Calculus 2 Final Exam: Key Formulas & Theorems
Chapter 9: Parametric & Polar Coordinates (Highest Priority)
Parametric Derivative (Slope)
$$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$
[cite_start]
Finds the slope of the tangent line for a parametric curve[cite: 1828]. Remember this as the change in y over the change in x, both with respect to the parameter t.
Parametric Arc Length
$$ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$
This is the Chapter 9 version of the arc length formula your professor said you need to know.
Polar Area
$$ A = \int_{a}^{b} \frac{1}{2} r^2 \, d\theta $$
[cite_start]
Used to find the area of a region bounded by a polar curve[cite: 1426]. Remember the $\frac{1}{2}$ and that the radius is squared. To find the integration limits for one "petal," set $r=0$ and solve for $\theta$.
Polar Arc Length
$$ L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta $$
Coordinate Conversion Formulas
Polar to Cartesian:
$$ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) $$
Cartesian to Polar:
$$ r^2 = x^2 + y^2 \quad \text{and} \quad \tan(\theta) = \frac{y}{x} $$
Be mindful of the correct quadrant when finding $\theta$ from $\tan^{-1}(y/x)$.
Chapter 8: Series, Power Series, & Taylor Series
Geometric Series
$$ S = \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} $$
This formula for the sum only works if the series converges, which happens when $|r| [cite_start]< 1$[cite: 1984, 1963]. [cite_start]Here, 'a' is the first term of the series[cite: 1967].
The Ratio Test
$$ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$
[cite_start]If $L < 1$, the series is absolutely convergent[cite: 2071].
[cite_start]If $L > 1$, the series is divergent[cite: 2072].
[cite_start]If $L = 1$, the test is inconclusive[cite: 2077].
This is the primary tool for finding the Radius of Convergence for power series.
Taylor & Maclaurin Series
Taylor Series centered at $x=a$: (A question on this is guaranteed)
$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $$
Maclaurin Series (centered at $x=0$):
$$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots $$
Key Maclaurin Series to Memorize
$$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots \quad (|x|<1) $$
$$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots $$
$$ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots $$
$$ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots $$
$$ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (|x|<1) $$
$$ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots \quad (|x|\le1) $$
Your professor explicitly recommended memorizing these.
Chapters 5-7: Integration, Area, and Volume Setups
Trigonometric Identity
$$ \sin^2(x) + \cos^2(x) = 1 $$
Your professor stated this is the "one and only trig identity you need to know." From this, you can derive others, like $1 + \tan^2(x) = \sec^2(x)$.
Area Between Curves
Integrating with respect to x:
$$ A = \int_{a}^{b} (y_{\text{top}} - y_{\text{bottom}}) \, dx $$
Integrating with respect to y:
$$ A = \int_{c}^{d} (x_{\text{right}} - x_{\text{left}}) \, dy $$
[cite_start]
Sketching the region is the most important step! [cite: 477] Remember you can use Desmos for the final calculation once the integral is correctly set up.
Volumes of Revolution
Disc Method: (Slice is perpendicular to axis, no gap)
$$ V = \int_{a}^{b} \pi [R(x)]^2 \, dx $$
Washer Method: (Slice is perpendicular to axis, with a gap)
$$ V = \int_{a}^{b} \pi \left( [R(x)]^2 - [r(x)]^2 \right) \, dx $$
Cylindrical Shell Method: (Slice is parallel to axis)
$$ V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dx $$
[cite_start]
Getting the radius and height correct is 90% of the battle[cite: 1759]. Pay attention to the axis of rotation!