1
hard
Sequences
Real & complex analysis
Determine whether \(\sum_{n=1}^{\infty} \frac{n^2}{e^n}\) is convergent.
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Convergence of n^2/e^n Using the Ratio Test →
2
hard
Antiderivatives
Calculus
Find \( \int (2-x^3)^2 x \, dx \).
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Integrating a Polynomial by Expansion →
3
warmup
The definite integral
Calculus
\(\int_{0}^{1} \sqrt{x^2 - 6x + 9} \, dx.\)
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Evaluating a Definite Integral with a Hidden Absolute Value →
4
core
Integrals
Calculus
Investigate \( \int_{0}^{1} \frac{1}{x^{2}} \, dx \).
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Divergence of the Integral of 1/x^2 from 0 to 1 →
5
warmup
Multivariable calculus
Calculus
Evaluating a line integral: area under a surface over a curve.
Find the area under the surface \(f(x,y) = \cos(x) + \sin(y) + 2\) over the curve \(C\), which is the segment of the line \(y = 2x + 1\) on \(-1 \leq x \leq 1,\) as shown in Figure 14.1.2.
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Evaluating a Scalar Line Integral Over a Line Segment →
6
warmup
Parametric equations and vector functions
Calculus
Find the total arc length of \( r = 3 \sin \theta \).
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Arc Length of a Polar Circle →
7
core
Logarithmic and exponential functions
Calculus
Evaluate \( \lim_{x\to 3} \frac{\ln (1+2x) - \ln 7}{x-3} \).
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Evaluating a Logarithmic Limit Using the Definition of the Derivative →
8
warmup
Trigonometric and inverse trigonometric functions
Calculus
Calculate \( \frac{d}{dx}(\sin 3x) \).
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Derivative of sin(3x) →
9
hard
Systems of linear equations
Linear algebra
Row reduce each of the following matrices to echelon form:
(a) $A = \begin{pmatrix} 1 & 2 & -3 & 0 \\ 2 & 4 & -2 & 2 \\ 3 & 6 & -4 & 3 \end{pmatrix}$, (b) $B = \begin{pmatrix} -4 & 1 & -6 \\ 1 & 2 & -5 \\ 6 & 3 & -4 \end{pmatrix}$
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10
core
Vector spaces
Linear algebra
Find \( z\overline{z} \) and \( |z| \) when \( z = 3+4i \).
For \( z = a + bi \), use \( z\overline{z} = a^2 + b^2 \) and \( z = \sqrt{z\overline{z}} = \sqrt{a^2 + b^2} \).
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Modulus and Conjugate Product of a Complex Number →