1
hard
Differential equations
Calculus
Find all solutions of
\[ y' = \frac{1}{2}x(1-y^2). \]
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2
hard
Multivariable calculus
Calculus
Find a vector tangent at the point $(2, 1, 4)$ to the curve of intersection of the cone $z^2 = 3x^2 + 4y^2$ and the plane $3x-2y+z= 8$.
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3
core
Multivariable calculus
Calculus
Use Equation 6.18 to find the area of the surface of revolution obtained by rotating curve
$y = \sin x$, $0 \leq x \leq \pi$ about the x-axis.
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4
hard
Parametric equations and vector functions
Calculus
\( r = \frac{3}{-4 + 2 \sin \theta} \)
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5
core
Parametric equations and vector functions
Calculus
\(x = a \cos^3 \theta, y = a \sin^3 \theta \text{ on the interval } [0, 2\pi)\) (the hypocycloid)
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6
core
Parametric equations and vector functions
Calculus
Find the area of the region bounded by \(x = 2 \sin^2 \theta, y = 2 \sin^2 \theta \tan \theta, \text{ for } 0 \le \theta \le \frac{\pi}{2}\).
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7
core
Parametric equations and vector functions
Calculus
\( 52x^2 - 72xy + 73y^2 + 40x + 30y - 75 = 0 \)
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8
core
Parametric equations and vector functions
Calculus
\( r = \frac{1}{1 + \sin \theta} \)
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9
core
Parametric equations and vector functions
Calculus
\begin{align*} x &= -5t+7, & y &= 3t-1 \end{align*}
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10
hard
Vector spaces
Linear algebra
\(\left(0, 3, 0\right)\)
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