Calc 2 Final Exam Study Guide

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Calc 2 Final Exam Study Guide

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].…

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<div class="container">
    <h1>Calculus 2 Final Exam: Key Formulas & Theorems</h1>

    <hr>
    
    <h2>Chapter 9: Parametric & Polar Coordinates (Highest Priority)</h2>

    <div class="card card-blue">
        <h3>Parametric Derivative (Slope)</h3>
        <div class="formula">
            $$ \frac{dy}{dx} = \frac{dy/dt}{dx/dt} $$
        </div>
        [cite_start]<p class="notes">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.</p>
    </div>

    <div class="card card-blue">
        <h3>Parametric Arc Length</h3>
        <div class="formula">
            $$ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt $$
        </div>
        <p class="notes">This is the Chapter 9 version of the arc length formula your professor said you need to know.</p>
    </div>

    <div class="card card-green">
        <h3>Polar Area</h3>
        <div class="formula">
            $$ A = \int_{a}^{b} \frac{1}{2} r^2 \, d\theta $$
        </div>
        [cite_start]<p class="notes">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 $\theta$.</p>
    </div>
    
    <div class="card card-green">
        <h3>Polar Arc Length</h3>
        <div class="formula">
            $$ L = \int_{a}^{b} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta $$
        </div>
    </div>
    
    <div class="card card-green">
        <h3>Coordinate Conversion Formulas</h3>
        <p><strong>Polar to Cartesian:</strong></p>
        <div class="formula">
            $$ x = r \cos(\theta) \quad \text{and} \quad y = r \sin(\theta) $$
        </div>
        <p><strong>Cartesian to Polar:</strong></p>
        <div class="formula">
            $$ r^2 = x^2 + y^2 \quad \text{and} \quad \tan(\theta) = \frac{y}{x} $$
        </div>
        <p class="notes">Be mindful of the correct quadrant when finding $\theta$ from $\tan^{-1}(y/x)$.</p>
    </div>

    <hr>

    <h2>Chapter 8: Series, Power Series, & Taylor Series</h2>

    <div class="card card-purple">
        <h3>Geometric Series</h3>
        <div class="formula">
            $$ S = \sum_{n=0}^{\infty} ar^n = \frac{a}{1-r} $$
        </div>
        <p class="notes">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].</p>
    </div>

    <div class="card card-purple">
        <h3>The Ratio Test</h3>
        <div class="formula">
            $$ L = \lim_{n\to\infty} \left| \frac{a_{n+1}}{a_n} \right| $$
        </div>
        <p class="notes">
            [cite_start]If $L < 1$, the series is absolutely convergent[cite: 2071].<br>
            [cite_start]If $L > 1$, the series is divergent[cite: 2072].<br>
            [cite_start]If $L = 1$, the test is inconclusive[cite: 2077].<br>
            This is the primary tool for finding the Radius of Convergence for power series.
        </p>
    </div>

    <div class="card card-red">
        <h3>Taylor & Maclaurin Series</h3>
        <p><strong>Taylor Series centered at $x=a$:</strong> (A question on this is guaranteed)</p>
        <div class="formula">
            $$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n $$
        </div>
        <p><strong>Maclaurin Series (centered at $x=0$):</strong></p>
        <div class="formula">
            $$ f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \dots $$
        </div>
    </div>
    
    <div class="card card-red">
        <h3>Key Maclaurin Series to Memorize</h3>
        <div class="formula">
            $$ \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n = 1 + x + x^2 + x^3 + \dots \quad (|x|<1) $$
            $$ e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots $$
            $$ \sin(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots $$
            $$ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots $$
            $$ \ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots \quad (|x|<1) $$
            $$ \arctan(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \dots \quad (|x|\le1) $$
        </div>
        <p class="notes">Your professor explicitly recommended memorizing these.</p>
    </div>

    <hr>
    
    <h2>Chapters 5-7: Integration, Area, and Volume Setups</h2>

    <div class="card card-orange">
        <h3>Trigonometric Identity</h3>
        <div class="formula">
            $$ \sin^2(x) + \cos^2(x) = 1 $$
        </div>
        <p class="notes">Your professor stated this is the "one and only trig identity you need to know." From this, you can derive others, like $1 + \tan^2(x) = \sec^2(x)$.</p>
    </div>
    
    <div class="card card-orange">
        <h3>Area Between Curves</h3>
        <p><strong>Integrating with respect to x:</strong></p>
        <div class="formula">
            $$ A = \int_{a}^{b} (y_{\text{top}} - y_{\text{bottom}}) \, dx $$
        </div>
        <p><strong>Integrating with respect to y:</strong></p>
        <div class="formula">
            $$ A = \int_{c}^{d} (x_{\text{right}} - x_{\text{left}}) \, dy $$
        </div>
        [cite_start]<p class="notes">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.</p>
    </div>

    <div class="card card-orange">
        <h3>Volumes of Revolution</h3>
        <p><strong>Disc Method:</strong> (Slice is <strong>perpendicular</strong> to axis, no gap)</p>
        <div class="formula">
            $$ V = \int_{a}^{b} \pi [R(x)]^2 \, dx $$
        </div>
        <p><strong>Washer Method:</strong> (Slice is <strong>perpendicular</strong> to axis, with a gap)</p>
        <div class="formula">
            $$ V = \int_{a}^{b} \pi \left( [R(x)]^2 - [r(x)]^2 \right) \, dx $$
        </div>
        <p><strong>Cylindrical Shell Method:</strong> (Slice is <strong>parallel</strong> to axis)</p>
        <div class="formula">
            $$ V = \int_{a}^{b} 2\pi (\text{radius})(\text{height}) \, dx $$
        </div>
        [cite_start]<p class="notes">Getting the radius and height correct is 90% of the battle[cite: 1759]. Pay attention to the axis of rotation!</p>
    </div>

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