1
hard
Sequences
Real & complex analysis
Determine whether \( \sum_{n=1}^{\infty} n\left(\frac{3}{4}\right)^n \) converges.
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Convergence of a Series with Polynomial and Exponential Terms →
2
core
Antiderivatives
Calculus
Evaluate \( \int \sqrt{7x+4} \, dx \).
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Integral of a Square Root with a Linear Inner Function →
3
core
The definite integral
Calculus
If the average value of \( f(x) = x^{3}+bx-2 \) on \( [0, 2] \) is 4, find \( b \).
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Finding a Parameter Given the Average Value of a Polynomial →
4
core
Multiple integrals
Calculus
The surface area of a sphere.
Find the surface area of the sphere with radius $a$ centered at the origin, whose top hemisphere has equation $f(x, y) = \sqrt{a^2 - x^2 - y^2}$.
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Surface Area of a Sphere via Double Integration →
5
warmup
Multivariable calculus
Calculus
Is vector field \( G(x, y, z) = \langle y, x, xyz \rangle \) conservative?
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Determining if a 3D Vector Field is Conservative →
6
core
Parametric equations and vector functions
Calculus
Determine the concavity of the curve \(x = 2t + \ln t, y = 2t - \ln t\).
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Concavity of a Parametric Curve with Logarithms →
7
hard
Logarithmic and exponential functions
Calculus
Find \( \int x \tanh x^2 \, dx \).
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Integral of x tanh(x^2) →
8
core
Trigonometric and inverse trigonometric functions
Calculus
Evaluate \( \lim_{x\to 0} \frac{\tan x}{x} \).
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Limit of tan(x)/x as x approaches 0 →
9
core
Linear transformations
Linear algebra
Consider the linear transformation $F$ on $\mathbb{R}^2$ defined by $F(x,y) = (5x - y, 2x + y)$ and the
following bases of $\mathbb{R}^2$:
$E = \{e_1, e_2\} = \{(1,0), (0,1)\}$
and $S = \{u_1, u_2\} = \{(1,4), (2,7)\}$
(a) Find the change-of-basis matrix $P$ from $E$ to $S$ and the change-of-basis matrix $Q$ from $S$ back
to $E$.
(b) Find the matrix $A$ that represents $F$ in the basis $E$.
(c) Find the matrix $B$ that represents $F$ in the basis $S$.
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10
core
Vector spaces
Linear algebra
Let $C$ be the curve $F(t) = (t^2, 3t - 2, t^3, t^2 + 5)$ in $\mathbf{R}^4$, where $0 \le t \le 4$.
(a) Find the point $P$ on $C$ corresponding to $t = 2$.
(b) Find the initial point $Q$ and terminal point $Q'$ of $C$.
(c) Find the unit tangent vector $T$ to the curve $C$ when $t = 2$.
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Points and Unit Tangent Vector of a Parametric Curve in R^4 →