1
core
Series of numbers
Real & complex analysis
If $ \mathbf{A} \times \mathbf{B} = \mathbf{A} \times \mathbf{C} $ and $ \mathbf{A} \neq \mathbf{0} $, does it follow that $ \mathbf{B} = \mathbf{C} $?
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2
warmup
Differential equations
Calculus
$$(3+x)y'' + (1+2x)y' - (2-x)y = 0, \quad y(-1) = 2, \quad y'(-1) = -3.$$
(7.3.4)
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3
core
Multivariable calculus
Calculus
Evaluate \( \int_C (x^2 + y^2 + z) ds \), where \( C \) is the curve with parameterization \( \mathbf{r}(t) = \langle \sin(3t), \cos(3t), 0 \rangle \), \( 0 \le t \le \frac{\pi}{4} \).
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4
core
Multivariable calculus
Calculus
Finding the curvature of a circle
Find the curvature of a circle with radius $r$, defined by $\vec{c}(t) = \langle r \cos t, r \sin t \rangle$.
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5
core
Parametric equations and vector functions
Calculus
\(r = 1 + \sin \theta\)
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6
warmup
Parametric equations and vector functions
Calculus
Find \(dy/dx\) and \(d^2y/dx^2\) along the general curve \(x = x(t)\), \(y = y(t)\).
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7
core
Parametric equations and vector functions
Calculus
\[\begin{aligned}[T] Use the results of the preceding two problems to
explore the graphs of r &= e^{-0.001\theta} and r \\ &= e^{-0.0001\theta} for
|\theta| \\ &> 100.\end{aligned}\]
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8
warmup
Parametric equations and vector functions
Calculus
$$x=3-2 \cos \theta, \quad y=-5+3 \sin \theta$$
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9
core
Counting
Discrete mathematics
What is the coefficient of \(x^{99} y^{60} z^{14}\) in \((2x^3 + y - z^2)^{100}\)? What about \(x^{99} y^{61} z^{13}\)?
multinomial-theoremcombinatoricsalgebraic-expansioncoefficientsdiscrete-mathematicspolynomialsexponentsfind-coefficient
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Coefficients in a Multinomial Expansion →
10
core
Vector spaces
Linear algebra
Find the unit vector that has the same direction as vector \(\mathbf{v}\) that begins at \((1, 4, 10)\) and ends at \((3, 0, 4)\).
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