Calculus 2 Exam Key Concepts

Direct Comparison Test: This test is used to determine the convergence or divergence of a series. If you have two series with positive terms, $a_n$ and $b_n$ : - If $b_n$ converges, then $a_n$ also converges. - If $b_n$ diverges, then $a_n$ also diverges. Example: Consider the series $a_n = rac{1}{n^2}$ and $b_n = rac{1}{n}$ . - We know that $rac{1}{n^2}$ converges (p-series with $p = 2$ ) and $rac{1}{n}$ diverges (p-series with $p = 1$ ). - If we compare $a_n = rac{1}{n^2}$ with $b_n$ , since $b_n$ diverges, we can conclude nothing about $a_n$ from this test alone, as we need both series to converge or diverge.

Alternating Series Test: This test is applicable for series that alternate in sign. The series $ext{lim}{n o ext{∞}} a_n = 0$ indicates convergence. Example: In the series $ext{S} = ext{ } rac{(-1)^n}{n}$ , - Here, $a_n = rac{1}{n}$ and as $ext{lim}{n o ext{∞}} a_n = 0$ , hence this series converges.

Root Test: The root test uses the limit ext{lim}{n o ext{∞}} oot n a_n : - If the limit is less than 1, the series converges. - If the limit is greater than 1, it diverges. - If it equals 1, the test is inconclusive. Example: For the series $a_n = rac{n^n}{n!}$ , we can calculate: - $ext{lim}{n o ext{∞}} oot n n^n/n!$ .
Ratio Test: The ratio test considers $L = ext{lim}{n o ext{∞}} rac{|a{n+1}|}{|a_n|}$ : - If L < 1 , series converges. - If L > 1 , series diverges. - If $L = 1$ , test is inconclusive. Example: For the series $a_n = rac{n!}{n^n}$ , - We find $L = ext{lim}_{n o ext{∞}} rac{(n+1)!/(n+1)^{n+1}}{n!/n^n}$ .

A power series takes the form $ext{Σ} a_n (x - c)^n$ where: - The center of the series is at $c$ , and $a_n$ are coefficients. - Radius of Convergence (R): Determined by the ratio test. Example: For the series $ext{Σ} rac{n}{2^n} x^n$ , calculate: - Using the ratio test, you can derive the radius of convergence $R = 2$ .

Functions can be expanded using Taylor or Maclaurin series. Example: For the function $f(x) = e^x$ , - The Maclaurin series is given by $f(x) = ext{Σ} rac{x^n}{n!}$ .

The formula for a Taylor series expansion at point $c$ is: - $f(x) = ext{Σ}_{n=0}^{ ext{∞}} rac{f^{(n)}(c)}{n!} (x - c)^n$ .
Example: For $f(x) = ext{sin}(x)$ expanded around $c = 0$ , - Gives the series $f(x) = x - rac{x^3}{3!} + rac{x^5}{5!} - rac{x^7}{7!} + …$ .

A Taylor polynomial approximates function $f(x)$ near $c$ up to degree $n$ and is given by: - $P_n(x) = ext{Σ}_{k=0}^{n} rac{f^{(k)}(c)}{k!} (x - c)^k$ . Example: For the function $f(x) = e^x$ at $c = 0$ , - The polynomial of degree 2 is $P_2(x) = 1 + x + rac{x^2}{2}$ .

Parametric equations define a curve with: - $x = f(t), y = g(t)$ and allow representation of shapes not provided by functions alone.

The slope of the tangent line is given by: - $rac{dy}{dx} = rac{ rac{dy}{dt}}{ rac{dx}{dt}}$ .
Example: For the curve defined by $x(t) = t^2, y(t) = t^3$ , - Find $rac{dy}{dx}$ by calculating derivatives with respect to $t$ .

Conversion processes are as follows: - Polar to Rectangular: $x = r ext{cos}( heta), y = r ext{sin}( heta)$ . - Rectangular to Polar: $r = ext{√}(x^2 + y^2), heta = an^{-1} rac{y}{x}$ . Example: For the point (3, rac{ ext{}}{4}) in polar coordinates: - Convert to rectangular gives (x, y) = (3 ext{cos}( rac{ ext{}}{4}), 3 ext{sin}( rac{ ext{}}{4})) . These steps illustrate how each concept of calculus is utilized, along with specific series and transformation techniques.