometric basis for polar integration is the area of a circle sector, given by $\text{Area} = \frac{1}{2} r^2 \theta$ .
Polar Integration: To find the area under a curve $r = f(\theta)$ between angles $\alpha$ and $\beta$ , the integral is: $\int_{\alpha}^{\beta} \frac{1}{2} [f(\theta)]^2 \,d\theta$

Question: How do we distinguish between derivative notation and exponents?
Response: Notation that looks like an exponent but is in parentheses, such as $f^{(n)}(x)$ , denotes a derivative. Apostrophes or parentheses are placed between the function $f$ and the argument $x$ . Standard digits without parentheses, like $f(x)^n$ , denote exponents.
Question: Is the radius of convergence smaller if the function is "faster"?
Response: It depends on how fast the derivatives are moving and the resulting coefficients. For example, in trigonometric functions like $\sin(x)$ or $\cos(x)$ , half the derivatives are lost (become zero), but they converge everywhere. In contrast, functions where derivatives grow very rapidly without a strong denominator in the series terms may have smaller radii of convergence.
Question: Can we change the interval of convergence for $\ln(x)$ ?
Response: The interval of length $2$ is fixed for that specific series at that center. To approximate values further away (like $x = 3$ ), the center $a$ must be shifted (e.g., to $a = 2$ ), though the derivatives may become more complex to calculate.