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Calc 2 Final - Cheat Sheet

Calc 2 Final - Cheat Sheet






















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|