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MAT 266 - Interactive Final Exam Study Guide
MAT 266 - Interactive Final Exam Study Guide
MAT 266 Final Exam Interactive Study Guide
Visualize concepts, interact with formulas, and master Calculus II.
Integration by Substitution
Integration by Parts
Trigonometric Integrals
Partial Fractions
Numerical Approximation
Improper Integrals
Area Between Curves
Volumes of Revolution
Arc Length
Work
Sequences & Series
Power & Taylor Series
Parametric & Polar
5.5 Integration by Substitution (u-Substitution)
This is the reverse of the chain rule. The goal is to find a part of the integrand, which we'll call $u$, whose derivative $du$ is also present (or can be formed with a constant multiple).
Key Formula
If $u = g(x)$, then $du = g'(x)dx$, and the integral transforms:
$\int f(g(x))g'(x)dx = \int f(u)du$
Example from Review (#1)
Evaluate $\int\frac{(\ln x)^{4}}{x}dx.$
Step 1: Identify u. Notice that the derivative of $\ln x$ is $\frac{1}{x}$, which is in the integrand. This makes $\ln x$ a great choice for $u$.
Let $u = \ln x$
Step 2: Find du.
Then $du = \frac{1}{x}dx$
Step 3: Substitute. Replace $\ln x$ with $u$ and $\frac{1}{x}dx$ with $du$.
$\int u^4 du$
Step 4: Integrate with respect to u.
$\frac{u^5}{5} + C$
Step 5: Substitute back to x.
$\frac{(\ln x)^5}{5} + C$
6.1 Integration by Parts
This technique is used for integrating products of functions. It's derived from the product rule for differentiation.
Key Formula
$\int u \, dv = uv - \int v \, du$
A helpful mnemonic for choosing $u$ is LIATE:
Logarithmic functions (e.g., $\ln x$)
Inverse trigonometric functions (e.g., $\arctan x$)
Algebraic functions (e.g., $x^2, 3x$)
Trigonometric functions (e.g., $\sin x, \cos x$)
Exponential functions (e.g., $e^x$)
Choose the function that comes first in LIATE as your $u$.
Example from Review (#8)
Evaluate $\int te^{-3t}dt$.
Step 1: Choose u and dv. Following LIATE, 't' is Algebraic and 'e-3t' is Exponential.…
MAT 266 - Interactive Final Exam Study Guide
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<h1 class="text-4xl font-bold text-white">MAT 266 Final Exam Interactive Study Guide</h1>
<p class="text-lg text-gray-400 mt-2">Visualize concepts, interact with formulas, and master Calculus II.</p>
</header>
<div class="flex flex-col lg:flex-row gap-8">
<!-- Navigation -->
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<a class="nav-link active" data-target="integration-by-substitution">Integration by Substitution</a>
<a class="nav-link" data-target="integration-by-parts">Integration by Parts</a>
<a class="nav-link" data-target="trig-integrals">Trigonometric Integrals</a>
<a class="nav-link" data-target="partial-fractions">Partial Fractions</a>
<a class="nav-link" data-target="numerical-approximation">Numerical Approximation</a>
<a class="nav-link" data-target="improper-integrals">Improper Integrals</a>
<a class="nav-link" data-target="area-between-curves">Area Between Curves</a>
<a class="nav-link" data-target="volumes-revolution">Volumes of Revolution</a>
<a class="nav-link" data-target="arc-length">Arc Length</a>
<a class="nav-link" data-target="work">Work</a>
<a class="nav-link" data-target="sequences-series">Sequences & Series</a>
<a class="nav-link" data-target="power-series">Power & Taylor Series</a>
<a class="nav-link" data-target="parametric-polar">Parametric & Polar</a>
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<!-- Content -->
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<!-- Integration by Substitution -->
<section id="integration-by-substitution" class="content-section active">
<div class="card">
<h2 class="card-title">5.5 Integration by Substitution (u-Substitution)</h2>
<p class="mb-4">This is the reverse of the chain rule. The goal is to find a part of the integrand, which we'll call $u$, whose derivative $du$ is also present (or can be formed with a constant multiple).</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Key Formula</h3>
<p>If $u = g(x)$, then $du = g'(x)dx$, and the integral transforms:</p>
<span>$\int f(g(x))g'(x)dx = \int f(u)du$</span>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#1)</h3>
<p class="mb-2">Evaluate $\int\frac{(\ln x)^{4}}{x}dx.$</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Identify u.</strong> Notice that the derivative of $\ln x$ is $\frac{1}{x}$, which is in the integrand. This makes $\ln x$ a great choice for $u$.</p>
<p class="mt-2">Let $u = \ln x$</p>
<p><strong>Step 2: Find du.</strong></p>
<p class="mt-2">Then $du = \frac{1}{x}dx$</p>
<p><strong>Step 3: Substitute.</strong> Replace $\ln x$ with $u$ and $\frac{1}{x}dx$ with $du$.</p>
<p class="mt-2">$\int u^4 du$</p>
<p><strong>Step 4: Integrate with respect to u.</strong></p>
<p class="mt-2">$\frac{u^5}{5} + C$</p>
<p><strong>Step 5: Substitute back to x.</strong></p>
<p class="mt-2">$\frac{(\ln x)^5}{5} + C$</p>
</div>
</div>
</section>
<!-- Integration by Parts -->
<section id="integration-by-parts" class="content-section">
<div class="card">
<h2 class="card-title">6.1 Integration by Parts</h2>
<p class="mb-4">This technique is used for integrating products of functions. It's derived from the product rule for differentiation.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Key Formula</h3>
<span>$\int u \, dv = uv - \int v \, du$</span>
</div>
<p class="mb-4">A helpful mnemonic for choosing $u$ is <strong>LIATE</strong>:</p>
<ul class="list-disc list-inside bg-gray-900 p-4 rounded-lg">
<li><strong>L</strong>ogarithmic functions (e.g., $\ln x$)</li>
<li><strong>I</strong>nverse trigonometric functions (e.g., $\arctan x$)</li>
<li><strong>A</strong>lgebraic functions (e.g., $x^2, 3x$)</li>
<li><strong>T</strong>rigonometric functions (e.g., $\sin x, \cos x$)</li>
<li><strong>E</strong>xponential functions (e.g., $e^x$)</li>
</ul>
<p class="mt-4">Choose the function that comes first in LIATE as your $u$.</p>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#8)</h3>
<p class="mb-2">Evaluate $\int te^{-3t}dt$.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Choose u and dv.</strong> Following LIATE, 't' is Algebraic and 'e<sup>-3t</sup>' is Exponential. So, we choose:</p>
<p class="mt-2">Let $u = t$ and $dv = e^{-3t}dt$</p>
<p><strong>Step 2: Find du and v.</strong></p>
<p class="mt-2">$du = dt$</p>
<p>$v = \int e^{-3t}dt = -\frac{1}{3}e^{-3t}$</p>
<p><strong>Step 3: Apply the formula.</strong></p>
<p class="mt-2">$\int te^{-3t}dt = (t)(-\frac{1}{3}e^{-3t}) - \int (-\frac{1}{3}e^{-3t})dt$</p>
<p class="mt-2">$= -\frac{t}{3}e^{-3t} + \frac{1}{3}\int e^{-3t}dt$</p>
<p><strong>Step 4: Solve the remaining integral.</strong></p>
<p class="mt-2">$= -\frac{t}{3}e^{-3t} + \frac{1}{3}(-\frac{1}{3}e^{-3t}) + C$</p>
<p class="mt-2">$= -\frac{t}{3}e^{-3t} - \frac{1}{9}e^{-3t} + C$</p>
</div>
</div>
</section>
<!-- Trig Integrals -->
<section id="trig-integrals" class="content-section">
<div class="card">
<h2 class="card-title">6.2 Trigonometric Integrals</h2>
<p class="mb-4">These integrals involve powers of trigonometric functions. The strategy depends on the powers of the functions.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Key Identities to Memorize</h3>
<span>$\sin^2 x + \cos^2 x = 1$</span><br>
<span>$\tan^2 x + 1 = \sec^2 x$</span><br>
<span>$\sin^2 x = \frac{1 - \cos(2x)}{2}$</span><br>
<span>$\cos^2 x = \frac{1 + \cos(2x)}{2}$</span><br>
<span>$\sin(2x) = 2\sin x \cos x$</span>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Strategy for $\int \sin^m x \cos^n x \, dx$</h3>
<ul class="list-disc list-inside bg-gray-900 p-4 rounded-lg">
<li>If power of $\cos x$ is odd ($n=2k+1$), save one $\cos x$ factor, convert the rest to sines using $\cos^2 x = 1 - \sin^2 x$, then use $u = \sin x$.</li>
<li>If power of $\sin x$ is odd ($m=2k+1$), save one $\sin x$ factor, convert the rest to cosines using $\sin^2 x = 1 - \cos^2 x$, then use $u = \cos x$.</li>
<li>If both powers are even, use the half-angle identities repeatedly.</li>
</ul>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#14)</h3>
<p class="mb-2">Evaluate $\int \sin^6 x \cos^3 x \, dx$.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Strategy:</strong> Power of cosine is odd.</p>
<p class="mt-2">$\int \sin^6 x \cos^2 x \cos x \, dx$</p>
<p class="mt-2">$= \int \sin^6 x (1 - \sin^2 x) \cos x \, dx$</p>
<p><strong>Let $u = \sin x$, so $du = \cos x \, dx$</strong></p>
<p class="mt-2">$= \int u^6(1 - u^2) \, du = \int (u^6 - u^8) \, du$</p>
<p class="mt-2">$= \frac{u^7}{7} - \frac{u^9}{9} + C$</p>
<p class="mt-2">$= \frac{\sin^7 x}{7} - \frac{\sin^9 x}{9} + C$</p>
</div>
</div>
</section>
<!-- Partial Fractions -->
<section id="partial-fractions" class="content-section">
<div class="card">
<h2 class="card-title">6.3 Integration by Partial Fractions</h2>
<p class="mb-4">Used to integrate rational functions $\frac{P(x)}{Q(x)}$. The first step is to decompose the rational function into simpler fractions.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Decomposition Rules</h3>
<p>1. <strong>Distinct Linear Factors:</strong> $\frac{P(x)}{(ax+b)(cx+d)} = \frac{A}{ax+b} + \frac{B}{cx+d}$</p>
<p>2. <strong>Repeated Linear Factors:</strong> $\frac{P(x)}{(ax+b)^k} = \frac{A_1}{ax+b} + \frac{A_2}{(ax+b)^2} + \dots + \frac{A_k}{(ax+b)^k}$</p>
<p>3. <strong>Irreducible Quadratic Factors:</strong> $\frac{P(x)}{ax^2+bx+c} = \frac{Ax+B}{ax^2+bx+c}$</p>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#20)</h3>
<p class="mb-2">Evaluate $\int\frac{1}{(x+6)(x-2)}dx$.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Decompose.</strong></p>
<p>$\frac{1}{(x+6)(x-2)} = \frac{A}{x+6} + \frac{B}{x-2}$</p>
<p><strong>Step 2: Solve for A and B.</strong> Multiply by the common denominator:</p>
<p>$1 = A(x-2) + B(x+6)$</p>
<p>Let $x=2$: $1 = A(0) + B(8) \implies B = \frac{1}{8}$</p>
<p>Let $x=-6$: $1 = A(-8) + B(0) \implies A = -\frac{1}{8}$</p>
<p><strong>Step 3: Integrate.</strong></p>
<p>$\int (\frac{-1/8}{x+6} + \frac{1/8}{x-2}) dx$</p>
<p>$= -\frac{1}{8}\ln|x+6| + \frac{1}{8}\ln|x-2| + C = \frac{1}{8}\ln|\frac{x-2}{x+6}| + C$</p>
</div>
</div>
</section>
<!-- Numerical Approximation -->
<section id="numerical-approximation" class="content-section">
<div class="card">
<h2 class="card-title">6.5 Numerical Approximation</h2>
<p class="mb-4">When an integral is difficult or impossible to solve analytically, we can approximate its value.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Trapezoidal Rule</h3>
<span>$\int_a^b f(x)dx \approx \frac{\Delta x}{2}[f(x_0) + 2f(x_1) + \dots + 2f(x_{n-1}) + f(x_n)]$</span>
<p>where $\Delta x = \frac{b-a}{n}$</p>
</div>
<div class="formula-box mt-4">
<h3 class="font-semibold text-lg mb-2">Simpson's Rule (n must be even)</h3>
<span>$\int_a^b f(x)dx \approx \frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + \dots + 4f(x_{n-1}) + f(x_n)]$</span>
<p>where $\Delta x = \frac{b-a}{n}$</p>
</div>
<div class="card mt-6">
<h3 class="card-title">Interactive Calculator</h3>
<div class="grid grid-cols-1 md:grid-cols-2 gap-4 mb-4">
<div><label for="na-func" class="block mb-1">Function f(x):</label><input type="text" id="na-func" value="sin(x**2)"></div>
<div><label for="na-a" class="block mb-1">Lower Limit (a):</label><input type="number" id="na-a" value="0"></div>
<div><label for="na-b" class="block mb-1">Upper Limit (b):</label><input type="number" id="na-b" value="1"></div>
<div><label for="na-n" class="block mb-1">Subintervals (n):</label><input type="number" id="na-n" value="4"></div>
</div>
<button id="na-calculate">Calculate</button>
<div id="na-results" class="mt-4 bg-gray-900 p-4 rounded-lg hidden"></div>
<canvas id="na-chart" class="mt-4 bg-gray-800 rounded-lg"></canvas>
</div>
</div>
</section>
<!-- Improper Integrals -->
<section id="improper-integrals" class="content-section">
<div class="card">
<h2 class="card-title">6.6 Improper Integrals</h2>
<p class="mb-4">Integrals with infinite limits of integration or with an infinite discontinuity within the interval.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Types</h3>
<p><strong>Type 1 (Infinite Interval):</strong> $\int_a^\infty f(x)dx = \lim_{t\to\infty} \int_a^t f(x)dx$</p>
<p><strong>Type 2 (Discontinuous Integrand):</strong> If $f(x)$ is discontinuous at $b$, then $\int_a^b f(x)dx = \lim_{t\to b^-} \int_a^t f(x)dx$</p>
<p>An integral <strong>converges</strong> if the limit exists and is finite. Otherwise, it <strong>diverges</strong>.</p>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#32)</h3>
<p class="mb-2">Determine if $\int_{e}^{\infty}\frac{79}{x(\ln x)^{3}}dx$ converges or diverges.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Set up the limit.</strong></p>
<p>$\lim_{t\to\infty} \int_e^t \frac{79}{x(\ln x)^3} dx$</p>
<p><strong>Step 2: Solve the integral (use u-sub).</strong></p>
<p>Let $u = \ln x$, $du = \frac{1}{x}dx$.</p>
<p>$\int \frac{79}{u^3} du = 79 \int u^{-3} du = 79 \frac{u^{-2}}{-2} = -\frac{79}{2u^2} = -\frac{79}{2(\ln x)^2}$</p>
<p><strong>Step 3: Evaluate the limit.</strong></p>
<p>$\lim_{t\to\infty} [-\frac{79}{2(\ln x)^2}]_e^t = \lim_{t\to\infty} (-\frac{79}{2(\ln t)^2} - (-\frac{79}{2(\ln e)^2}))$</p>
<p>$= \lim_{t\to\infty} (-\frac{79}{2(\ln t)^2} + \frac{79}{2(1)^2})$</p>
<p>As $t \to \infty$, $\ln t \to \infty$, so $\frac{79}{2(\ln t)^2} \to 0$.</p>
<p>$= 0 + \frac{79}{2} = \frac{79}{2}$</p>
<p><strong>Conclusion:</strong> The integral converges to $\frac{79}{2}$.</p>
</div>
</div>
</section>
<!-- Area Between Curves -->
<section id="area-between-curves" class="content-section">
<div class="card">
<h2 class="card-title">7.1 Area Between Curves</h2>
<div class="formula-box">
<p>If $f(x) \ge g(x)$ on $[a, b]$, the area is: $A = \int_a^b [f(x) - g(x)] dx$</p>
<p>If $f(y) \ge g(y)$ on $[c, d]$, the area is: $A = \int_c^d [f(y) - g(y)] dy$</p>
</div>
<div class="card mt-6">
<h3 class="card-title">Interactive Visualizer</h3>
<div class="grid grid-cols-1 md:grid-cols-2 gap-4 mb-4">
<div><label for="ac-f1" class="block mb-1">Top/Right Function f(x):</label><input type="text" id="ac-f1" value="7*x - x**2"></div>
<div><label for="ac-f2" class="block mb-1">Bottom/Left Function g(x):</label><input type="text" id="ac-f2" value="2*x"></div>
</div>
<button id="ac-visualize">Visualize Area</button>
<p class="mt-4 text-sm text-gray-400">Note: Intersection points are calculated automatically. Visualization range is based on these points.</p>
<div id="ac-results" class="mt-4 bg-gray-900 p-4 rounded-lg hidden"></div>
<canvas id="ac-chart" class="mt-4 bg-gray-800 rounded-lg"></canvas>
</div>
</div>
</section>
<!-- Volumes of Revolution -->
<section id="volumes-revolution" class="content-section">
<div class="card">
<h2 class="card-title">7.2 & 7.3 Volumes of Revolution</h2>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Disk Method</h3>
<p>Rotation about x-axis: $V = \int_a^b \pi [R(x)]^2 dx$</p>
<p>Rotation about y-axis: $V = \int_c^d \pi [R(y)]^2 dy$</p>
</div>
<div class="formula-box mt-4">
<h3 class="font-semibold text-lg mb-2">Washer Method</h3>
<p>Rotation about x-axis: $V = \int_a^b \pi ([R(x)]^2 - [r(x)]^2) dx$</p>
<p>Rotation about y-axis: $V = \int_c^d \pi ([R(y)]^2 - [r(y)]^2) dy$</p>
</div>
<div class="formula-box mt-4">
<h3 class="font-semibold text-lg mb-2">Cylindrical Shells Method</h3>
<p>Rotation about y-axis: $V = \int_a^b 2\pi x h(x) dx$</p>
<p>Rotation about x-axis: $V = \int_c^d 2\pi y h(y) dy$</p>
</div>
<p class="mt-4 text-sm text-gray-400">Interactive visualizer coming soon!</p>
</div>
</section>
<!-- Arc Length -->
<section id="arc-length" class="content-section">
<div class="card">
<h2 class="card-title">7.4 Arc Length</h2>
<div class="formula-box">
<p>For a function $y=f(x)$ from $x=a$ to $x=b$: $L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$</p>
<p>For a function $x=g(y)$ from $y=c$ to $y=d$: $L = \int_c^d \sqrt{1 + [g'(y)]^2} \, dy$</p>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#52)</h3>
<p class="mb-2">Find the length of $y=1+2x^{3/2}$ for $0 \le x \le 1$.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Find the derivative.</strong></p>
<p>$y' = \frac{dy}{dx} = 2 \cdot \frac{3}{2}x^{1/2} = 3\sqrt{x}$</p>
<p><strong>Step 2: Square the derivative.</strong></p>
<p>$(y')^2 = (3\sqrt{x})^2 = 9x$</p>
<p><strong>Step 3: Set up the integral.</strong></p>
<p>$L = \int_0^1 \sqrt{1 + 9x} \, dx$</p>
<p><strong>Step 4: Solve the integral (u-sub).</strong></p>
<p>Let $u = 1+9x$, $du = 9dx \implies dx = \frac{1}{9}du$.</p>
<p>Change bounds: $x=0 \to u=1$, $x=1 \to u=10$.</p>
<p>$L = \int_1^{10} \sqrt{u} \frac{1}{9}du = \frac{1}{9} [\frac{2}{3}u^{3/2}]_1^{10}$</p>
<p>$= \frac{2}{27}[10^{3/2} - 1^{3/2}] = \frac{2}{27}(10\sqrt{10} - 1)$</p>
</div>
</div>
</section>
<!-- Work -->
<section id="work" class="content-section">
<div class="card">
<h2 class="card-title">7.6 Work</h2>
<p class="mb-4">Work done by a variable force is the integral of the force. For pumping liquid, we integrate the work done to lift thin horizontal slices of the liquid.</p>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Work Pumping Liquid</h3>
<p>$W = \int_a^b (\text{density}) \cdot g \cdot A(y) \cdot D(y) \, dy$</p>
<ul class="list-disc list-inside mt-2">
<li>$A(y)$: Area of a horizontal slice at height $y$.</li>
<li>$D(y)$: Distance the slice at height $y$ must be lifted.</li>
<li>$g$: acceleration due to gravity ($9.8 \, m/s^2$ or $32 \, ft/s^2$).</li>
<li>Density of water: $1000 \, kg/m^3$ or $62.5 \, lb/ft^3$.</li>
</ul>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#55a)</h3>
<p class="mb-2">Inverted cone, height 8m, radius 4m. Full of water. Pump all water out (no spout).</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Setup:</strong> Place origin at the bottom vertex. The y-axis goes up.</p>
<p><strong>Find radius of a slice:</strong> By similar triangles, $\frac{r}{y} = \frac{4}{8} \implies r = \frac{y}{2}$.</p>
<p><strong>Area of a slice $A(y)$:</strong> $A(y) = \pi r^2 = \pi (\frac{y}{2})^2 = \frac{\pi y^2}{4}$.</p>
<p><strong>Distance to lift $D(y)$:</strong> A slice at height $y$ must travel to the top (height 8). So, $D(y) = 8-y$.</p>
<p><strong>Integral bounds:</strong> The water is from $y=0$ to $y=8$.</p>
<p><strong>Work Integral:</strong></p>
<p>$W = \int_0^8 (1000)(9.8) \pi (\frac{y^2}{4})(8-y) dy$</p>
</div>
</div>
</section>
<!-- Sequences & Series -->
<section id="sequences-series" class="content-section">
<div class="card">
<h2 class="card-title">8.1 & 8.2 Sequence and Series Convergence</h2>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Key Tests</h3>
<p><strong>Test for Divergence:</strong> If $\lim_{n\to\infty} a_n \neq 0$ or the limit DNE, then the series $\sum a_n$ diverges.</p>
<p><strong>Geometric Series:</strong> $\sum_{n=1}^\infty ar^{n-1}$ converges to $\frac{a}{1-r}$ if $|r| < 1$. It diverges if $|r| \ge 1$.</p>
<p><strong>Ratio Test:</strong> Let $L = \lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|$.</p>
<ul class="list-disc list-inside mt-2">
<li>If $L < 1$, the series is absolutely convergent.</li>
<li>If $L > 1$, the series is divergent.</li>
<li>If $L = 1$, the test is inconclusive.</li>
</ul>
</div>
<h3 class="font-semibold text-lg mb-2 mt-6">Example from Review (#59)</h3>
<p class="mb-2">Determine if $\sum_{n=1}^{\infty}\frac{(-7)^{n-1}}{9^{n}}$ converges or diverges.</p>
<div class="bg-gray-900 p-4 rounded-lg">
<p><strong>Step 1: Rewrite as a geometric series.</strong></p>
<p>$\sum_{n=1}^{\infty}\frac{(-7)^{n-1}}{9 \cdot 9^{n-1}} = \sum_{n=1}^{\infty} \frac{1}{9} (\frac{-7}{9})^{n-1}$</p>
<p><strong>Step 2: Identify a and r.</strong></p>
<p>This is a geometric series with $a = \frac{1}{9}$ and $r = -\frac{7}{9}$.</p>
<p><strong>Step 3: Check for convergence.</strong></p>
<p>$|r| = |-\frac{7}{9}| = \frac{7}{9} < 1$. So, the series converges.</p>
<p><strong>Step 4: Find the sum.</strong></p>
<p>$S = \frac{a}{1-r} = \frac{1/9}{1 - (-7/9)} = \frac{1/9}{16/9} = \frac{1}{16}$.</p>
</div>
</div>
</section>
<!-- Power & Taylor Series -->
<section id="power-series" class="content-section">
<div class="card">
<h2 class="card-title">8.5-8.7 Power, Taylor, & Maclaurin Series</h2>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Definitions</h3>
<p><strong>Power Series (centered at a):</strong> $\sum_{n=0}^\infty c_n (x-a)^n$</p>
<p><strong>Taylor Series (centered at a):</strong> $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!} (x-a)^n$</p>
<p><strong>Maclaurin Series (Taylor series at a=0):</strong> $f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} x^n$</p>
</div>
<div class="formula-box mt-4">
<h3 class="font-semibold text-lg mb-2">Important Maclaurin Series to Memorize</h3>
<p>$e^x = \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \dots$</p>
<p>$\sin x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$</p>
<p>$\cos x = \sum_{n=0}^\infty \frac{(-1)^n x^{2n}}{(2n)!} = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$</p>
<p>$\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + \dots$, for $|x|<1$</p>
</div>
</div>
</section>
<!-- Parametric & Polar -->
<section id="parametric-polar" class="content-section">
<div class="card">
<h2 class="card-title">9.1-9.4 Parametric & Polar Equations</h2>
<div class="formula-box">
<h3 class="font-semibold text-lg mb-2">Calculus with Parametric Equations</h3>
<p>Curve: $x=f(t), y=g(t)$</p>
<p>Slope: $\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$</p>
<p>Arc Length: $L = \int_\alpha^\beta \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \, dt$</p>
</div>
<div class="formula-box mt-4">
<h3 class="font-semibold text-lg mb-2">Calculus with Polar Equations</h3>
<p>Curve: $r=f(\theta)$</p>
<p>Area: $A = \int_a^b \frac{1}{2} [f(\theta)]^2 \, d\theta = \int_a^b \frac{1}{2} r^2 \, d\theta$</p>
<p>Arc Length: $L = \int_a^b \sqrt{r^2 + (\frac{dr}{d\theta})^2} \, d\theta$</p>
</div>
</div>
</section>
</main>
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