1
core
Differential equations
Calculus
Let $a$ be a constant.
(a) Find the general solution of
\[ y' - ay = 0. \]
(b) Solve the initial value problem
\[ y' - ay = 0, \quad y(x_0) = y_0. \]
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2
hard
Multivariable calculus
Calculus
Calculating a Surface Integral
Calculate surface integral \(\iint_S 5 dS\), where \(S\) is the surface with parameterization \(\mathbf{r}(u, v) = \langle u, u^2, v \rangle\) for
\(0 \le u \le 2\) and \(0 \le v \le u\).
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3
warmup
Multivariable calculus
Calculus
\(Applying Green’s Theorem over a Rectangle
Calculate the line integral
\oint_{C} x^2 y dx + (y - 3)dy,
where C is a rectangle with vertices (1, 1), (4, 1), (4, 5), and (1, 5) oriented counterclockwise.\)
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4
core
Parametric equations and vector functions
Calculus
\([T] Use technology to graph
r = e^{\sin(\theta)} - 2 \cos(4\theta).\)
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5
core
Parametric equations and vector functions
Calculus
\(r = 2 - 2 \sin \theta\)
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6
hard
Parametric equations and vector functions
Calculus
Given \(x = f(t)\) and \(y = g(t)\), if \(\frac{dx}{dy} = \frac{dy}{dx}\), then \(f(t) = g(t) + C\), where \(C\) is a constant.
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7
core
Parametric equations and vector functions
Calculus
Foci located at (6, -0), (6, 0) and eccentricity of 3
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8
core
Parametric equations and vector functions
Calculus
Enclosed by one petal of $r=3 \cos (2\theta)$
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9
hard
Parametric equations and vector functions
Calculus
$r = \frac{3}{2 - 6 \sin \theta}$
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10
core
Vector spaces
Linear algebra
Describe the relationship between the lines with the following parametric equations:
\[ x = 1 - 4t, y = 3 + t, z = 8 - 6t \]
\[ x = 2 + 3s, y = 2s, z = -1 - 3s. \]
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