1
warmup
Series of numbers
Real & complex analysis
Find the first three nonzero terms of the Maclaurin series for \(\tan x\).
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Maclaurin Series for Tangent →
2
core
Antiderivatives
Calculus
Evaluate \( \int (4-2t^2)^7 t \, dt \).
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Evaluating an Indefinite Integral Using Substitution →
3
core
Differential equations
Calculus
Rewrite the system
\[ x'' = f(t, x, x', y, y', y'') \]
\[ y''' = g(t, x, x', y, y', y'') \]
as a first order system.
differential-equationssystems-of-differential-equationsfirst-order-systemhigher-order-differential-equationsreduction-of-orderstate-variableschange-of-variablesordinary-differential-equations
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Rewriting a Higher-Order System as a First-Order System →
4
hard
Limits
Calculus
\(\lim_{x\to 0} \frac{\cos 2x - \cos x}{\sin^2 x}\)
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Limit of a Trigonometric Fraction Using L'Hôpital's Rule →
5
core
Multivariable calculus
Calculus
Let \(\vec{r}(t) = \langle \cos t, \sin t, t \rangle\). Compute \(\vec{r}'(t)\) and \(\vec{r}'(\pi/2)\). Sketch \(\vec{r}'(\pi/2)\) with its initial point at the origin and at \(\vec{r}(\pi/2)\).
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Derivative and Tangent Vector of a Helix →
6
core
Parametric equations and vector functions
Calculus
Find all points on the curve \( x = \sec \theta, y = \tan \theta \) at which horizontal and vertical tangents exist.
parametric-equationsderivativeshorizontal-tangentvertical-tangenttrigonometric-functionssecanttangentcalculus-iicurve-sketchingfind-pointsevaluate-derivative
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Horizontal and Vertical Tangents of a Parametric Curve →
7
core
Polar coordinates
Calculus
\( a_{n}=\frac{n^{2}+5}{\sqrt{4n^{4}+n}} \)
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Limit of a Sequence with a Square Root →
8
warmup
Techniques of integration
Calculus
Let \( \mathcal{R} \) be the region bounded by the curve \( y = \frac{\ln x}{x} \), the \( x \)-axis, and the line \( x = e \). Find the area of \( \mathcal{R} \).
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Area Under the Curve of ln(x)/x →
9
core
Eigenvalues and eigenvectors
Linear algebra
Subtract \(\lambda = 7\) down the diagonal of A to obtain
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10
core
Vector spaces
Linear algebra
Suppose \( z = 5 + 3i \) and \( w = 2 - 4i \). Find: (a) \( z+w \), (b) \( z- w \), (c) \( zw \).
Use the ordinary rules of algebra together with \( i^2 = -1 \) to obtain a result in the standard form \( a + bi \).
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Basic Arithmetic with Complex Numbers →