1
core
Series of numbers
Real & complex analysis
Prove \( (A - B) \times (A + B) = 2(A \times B) \).
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Cross Product of Sum and Difference of Vectors →
2
warmup
Lines, circles and conics
Calculus
Describe geometrically the family of lines \( y = 3x + b \).
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Geometric Description of a Family of Lines with Constant Slope →
3
warmup
Antiderivatives
Calculus
Evaluate \( \int \left( \csc^2 x + 3x^2 \right) dx \).
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Integral of Cosecant Squared Plus a Quadratic →
4
core
Functions and graphs
Calculus
Let $f(x, y) = \frac{5x^2 y^2}{x^2 + y^2}$. Find $\lim_{(x,y) \to (0,0)} f(x, y)$.
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Epsilon-Delta Proof for a Multivariable Limit →
5
warmup
Inequalities and absolute value
Calculus
Solve
\[
\frac{x-7}{x+3} > 2
\]
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Solving a Rational Inequality by Cases →
6
hard
Integrals
Calculus
Evaluate \( \int_e^\infty \frac{dx}{x \ln x \sqrt{\ln x}} \).
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Evaluating an Improper Integral with a Logarithm →
7
hard
Limits
Calculus
\(\lim_{x\to 1} \csc \pi x \cdot \ln x\)
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Limit of Cosecant Times Natural Logarithm →
8
core
Multivariable calculus
Calculus
Calculate the mass of a spring in the shape of a helix parameterized by \( \mathbf{r}(t) = \langle \cos t, \sin t, t \rangle, 0 \le t \le 6\pi \), with a density function given by \( \rho(x, y, z) = x + y + z \) kg/m.
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Mass of a Helical Spring →
9
hard
Matrix operations
Linear algebra
Find an upper triangular matrix \( A \) such that \( A^3 = \begin{pmatrix} 8 & -57 \\ 0 & 27 \end{pmatrix} \).
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Finding an Upper Triangular Matrix from its Cube →
10
core
Vector spaces
Linear algebra
Find parametric equations of the line segment between points \(P(-1, 3, 6)\) and \(Q(-8, 2, 4)\).
parametric-equationsline-segment3d-geometrydirection-vectorvector-subtractionmultivariable-calculusanalytic-geometrypoints-and-linesfind-parametric-equations
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Parametric Equations of a Line Segment in 3D →