1
hard
Sequences
Real & complex analysis
Determine whether \( \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sin (\pi/n)}{n^2} \) converges.
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Convergence of an Alternating Series with Sine →
2
core
Antiderivatives
Calculus
Evaluate \( \int \sec^2 4x \tan 4x \, dx \).
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Integral of sec^2(4x) tan(4x) →
3
hard
The definite integral
Calculus
Evaluate \(\lim_{n\to\infty} \frac{1}{n} \left( \sin \frac{\pi}{n} + \sin \frac{2\pi}{n} + \dots + \sin \frac{n\pi}{n} \right)\).
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Evaluating a Limit of a Sum Using a Definite Integral →
4
hard
Differential equations
Calculus
Use (8.4.12) to find
\[ \mathcal{L}^{-1}\left(\frac{e^{-2s}}{s^2}\right). \]
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Inverse Laplace Transform with a Time Delay →
5
core
Limits
Calculus
\(\lim_{x\to 0} \frac{e^x}{x^2}\)
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Limit of e^x / x^2 as x approaches 0 →
6
core
Parametric equations and vector functions
Calculus
Find the distance traveled by a particle with position \((x, y)\) as \(t\) varies in the given time interval: \(x = \sin^2 t, y = \cos^2 t, 0 \le t \le 3\pi\).
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Distance Traveled by a Particle on a Parametric Curve →
7
core
Logarithmic and exponential functions
Calculus
Evaluate \( \lim_{x\to 0} \frac{\ln (1+x)}{x} \).
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Limit of ln(1+x)/x as x approaches 0 →
8
warmup
Trigonometric and inverse trigonometric functions
Calculus
Find \( \frac{d}{dx}(\sqrt{\sin x}) \).
derivativechain-rulesquare-rootsinetrigonometric-functionscalculus-ievaluate-derivativepower-rulecomposite-function
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Derivative of the Square Root of Sine →
9
core
Linear transformations
Linear algebra
Let $F$ and $G$ be the linear operators on $\mathbb{R}^2$ defined by $F(x,y) = (y,x)$ and $G(x,y) = (0,x)$. Find
formulas defining the following operators:
(a) $F + G$, (b) $2F - 3G$, (c) $FG$, (d) $GF$, (e) $F^2$, (f) $G^2$.
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10
core
Vector spaces
Linear algebra
Find an equation of the plane containing the lines $L_1$ and $L_2$:
\[ L_1: x = -y = z \]
\[ L_2: \frac{x-3}{2} = y = z - 2 \]
analytic-geometrycross-productdirection-vectorintersecting-lineslinear-algebramultivariable-calculusnormal-vectorparametric-equationsplane-equationsymmetric-equationsthree-dimensional-geometryvector-geometry
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Equation of a Plane Containing Two Intersecting Lines →