1
core
Differential equations
Calculus
Solve the initial-value problem \(y'' - 3y' - 10y = 0, y(0) = 0, y'(0) = 7\).
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2
core
Differential equations
Calculus
An archaeologist investigating the site of an ancient village finds a burial ground where
the amount of carbon-14 present in individual remains is between 42 and 44% of the amount present in
live individuals. Estimate the age of the village and the length of time for which it survived.
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3
warmup
Differential equations
Calculus
Solve the Bernoulli equation
$y' - y = xy^2$.
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4
warmup
Differential equations
Calculus
$$x^2y' = y^2 + xy - x^2, \quad y(1) = 2.$$
(2.4.10)
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5
core
Multivariable calculus
Calculus
Find \( \text{div } \mathbf{F} \) for \( \mathbf{F}(x, y, z) = \langle xy, 5 - z^2 y, x^2 + y^2 \rangle \).
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6
core
Parametric equations and vector functions
Calculus
Enclosed by $r=6 \sin \theta$
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7
warmup
Parametric equations and vector functions
Calculus
Area of a polar region
Find the area of the circle defined by \( r = \cos \theta \). (Recall this circle has radius \( 1/2 \).)
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8
core
Parametric equations and vector functions
Calculus
Find a general solution of $y'' - y' - 6y = 2(\sin 4x + \cos 4x)$.
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9
core
Parametric equations and vector functions
Calculus
For the curve $x = 4t, y = 3t-2$, find the slope and
concavity of the curve at $t=3$.
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10
hard
Vector spaces
Linear algebra
\(x^2 = y^2 + z^2\)
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