Calc 2 Final - The Blueprint

Chapter 9: Parametric & Polar (Focus Here!)

Parametric Calculus

• Derivative (Slope): The slope of the tangent line is found by the ratio of the derivatives with respect to \(t\). \[ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \]
• Horizontal & Vertical Tangents:
  • Horizontal Tangent: Set the numerator to zero: \( \frac{dy}{dt} = 0 \).
  • Vertical Tangent: Set the denominator to zero: \( \frac{dx}{dt} = 0 \).
• Arc Length (Parametric): Remember this version! It comes from the Pythagorean theorem. \[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt \]

Polar Calculus

• Coordinate Conversion:
  • Polar to Cartesian: \( x = r\cos\theta \), \( y = r\sin\theta \)
  • Cartesian to Polar: \( r^2 = x^2 + y^2 \), \( \tan\theta = \frac{y}{x} \) (Watch the quadrant!)
• Area in Polar Coordinates: This is for the area of "wedges" from the origin. \[ A = \int_{a}^{b} \frac{1}{2} r^2 \, d\theta \]
• Finding Area Limits (a, b): For one "petal" or an "inner loop," find where the curve is at the origin by setting \(r=0\) and solving for \(\theta\).
• Arc Length (Polar): \[ L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta \]

Chapter 8: Series

• Nth-Term Test for Divergence: Your first check! If \( \lim_{n\to\infty} a_n \neq 0 \), the series DIVERGES. If the limit is 0, the test is inconclusive.
• Geometric Series: \( \sum_{n=0}^{\infty} ar^n \)
  • Converges if \(|r| < 1\).
  • Sum Formula: If it converges, the sum is \( S = \frac{a}{1-r} \), where \(a\) is the first term.
• Ratio Test: The workhorse for convergence. Compute \( L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| \).
  • If \(L < 1\), the series is absolutely convergent.
  • If \(L > 1\), the series is divergent.
  • If \(L = 1\), the test is inconclusive.
• Taylor Series at \(x=a\): Guaranteed question on a non-zero center! \[ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n \]

Must-Memorize Maclaurin Series (\(a=0\))

• \( \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots \) for \(|x|<1\)
• \( e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \) for all \(x\)
• \( \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots \) for all \(x\)
• \( \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots \) for all \(x\)
• \( \ln(1+x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{n+1}}{n+1} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \) for \(|x|<1\)
• \( \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots \) for \(|x|\le1\)

Old Material Review (Ch 5-7) - Just the Setup!

• The ONE Trig Identity: \( \sin^2(x) + \cos^2(x) = 1 \). (Divide by \(\cos^2x\) to get \( \tan^2(x) + 1 = \sec^2(x) \)).
• Area Between Curves: Sketch it first! \( A = \int_{a}^{b} (y_{top} - y_{bottom}) \, dx \) or \( A = \int_{c}^{d} (x_{right} - x_{left}) \, dy \).
• Volume - Washer Method: Slice is PERPENDICULAR to axis of rotation. \[ V = \pi \int_{a}^{b} (R_{outer}^2 - r_{inner}^2) \, dx \]
• Volume - Shell Method: Slice is PARALLEL to axis of rotation. \[ V = 2\pi \int_{a}^{b} (\text{radius})(\text{height}) \, dx \]