1
hard
Series of numbers
Real & complex analysis
Find the first five terms of the power series for \( e^x \cos x \) by multiplication of power series.
power-seriesmaclaurin-seriesseries-multiplicationexponential-functioncosine-functioncalculus-iitaylor-seriesfind-series-expansion
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Power Series of e^x cos x by Multiplication →
2
core
Antiderivatives
Calculus
Find \( \int \sin^4 x \cos x \, dx \).
integrationsubstitutionu-substitutiontrigonometric-functionssinecosineindefinite-integralpower-rulecalculus-iantiderivativesevaluate-integral
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Integrating a Power of Sine Multiplied by Cosine →
3
core
Functions and graphs
Calculus
Implicit Differentiation
Given the implicitly defined function $\sin (x^2 y^2) + y^3 = x + y$, find $y'$. Note: this is the same problem as given in Example 2.6.4 of Section 2.6, where the solution took about a full page to find.
implicit-differentiationpartial-derivativesmultivariable-calculuschain-ruleimplicit-function-theoremcalculus-iiievaluate-derivativetrigonometric-functionsfunctions-of-two-variables
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Implicit Differentiation via Partial Derivatives →
4
warmup
Integrals
Calculus
Evaluate \( \int_{1}^{\infty} \frac{e^{-\sqrt{x}}}{\sqrt{x}} \, dx \).
improper-integralintegration-by-substitutionexponential-functionlimits-at-infinityevaluate-integralcalculus-iidefinite-integralfundamental-theorem-of-calculus
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Evaluating an Improper Integral with an Exponential Function →
5
core
Limits
Calculus
Evaluate \(\lim_{x\to-\infty} (3x^4 - x^2 + x - 7)\).
limitpolynomialinfinitynegative-infinityleading-termfactoringend-behaviorcalculus-ievaluate-limit
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Limit of a Quartic Polynomial at Negative Infinity →
6
core
Multiple integrals
Calculus
Find the area of a region bounded above by the curve \(y = x^3\) and below by \(y = 0\) over the interval \([0, 3]\).
calculus-iiimultiple-integralsdouble-integralarea-using-double-integralsiterated-integralpolynomial-integrationdefinite-integralevaluate-areapower-rulefundamental-theorem-of-calculustype-i-regionintegration-bounds
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Area Under a Cubic Curve Using Double Integrals →
7
core
Multivariable calculus
Calculus
Give a parameterization for the portion of cone \(x^2 + y^2 = z^2\) lying in the first octant.
parameterizationconefirst-octantmultivariable-calculusparametric-equationssurface-parameterizationcylindrical-coordinatesvector-functionscalculus-iii
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Parameterizing a Cone in the First Octant →
8
hard
Parametric equations and vector functions
Calculus
Vertices located at \( (5, 0) \), \( (-5, 0) \) and foci located at \( (6, 0) \), \( (-6, 0) \)
analytic-geometryconic-sectionshyperbolaverticesfocistandard-equationfind-equationalgebraprecalculus
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Equation of a Hyperbola from Vertices and Foci →
9
hard
Inner product spaces and orthogonality
Linear algebra
Consider the function $f(t) = t^2 – 4t$ in $C[0, 3]$.
(a) Find $|| f ||_{\infty}$, (b) Plot $f(t)$ in the plane $\mathbb{R}^2$, (c) Find $|| f ||_1$, (d) Find $|| f ||_2$.
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10
core
Vector spaces
Linear algebra
Are the vectors \(\mathbf{a} = \mathbf{i} + \mathbf{j} - \mathbf{k}\), \(\mathbf{b} = \mathbf{i} - \mathbf{j} + \mathbf{k}\), and \(\mathbf{c} = \mathbf{i} + \mathbf{j} + \mathbf{k}\) coplanar?
vectorsthree-dimensionscoplanar-vectorstriple-scalar-productdeterminantparallelepipedvolumecross-productdot-productlinear-independencemultivariable-calculusanalytic-geometry
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Determining if Three Vectors are Coplanar →