1
core
Series of numbers
Real & complex analysis
If \( \mathbf{B} \times \mathbf{C} = \mathbf{0} \), what can be concluded about \( \mathbf{B} \) and \( \mathbf{C} \)?
cross-productvectorsparallel-vectorslinear-dependencezero-vectormagnitudeangle-between-vectorsmultivariable-calculusanalytic-geometryvector-algebra
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Conditions for a Zero Cross Product →
2
hard
Antiderivatives
Calculus
Find \( \int \tan^2 \theta \sec^4 \theta \, d\theta \).
calculus-icalculus-iiindefinite-integraltrigonometric-integralsintegration-by-substitutiontangentsecantpythagorean-identitypower-ruleantiderivativesevaluate-integral
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Integral of tan^2(theta) sec^4(theta) →
3
hard
The definite integral
Calculus
Find \( \lim_{h \to 0} \frac{1}{h} \int_{2}^{2+h} \sqrt{t^{2}+2} \, dt \).
calculus-idefinite-integrallimitderivative-definitionfundamental-theorem-of-calculusevaluate-limitdifference-quotientintegration
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Limit of an Integral Using the Definition of the Derivative →
4
warmup
Limits
Calculus
Evaluate the limit:
\[ \lim_{x\to 1} \frac{x^{4}-1}{x-1} \]
calculus-ilimitsl-hopitalindeterminate-formsderivativespower-ruleevaluate-limitrational-functiondirect-substitutionalgebraic-limitlimit-evaluationzero-over-zero
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Limit of (x^4 - 1)/(x - 1) as x approaches 1 →
5
core
Multiple integrals
Calculus
Consider the region in the first quadrant between the functions \(y = 2x\) and \(y = x^2\). Describe the region first as Type I and then as Type II.
multiple-integralsdouble-integralsregion-of-integrationtype-i-regiontype-ii-regionchange-of-order-of-integrationmultivariable-calculuscalculus-iiiinequalitiesfunctions-and-graphsintersection-of-curvesset-builder-notation
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Describing a Region as Type I and Type II →
6
warmup
Multivariable calculus
Calculus
Parameterize the surface $z = x^2 + 2y^2$ over the circular region $R$ enclosed by the circle of radius 2 that is centered at the origin.
surface-parameterizationparametric-equationsmultivariable-calculuspolar-coordinatescircular-regionparaboloidvector-valued-functionsdomain-mapping
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Parameterizing a Paraboloid over a Circular Disk →
7
warmup
Parametric equations and vector functions
Calculus
Solve \( \frac{dy}{dx} = ky \).
differential-equationsseparable-differential-equationsexponential-growthexponential-decayintegrationnatural-logarithmabsolute-valuearbitrary-constantcalculus-icalculus-iifind-general-solutionfirst-order-differential-equation
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Solving the Exponential Growth Differential Equation →
8
core
Logarithmic and exponential functions
Calculus
Prove that the only solutions of the differential equation \(f'(x) = f(x)\) are the functions \(Ce^x\), where \(C\) is a constant.
differential-equationsexponential-functionproduct-ruleuniqueness-of-solutioncalculus-ifirst-order-linearconstant-of-integrationprove-uniqueness
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Uniqueness of Solutions to f'(x) = f(x) →
9
core
Determinants
Linear algebra
Solve the system using determinants
\begin{cases}
x + y + z = 5 \\
x-2y-3z = -1 \\
2x + y - z = 3
\end{cases}
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10
warmup
Vector spaces
Linear algebra
Find the measure of the angle between planes \( x+y-z=3 \) and \( 3x-y+3z=5 \). Give the answer in radians and round to two decimal places.
analytic-geometryangle-between-planesarccoscalculus-iiidot-productfind-anglemultivariable-calculusnormal-vectorsplanesradiansvector-magnitudevectors
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Angle Between Two Planes →