data_source
stringclasses 9
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stringlengths 14
17
| problem
stringlengths 14
1.32k
| answer
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| license
stringclasses 8
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stringclasses 7
values | solution
stringlengths 1
991
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|---|---|---|---|---|---|---|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.1
|
Evaluate the given double integral: $\int_{0}^{1} \int_{\sqrt{x}}^{1} 24 x^{2} y d y d x$
|
$1 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.5
|
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$
|
0
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0
|
college_math.Corrals_Vector_Calculus
|
exercise.1.1.3
|
For the points $P=(0,0,0), Q=(1,3,2), R=(1,0,1), S=(2,3,4)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ?
|
No
|
GNU Free Documentation License
|
college_math.vector_calculus
|
No
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.9
|
Find all local maxima and minima of the function $f(x, y) = 4x^{2} - 4xy + 2y^{2} + 10x - 6y$.
|
local min. $(-1,1 / 2) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
local min. $(-1,1 / 2)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.6.3
|
Find the Laplacian of the function $f(x, y, z)=\left(x^{2}+y^{2}+z^{2}\right)^{3 / 2}$.
|
$12 \sqrt{x^{2}+y^{2}+z^{2}} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
12 \sqrt{x^{2}+y^{2}+z^{2}}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.31
|
Describe geometrically the set of points with position vector $\mathbf{x}$ satisfying the equation
$$(\mathbf{v} \times \mathbf{x}) \times \mathbf{x}=\mathbf{v}$$
for given vector $\mathbf{v} \neq \mathbf{0}$
|
A circle of radius $\frac{1}{\|\mathbf{v}\|}$ centered at the origin in the normal plane to $\mathbf{v}$.
|
GNU Free Documentation License
|
college_math.vector_calculus
|
A circle of radius $\frac{1}{\|\mathbf{v}\|}$ centered at the origin in the normal plane to $\mathbf{v}$.
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.3
|
Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=x^{3}+y^{2}, R=[0,1] \times[0,1]$
|
$\frac{7}{12} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{7}{12}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.3.3
|
Evaluate the given triple integral: $\int_{0}^{\pi} \int_{0}^{x} \int_{0}^{x y} x^{2} \sin z d z d y d x$
|
$\left(2 \cos \left(\pi^{2}\right)+\pi^{4}-2\right) / 4 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\left(2 \cos \left(\pi^{2}\right)+\pi^{4}-2\right) / 4
|
college_math.Corrals_Vector_Calculus
|
exercise.4.6.1
|
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\{(x, y): 0 \leq x \leq 2,0 \leq y \leq 4\}, \delta(x, y)=2 y$
|
$(1,8 / 3)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(1,8 / 3)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.5
|
State the domain and range of the given function: $f(x, y, z)=\sin (x y z)$
|
domain: $\mathbb{R}^{3}$, range: $[-1,1] $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
domain: $\mathbb{R}^{3}$, range: $[-1,1]
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.7
|
Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=(y-2 z) \mathbf{i}+x y \mathbf{j}+(2 x z+y) \mathbf{k} ; \quad C: x=t, y=2 t, z=t^{2}-1,0 \leq t \leq 1$
|
$67 / 15 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
67 / 15
|
college_math.Corrals_Vector_Calculus
|
exercise.4.3.7
|
Evaluate the given triple integral: $\int_{1}^{2} \int_{2}^{4} \int_{0}^{3} 1 d x d y d z$
|
6
|
GNU Free Documentation License
|
college_math.vector_calculus
|
6
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.13
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(1,1)} \frac{x^{2}-y^{2}}{x-y}$
|
$2 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.17
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \frac{x}{y}$
|
does not exist
|
GNU Free Documentation License
|
college_math.vector_calculus
|
does not exist
|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.5
|
Evaluate the given double integral: $\int_{0}^{\pi / 2} \int_{0}^{y} \cos x \sin y d x d y$
|
$\frac{\pi}{4} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{\pi}{4}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.7
|
Evaluate the double integral: $\int_{0}^{2} \int_{0}^{1}(x+2) d x d y$
|
$5 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
5
|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.3
|
Evaluate the given double integral: $\int_{1}^{2} \int_{0}^{\ln x} 4 x d y d x$
|
$8 \ln 2-3$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
8 \ln 2-3
|
college_math.Corrals_Vector_Calculus
|
exercise.5.1.3
|
Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=2 x+y ; \quad C$ : polygonal path from $(0,0)$ to $(3,0)$ to $(3,2)$
|
23
|
GNU Free Documentation License
|
college_math.vector_calculus
|
23
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.1
|
Find the volume under the surface $z=f(x, y)$ over the rectangle $R$: $f(x, y)=4 x y, R=[0,1] \times[0,1]$
|
$1 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.13
|
Find three positive numbers $x, y, z$ whose sum is 10 such that $x^{2}y^{2}z$ is a maximum.
|
$x=y=4, z=2$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
x=y=4, z=2
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.7
|
Find all local maxima and minima of the function $f(x, y) = \sqrt{x^{2} + y^{2}}$.
|
local min. $(0,0) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
local min. $(0,0)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.3
|
Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{3} - 3y$.
|
local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
local min. $(1,1)$; local max. $(-1,-1)$; saddle pts. $(1,-1),(-1,1)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.1.7
|
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$
|
$-2 \pi$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
-2 \pi
|
college_math.Corrals_Vector_Calculus
|
exercise.4.7.2
|
For $\sigma>0$ and $\mu>0$, evaluate
$$
\int_{-\infty}^{\infty} \frac{1}{\sigma \sqrt{2 \pi}} e^{-(x-\mu)^{2} / 2 \sigma^{2}} d x
$$
|
1
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1
|
college_math.Corrals_Vector_Calculus
|
exercise.4.6.5
|
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq 1\right\}, \delta(x, y)=y$
|
$(0,3 \pi / 16) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(0,3 \pi / 16)
|
college_math.Corrals_Vector_Calculus
|
exercise.4.6.9
|
Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\{(x, y, z): 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$
|
$(7 / 12,7 / 12,7 / 12)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(7 / 12,7 / 12,7 / 12)
|
college_math.Corrals_Vector_Calculus
|
exercise.2.3.1
|
Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (\cos t, \sin t, t)$.
|
$\frac{3 \pi \sqrt{5}}{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{3 \pi \sqrt{5}}{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.3
|
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(2,1,4), \mathbf{w}=(1,-2,0)$
|
$(8,4,-5) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(8,4,-5)
|
college_math.Corrals_Vector_Calculus
|
exercise.2.1.5
|
Find the velocity $\mathbf{v}(t)$ and acceleration $\mathbf{a}(t)$ of an object with the given position vector $\mathbf{r}(t)$: $\mathbf{r}(t)=(t, t-\sin t, 1-\cos t)$
|
$\mathbf{v}(t)=(1,1-\cos t, \sin t)$, $\mathbf{a}(t)=(0, \sin t, \cos t) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\mathbf{v}(t)=(1,1-\cos t, \sin t)$, $\mathbf{a}(t)=(0, \sin t, \cos t)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.3.9
|
Find the equation of the tangent plane to the given surface at the point $P$: $x^{2}+y^{2}-z^{2}=0$, $P=(3,4,5)$.
|
$3 x+4 y-5 z=0$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
3 x+4 y-5 z=0
|
college_math.Corrals_Vector_Calculus
|
exercise.4.3.1
|
Evaluate the given triple integral: $\int_{0}^{3} \int_{0}^{2} \int_{0}^{1} x y z d x d y d z$
|
$\frac{9}{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{9}{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.3.1
|
Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2}+y^{3}$, $P=(1,1,2)$.
|
$2 x+3 y-z-3=0 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 x+3 y-z-3=0
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.7
|
Calculate the area of the triangle $\triangle P Q R$:
$P=(5,1,-2), Q=(4,-4,3), R=(2,4,0)$
|
$16.72 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
16.72
|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.10
|
Evaluate the double integral: $\iint_{R} f(x, y) d A$, where $f(x, y)=x^{2}+y$ and $R$ is the triangle with vertices $(0,0),(2,0)$ and $(0,1)$.
|
$\frac{6}{5} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{6}{5}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.6.7
|
Find the center of mass of the solid $S$ with the given density function $\delta(x, y, z)$:
$S=\left\{(x, y, z): z \geq 0, x^{2}+y^{2}+z^{2} \leq a^{2}\right\}, \delta(x, y, z)=x^{2}+y^{2}+z^{2}$
|
$(0,0,5 a / 12) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(0,0,5 a / 12)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.13
|
State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=x y \mathbf{i}-\left(x-y z^{2}\right) \mathbf{j}+y^{2} z \mathbf{k}$
|
No
|
GNU Free Documentation License
|
college_math.vector_calculus
|
No
|
college_math.Corrals_Vector_Calculus
|
exercise.4.5.7
|
Evaluate $\iint_{R} \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) d A$, where $R$ is the triangle with vertices $(0,0),(2,0)$ and $(1,1)$. (Hint: Use the change of variables $u=(x+y) / 2, v=(x-y) / 2$.)
|
$1-\frac{\sin 2}{2}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1-\frac{\sin 2}{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.3.9
|
Let $\mathbf{v}=(8,4,3)$ and $\mathbf{w}=(-2,1,4)$. Is $\mathbf{v} \perp \mathbf{w}$ ? Justify your answer.
|
Yes, since $\mathbf{v} \cdot \mathbf{w}=0
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes, since $\mathbf{v} \cdot \mathbf{w}=0
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.7
|
Compute the gradient $\nabla f$ for the function $f(x, y, z)=\sin (x y z)$.
|
$\quad(y z \cos (x y z), x z \cos (x y z), x y \cos (x y z))$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\quad(y z \cos (x y z), x z \cos (x y z), x y \cos (x y z))
|
college_math.Corrals_Vector_Calculus
|
exercise.5.7.5
|
For $\mathbf{f}(\rho, \theta, \phi)=\mathbf{e}_{\rho}+\rho \cos \theta \mathbf{e}_{\theta}+\rho \mathbf{e}_{\phi}$ in spherical coordinates, find $\operatorname{div} \mathbf{f}$ and curlf.
|
$\operatorname{div} \mathbf{f}=\frac{2}{\rho}-\frac{\sin \theta}{\sin \phi}+\cot \phi, \operatorname{curl} \mathbf{f}=\cot \phi \cos \theta \mathbf{e}_{\rho}+$ $2 \mathbf{e}_{\theta}-2 \cos \theta \mathbf{e}_{\phi} \mathbf{6}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\operatorname{div} \mathbf{f}=\frac{2}{\rho}-\frac{\sin \theta}{\sin \phi}+\cot \phi, \operatorname{curl} \mathbf{f}=\cot \phi \cos \theta \mathbf{e}_{\rho}+$ $2 \mathbf{e}_{\theta}-2 \cos \theta \mathbf{e}_{\phi} \mathbf{6}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.11
|
Evaluate the double integral: $\int_{0}^{2} \int_{1}^{4} x y d x d y$
|
$15 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
15
|
college_math.Corrals_Vector_Calculus
|
exercise.5.7.3
|
Let $f(x, y, z)=\frac{z}{x^{2}+y^{2}}$ in Cartesian coordinates. Find $\nabla f$ in cylindrical coordinates.
|
$-\frac{2 z}{r^{3}} \mathbf{e}_{r}+\frac{1}{r^{2}} \mathbf{e}_{z}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
-\frac{2 z}{r^{3}} \mathbf{e}_{r}+\frac{1}{r^{2}} \mathbf{e}_{z}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.9
|
Compute the gradient $\nabla f$ for the function $f(x, y, z)=x^{2}+y^{2}+z^{2}$.
|
$\quad(2 x, 2 y, 2 z) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\quad(2 x, 2 y, 2 z)
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.9
|
Evaluate the double integral: $\int_{0}^{\pi / 2} \int_{0}^{1} x y \cos \left(x^{2} y\right) d x d y$
|
$\frac{1}{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{1}{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.5.3.7
|
Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=(8 x y+3) \mathbf{i}+4\left(x^{2}+y\right) \mathbf{j}$ ? If so, find one.
|
Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes. $F(x, y)=4 x^{2} y+2 y^{2}+3 x
|
college_math.Corrals_Vector_Calculus
|
exercise.5.2.1
|
Evaluate $\oint_{C}\left(x^{2}+y^{2}\right) d x+2 x y d y$ for $C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$.
|
$0 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0
|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.7
|
Evaluate the given double integral: $\int_{0}^{2} \int_{0}^{y} 1 d x d y$
|
$2 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2
|
college_math.Corrals_Vector_Calculus
|
exercise.3.3.7
|
Find the equation of the tangent plane to the given surface at the point $P$: $\frac{x^{2}}{4}+\frac{y^{2}}{9}+\frac{z^{2}}{16}=1$, $P=\left(1,2, \frac{2 \sqrt{11}}{3}\right)$.
|
$\frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2 \sqrt{11}}{3})=0 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{1}{2}(x-1)+\frac{4}{9}(y-2)+\frac{\sqrt{11}}{12}(z-\frac{2 \sqrt{11}}{3})=0
|
college_math.Corrals_Vector_Calculus
|
exercise.2.3.3
|
Find the tangent line, the osculating plane, and the curvature at each point of the curve $\mathbf{f}(t)= (t \sin t, t \cos t)$.
|
$2\left(5^{3 / 2}-8\right) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2\left(5^{3 / 2}-8\right)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.3
|
Compute the gradient $\nabla f$ for the function $f(x, y)=\sqrt{x^{2}+y^{2}+4}$.
|
$\left(\frac{x}{\sqrt{x^{2}+y^{2}+4}}, \frac{y}{\sqrt{x^{2}+y^{2}+4}}\right) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\left(\frac{x}{\sqrt{x^{2}+y^{2}+4}}, \frac{y}{\sqrt{x^{2}+y^{2}+4}}\right)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.5
|
Compute the gradient $\nabla f$ for the function $f(x, y)=\ln (x y)$.
|
$\left(\frac{1}{x}, \frac{1}{y}\right)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\left(\frac{1}{x}, \frac{1}{y}\right)
|
college_math.Corrals_Vector_Calculus
|
exercise.1.6.9
|
Find the trace of the hyperbolic paraboloid $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=\frac{z}{c}$ in the $x y$-plane
|
lines $\frac{x}{a}=\frac{y}{b}, z=0$ and $\frac{x}{a}=-\frac{y}{b}, z=0$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
lines $\frac{x}{a}=\frac{y}{b}, z=0$ and $\frac{x}{a}=-\frac{y}{b}, z=0
|
college_math.Corrals_Vector_Calculus
|
exercise.4.5.1
|
Find the volume $V$ inside the paraboloid $z=x^{2}+y^{2}$ for $0 \leq z \leq 4$.
|
$8 \pi$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
8 \pi
|
college_math.Corrals_Vector_Calculus
|
exercise.1.3.7
|
Find the angle $\theta$ between the vectors $\mathbf{v}=-\mathbf{i}+2 \mathbf{j}+\mathbf{k}$ and $\mathbf{w}=-3 \mathbf{i}+6 \mathbf{j}+3 \mathbf{k}$.
|
$0^{\circ} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0^{\circ}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.1
|
State the domain and range of the given function: $f(x, y)=x^{2}+y^{2}-1$
|
domain: $\mathbb{R}^{2}$, range: $[-1, \infty) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
domain: $\mathbb{R}^{2}$, range: $[-1, \infty)
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.1
|
Calculate $\mathbf{v} \times \mathbf{w}$:
$\mathbf{v}=(5,1,-2), \mathbf{w}=(4,-4,3)$
|
$(-5,-23,-24) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(-5,-23,-24)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.3.3
|
Find the equation of the tangent plane to the surface $z=f(x, y)$ at the point $P$: $f(x, y)=x^{2} y$, $P=(-1,1,1)$.
|
$-2 x+y-z-2=0$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
-2 x+y-z-2=0
|
college_math.Corrals_Vector_Calculus
|
exercise.5.2.3
|
Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=y \mathbf{i}-x \mathbf{j}$ ? If so, find one.
|
No
|
GNU Free Documentation License
|
college_math.vector_calculus
|
No
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.1
|
Find all local maxima and minima of the function $f(x, y) = x^{3} - 3x + y^{2}$.
|
local min. $(1,0)$; saddle pt. $(-1,0)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
local min. $(1,0)$; saddle pt. $(-1,0)
|
college_math.Corrals_Vector_Calculus
|
exercise.1.3.1
|
Let $\mathbf{v}=(5,1,-2)$ and $\mathbf{w}=(4,-4,3)$. Calculate $\mathbf{v} \cdot \mathbf{w}$.
|
$10 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
10
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.11
|
For a rectangular solid of volume 1000 cubic meters, find the dimensions that will minimize the surface area. (Hint: Use the volume condition to write the surface area as a function of just two variables.)
|
width $=$ height $=\operatorname{depth}=10$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
width $=$ height $=\operatorname{depth}=10
|
college_math.Corrals_Vector_Calculus
|
exercise.1.6.1
|
Determine if the given equation describes a sphere. If so, find its radius and center: $x^{2}+y^{2}+z^{2}-4 x-6 y-10 z+37=0$
|
radius: 1 , center: $(2,3,5)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
radius: 1 , center: $(2,3,5)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.1
|
Compute the gradient $\nabla f$ for the function $f(x, y)=x^{2}+y^{2}-1$.
|
$(2 x, 2 y)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(2 x, 2 y)
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.13
|
Find the directional derivative of $f(x, y)=\sqrt{x^{2}+y^{2}+4}$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$.
|
$\quad \frac{1}{\sqrt{3}}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\quad \frac{1}{\sqrt{3}}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.15
|
Find the directional derivative of $f(x, y, z)=\sin (x y z)$ at the point $P=(1,1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right)$.
|
$\sqrt{3} \cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\sqrt{3} \cos (1) ; 1$. increase: $(45,20)$, decrease: $(-45,-20)
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.11
|
Find the volume of the parallelepiped with adjacent sides $\mathbf{u}, \mathbf{v}, \mathbf{w}$:
$\mathbf{u}=(1,1,3), \mathbf{v}=(2,1,4), \mathbf{w}=(5,1,-2)$
|
$9 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
9
|
college_math.Corrals_Vector_Calculus
|
exercise.3.5.5
|
Find all local maxima and minima of the function $f(x, y) = 2x^{3} + 6xy + 3y^{2}$.
|
local min. $(1,-1)$, saddle pt. $(0,0) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
local min. $(1,-1)$, saddle pt. $(0,0)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.2.5
|
Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x y^{2} \mathbf{i}+x^{3} y \mathbf{j}$ ? If so, find one.
|
No
|
GNU Free Documentation License
|
college_math.vector_calculus
|
No
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.9
|
Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=x y \mathbf{i}+(z-x) \mathbf{j}+2 y z \mathbf{k} ; \quad C$ : the polygonal path from $(0,0,0)$ to $(1,0,0)$ to $(1,2,0)$ to $(1,2,-2)$
|
6
|
GNU Free Documentation License
|
college_math.vector_calculus
|
6
|
college_math.Corrals_Vector_Calculus
|
exercise.4.5.3
|
Find the volume $V$ of the solid inside both $x^{2}+y^{2}+z^{2}=4$ and $x^{2}+y^{2}=1$.
|
$\frac{4 \pi}{3}\left(8-3^{3 / 2}\right)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{4 \pi}{3}\left(8-3^{3 / 2}\right)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.2.4
|
Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=x \mathbf{i}-y \mathbf{j}$ ? If so, find one.
|
Yes. $F(x, y)=\frac{x^{2}}{2}-\frac{y^{2}}{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes. $F(x, y)=\frac{x^{2}}{2}-\frac{y^{2}}{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.2.1.1
|
Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=\left(t+1, t^{2}+1, t^{3}+1\right)$
|
$\mathbf{f}^{\prime}(t)=\left(1,2 t, 3 t^{2}\right), \quad x=1+t, \quad y=z=$ $1 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\mathbf{f}^{\prime}(t)=\left(1,2 t, 3 t^{2}\right), \quad x=1+t, \quad y=z=$ $1
|
college_math.Corrals_Vector_Calculus
|
exercise.5.1.11
|
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}+y^{2}\right) \mathbf{i} ; \quad C: x=2+\cos t, y=\sin t, 0 \leq t \leq 2 \pi$
|
0
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0
|
college_math.Corrals_Vector_Calculus
|
exercise.3.4.11
|
Find the directional derivative of $f(x, y)=x^{2}+y^{2}-1$ at the point $P=(1,1)$ in the direction of $\mathbf{v}=\left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$.
|
$\quad 2 \sqrt{2} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\quad 2 \sqrt{2}
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.5
|
Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=y \mathbf{i}-x \mathbf{j}+z \mathbf{k} ; \quad C: x=\cos t, y=\sin t, z=t, 0 \leq t \leq 2 \pi$
|
$2 \pi(\pi-1)$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 \pi(\pi-1)
|
college_math.Corrals_Vector_Calculus
|
exercise.2.1.3
|
Calculate $\mathbf{f}^{\prime}(t)$ and find the tangent line at $\mathbf{f}(0)$ for the following function: $\mathbf{f}(t)=(\cos 2 t, \sin 2 t, t)$
|
$\mathbf{f}^{\prime}(t)=(-2 \sin 2 t, 2 \cos 2 t, 1) ; \quad x=1$, $y=2 t, \quad z=t $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\mathbf{f}^{\prime}(t)=(-2 \sin 2 t, 2 \cos 2 t, 1) ; \quad x=1$, $y=2 t, \quad z=t
|
college_math.Corrals_Vector_Calculus
|
exercise.1.1.2
|
For the points $P=(1,-1,1), Q=(2,-2,2), R=(2,0,1), S=(3,-1,2)$, does $\overrightarrow{P Q}=\overrightarrow{R S}$ ?
|
Yes
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes
|
college_math.Corrals_Vector_Calculus
|
exercise.5.6.5
|
Find the Laplacian of the function $f(x, y, z)=x^{3}+y^{3}+z^{3}$.
|
$6(x+y+z) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
6(x+y+z)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.2
|
Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=\frac{x}{y}+y+2 y z ; \quad C: x=t^{2}, y=t, z=1,1 \leq t \leq 2$
|
$(17 \sqrt{17}-5 \sqrt{5}) / 3 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
(17 \sqrt{17}-5 \sqrt{5}) / 3
|
college_math.Corrals_Vector_Calculus
|
exercise.4.1.5
|
Evaluate the double integral: $\int_{0}^{1} \int_{1}^{2}(1-y) x^{2} d x d y$
|
$\frac{7}{6} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{7}{6}
|
college_math.Corrals_Vector_Calculus
|
exercise.5.6.1
|
Find the Laplacian of the function $f(x, y, z)=x+y+z$.
|
$0 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
0
|
college_math.Corrals_Vector_Calculus
|
exercise.3.7.1
|
Find the constrained maxima and minima of $f(x, y)=2 x+y$ given that $x^{2}+y^{2}=4$.
|
$\max .\left(\frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\max .\left(\frac{4}{\sqrt{5}}, \frac{2}{\sqrt{5}}\right)
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.4
|
Calculate $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y, z)$ and curve $C$: $\mathbf{f}(x, y, z)=\mathbf{i}-\mathbf{j}+\mathbf{k} ; \quad C: x=3 t, y=2 t, z=t, 0 \leq t \leq 1$
|
2
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2
|
college_math.Corrals_Vector_Calculus
|
exercise.5.1.1
|
Calculate the line integral $\int_{C} f(x, y) d s$ for the given function $f(x, y)$ and curve $C$.
$f(x, y)=x y ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq \pi / 2$
|
$1 / 2$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1 / 2
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.3
|
State the domain and range of the given function: $f(x, y)=\sqrt{x^{2}+y^{2}-4}$
|
domain: $\left\{(x, y): x^{2}+y^{2} \geq 4\right\}$, range: $[0, \infty) $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
domain: $\left\{(x, y): x^{2}+y^{2} \geq 4\right\}$, range: $[0, \infty)
|
college_math.Corrals_Vector_Calculus
|
exercise.4.3.10
|
Find the volume $V$ of the solid $S$ bounded by the three coordinate planes, bounded above by the plane $x+y+z=2$, and bounded below by the plane $z=x+y$.
|
$\frac{1}{3}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{1}{3}
|
college_math.Corrals_Vector_Calculus
|
exercise.5.3.5
|
Is there a potential $F(x, y)$ for $\mathbf{f}(x, y)=\left(y^{2}+3 x^{2}\right) \mathbf{i}+2 x y \mathbf{j}$ ? If so, find one.
|
Yes. $F(x, y)=x y^{2}+x^{3}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes. $F(x, y)=x y^{2}+x^{3}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.6.3
|
Find the center of mass of the region $R$ with the given density function $\delta(x, y)$:
$R=\left\{(x, y): y \geq 0, x^{2}+y^{2} \leq a^{2}\right\}, \delta(x, y)=1$
|
$\left(0, \frac{4 a}{3 \pi}\right) \quad$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\left(0, \frac{4 a}{3 \pi}\right) \quad
|
college_math.Corrals_Vector_Calculus
|
exercise.4.2.6
|
Evaluate the given double integral: $\int_{0}^{\infty} \int_{0}^{\infty} x y e^{-\left(x^{2}+y^{2}\right)} d x d y$
|
$\frac{1}{4} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{1}{4}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.11
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(1,-1)} \frac{x^{2}-2 x y+y^{2}}{x-y}$
|
$2 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2
|
college_math.Corrals_Vector_Calculus
|
exercise.3.1.7
|
Evaluate the limit: $\lim _{(x, y) \rightarrow(0,0)} \cos (x y)$
|
$1 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
1
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.3
|
Calculate $\int_{C} f(x, y, z) d s$ for the given function $f(x, y, z)$ and curve $C$: $f(x, y, z)=z^{2} ; \quad C: x=t \sin t, y=t \cos t, z=\frac{2 \sqrt{2}}{3} t^{3 / 2}, 0 \leq t \leq 1$
|
$2 / 5 $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 / 5
|
college_math.Corrals_Vector_Calculus
|
exercise.5.7.2
|
Let $f(x, y, z)=e^{-x^{2}-y^{2}-z^{2}}$ in Cartesian coordinates. Find the Laplacian of the function in spherical coordinates.
|
$\left(4 \rho^{2}-6\right) e^{-\rho^{2}} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\left(4 \rho^{2}-6\right) e^{-\rho^{2}}
|
college_math.Corrals_Vector_Calculus
|
exercise.1.3.5
|
Find the angle $\theta$ between the vectors $\mathbf{v}=(2,1,4)$ and $\mathbf{w}=(1,-2,0)$.
|
$90^{\circ} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
90^{\circ}
|
college_math.Corrals_Vector_Calculus
|
exercise.3.7.5
|
Find the constrained maxima and minima of $f(x, y, z)=x+y^{2}+2 z$ given that $4 x^{2}+9 y^{2}-36 z^{2}=36$.
|
$\frac{8 a b c}{3 \sqrt{3}}$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{8 a b c}{3 \sqrt{3}}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.5.9
|
Find the volume inside the elliptic cylinder $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}} \leq 1$ for $0 \leq z \leq 2$.
|
$2 \pi a b$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 \pi a b
|
college_math.Corrals_Vector_Calculus
|
exercise.1.4.9
|
Calculate the area of the parallelogram $P Q R S$:
$P=(2,1,3), Q=(1,4,5), R=(2,5,3), S=(3,2,1)$
|
4 \sqrt{5}
|
GNU Free Documentation License
|
college_math.vector_calculus
|
4 \sqrt{5}
|
college_math.Corrals_Vector_Calculus
|
exercise.4.3.5
|
Evaluate the given triple integral: $\int_{1}^{e} \int_{0}^{y} \int_{0}^{1 / y} x^{2} z d x d z d y$
|
$\frac{1}{6} $
|
GNU Free Documentation License
|
college_math.vector_calculus
|
\frac{1}{6}
|
college_math.Corrals_Vector_Calculus
|
exercise.5.5.11
|
State whether or not the vector field $\mathbf{f}(x, y, z)$ has a potential in $\mathbb{R}^{3}$ (you do not need to find the potential itself): $\mathbf{f}(x, y, z)=a \mathbf{i}+b \mathbf{j}+c \mathbf{k}(a, b, c$ constant $)$
|
Yes
|
GNU Free Documentation License
|
college_math.vector_calculus
|
Yes
|
college_math.Corrals_Vector_Calculus
|
exercise.5.1.9
|
Calculate the line integral $\int_{C} \mathbf{f} \cdot d \mathbf{r}$ for the given vector field $\mathbf{f}(x, y)$ and curve $C$.
$\mathbf{f}(x, y)=\left(x^{2}-y\right) \mathbf{i}+\left(x-y^{2}\right) \mathbf{j} ; \quad C: x=\cos t, y=\sin t, 0 \leq t \leq 2 \pi$
|
$2 \pi$
|
GNU Free Documentation License
|
college_math.vector_calculus
|
2 \pi
|
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