1
core
Sequences
Real & complex analysis
Test the convergence of \( 3+\sqrt[3]{3}+\sqrt[4]{3}+\dots \)
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Divergence of a Series with Roots of 3 →
2
hard
Series of numbers
Real & complex analysis
Evaluate \(\psi(x) = \sum_{n=0}^\infty (n + 4) x^n\).
power-seriesgeometric-seriesdifferentiation-of-seriesevaluate-seriescalculus-iiinfinite-seriesrational-functionsclosed-form
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Evaluating a Power Series with a Linear Coefficient →
3
core
The chain rule
Calculus
If \( y = x^3-2 \) and \( x = 3z^2+5 \), then \( y \) can be considered a function of \( z \). Express \( \frac{dy}{dz} \) in terms of \( z \).
chain-ruledifferentiationcomposite-functionssubstitutioncalculus-iderivativespolynomial-differentiationpower-rule
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Derivative of a Composite Polynomial Function →
4
core
Directional derivatives and the gradient
Calculus
Evaluate \( I = \int_{\mathcal{R}} y^2 \, dA \), where \( \mathcal{R} \) is the region bounded by \( y=2x \), \( y=5x \), and \( x=2 \).
double-integraliterated-integralmultiple-integralspolynomialpower-ruleevaluate-integralcalculus-iiimultivariable-calculusregion-of-integration
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Evaluating a Double Integral over a Triangular Region →
5
core
Integrals
Calculus
Find \(\int_{0}^{\infty} x^{3}e^{-x} \, dx\).
improper-integralintegration-by-partsexponential-functionpolynomiall-hopitallimits-at-infinitygamma-functionreduction-formulaevaluate-integralcalculus-ii
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Integral of x^3 e^(-x) from 0 to Infinity →
6
core
Multivariable calculus
Calculus
Find equations of the normal line to the surface $x^2 + 4y^2 = z^2$ at $(3,2,5)$.
multivariable-calculusnormal-linegradientlevel-surfacepartial-derivativessymmetric-equationsparametric-equationsfind-normal-line
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Normal Line to a Surface →
7
warmup
Partial derivatives
Calculus
Find the relative maxima and minima of \( f(x, y) = x^3 + y^3 - 3xy \).
multivariable-calculuspartial-derivativescritical-pointsrelative-extremasecond-derivative-testhessianoptimizationsaddle-pointpolynomial-functions
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Relative Extrema of a Cubic Function of Two Variables →
8
core
Volume
Calculus
The region inside the circle $x^{2} + (y - b)^{2} = a^{2}$ ($0<a<b$); about the x-axis. (This yields the volume of a doughnut.)
calculus-ivolumesolid-of-revolutioncylindrical-shellstorusodd-functioneven-functionintegration-by-substitutiondefinite-integralgeometric-interpretation-of-integralsymmetry
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Volume of a Torus Using Cylindrical Shells →
9
hard
Linear transformations
Linear algebra
The linear map \( F : \mathbb{R}^2 \to \mathbb{R}^2 \) defined by \( F(x,y) = (x - y, x - 2y) \) is nonsingular by the previous Problem 5.24. Find a formula for \( F^{-1} \).
linear-algebralinear-mapslinear-transformationsinverse-functionsystem-of-linear-equationsnonsingularfind-inversevector-spaces
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Finding the Inverse of a Linear Map in R^2 →
10
core
Matrix operations
Linear algebra
Given \( A = \begin{pmatrix} 1 & -2 & 3 \\ 4 & 5 & -6 \end{pmatrix} \) and \( B = \begin{pmatrix} 3 & 0 & 2 \\ -7 & 1 & 8 \end{pmatrix} \), find:
(a) \( A+B \), (b) \( 2A-3B \).
matrix-additionscalar-multiplicationlinear-combinationmatriceslinear-algebraevaluate-expressionarithmeticmatrix-operations
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Basic Matrix Addition and Scalar Multiplication →