1
core
Series of numbers
Real & complex analysis
Find the first five terms of the power series for \(e^x \sin x\).
power-seriesmaclaurin-seriesseries-multiplicationexponential-functionsine-functioncauchy-productcalculus-iitaylor-series
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Power Series for e^x sin x →
2
core
Area and arc length
Calculus
Find the arc length of the curve $y = \frac{x^4}{8} + \frac{1}{4x^2}$ from $x=1$ to $x=2$.
calculus-iarc-lengthdefinite-integralintegrationalgebraic-simplificationperfect-squarepolynomial-fractionscurve-lengthpower-rulefundamental-theorem-of-calculus
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Arc Length of a Rational Function →
3
core
The definite integral
Calculus
\[
\int_{-1}^1 \sqrt{3x^2-2x+3} (3x-1) \, dx
\]
definite-integralintegration-by-substitutionu-substitutionpower-rulefundamental-theorem-of-calculuspolynomialsfractional-exponentsevaluate-integralcalculus-i
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Definite Integral with a Square Root and Substitution →
4
warmup
Derivatives
Calculus
If \( f \) is differentiable and \( f(x_0) = 0 \), find \( \lim_{h\to 0} \frac{f(x_0 + h)}{7h} \).
calculus-iderivativeslimit-definition-of-derivativelimit-lawsdifferentiabilityevaluate-limitalgebraic-manipulationdifference-quotient
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Evaluating a Limit Using the Definition of the Derivative →
5
core
Functions and graphs
Calculus
Let \(f(x,y) = \sin(x^2 \cos y)\). Show \(f\) is continuous everywhere.
multivariable-calculuscontinuityfunctions-of-two-variablescomposition-of-functionsproduct-of-functionstrigonometric-functionspolynomialsprove-continuityelementary-functions
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Continuity of a Multivariable Trigonometric Function →
6
hard
Rectilinear and curvilinear motion
Calculus
A ball is thrown vertically upward. Its height \( s \) (in feet) after \( t \) seconds is given by \( s = 40t - 16t^2 \). Find
(a) when the ball hits the ground, (b) the instantaneous velocity at \( t=1 \), (c) the maximum height.
calculus-ikinematicsrectilinear-motionvelocityaccelerationmaximum-heightquadratic-functionderivativessecond-derivative-testcritical-pointsoptimizationpolynomial-differentiation
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Analyzing the Vertical Motion of a Ball →
7
core
Multiple integrals
Calculus
Find the volume under the surface $f(x, y) = \frac{1}{x^2 + y^2 + 1}$ over the sector of the circle with radius $a$ centered at the origin in the first quadrant, as shown in Figure 13.3.4.
double-integralpolar-coordinatesvolumesubstitutionnatural-logarithmcalculus-iiimultivariable-calculusfind-volumeevaluate-integral
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Volume Under a Surface Using Polar Coordinates →
8
warmup
Parametric equations and vector functions
Calculus
Solve \( \frac{dy}{dx} = \frac{x^2}{y^3} \).
differential-equationsseparable-differential-equationsseparation-of-variablesintegrationpower-ruleconstant-of-integrationfirst-order-differential-equationsordinary-differential-equationsimplicit-solutioncalculus-icalculus-iisolve-differential-equation
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Solving a Simple Separable Differential Equation →
9
hard
Determinants
Linear algebra
Let \( A = [a_{ij}] \) be a 5-square matrix, and let \( I = \{1, 2, 4\} \) and \( J = \{2, 3, 5\} \). Then \( I' = \{3, 5\} \) and \( J' = \{1, 4\} \), and the corresponding minor \( |M| \) and complementary minor \( |M'| \) are as follows:
linear-algebradeterminantsminorssubmatricescomplementary-minorsmatrix-entriesindex-setsmatrix-theory
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Finding a Minor and its Complementary Minor →
10
warmup
Vector spaces
Linear algebra
Find the volume of the parallelepiped formed by the vectors \(\mathbf{a} = 3\mathbf{i} + 4\mathbf{j} - \mathbf{k}\), \(\mathbf{b} = 2\mathbf{i} - \mathbf{j} - \mathbf{k}\), and \(\mathbf{c} = 3\mathbf{j} + \mathbf{k}\).
vectorsscalar-triple-productvolume-of-parallelepipeddeterminantcross-productdot-product3d-geometrymultivariable-calculuslinear-algebra
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Volume of a Parallelepiped Using Vectors →