uuid stringlengths 36 36 | subject stringclasses 6
values | has_image bool 2
classes | image stringclasses 160
values | problem_statement stringlengths 32 784 | golden_answer stringlengths 7 1.13k |
|---|---|---|---|---|---|
eb55c837-bb39-44bd-9cc9-e835a6446582 | multivariable_calculus | false | null | Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = y \cdot z + x \cdot y$ subject to the constraints $x \cdot y = 1$ and $y^2 + z^2 = 1$. | A minimum of $f(x,y,z)$ is $\frac{1}{2}$
A maximum of $f(x,y,z)$ is $\frac{3}{2}$ |
eb966639-1939-4210-945f-05a52370964f | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAmYAAAFfCAIAAABr05clAADKjklEQVR4nOy9d2Acx3X4/97sXkHvvYMgAYK9ir0XkZJIierNktwlO26JncT5/uwkTrMt23Jiy7Ik21SXVaxGsXcS7A1sAAt67/X67rzfH7O3PAJgEwmABOcTRzzs7e3O7s7Om/fmFSQikEgkF0NEiDjYrZBIJDcXbLAbIJFIrgznfLCbIJFIAKWWKZFIJBLJ1SC1TInEIHD6eLNpdXJqK5HcDEiRKZEYmIuXnHPG5KshkUh6Ig2zEolEIp... | Estimate the intervals where the function is increasing or decreasing: | The final answer:
1. Interval(s) of increase: $(-\infty,-3)\cup(3,\infty)$
2. Interval(s) of decrease: $(-3,3)$ |
ebce3086-94ba-4f5d-a9eb-d41c4104e62a | differential_calc | false | null | Sketch the curve:
$$
y = 25 \cdot x^2 \cdot e^{\frac{ 1 }{ 5 \cdot x }}
$$
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals where... | 1. The domain (in interval notation): $(-\infty,0)\cup(0,\infty)$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(\frac{1}{10},\infty\right)$
6. Intervals where the function is decreasing: $(-\infty,0)$, $\left(0,\frac{1}{10}\r... |
ebd03078-271e-4a68-af4e-459266d93331 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAg8AAAHdCAYAAACaO5obAADOvElEQVR4nOz9eXBU57rne37XyjlTSik1D0gIJBDzPM8Gg7ENBmOwvc8++5yqujVFdFdX3Oj+43ZURHd0R3TE7bgRHdHRdStqOmfvXccz2AbbgAEzG2xmMUgIJCEJzbOUynlYq/9YzkRCAjMjwfOJIGykJHNlKpXrt973eZ9X0XVdRwghhBDiEakv+wCEEEIIMb5IeBBCCCHEY5HwIIQQQojHIuFBCCGEEI9FwoMQQgghHouEByGEEEI8Fg... | Find the areas enclosed by the curves $y^2 = 4 \cdot x$ and $y^2 = 8 \cdot x - x^2$ using integration with respect to $x$:
1. The area of the smaller, shaded region.
2. The area of the larger, yellow region. | 1. The area of the smaller, shaded region is $8\cdot\left(\pi-\frac{8}{3}\right)$
2. The area of the larger, yellow region is $8\cdot\left(\pi+\frac{8}{3}\right)$ |
ecc3ece9-4782-446c-a583-bf68cbb20da0 | algebra | false | null | 1 cup = 8 fluid oz
1 pint = 2 cups = 16 fl oz
1 quart = 2 pints = 4 cups = 32 fluid oz
1 gal = 4 quarts = 8 pints = 16 cups = 128 fl oz
1 lb = 16 oz
1 ton = 2,000 lbs
1 kg = 1,000 g
1 metric ton = 1,000 kg
1 m = 100 cm
1 km = 1,000 m
1. Miles has a bag of peanuts that weighs 64 oz. How many lbs of peanuts does he h... | 1. $4$
2. $4$
3. $4$
4. $6,000$
5. $10.25$ |
ed054e0e-5292-4e76-a784-4603be66e688 | multivariable_calculus | false | null | Evaluate the integral by interchanging the order of integration:
$$
\int_{1}^{27} \int_{1}^2 \left(\sqrt[3]{x} + \sqrt[3]{y}\right) \, dy \, dx
$$ | $\int_{1}^{27} \int_{1}^2 \left(\sqrt[3]{x} + \sqrt[3]{y}\right) \, dy \, dx$ = $39\cdot\sqrt[3]{2}+\frac{81}{2}$ |
ed1c56f4-e389-41c3-a2dd-34b41b6cfc41 | differential_calc | false | null | For the function $y = \ln\left(x^2 - \frac{ 3 }{ 2 }\right) - 4 \cdot x$ determine the intervals, where the function is increasing and decreasing.
Submit as your final answer:
1. Interval(s) where the function is increasing
2. Interval(s) where the function is decreasing | 1. Interval(s) where the function is increasing: $\left(\sqrt{\frac{3}{2}},\frac{3}{2}\right)$
2. Interval(s) where the function is decreasing: $\left(-\infty,-\sqrt{\frac{3}{2}}\right)$, $\left(\frac{3}{2},\infty\right)$ |
ed2645bb-1dfa-4a91-ae3e-019819072602 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAA0YAAAE5CAIAAAA7radyAABCV0lEQVR4nO3dd2AUZeI+8Hd7z256Nh1IQkISWgKEKiBwdILIISKKcionyNd+FsR2Uu5EEBTvxI6CFRRBpSgQiLQkkIQQ0nvPZnvfnfn9sZrjp4BJWJjM8nz+2p3Mjs/6ssmzU97h0DRNAAAAAIDNuEwHAAAAAIBrhUoHAAAAwHqodAAAAACsh0oHAAAAwHqodAAAAACsh0oHAAAAwHqodAAAAACsh0oHAAAAwHqodAAAAACsh0oHAAAAwH... | Given $f(t) = 2 \cdot t^3 \cdot h(t)$, find $f'(2)$ using the table below: | $f'(2)$ = $-28$ |
ed6590ad-a3e5-4993-8bb9-046bd2baed10 | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^\pi \sin\left(\frac{ 1 }{ 2 } \cdot x\right) \cdot \cos\left(\frac{ 1 }{ 4 } \cdot x\right) \, dx
$$ | $\int_{0}^\pi \sin\left(\frac{ 1 }{ 2 } \cdot x\right) \cdot \cos\left(\frac{ 1 }{ 4 } \cdot x\right) \, dx$ = $\frac{8-2\cdot\sqrt{2}}{3}$ |
edd9644b-fd02-4a1a-a1f2-e7bca71a8d16 | multivariable_calculus | false | null | Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)
$f(x,y,z) = \arctan(x \cdot y \cdot z)$. | $f_{xx}(x,y,z)$ = $\frac{-2\cdot x\cdot y^3\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$
$f_{xy}(x,y,z)$ = $f_{yx}(x,y,z)$ = $\frac{z-x^2\cdot y^2\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$
$f_{yy}(x,y,z)$ = $\frac{-2\cdot x^3\cdot y\cdot z^3}{\left(1+x^2\cdot y^2\cdot z^2\right)^2}$
$f_{yz}(x,y,z)$ = $... |
ede288d5-dc9e-40cc-8a71-19739dead514 | differential_calc | false | null | For the function $f(x) = x^3 + x^4$, determine:
1. Intervals where $f$ is increasing or decreasing,
2. Local minima and maxima of $f$,
3. Intervals where $f$ is concave up and concave down, and
4. The inflection points of $f$. | 1. Increasing over $\left(-\frac{3}{4},0\right)\cup(0,\infty)$; decreasing over $\left(-\infty,-\frac{3}{4}\right)$
2. Local maxima at None; local minima at $x=-\frac{3}{4}$
3. Concave up for $x<-\frac{1}{2}$, $x>0$; concave down for $-\frac{1}{2}<x<0$
4. Inflection points at $P\left(-\frac{1}{2},-\frac{1}{16}\right)$ |
ee44cdc7-c5a2-4805-89a4-1ab2303c9f5f | integral_calc | false | null | Find the area of the figure enclosed between the curves $y = 3 \cdot x^2$, $y = \frac{ x^2 }{ 6 }$, and $y = 2$. | Area: $\frac{48-8\cdot\sqrt{2}}{3\cdot\sqrt{3}}$ |
ee5612f3-dfe9-48e0-923c-6283b47e7047 | sequences_series | false | null | Compute $\sqrt[5]{1000}$ with accuracy $10^{-5}$. | The final answer: $3.981072$ |
ee885838-2669-4456-821d-5bf3b8ec639c | differential_calc | false | null | Given the curve $y = 2 \cdot x - \arccos\left(\frac{ 1 }{ x }\right)$, find all
1. horizontal asymptotes,
2. vertical asymptotes,
3. slant asymptotes. | 1. horizontal asymptotes: None
2. vertical asymptotes: None
3. slant asymptotes: $y=2\cdot x-\frac{\pi}{2}$ |
ef10d9c3-512a-450c-af52-3c11bd8c7b53 | algebra | false | null | Find the area of a triangle bounded by the x-axis, the line $f(x) = 12 - \frac{ 1 }{ 3 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin. | The area of the triangle is $\frac{972}{5}$ |
ef27aab9-82a3-4029-ad59-88207a0aa73d | precalculus_review | false | null | Use properties of the natural logarithm to write the expression $\ln\left(\frac{ 6 }{ \sqrt{e^3} }\right)$ as an expression of $\ln(6)$. | $\ln\left(\frac{ 6 }{ \sqrt{e^3} }\right)$ = $\ln(6)-\frac{3}{2}$ |
ef4c1961-77eb-4ec9-9674-8d37ac5fb76d | sequences_series | false | null | Find the Fourier integral of the function $q(x) = \begin{cases} 0, & x < 0 \\ \pi \cdot x, & 0 \le x \le 2 \\ 0, & x > 2 \end{cases}$ | $q(x) = $\int_0^\infty\left(\frac{\left(2\cdot\alpha\cdot\sin\left(2\cdot\alpha\right)+\cos\left(2\cdot\alpha\right)-1\right)\cdot\cos\left(\alpha\cdot x\right)+\left(\sin\left(2\cdot\alpha\right)-2\cdot\alpha\cdot\cos\left(2\cdot\alpha\right)\right)\cdot\sin\left(\alpha\cdot x\right)}{\alpha^2}\right)d\alpha$ |
ef891729-6f14-49d4-9e8b-5359940d9704 | algebra | false | null | The population $P$ of a koi pond over $x$ months is modeled by the function $P(x) = \frac{ 68 }{ 1 + 16 \cdot e^{-0.28 \cdot x} }$. How many months will it take before there are $20$ koi in the pond? | $x$: $\frac{\ln(0.15)}{-0.28}$ |
ef9db5b2-252c-4e24-b0bc-60680edcc39f | sequences_series | false | null | Find the Fourier series of the function $\varphi(x) = \frac{ x }{ 2 }$ in the interval $(0, 2 \cdot \pi)$. | The Fourier series is: $\frac{\pi}{2}-\frac{\sin(x)}{1}-\frac{\sin(2\cdot x)}{2}-\frac{\sin(3\cdot x)}{3}-\cdots$ |
efa9f4b4-8a9e-4f58-9671-8a0e93572e95 | algebra | false | null | Find the area of a triangle bounded by the x-axis, the line $f(x) = 12 - \frac{ 1 }{ 5 } \cdot x$, and the line perpendicular to $f(x)$ that passes through the origin. | The area of the triangle is $\frac{4500}{13}$ |
efadedf0-ebaf-4ce0-b95e-58234f0c05d5 | differential_calc | false | null | Find the derivative of $y = \sin(2 \cdot x) \cdot \cos(3 \cdot x) - \frac{ \ln(x-1) }{ \ln(x+1) } + c$ | The final answer: $y'=2\cdot\cos(5\cdot x)-\sin(3\cdot x)\cdot\sin(2\cdot x)-\frac{(x+1)\cdot\ln(x+1)-(x-1)\cdot\ln(x-1)}{(x-1)\cdot(x+1)\cdot\left(\ln(x+1)\right)^2}$ |
efdc4110-cf56-4f37-bf54-40fdd5d58145 | differential_calc | false | null | Compute the limit using L'Hopital's Rule:
$$
\lim_{x \to \infty} \left(x - x^2 \cdot \ln\left(1 + \frac{ 1 }{ x }\right)\right)
$$ | The final answer: $\frac{1}{2}$ |
efe8b07b-20ef-4acb-a3a8-924f76bbb728 | differential_calc | false | null | Find all values of the constant $c$ such that the limit
$$
\lim_{x \to -\infty} \left(\frac{ 3^{c \cdot x}+1 }{ 3^{2 \cdot x}+1 }\right)
$$
exists. Enter the range for the constant $c$ as an interval of the real line. | The final answer: $[0,\infty)$ |
f04a641b-4ad9-445a-99a6-43a8def62a2f | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 1 }{ (x+4) \cdot \sqrt{x^2+2 \cdot x+5} } \, dx
$$ | $\int \frac{ 1 }{ (x+4) \cdot \sqrt{x^2+2 \cdot x+5} } \, dx$ = $C+\frac{1}{\sqrt{13}}\cdot\ln\left(\sqrt{13}-4-x-\sqrt{x^2+2\cdot x+5}\right)-\frac{1}{\sqrt{13}}\cdot\ln\left(4+\sqrt{13}+x+\sqrt{x^2+2\cdot x+5}\right)$ |
f05e9aaf-c4af-4097-b180-bbbce6a70c0a | precalculus_review | false | null | Use the Rational Zero Theorem to find all real zeros of the following polynomial:
$p(x) = x^3 - 8 \cdot x^2 + 17 \cdot x - 10$ | The real zeros are $1$, $2$, $5$ |
f06583ca-72cf-40eb-a419-128a961eea6e | integral_calc | false | null | Compute the integral:
$$
-\int \frac{ \sin\left(\frac{ x }{ 3 }\right)^4 }{ \cos\left(\frac{ x }{ 3 }\right)^2 } \, dx
$$ | $-\int \frac{ \sin\left(\frac{ x }{ 3 }\right)^4 }{ \cos\left(\frac{ x }{ 3 }\right)^2 } \, dx$ = $-\frac{3\cdot\sin\left(\frac{x}{3}\right)^3}{\cos\left(\frac{x}{3}\right)}+\frac{3}{2}\cdot x-\frac{9}{4}\cdot\sin\left(\frac{2\cdot x}{3}\right)+C$ |
f0abc5d4-cb46-4522-a5e4-08a47caf8212 | integral_calc | false | null | Compute the integral:
$$
\int \sqrt[3]{x \cdot \left(8-x^2\right)} \, dx
$$ | $\int \sqrt[3]{x \cdot \left(8-x^2\right)} \, dx$ = $C+\frac{1}{3}\cdot\left(2\cdot\ln\left(\left|1+\sqrt[3]{\frac{8}{x^2}-1}^2-\sqrt[3]{\frac{8}{x^2}-1}\right|\right)-4\cdot\ln\left(\left|1+\sqrt[3]{\frac{8}{x^2}-1}\right|\right)-4\cdot\sqrt{3}\cdot\arctan\left(\frac{1}{\sqrt{3}}\cdot\left(2\cdot\sqrt[3]{\frac{8}{x^2}... |
f0b46da5-b7cf-4d8a-81ae-e3ff591ba59c | precalculus_review | false | null | Solve $|3 \cdot x - 11| = 4 \cdot x - 3$. | The final answer: $x=2$ |
f12059b7-639d-4813-96ca-8ce7814a6f55 | multivariable_calculus | false | null | Find points on the curve at which the tangent line is horizontal or vertical:
$$
x = \frac{ 3 \cdot t }{ 1+t^3 }, \quad y = \frac{ 3 \cdot t^2 }{ 1+t^3 }
$$ | 1. Horizontal: $P(0,0)$, $P\left(\sqrt[3]{2},\sqrt[3]{4}\right)$
2. Vertical: $P\left(\sqrt[3]{4},\sqrt[3]{2}\right)$ |
f124cb60-bc4d-44b2-b851-fd1aef9a1bfb | integral_calc | false | null | Solve the integral:
$$
\int 22 \cdot \cot(-11 \cdot x)^5 \, dx
$$ | $\int 22 \cdot \cot(-11 \cdot x)^5 \, dx$ = $C+\frac{1}{2}\cdot\left(\cot(11\cdot x)\right)^4+\ln\left(1+\left(\cot(11\cdot x)\right)^2\right)-\left(\cot(11\cdot x)\right)^2$ |
f1283a97-29cb-42c8-a96e-42213e587522 | differential_calc | false | null | Calculate the derivative of the function $r = 2 \cdot \ln\left(\sqrt[6]{\frac{ 1+\tan(3 \cdot \varphi) }{ 1-\tan(3 \cdot \varphi) }}\right)$. | $r'$ = $2\cdot\sec(6\cdot\varphi)$ |
f13796c5-a85c-4895-983c-ead174a245bc | multivariable_calculus | false | null | Find the moment of inertia of one arch of the cycloid $x = a \cdot \left(t - \sin(t)\right)$, $y = a \cdot \left(1 - \cos(t)\right)$ relative to the x-axis. | Moment of Inertia: $\frac{256}{15}\cdot a^3$ |
f1e83625-9780-41d5-9516-ace9a7d12801 | precalculus_review | false | null | Suppose a function $f(x)$ is periodic with the period $T$. What is the period of the function $f(a \cdot x + b)$, when $a > 0$? | The final answer: $\frac{T}{a}$ |
f1efd818-8385-44fb-ac8f-8bf513dfc2e8 | algebra | false | null | Solve the equation for $x$:
$$
\frac{ 3 }{ \log_{2}(10) }-\log_{10}(x-9)=\log_{10}(44)
$$ | $x$ = $\frac{101}{11}$ |
f22f4c69-3c17-40de-9b09-9d9e2366f174 | sequences_series | false | null | Suppose that $\sum_{n=0}^\infty\left(a_{n} \cdot x^n\right)$ converges to a function $y$ such that $y'' - y' + y = 0$ where $y(0) = 1$ and $y'(0) = 0$. Compute $a_{0}$, ..., $a_{5}$. | The final answer:
$a_{0}$: $1$
$a_{1}$: $0$
$a_{2}$: $\frac{1}{2}$
$a_{3}$: $\frac{1}{6}$
$a_{4}$: $0$
$a_{5}$: $-\frac{1}{120}$ |
f26b7d10-ebe6-4a28-b7ac-df7c888f2ccf | precalculus_review | false | null | Solve the logarithmic equation exactly, if possible: $\ln(x) + \ln(x-2) = \ln(4)$
If the solution does not exist, enter undefined. | $x$ = $1+\sqrt{5}$ |
f29b52bc-52d0-49f7-977d-cf9effc0cd00 | algebra | false | null | Given the rational function $f(x) = \frac{ x^2-1 }{ x^3+9 \cdot x^2+14 \cdot x }$, find
1. the domain (in interval notation),
2. vertical asymptotes (in the form $x=a$),
3. horizontal asymptotes (in the form $y=c$). | 1. The domain in interval notation is $(-\infty,-7)\cup(-7,-2)\cup(-2,0)\cup(0,\infty)$
2. Vertical asymptotes of $f(x)$: $x=-7$, $x=0$, $x=-2$
3. Horizontal asymptotes of $f(x)$: $y=0$ |
f2ac69f4-7404-4834-a8cc-1658a2cdda96 | sequences_series | false | null | Evaluate $\int_{0}^{\frac{ \pi }{ 2 }}{\sin\left(\theta\right)^4 d \theta}$ and $\int_{0}^{\frac{ \pi }{ 2 }}{\sin\left(\theta\right)^2 d \theta}$ in the approximation $T=4 \cdot \sqrt{\frac{ L }{ g }} \cdot \int_{0}^{\frac{ \pi }{ 2 }}{\left(1+\frac{ 1 }{ 2 } \cdot k^2 \cdot \sin\left(\theta\right)^2+\frac{ 3 }{ 8 } \... | $T$: $2\cdot\pi\cdot\sqrt{\frac{L}{g}}\cdot\left(1+\frac{k^2}{4}+\frac{9\cdot k^4}{64}\right)$ |
f3483e15-54ea-41ca-9681-11eb71d774d0 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAArUAAADmCAIAAABER+pmAACEPklEQVR4nO29ebykRXXwf2p51t7uOhsMi4BARKNEVFxAI4oYxeVNopj4JqIiBmP8EYNLYqJGUeOSSDSaaDSaKGCMS5LXiEmIiooYIiDBBWQbmPVuvT17VZ3fH6f7mZ47MzAMc29331vfD5+Hvt1979RT56mqU6fOwtAUAACMAwAARwADe0EAAGAD73CwHEmotw/Wq/QpDrwjVrY5lgfBPPhX9sGOlyML229Kojf2HykGgAMwK4LhsUwoJL... | Given $f(t) = g(t) \cdot h(t)$, find $f'(2)$ using the table below: | $f'(2)$ = $-15$ |
f38c8513-234c-4b70-b306-880df56aa4e9 | integral_calc | false | null | Evaluate the integral:
$$
I = \int 3 \cdot \ln\left(\sqrt{2-x}+\sqrt{2+x}\right) \, dx
$$ | The final answer: $3\cdot x\cdot\ln\left(\sqrt{2-x}+\sqrt{2+x}\right)-\frac{3}{2}\cdot\left((C+x)-2\cdot\arcsin\left(\frac{x}{2}\right)\right)$ |
f395b144-92c4-40f4-8eb6-84b365bd4f7c | algebra | false | null | Use the vertex $P(h,k) = P(-4,-5)$ and a point $P(x,y) = P(-2,7)$ on the graph of $f(x)$ to find the general form of the quadratic function. | The final answer: $f(x)=3\cdot x^2+24\cdot x+43$ |
f3c9a461-3a76-4213-94ae-ea4760b8a013 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABJ/ElEQVR4nO3deZjd89n48XvMJLKJrbQIqWqKlmgNQiSSTCJi60apiqW0sVQpU7SW2rcwpVVr7UKssSSIhCRqebjaPC2e/oIqTQQVEiG7iPn9kWaayZwZmeV8vufMeb2uy3Wl3zOTc/fjiHk79zmnrLa2tjby6Prrr4+IiBEjRuTzborO1KlTIyKisrIy40kKg/PIzbnU5zzqq6mpiZqamoiIqK6ujurq6ownKgweJ7k5l/qcR27OpT4/x+... | | | $x$ | $f$ | $f'$ | | --- | --- | --- | --- | | $0$ | $3$ | $-1$ | The function $f$ is a differentiable function. The table above shows values of $f$ and $f'$, the derivative of $f$, at selected values of $x$. The graph of $g$ is shown at right. Given $h(x) = (3 \cdot g(x) + 2 \cdot x) \cdot (f(x) - 2)$, find $h'(0... | $h'(0)$ = $\frac{1}{2}$ |
f3e17598-93ba-4014-916f-e790a5525c7a | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAfwAAAD4CAYAAAAJtFSxAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | The graph below illustrates the decay of a radioactive substance over $t$ days:
Use the graph to estimate the average decay rate from $t=5$ to $t=10$. | The final answer: $-\frac{4}{5}$ |
f4126597-791b-4deb-99db-5c16af183608 | multivariable_calculus | false | null | Consider points $A(3,-1,2)$, $B(2,1,5)$, and $C(1,-2,-2)$.
1. Find the area of parallelogram $ABCD$ with adjacent sides $\vec{AB}$ and $\vec{AC}$.
2. Find the area of triangle $ABC$.
3. Find the distance from point $A$ to line $BC$. | 1. $A$ = $5\cdot\sqrt{6}$
2. $A$ = $\frac{5\cdot\sqrt{6}}{2}$
3. $d$ = $\frac{5\cdot\sqrt{6}}{\sqrt{59}}$ |
f46c0483-d278-4a55-af5b-bc6b958126ff | integral_calc | false | null | Solve the integral:
$$
\int \left(\frac{ x+2 }{ x-2 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$ | $\int \left(\frac{ x+2 }{ x-2 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+2}{x-2}}\cdot(x-10)-6\cdot\ln\left(\left|\frac{\sqrt{x-2}-\sqrt{x+2}}{\sqrt{x-2}+\sqrt{x+2}}\right|\right)$ |
f47cb838-a60c-461f-8c9b-9143e021222f | integral_calc | false | null | Use integration by substitution and/or by parts to compute the integral:
$$
\int x \cdot \ln(5+x) \, dx
$$ | The final answer: $D+5\cdot(x+5)+\left(\frac{1}{2}\cdot(x+5)^2-5\cdot(x+5)\right)\cdot\ln(x+5)-\frac{1}{4}\cdot(x+5)^2$ |
f4aa6ca1-ce0a-4dd9-bd2d-056d17e9d173 | precalculus_review | false | null | Find the period of the function $f(x) = \left(\sin(x)\right)^4 + \left(\cos(x)\right)^4$. | The final answer: $T=\frac{\pi}{2}$ |
f4bc1799-6bb0-4f5b-8914-a951b5560142 | multivariable_calculus | false | null | Find the moment of inertia of one arch of the cycloid $x=a \cdot \left(3 \cdot t-\sin(3 \cdot t)\right)$, $y=a \cdot \left(1-\cos(3 \cdot t)\right)$ relative to the x-axis. | Moment of Inertia: $\frac{256}{15}\cdot a^3$ |
f50abb06-3b30-462d-b87a-b921c7c38f0c | multivariable_calculus | false | null | For the function $f(x,y) = 2 \cdot x \cdot y \cdot e^{-x^2-y^2}$, use the second derivative test to identify any critical points and determine whether each critical point is a local minimum, local maximum, saddle point, or none of these. | $f$ has a local minimum at $P\left(-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$, $P\left(\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$ .
$f$ has a local maximum at $P\left(-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right)$, $P\left(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}}\right)$ .
$f$ has a saddle point at $P(0,0)$ .
... |
f61c3d0e-e73d-42ec-9f3a-0c30d2022bef | precalculus_review | false | null | Identify the set of real numbers for which $x \cdot (x-1) \cdot (x-4) \le 0$. | The final answer: $(-\infty,0]\cup[1,4]$ |
f6a631ca-4b2b-4716-be3b-538b249f543e | multivariable_calculus | false | null | Calculate the average lengths of all vertical chords of the parabola $\frac{ x^2 }{ a^2 } - \frac{ y^2 }{ b^2 } = 1$ over the interval $a \le x \le 2 \cdot a$. | The final answer: $\mu=b\left(2\cdot\sqrt{3}-\ln\left(2+\sqrt{3}\right)\right)$ |
f6ce3b4f-29e4-44a8-832c-9912388bdc0f | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAAB/qElEQVR4nO3dd3RU5fr28QsSOpFqkKKiIiAoKgonFkRCEwkqKAgKioqxABaCYAHEjsgc7EJU5EhUjqgoCVJNkKJBFA+gNBUQEATpgdAS5v2Dd/Yvw5S0mb1n7/l+1mKtmWd2JvfDJJNcee797DJut9stB0lNTZUkJScnh/1z3XfffUpPT/cau/vuu/X888+H7XOaOT8rOX2eTp+fy+WSy+WSJKWkpCglJcXiikLL6a+fh9Pnyfzszenz83... | Graph of $f$:
Let $f$ be a function whose graph, consisting of four line segments, is shown in the figure above.
Let $g$ be the function defined by $g(x) = x + f(x)$.
At what value of $x$ does $g(x)$ attain its absolute maximum on the interval $[-3,5]$? | The function $g(x)$ attains its absolute maximum on the interval $[-3,5]$ at $x$ = $5$ |
f6f3fc6f-7ebb-4b8d-8bf9-3dee49a8e6c6 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACxi0lEQVR4nOzdd3yV5f3/8ffJJoGEEQibsPceshFUKCNsRREBra2rjmqt9adWW1urtmrrah0VN0PZCMreO+wQNmETZhISkpD1+8MvpxzuG0jIOec+9zmv5+PRx7d8csaHbyHkfa7PdV2OoqKiIgEAfMZbb72lt956S5L0zDPP6JlnnrG4IwDwTWfOnFGrVq1catHR0dq1a5dFHaE4gqxuAAAAALgZJ0+eNNSqVq1qQScoCQIIAAAAbOnEiR... | Let $R$ and $S$ in the figure above be defined as follows: $R$ is the region in the first and second quadrants bounded by the graphs of $y = (x-1)^2 \cdot (x+2)$ and $y = \ln(x+3)$. $S$ is the shaded region in the first quadrant bounded by the two graphs, the $x$-axis, and the $y$-axis.
The region $R$ is the base of a... | $V$ = $\frac{1}{2}\cdot\int_{-2}^{0.278}\left((x-1)^2\cdot(x+2)-\ln(x+3)\right)^2dx$ |
f715864e-78fd-4609-a76c-00ec0430b38f | multivariable_calculus | false | null | Given $\vec{r}(t) = \left\langle e^t \cdot \cos(t), e^t \cdot \sin(t), e^t \right\rangle$. Find the tangential and normal components of acceleration. | The final answer:
1. The tangential component of acceleration, $a_{t}$, is: $\sqrt{3}\cdot e^t$
2. The normal component of acceleration, $a_{N}$, is: $\sqrt{2}\cdot e^t$ |
f73136c7-5fe8-4f38-8765-b1ec63b449af | sequences_series | false | null | Expand the function $f(x) = e^x$ in terms of powers of $x + 1$. | The final answer: $e^x=\frac{1}{e}+(x+1)\cdot\frac{1}{e}+\frac{(x+1)^2}{2!}\cdot\frac{1}{e}+\frac{(x+1)^3}{3!}\cdot\frac{1}{e}+\frac{(x+1)^4}{4!}\cdot e^{\xi}$ |
f75e7bcb-5148-4043-96cb-45ed33624be4 | precalculus_review | false | null | If $y=100$ at $t=4$ and $y=10$ at $t=8$, when does $y=1$? Use $y=y_{0} \cdot e^{k \cdot t}$, where $y_{0}$ is the beginning amount, $y$ is the ending amount, $k$ is the growth or decay rate, and $t$ is time. | $y$ = 1 at $t$ = $12$ |
f79b682d-3c4b-4608-9f46-0f999d9bc010 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACWoElEQVR4nOzdZ3RU57n28UsNEEKi994RHQQIMNX0jluaS47juCZvchzHyUli5zixEycu58SxY8eO7Rz3OI5ji16N6IheDKIjgehGgAQIUHs/ECkM+xGozMyz98z/t5bXMrdG0sUwmq17Py2iuLi4WB6yYcMGSVJSUpLlJOXjtbxvvPGGJOmBBx6wnKR8vPb8lvBabvIG1osvvqgXX3xRkvTYY4/pscces5yofLz2PJM3sLyW12vXuxJee5... | Let $g(x) = \frac{ |x-3| }{ x-3 }$. Values of $h(x)$ are shown in the table below and $f(x)$ is the graph shown below.
| | $x$ | $2.9$ | $2.99$ | $2.999$ | $3.001$ | $3.01$ | $3.1$ | | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| $h(x)$ | $-5.58$ | $-5.959$ | $-5.9... | $\lim_{x \to 3^{-}}\left(\frac{ 2 \cdot f(x) + g(x) }{ \left(h(x)\right)^2 }\right)$ = $-\frac{5}{36}$ |
f7a7e2da-76ef-463e-b7ab-23d607c12f2d | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAa4AAAGpCAIAAABakkV1AABhf0lEQVR4nO39d1RbaZ4n/j/PvVcR5UCONrbJGZtgjMEB52yXq6a7enp6enpnZue7e3bPmT/3390z58zZ3+5Od3Wscjl2OSfAAedIBoNxwAGDySCBQFm6z++PK2QZHABLgODz+sOFVVi6kq7eesLnPg8mhCAAAFjYqNk+AAAAmH0QhQAAAFEIAAAQhQAAgCAKAQAAQRQCAACCKAQAAARRCAAACKIQAAAQRCEAACCIQgAAQBCFAACAIAoBAA... | Use the following graph and find: $\lim_{x \to 1}\left(f(x)\right)$ | $\lim_{x \to 1}\left(f(x)\right)$: None |
f7adf486-9276-4aad-b8ab-9ff6c2894646 | precalculus_review | false | null | Simplify the radical expression:
$$
\frac{ 4 \cdot \sqrt{2 \cdot n} }{ \sqrt{16 \cdot n^4} }
$$ | The final answer: $\frac{\sqrt{2}}{n^{\frac{3}{2}}}$ |
f7c85f4d-2628-4193-9579-41041b20f083 | differential_calc | false | null | Differentiate the function
$$
f(x) = \frac{ 5 \cdot x^3-\sqrt[3]{x}+6 \cdot x+2 }{ \sqrt{x} }
$$ | $\frac{ d }{d x}\left(\frac{ 5 \cdot x^3-\sqrt[3]{x}+6 \cdot x+2 }{ \sqrt{x} }\right)$: $\frac{18\cdot x\cdot\sqrt[6]{x}+75\cdot x^3\cdot\sqrt[6]{x}+\sqrt{x}-6\cdot\sqrt[6]{x}}{6\cdot x\cdot x^{\frac{2}{3}}}$ |
f85314b7-0e73-481b-aadb-9b48f8cf8fbc | algebra | false | null | Rewrite the quadratic expression $2 \cdot x^2 - 6 \cdot x - 9$ by completing the square. | $2 \cdot x^2 - 6 \cdot x - 9$ = $2\cdot(x-1.5)^2-13.5$ |
f89bd354-18c9-4f31-b91f-cf6421e24921 | sequences_series | false | null | Compute the first $6$ nonzero terms (not necessarily a quadratic polynomial) of the Maclaurin series of $f(x) = \sin(x) \cdot \cos\left(\frac{ \pi }{ 4 }\right) + \cos(x) \cdot \sin\left(\frac{ \pi }{ 4 }\right)$. | $f(x)$ = $\frac{1}{34560\cdot\sqrt{2}}\cdot\left(288\cdot x^5+1440\cdot x^4-5760\cdot x^3-17280\cdot x^2+34560\cdot x+34560\right)+\cdots$ |
f8d1def5-7d5c-4ab3-a9ca-2baa0bc33b11 | precalculus_review | false | null | $P = \left(x, \frac{ \sqrt{7} }{ 3 }\right)$, $x<0$ is a point on the unit circle.
1. Find the (exact) missing coordinate value of the point.
2. Find the values of the three trigonometric functions $\sin\left(\theta\right)$, $\cos\left(\theta\right)$, $\tan\left(\theta\right)$ for the angle $\theta$ with a terminal si... | 1. The (exact) missing coordinate value of the point is: $-\frac{\sqrt{2}}{3}$
2. The values of the three trigonometric functions are:
* $\sin\left(\theta\right) = $\frac{\sqrt{7}}{3}$$
* $\cos\left(\theta\right) = $-\frac{\sqrt{2}}{3}$$
* $\tan\left(\theta\right) = $-\frac{\sqrt{14}}{2}$$ |
f8d9c275-c54e-423d-b660-60beec9f19d0 | algebra | false | null | Find the dimensions of the box whose length is twice as long as the width, whose height is $2$ inches greater than the width, and whose volume is $192$ cubic inches. | The dimensions of the box are $4$, $6$, $8$ |
f8f1b940-4771-4e79-b29b-64ddff634da6 | differential_calc | false | null | Given $y = \frac{ 2 }{ 5 } \cdot x^6 + \frac{ 6 }{ 5 } \cdot x^5 + x^4 + 2$, find where the function is
1. concave up
2. concave down
3. point(s) of inflection | 1. Concave up: $(-\infty,-1)$, $(-1,0)$, $(0,\infty)$
2. Concave down: None
3. Point(s) of Inflection: None |
f928de60-dedf-4c5b-8ffc-36c6b402744a | differential_calc | false | null | Sketch the curve:
$$
y = 9 \cdot x^2 \cdot e^{\frac{ 1 }{ 3 \cdot x }}
$$
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes
3. Horizontal asymptotes
4. Slant asymptotes
5. Intervals where the function is increasing
6. Intervals where the function is decreasing
7. Intervals wher... | 1. The domain (in interval notation): $(-\infty,0)\cup(0,\infty)$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(\frac{1}{6},\infty\right)$
6. Intervals where the function is decreasing: $(-\infty,0)$, $(0,\infty)$
7. Interval... |
f9845500-b43b-4187-9da5-1babc7852f25 | differential_calc | false | null | Calculate the derivative of the function $r = \ln\left(\sqrt[4]{\frac{ 1+\tan(\varphi) }{ 1-\tan(\varphi) }}\right)$. | $r'$ = $\frac{1}{2}\cdot\sec(2\cdot\varphi)$ |
f98550db-7106-4a40-9fca-348aa446b703 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADdCAIAAADl1CnLAACf5UlEQVR4nO2ddXwT2ff3b5K6u7dIcSlWCou77uKyy7KLuy6wsLg7i9vi0AJFi0sFaQt16t7UvUnaxm0yzx+zvzz5liZNcwKhy7z/4FWa6aefzmTu3Jx77jkUHMcRCQkJCQkJCYlOoeraAAkJCQkJCQkJOSMhISEhISEh+QYgZyQkJCQkJCQkuoeckZCQkJCQkJDoHnJGQkJCQkJCQqJ7yBkJCQkJCQkJie4hZyQkJCQkJCQkuoeckZCQkJ... | Using the given graphs of $f(x)$ and $g(x)$, find $g\left(g(2)\right)$. | The final answer:
$g\left(g(2)\right)$ = $2$ |
f9e23b20-cd31-4e53-bba3-62353f283cd3 | algebra | false | null | The radius of the right circular cylinder is $\frac{ 1 }{ 3 }$ meter greater than the height. The volume is $\frac{ 74 \cdot \pi }{ 9 }$ cubic meters. Find the dimensions. | $r$: $\frac{1+5.40740373}{3}$ $h$: $1.80246791$ |
fa6de80d-c556-45a5-ba55-741e46642251 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 3 }{ 3+\sin(2 \cdot x)+\cos(2 \cdot x) } \, dx
$$ | $\int \frac{ 3 }{ 3+\sin(2 \cdot x)+\cos(2 \cdot x) } \, dx$ = $C+\frac{3}{\sqrt{7}}\cdot\arctan\left(\frac{1}{\sqrt{7}}\cdot\left(1+2\cdot\tan(x)\right)\right)$ |
fa98708a-a7a3-4056-bfa4-2b0a16fea068 | multivariable_calculus | false | null | Use the second derivative test to identify any critical points of the function $f(x,y) = x^2 \cdot y^2$ and determine whether each critical point is a maximum, minimum, saddle point, or none of these. | Maximum: None
Minimum: $P(0,0)$
Saddle point: None
The second derivative test is inconclusive at: None |
fa9976dd-852a-4d21-bce4-318821b7397d | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACNm0lEQVR4nOzdd3hUdfr+8XtSCUmGBAggiBQLhGIMVQQR6SJSFLsoWLAgKqCua/mtZXVtiIoF0RUXRQRBEFABC6IUQVqk994hlPRkyu8PvnPM5CRAksmczOT9ui6vzTxzZs4Di+PcfJrN7Xa7BQBABTVq1CiNGjVKkjRy5EiNHDnS4o4AILiFWN0AAAAAgIqDAAIAAADAbwggAAAAAPyGAAIAAADAbwggAAAAAPyGAAIAAADAbwggAAAAAP... | Let $A$ and $B$ be regions bounded by the graph of $f(x) = -3 \cdot \cos(x)$ and the $x$-axis for $-\pi \le x \le 0$.
1. Find the volume of the solid generated when $A$ is revolved about the $x$-axis.
2. Find the volume of the solid generated when $B$ is revolved about the $y$-axis. | 1. $22.207$ units³
2. $10.759$ units³ |
fab49e1b-7b62-4ff1-9344-915c84b6cf60 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARsAAAC1CAIAAADgLsxcAABGC0lEQVR4nO2dZ1wT2ffw70x6CBB676IooiIiIqsoKmLXFXtZe++99y32Lpa1YcXeVte+ir0giiAi0qTX0NJn5nlxnp1f/oCuYgpgvi/8xJDMnJnMuffcc0/BKIpCenSBQqHYu3evmZnZgAEDMAzTtTh61AOm1yiNIhKJNm/enJubu2PHDgaDQb9PUdTcuXM3btzIYrHi4+NdXV11KKQedULp0STPnz9HCGEYtnbtWtX3r169Ss9Lc+bM0Z... | Find the surface area of the lemniscate: $\rho^2 = a^2 \cdot \cos(2 \cdot \varphi)$. | $A$ = $a^2$ |
fad0653d-12d0-4153-8e85-7fbd01db1c0f | multivariable_calculus | false | null | Use the second derivative test to identify any critical points of the function $f(x,y) = 8 \cdot x \cdot y \cdot (x+y) + 7$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these. | Maximum: None
Minimum: None
Saddle point: None
The second derivative test is inconclusive at: $P(0,0)$ |
fb6418ae-3440-4258-9388-89d799fd859a | sequences_series | false | null | Find the power series of $f(x) \cdot g(x)$ for given $f(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$ and $g(x) = \sum_{n=1}^\infty \left(n \cdot x^n\right)$. | $f(x) \cdot g(x)$ = $\sum_{n=2}^\infty\left(\frac{1}{6}\cdot n\cdot\left(n^2-1\right)\cdot x^n\right)$ |
fbfe5f15-e991-40c9-b306-3e83535db351 | multivariable_calculus | false | null | Find the equation of the tangent line to the curve: $r = 3 + \cos(2 \cdot t)$, $t = \frac{ 3 \cdot \pi }{ 4 }$. | $y$ = $\frac{1}{5}\cdot\left(x+\frac{3}{\sqrt{2}}\right)+\frac{3}{\sqrt{2}}$ |
fc9fb6e0-d325-42da-b6b4-4f51c745ed76 | multivariable_calculus | false | null | Find and classify all critical points of the function $f(x,y) = x \cdot y \cdot (1-7 \cdot x-9 \cdot y)$. | Points of local minima: None
Points of local maxima: $P\left(\frac{1}{21},\frac{1}{27}\right)$
Saddle points: $P(0,0)$, $P\left(\frac{1}{7},0\right)$, $P\left(0,\frac{1}{9}\right)$ |
fcada4da-798e-438e-8bfd-3efa21ce1322 | integral_calc | false | null | Compute the integral:
$$
\int x \cdot \arctan(2 \cdot x)^2 \, dx
$$ | $\int x \cdot \arctan(2 \cdot x)^2 \, dx$ = $\frac{1}{16}\cdot\left(2\cdot\left(\arctan(2\cdot x)\right)^2+2\cdot\ln\left(4\cdot x^2+1\right)+8\cdot x^2\cdot\left(\arctan(2\cdot x)\right)^2-8\cdot x\cdot\arctan(2\cdot x)\right)+C$ |
fcbb2928-b8f4-4a3c-9fd8-1e095d95bd28 | algebra | false | null | When hired at a new job selling electronics, you are given two pay options:
Option A: Base salary of $20,000$ USD a year with a commission of $12\%$ of your sales.
Option B: Base salary of $26,000$ USD a year with a commission of $3\%$ of your sales.
How much electronics would you need to sell for Option A to produc... | The final answer: $66666.67$ |
fce0ff42-571f-457d-bd35-3be8611edff9 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAaoAAAEbCAIAAAAEe1dxAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | Use the graph in the figure, which shows the profit, $y$, in thousands of dollars, of a company in a given year, $t$, where $t$ represents the number of years since $1980$:
Find the $y$-intercept. | $y$-intercept: $P(0,-300)$ |
fcf8b9bc-dc7a-4385-82e3-98bd3ce3841e | precalculus_review | false | null | Use the double-angle formulas to evaluate the integral:
$$
\int_{0}^\pi \sin(x)^4 \, dx
$$ | $\int_{0}^\pi \sin(x)^4 \, dx$ = $\frac{3\cdot\pi}{8}$ |
fd219d83-cf3d-41b2-9d26-f01e81827b91 | integral_calc | false | null | Compute the integral:
$$
3 \cdot \int \cos(2 \cdot x)^6 \, dx
$$ | $3 \cdot \int \cos(2 \cdot x)^6 \, dx$ = $\frac{3}{8}\cdot\sin(4\cdot x)+\frac{9}{128}\cdot\sin(8\cdot x)+\frac{15}{16}\cdot x-\frac{1}{32}\cdot\left(\sin(4\cdot x)\right)^3+C$ |
fd3587b1-b0c7-48de-8d35-280390009cb4 | precalculus_review | false | null | Use the double-angle formulas to evaluate the integral:
$$
\int \sin(x)^2 \cdot \cos(2 \cdot x)^2 \, dx
$$ | $\int \sin(x)^2 \cdot \cos(2 \cdot x)^2 \, dx$ = $\frac{x}{4}-\frac{3}{16}\cdot\sin(2\cdot x)+\frac{1}{16}\cdot\sin(4\cdot x)-\frac{1}{48}\cdot\sin(6\cdot x)+C$ |
fd553420-a490-4c12-a21d-7d301ef75783 | multivariable_calculus | false | null | Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)
$f(x,y,z) = \sin\left(x+z^y\right)$. | $f_{xx}(x,y,z)$ = $-\sin\left(x+z^y\right)$
$f_{xy}(x,y,z)$ = $f_{yx}(x,y,z)$ = $-z^y\cdot\ln(z)\cdot\sin\left(x+z^y\right)$
$f_{yy}(x,y,z)$ = $-z^y\cdot\left(\ln(z)\right)^2\cdot\left(-\cos\left(x+z^y\right)+z^y\cdot\sin\left(x+z^y\right)\right)$
$f_{yz}(x,y,z)$ = $f_{zy}(x,y,z)$ = $z^{-1+y}\cdot\left(\cos\left(x+z^... |
fd5e835f-ff53-4b16-9353-bc32e4289773 | multivariable_calculus | false | null | Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y) = x^2 - y^2$ with the constraint $x + 6 \cdot y = 4$. | Minimum value of the function $f(x,y)$ is $-\frac{16}{35}$
Maximum value of the function $f(x,y)$ is None |
fd93bc12-3ce3-49a6-bcc2-67359cb0d155 | differential_calc | false | null | Find $\frac{d^3}{dx^3}f(x)$, given $f(x) = \ln\left(\frac{ x+7 }{ x-7 }\right)$. | The final answer: $\frac{d^3}{dx^3}f(x)=-\frac{84\cdot x^2+1372}{\left(x^2-49\right)^3}$ |
fd9d9457-b5dd-48b4-abae-3ca1024ec63c | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAoAAAADNCAIAAACn9lbvAACuGklEQVR4nOx9d5wkV3Xud869VdVhZjZolYWEEEiACCJLQkIgJIJJNsGAbZKxDA+DDcYPg4EHhscjmBxNsAGDSQ9MxgQ9ZGxsggCJIJDAQlkr7WrjhO6uuuec98etqq4JG2Z3Z6ZHmu+3v96ZnurqW+eee+65J9J0rt4TgjHDUBDUEQMweCOQKQCCAoB5ACDFGvYMAwBQ4x0yGJV/MgLUmEhVCc4IZuYYFgRQTpyYMXlVZTLAyMgIBg8om4... | Given $h(x) = f(x) \cdot g(x)$, find $h'(1)$ using the table below: | $h'(1)$ = $16$ |
fe42fdde-3ec3-49f2-af06-ec452da37893 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ -\sqrt[3]{2 \cdot x} }{ \sqrt[3]{(2 \cdot x)^2}-\sqrt{2 \cdot x} } \, dx
$$ | $\int \frac{ -\sqrt[3]{2 \cdot x} }{ \sqrt[3]{(2 \cdot x)^2}-\sqrt{2 \cdot x} } \, dx$ = $C-3\cdot\left(\frac{1}{2}\cdot\sqrt[6]{2\cdot x}^2+\frac{1}{3}\cdot\sqrt[6]{2\cdot x}^3+\frac{1}{4}\cdot\sqrt[6]{2\cdot x}^4+\sqrt[6]{2\cdot x}+\ln\left(\left|\sqrt[6]{2\cdot x}-1\right|\right)\right)$ |
fe510a69-db18-4182-a55e-c5e84eba20f5 | multivariable_calculus | false | null | Calculate the double integral $\int\int_{R}{\left(x^2+y^2\right) d A}$, where $R$ is the parallelogram with the sides $y=x$, $y=x+2$, $y=2$, and $y=6$. | The final answer: $224$ |
fe62dec1-a9c2-4d3b-a1e6-e24baaf52c55 | multivariable_calculus | false | null | Find the curvature for the vector function: $\vec{r}(t) = \left\langle \sqrt{2} \cdot e^t, \sqrt{2} \cdot e^{-t}, 2 \cdot t \right\rangle$. | The final answer: $\frac{2\cdot e^{2\cdot t}}{2+2\cdot e^{4\cdot t}+4\cdot e^{2\cdot t}}$ |
fe917e4b-795e-455a-926b-e27828a7f48d | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPwAAAEDCAIAAACeev5IAABYXklEQVR4nO29d5hcR5U2fuqmzjlOTq0wCqNsW7bknHACPts4LmuigV0T91vYNWDADyz74SXzA74ftgkGDDYY44BxkpNsS1YOM5qgyalzzt33fH9Ud0/PTI800z3TEp77Praenu57q+pWvffUqVPnnCKICBIkLCcwZ7oB71pks1lRFOlnURQzmUyxfCn8JKH6kEi/JBBF8ctf/vIDDzxA//zZz372uc99zufz0T+z2exTTz0l8f5MQSL9Uq... | The given graph is of the form $y = A \cdot \sin(B \cdot x)$ or $y = A \cdot \cos(B \cdot x)$, where $B > 0$. Write the equation of the graph. | The equation of the graph is: $y=\cos(2\cdot\pi\cdot x)$ |
fed13bfa-aef9-4bdd-9b1b-538a293f369e | precalculus_review | false | null | Multiply the rational expressions and express the product in simplest form:
$$
\frac{ x^2-x-6 }{ 2 \cdot x^2+x-6 } \cdot \frac{ 2 \cdot x^2+7 \cdot x-15 }{ x^2-9 }
$$ | The final answer: $\frac{x+5}{x+3}$ |
fed9d0b7-506f-441f-b52a-8f0e24292fd1 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 2 \cdot x+\sqrt{2 \cdot x-3} }{ 3 \cdot \sqrt[4]{2 \cdot x-3}+\sqrt[4]{(2 \cdot x-3)^3} } \, dx
$$ | $\int \frac{ 2 \cdot x+\sqrt{2 \cdot x-3} }{ 3 \cdot \sqrt[4]{2 \cdot x-3}+\sqrt[4]{(2 \cdot x-3)^3} } \, dx$ = $C+2\cdot\left(9\cdot\sqrt[4]{2\cdot x-3}+\frac{1}{5}\cdot\sqrt[4]{2\cdot x-3}^5-\frac{2}{3}\cdot\sqrt[4]{2\cdot x-3}^3-\frac{27}{\sqrt{3}}\cdot\arctan\left(\frac{1}{\sqrt{3}}\cdot\sqrt[4]{2\cdot x-3}\right)\... |
fef6c9b8-267b-4fd6-83ad-75e555451080 | sequences_series | false | null | Expand the function $f(x) = \ln\left(1+\frac{ x }{ 5 }\right)$ given on the interval $[0,1]$ in powers of $x$ using the Maclaurin formula. Estimate the error allowed with the retention of the first ten members.
Submit as your final answer:
1. the resulting expansion of the function (the first ten terms)
2. the estimat... | 1. $\frac{x}{5}-\frac{x^2}{5^2\cdot2}+\frac{x^3}{5^3\cdot3}-\frac{x^4}{5^4\cdot4}+\frac{x^5}{5^5\cdot5}-\frac{x^6}{5^6\cdot6}+\frac{x^7}{5^7\cdot7}-\frac{x^8}{5^8\cdot8}+\frac{x^9}{5^9\cdot9}$
2. $\left|R_{10}(x)\right|<\frac{1}{5^{10}\cdot10}$ |
ff08d451-e6d0-4570-b370-cb5504ceda3d | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASgAAAEhCAIAAABDaqqGAAAwcklEQVR4nO3deXhb1Z038N+RLNuSLcuLvMn7EtuJE6+J4+wLTiANZIEQptAwtClLocPDZIb3YTp9n/I+fWdgeBkGAgydshQIbUIKJQkhi53EcZzYceLY8e543+RFljfJsnad948rC+EstrXkXoXfp32CfSUdHcn66px77rnnEkopIITuLh7bFUDoxwiDhxALMHgIsQCDhxALMHieBEfC7hkYPE9CCLl5I6bRE2HwuMtgMDQ3N2s0Gubn1t... | Use the graph of the function $y = g(x)$ shown here to find $\lim_{x \to 0}\left(g(x)\right)$, if possible. Estimate when necessary. | $\lim_{x \to 0}\left(g(x)\right)$ = None |
ff11c6e0-574a-477e-9ea3-0232aa803f21 | differential_calc | false | null | Find the extrema of a function $y = \frac{ x^4 }{ 4 } - \frac{ 2 \cdot x^3 }{ 3 } - \frac{ x^2 }{ 2 } + 2$. Then determine the largest and smallest value of the function when $-2 \le x \le 4$. | 1. Extrema points: $P\left(\frac{2-2\cdot\sqrt{2}}{2},1.969\right)$, $P(0,2)$, $P\left(\frac{2+2\cdot\sqrt{2}}{2},-1.8023\right)$
2. The largest value: $\frac{46}{3}$
3. The smallest value: $-1.8023$ |
ff6817e6-7ee7-450d-981d-fa96e8f2d0d4 | differential_calc | false | null | Given $y = \frac{ 1 }{ \sqrt{a \cdot x^2 + b \cdot x + c} }$, evaluate $D = y \cdot y'' - 3 \cdot \left(y' \right)^2 + y^4$. | The final answer: $D=\frac{1-a}{\left(a\cdot x^2+b\cdot x+c\right)^2}$ |
ff7e301b-1784-4bb0-8251-948887a6254b | multivariable_calculus | false | null | The distances of all the points of a curve from two fixed points $M$ and $N$ with coordinates $(c,0)$ and $(-c,0)$ are equal to $16$. Find the equation of the curve. | The final answer: $\frac{x^2}{64}+\frac{y^2}{\left(64-c^2\right)}=1$ |
ffeb1d68-82f1-4622-8c63-c49ecaa82c66 | precalculus_review | false | null | A professor asks her class to report the amount of time $t$ they spent writing two assignments. Most students report that it takes them about $45$ minutes to type a four-page assignment and about $90$ minutes to type a nine-page assignment.
1. Find the linear function $y = N(t)$ that models this situation, where $N$ i... | 1. $N(t)$ = $\frac{1}{9}\cdot t-1$
2. $12$
3. $189$ |
ffee282b-ad0f-40e6-a100-9ed7da950b5a | integral_calc | false | null | Compute the integral:
$$
-\int \cos(6 \cdot x)^6 \, dx
$$ | $-\int \cos(6 \cdot x)^6 \, dx$ = $C+\frac{1}{288}\cdot\left(\sin(12\cdot x)\right)^3-\frac{1}{24}\cdot\sin(12\cdot x)-\frac{1}{128}\cdot\sin(24\cdot x)-\frac{5}{16}\cdot x$ |
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Differential Calc Problems
Retrieves specific math problems related to differential calculus, providing basic filtering but limited analytical value beyond finding relevant entries.