1
hard
Differential equations
Calculus
Rewrite the equation
\[ y^{(4)} + 4y''' + 6y'' + 4y' + y = 0 \]
as a 4 \(\times\) 4 first order system.
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2
core
Differential equations
Calculus
Find the general solution of
\[ y' = \begin{pmatrix} 4 & -5 \\ 5 & -2 \end{pmatrix} y. \]
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3
core
Multivariable calculus
Calculus
Evaluate line integral \( \int_C (x^2 + yz)ds \), where \( C \) is the line with parameterization \( \mathbf{r}(t) = \langle 2t, 5t, -t \rangle \), \( 0 \le t \le 10 \). Reparameterize \( C \) with parameterization \( \mathbf{s}(t) = \langle 4t, 10t, -2t \rangle \), \( 0 \le t \le 5 \), recalculate line integral \( \int_C (x^2 + yz)ds \), and notice that the change of parameterization had no effect on the value of the integral.
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4
hard
Multivariable calculus
Calculus
Let \(\mathbf{v} = \left\langle \frac{x}{z}, \frac{y}{z}, 0 \right\rangle\) be the velocity field of a fluid. Let C be the solid cube given by
$1 \leq x \leq 4$, $2 \leq y \leq 5$, $1 \leq z \leq 4$, and let S be the boundary of this cube (see the following figure). Find the
flow rate of the fluid across S.
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5
core
Multivariable calculus
Calculus
Find the value of \(\int_{C} 4xdx + zdy + 4y^{2}dz\), where \(C\) is the curve parameterized by \( \mathbf{r}(t) = \langle 4\cos(2t), 2\sin(2t), 3 \rangle, 0 \le t \le \frac{\pi}{4} \).
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6
core
Parametric equations and vector functions
Calculus
If $F(u)=A$ for all $u$, show that $F'(u)=0$, and, conversely, if $F'(u)=0$ for all $u$, then $F(u)$ is a constant
vector.
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7
core
Parametric equations and vector functions
Calculus
Solve $xy'' + y' = x$.
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8
hard
Parametric equations and vector functions
Calculus
Region common to $r=2$ and $r=4 \cos \theta$
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9
core
Parametric equations and vector functions
Calculus
\(Endpoints of major axis at (0, 2), (0, \text{-}2) and foci
located at (3, 0), (\text{-}3, 0)\)
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10
hard
Vector spaces
Linear algebra
\(\mathbf{a} = 9\mathbf{i} - 2\mathbf{j}, \mathbf{b} = -3\mathbf{i} + \mathbf{j}\)
a. \(3\mathbf{a} + \mathbf{b}\)
b. \(|\mathbf{a}|\)
c. \(\mathbf{a} \times |\mathbf{b} \times \mathbf{a}|\)
d. \(\mathbf{b} \times |\mathbf{a}|\)
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