1
hard
Differential equations
Calculus
Find the orthogonal trajectories of the family of hyperbolas
\[ xy = c \quad (c \neq 0) \]
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2
core
Multivariable calculus
Calculus
Calculate integral \(\int_C \mathbf{F} \cdot d\mathbf{r}\), where \(\mathbf{F}(x, y) = \langle \sin x \sin y, 5 - \cos x \cos y \rangle\) and \(C\) is a semicircle with starting point \( (0, \pi) \) and endpoint \( (0, -\pi) \).
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3
warmup
Multivariable calculus
Calculus
Using the Divergence Theorem to compute flux
Let \(S\) be the cube bounded by the planes \(x = \pm 1, y = \pm 1, z = \pm 1\), and let
\(\vec{F} = \langle x^{2}y, 2yz, x^{2}z^{3} \rangle\). Compute the outward flux of \(\vec{F}\) over \(S\).
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4
core
Parametric equations and vector functions
Calculus
\( 25x^2 - 4y^2 = 100 \)
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5
hard
Parametric equations and vector functions
Calculus
\(r = 3 \cos(2\theta)\)
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6
warmup
Parametric equations and vector functions
Calculus
Focus: $(1,5)$; directrix: $x = 3$
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7
core
Parametric equations and vector functions
Calculus
\begin{align*} x &= \cos t, & y &= 8 \sin t, & t = \frac{\pi}{2} \end{align*}
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8
core
Parametric equations and vector functions
Calculus
Focus (0, 2) and directrix $y = 4$
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9
hard
Parametric equations and vector functions
Calculus
$x = 3t + 4$, $y = 9t - 2$, $0 \leq t \leq 3$
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10
warmup
Vector spaces
Linear algebra
Finding the Line of Intersection for Two Planes
Find parametric and symmetric equations for the line formed by the intersection of the planes given by
$x + y + z = 0$ and $2x - y + z = 0$ (see the following figure).
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