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 |
|---|---|---|---|---|---|
00143e2e-2d9f-4fc3-a9c5-9d2e6aba5cba | multivariable_calculus | false | null | Determine the coordinates, if any, for which $f(x,y) = 6 \cdot x^2 - 3 \cdot x^2 \cdot y + y^3 + 12$ has
1. a Relative Minimum(s)
2. a Relative Maximum(s)
3. a Saddle Point(s)
If a Relative Minimum or Maximum, find the Minimum or Maximum value. If none, enter None. | 1. The function $f(x,y)$ has Relative Minimum(s) at None with the value(s) None
2. The function $f(x,y)$ has Relative Maximum(s) at None with the value(s) None
3. The function $f(x,y)$ has a Saddle Point(s) at $P(-2,2)$, $P(2,2)$ |
00282b7b-1bf6-415d-9f17-d04e9c9700e2 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdAAAAGoCAYAAAD2LLSsAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Estimate the average rate of change from $x=2$ to $x=6$ for the following function $f(x)$ whose graph is given below:
Note that the average rate of change of a function from $x=a$ to $x=b$ is $\frac{ f(b)-f(a) }{ b-a }$. | The final answer: $0.5$ |
0075433c-0afb-41a9-a18a-986d8a81a653 | multivariable_calculus | false | null | Evaluate the integral by choosing the order of integration:
$$
\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx
$$ | $\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx$ = $\ln\left(\frac{25\cdot\sqrt{5}}{32}\right)$ |
00f6affb-905a-4109-a78e-2dde7a0b83ac | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx
$$ | $\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx$ = $C+\ln\left(\left|\tan(x)\right|\right)-\frac{3}{2\cdot\left(\tan(x)\right)^2}-\frac{3}{4\cdot\left(\tan(x)\right)^4}-\frac{1}{6\cdot\left(\tan(x)\right)^6}$ |
0118dbbc-db0e-4ab9-a2ed-ddb2edd0eefc | sequences_series | false | null | Find the Fourier series of the periodic function $f(x) = x^2$ in the interval $-2 \cdot \pi \leq x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$. | The Fourier series is: $\frac{4\cdot\pi^2}{3}+\sum_{n=1}^\infty\left(\frac{16\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$ |
0146d63a-d9e9-4910-9267-10f87812aff4 | multivariable_calculus | false | null | If $z = x \cdot y \cdot e^{\frac{ x }{ y }}$, $x = r \cdot \cos\left(\theta\right)$, $y = r \cdot \sin\left(\theta\right)$, find $\frac{ d z }{d r}$ and $\frac{ d z }{d \theta}$ when $r = 2$ and $\theta = \frac{ \pi }{ 6 }$. | The final answer:
$\frac{ d z }{d r}$: $\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$
$\frac{ d z }{d \theta}$: $2\cdot e^{\left(\sqrt{3}\right)}-4\cdot\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$ |
014e9af5-759a-4711-aa22-0f9c76acb502 | sequences_series | false | null | Consider the function $y = \left| \cos\left( \frac{ x }{ 8 } \right) \right|$.
1. Find the Fourier series of the function.
2. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$.
3. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty... | 1. The Fourier series is $\frac{2}{\pi}-\frac{4}{\pi}\cdot\sum_{n=1}^\infty\left(\frac{(-1)^n}{\left(4\cdot n^2-1\right)}\cdot\cos\left(\frac{n\cdot x}{4}\right)\right)$
2. The sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$ is $\frac{(2-\pi)}{4}$
3. The sum of the series $\sum_{n=1}^\infty \fr... |
01683c2c-a5b7-4bff-8e4f-d8fdad3cac45 | differential_calc | false | null | For the function $r = \arctan\left(\frac{ m }{ \varphi }\right) + \arccot\left(m \cdot \cot(\varphi)\right)$, find the derivative $r'(0)$ and $r'(2 \cdot \pi)$. Submit as your final answer:
1. $r'(0)$
2. $r'(2 \cdot \pi)$ | 1. $0$
2. $\frac{1}{m}-\frac{m}{m^2+4\cdot\pi^2}$ |
01961276-06fd-4869-8e58-eb15a4eb4034 | algebra | false | null | Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ by completing the square. | $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ = $\left(x+\frac{1}{3}\right)^2-\frac{4}{9}$ |
01c011c6-2528-46de-aa90-9a86ace5747a | sequences_series | false | null | Using the Taylor formula, decompose the function $f(x) = \ln(1+4 \cdot x)$ in powers of the variable $x$ on the segment $[0,1]$. Use the first nine terms.
Then estimate the accuracy obtained by dropping an additional term after the first nine terms. | 1. $\ln(1+4 \cdot x)$ = $4\cdot x-\frac{(4\cdot x)^2}{2}+\frac{(4\cdot x)^3}{3}-\frac{(4\cdot x)^4}{4}+\frac{(4\cdot x)^5}{5}-\frac{(4\cdot x)^6}{6}+\frac{(4\cdot x)^7}{7}-\frac{(4\cdot x)^8}{8}+\frac{(4\cdot x)^9}{9}$
2. Accuracy is not more than $\frac{4^{10}}{10}$ |
01e4ed04-1ed3-4cda-af81-35686c0a16d0 | differential_calc | false | null | Compute the limit:
$$
\lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right)
$$ | $\lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right)$ = $3$ |
023674a7-236e-4771-9662-77157e370160 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAAEYCAYAAAD1QYs6AABm4UlEQVR4nO39Z5AcV3anjT83s3x1ufbewXuCBAGCQ9AOSZAgh0Mzw1caiRppTcTGKnZjN2L1YT+8Efqwqw3F7sZfG1LEjEbi6F2NHw2HHA4tSNABhCU8YRvdQHtvqrury2Xd/4eszDLoBhpAo13lw2iiuyoz61bmzV+ee+455woppcTCwsJiEaAsdAMsFh/5z6jbfWYZ+1nPPIvZYgmShYkhHEKInNeFELckUtnHkVJedzwLi5mwBMnCxB... | Determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified. | Absolute minimum at: $x=3$
Absolute maximum at: $x=-2.4$
Local minimum at: $x=-2$, $x=1$
Local maximum at: $-1, 2$ |
0296afc7-857a-46e2-8497-63653d268ac8 | differential_calc | false | null | Compute the derivative of the function $y = \sqrt{\frac{ x^5 \cdot \left(2 \cdot x^6+3\right) }{ \sqrt[3]{1-2 \cdot x} }}$ by taking the natural log of both sides of the equation. | Derivative: $y'=\frac{-128\cdot x^7+66\cdot x^6-84\cdot x+45}{-24\cdot x^8+12\cdot x^7-36\cdot x^2+18\cdot x}\cdot\sqrt{\frac{x^5\cdot\left(2\cdot x^6+3\right)}{\sqrt[3]{1-2\cdot x}}}$ |
029deb6d-6866-4e2d-9d92-3a555ec4de2c | algebra | false | null | 1. On a map, the scale is 1/2 inch : 25 miles. What is the actual distance between two cities that are 3 inches apart?
2. What are the dimensions of a 15 foot by 10 foot room on a blueprint with a scale of 1.5 inches : 2 feet?
3. The distance between Newark, NJ and San Francisco, CA is 2,888 miles. How far would that... | 1. $150$ miles
2. $11.25$ feet by $7.5$ feet
3. $8.664$ cm
4. $8.3$ inches |
02a70481-c574-417a-bf03-50e5080f4d19 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT0AAAFlCAIAAAAWJKhQAABqgklEQVR4nO39Z5Acx5nnAWeWr+qu9nZmeryF956gAyk6iKKk5VFakZI2dlfa29uL996Ni/twEaeI/XBfLmIvbmMvNvb2vePuSjxJJ1KURNFDoAFJEADhzQCYwXjT3tvqqsr3Q3bX9BgYAgNO90z+JA56aqqrq6rzX8+TTz75JEQIAUJjghCCEN5i45I7EFYBkOiWQGg4mJU+AcJtQAiVy2VVVSmK4jiOoigAgKZppVIJAMBxHE3TtUZV13... | List all intervals where $f$ is increasing or decreasing and the minima and maxima are located. | The function is increasing at: $(-1,0)\cup(0,1)\cup(1,\infty)$
The function is decreasing at: $(-\infty,-1)$
The minimum is at: $x=-1$
The maximum is at: None |
02c57487-7f56-412a-8aeb-b1abc78ac85b | sequences_series | false | null | Find the radius of convergence $R$ and interval of convergence for the power series $\sum_{n=0}^\infty a_{n} \cdot x^n$:
$$
\sum_{n=1}^\infty (-1)^n \cdot \frac{ x^n }{ \ln(2 \cdot n) }
$$ | * $R$ = $1$
* $I$ = $(-1,1]$ |
02d22949-d7a2-46ca-8af8-4bda85bc8e0e | differential_calc | false | null | Sketch the curve:
$$
y = 16 \cdot x^2 \cdot e^{\frac{ 1 }{ 4 \cdot x }}
$$
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 whe... | 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: $\left(\frac{1}{8},\infty\right)$
6. Intervals where the function is decreasing: $\left(0,\frac{1}{8}\right)$, $(-\infty... |
02d80ab4-c222-493c-a91d-448fba115380 | multivariable_calculus | false | null | Evaluate $L=\lim_{P(x,y) \to P\left(1,\frac{ 1 }{ 2 }\right)}\left(f(x,y)\right)$ given $f(x,y) = \frac{ x^2 - 2 \cdot x^3 \cdot y - 2 \cdot x \cdot y^3 + y^2 }{ 1 + x + y - 2 \cdot x \cdot y - 2 \cdot x^2 \cdot y - 2 \cdot x \cdot y^2 }$. | The final answer: $L=\frac{1}{2}$ |
032cf885-c1e2-49f9-8584-c99b0161c08c | precalculus_review | false | null | Form the compositions $f\left(g(x)\right)$ and $g\left(f(x)\right)$ if $f(x) = \sin(3 \cdot x)$ and $g(x) = \frac{ 1 }{ \sqrt{2-x^2} }$.
1. Find $f\left(g(x)\right)$ and $g\left(f(x)\right)$.
2. Find the domain and range of $f\left(g(x)\right)$ and $g\left(f(x)\right)$. | 1. $f\left(g(x)\right)$ = $\sin\left(\frac{3}{\sqrt{2-x^2}}\right)$
$g\left(f(x)\right)$ = $\frac{1}{\sqrt{2-\sin(3\cdot x)^2}}$
2. Domain of $f\left(g(x)\right)$ is $\left(-\sqrt{2},\sqrt{2}\right)$
Range of $f\left(g(x)\right)$ is $[-1,1]$
Domain of $g\left(f(x)\right)$ is $(-\infty,\infty)$
Range of $g\l... |
03b87419-fe7c-481a-9396-1d0723fc2b15 | differential_calc | false | null | Sketch the curve:
$y = 5 \cdot x \cdot \sqrt{4-x^2}$.
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 the function is co... | 1. The domain (in interval notation): $[-2,2]$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(-\sqrt{2},\sqrt{2}\right)$
6. Intervals where the function is decreasing: $\left(-2,-\sqrt{2}\right)$, $\left(\sqrt{2},2\right)$
7. I... |
03c28ad4-cfe4-430f-ab38-cb3e56091616 | precalculus_review | false | null | Solve the following equation:
$$
x^2 - x + 1 = \frac{ 1 }{ 2 } + \sqrt{x - \frac{ 3 }{ 4 }}
$$ | The final answer: $x=1$ |
040dbb94-2747-4799-89f8-dd544c248a9c | integral_calc | false | null | Consider the function $f(x) = x^2$ on $[-1,1]$ and the partition $\left\{-1, -\frac{ 1 }{ 2 }, \frac{ 1 }{ 4 }, 1\right\}$. Find the upper and lower sums. | The upper sum is: $\frac{23}{16}$
The lower sum is: $\frac{11}{64}$ |
0429a27b-d694-4e42-a60c-d446ae515ed2 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABQIUlEQVR4nO3dfXDd1Zkf8EfBJhDDVYDFWbNXIQnZNZEEnkY7IRIQ0m5YIDNJs0PG7kxCPBCc7dIkDfVOt2leIKbTNjQE6CTsECekeduModl0mUljN25jCEhmi7drVsIlG0KEtLCGBqKLzZth3T88UpB/R7Ze7r2/l/v5zDADx7L0HF1J/L46zzmn6+DBgwejhL7yla9ERMRHPvKRnCtprV27dkVExMDAQM6VtFanzvO0007LvM1jjz3W1p... | The graph of $g'$ is given.
Let $g$ be a differentiable function with $g(1)=-4$. The graph of $g'(x)$, the derivative of $g$, is shown. Write an equation for the line tangent to the graph of $g$ at $x=1$. | The equation for the tangent line is $y+4=-3\cdot(x-1)$ |
04960c86-a731-4641-a3a8-fd0529de5a51 | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{(x+2 \cdot y \cdot z) \, dV}$, where $E = \left\{(x,y,z) | 0 \le x \le 1, 0 \le y \le x, 0 \le z \le 5-x-y \right\}$. | $I$ = $\frac{439}{120}$ |
04aefca7-201f-46bd-b434-5176136433fc | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAiwAAAFjCAYAAAAekVDNAABL1UlEQVR4nO3deXhTVf4/8HebpCsJJaUFUmjZhLIUpFAECqUssqq4sHwRZRwQHAccBX6Kyyibw4gLIorrqDMqosgAIiAwbEIBtdCy27J3L5SmG03apml+f5TeJG1pkzbJzfJ+PQ+PJ2mS87k5nuSTc889x+vYsWMGlUoFe9uwYQOmT59u1zqys7NR37F8tD8LO07lAwDiugfhhQnhNq/DlhxRByBum7CO+rFNnKsOAIiIiMCCBQvw3HPP2a... | Find a formula for $p(x)$, the cubic polynomial whose graph is shown below: | The final answer: $p(x)=-\frac{1}{9}\cdot(x+3)^2\cdot(x-2)$ |
04e91d7e-e8a6-4fb4-89e1-63402a89f8bc | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAY8AAAEbCAIAAACtD5/lAAB3EElEQVR4nO39Z5Qc55nniT5vuIz03pvyBmUBwgMEQJCgpyhSpltqaXb6ztzZ3dmdD3P2nN79OKd3zrkzH+b0nT6zZ/vce7pHLdeS2KREkRJJ0QAgQXjClEF5X5WZld77zHj3Q0RkJQogSICVVZmF+EkEClmR4eMfz/u4F2GMQULia4AxRgjt9F5IPL4QO70DEk1DVaru+4aTXnsS9UZSK4n78wD1QQhVf1v9ofZDCYl6IKmVxP2576CP16... | List all intervals where $f$ is increasing or decreasing. | The function is increasing at: $(-\infty,-2)\cup(-1,0)\cup(0,1)\cup(2,\infty)$
The function is decreasing at: $(-2,-1)\cup(1,2)$ |
05d5fbde-1eff-44af-8037-eca12fcd412f | sequences_series | false | null | Calculate $e$ with an estimated error of $0.001$, using the series expansion. | The final answer: $2.7181$ |
05ea9929-8cbb-432b-bbbb-ec1e74c9f401 | integral_calc | false | null | Compute the integral:
$$
-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }} \, dx
$$ | $-2 \cdot \int x^{-4} \cdot \left(4+x^2\right)^{\frac{ 1 }{ 2 }} \, dx$ = $C+\frac{1}{6}\cdot\left(\frac{4}{x^2}+1\right)\cdot\sqrt{\frac{4}{x^2}+1}$ |
05f8a29d-c47b-4532-a0ec-df1909c43460 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVAAAADECAIAAABGCu5vAABSe0lEQVR4nO1dd3xb1fU/5z5tee89YztxnOHE2QkhiyRACAQKKZC20EGhKw2U0f5oWkZbaFmFQgsUKIWyCQkjJJAdspcTx9m2472XJMuW9O75/XHfk5VBSLBsS7a+H5rKGu/d99499575PUhE0GO0trZu3LixoqJi/vz5Q4YMcb9vsVj27t07btw4s9nc87MEEEAAPQR6ReADCCAAvwDr7wEEEEAAfYeAwAcQwCBCQOAHCM5rml3YXgtYc4... | The graph of the polar rectangular region $D$ is given. Express $D$ in polar coordinates. | 1. The interval of $r$ is $[3,5]$
2. The interval of $\theta$ is $[0,\pi]$ |
064eaf70-e653-406c-9fdf-311c7d02c84e | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXMAAACwCAIAAABy0xlrAABiIklEQVR4nO29d3Qc2Xkv+N2q6pwzcs4MAAnmTE7m5KAZpZElzViytZal1cp6x8c+5/mct7t6b+31yse2JFsepQmazOEMOSSHaZhJMIBEBogcG91ooHPu+vaPCmiC5AwSAZCsnyhMd3V11VfVdX/3y5cgIkiQIEHCvIJabAEkSJBwD0JiFgkSJMw/JGaRIEHC/GMemEXy1EiQIGEK5oFZCCE3b5ToRoKE+xkzY5Z0vpC4Q4IECbfDDJgFEU... | Evaluate the integral of the functions graphed using the formula for circles: | The final answer: $7\cdot\pi$ |
065a1eeb-7581-4b18-bc52-69c80d37e90d | sequences_series | false | null | Use the ratio test to determine the radius of convergence of the series:
$$
\sum_{n=1}^\infty \left( \frac{ (2 \cdot n)! }{ n^{2 \cdot n} } \cdot x^n \right)
$$ | $R$ = $\frac{e^2}{4}$ |
068e40ce-9108-4ef8-8ee5-0d1471ebbe43 | sequences_series | false | null | Compute $\lim_{x \to 0}\left(\frac{ 2 \cdot \cos(x)+4 }{ 5 \cdot x^3 \cdot \sin(x) }-\frac{ 6 }{ 5 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series. | The final answer: $\frac{1}{150}$ |
06cd1a73-2eb9-4648-ab8c-d356cc78baff | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARcAAAI5CAIAAAAE/+lYAAB4EklEQVR4nO2dd1wUx/vHn9m726sc5ei9igVRQUWxABaMNRpLjN0YY0lMTKIxxpjkm95Ms8aUn8bE3nsvsReQYgNEAekIUg64uvP7Y4/zRETggLuDeX9fX3Ps7c4+u7efnZln5nkGKRQKIFgsGGOEkKmtaO1wKYoytQ2EelBaWkrTtEAg0Gq1crlcIpGwvyDGWKvVcrlcUxvYGkEajcbUNhDqBMYYACZMmBAcHLx48eKtW7euWbNm+/btdn... | Use the following graphs and the limit laws to evaluate the limit:
$$
\lim_{x \to -3^{-}}\left(f(x)-3 \cdot g(x)\right)
$$ | $\lim_{x \to -3^{-}}\left(f(x)-3 \cdot g(x)\right)$ = $6$ |
06dca8af-0c22-4f2e-8f47-719564cef1e5 | precalculus_review | false | null | Evaluate $\cos\left(2 \cdot \arctan(-7)\right)$. | The final answer: $\cos(a)=-\frac{24}{25}$ |
07416c4c-88df-4c48-9413-be3713fa7df6 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACGtklEQVR4nO3dd3hUhdru4WfSIQmGXpTei1RREBERBZEmiqAbu6CAoiICoh9uLNjY2GXbKxYEREGkCCIC0kGaEektEFqA9GSSOX9wMpthTWCSzKw15Xdf174Oeae95PAhT55VbA6HwyEAfm3SpEmaNGmSJGnkyJEaOXKkxRsBAAAUT5jVCwAAAAAIHQQQAAAAAKYhgAAAAAAwDQEEAAAAgGkIIAAAAABMQwABAAAAYBoCCAAAAADTEEAAAA... | The shaded region $R$ is the region in the first quadrant bounded by the graphs of $y=2 \cdot x$ and $y=x^2$.
1. Write, but do not evaluate, an integral expression that gives the volume of the solid generated from revolving the region $R$ about the vertical line $x=3$.
2. Write, but do not evaluate, an integral expres... | 1. $V$ = $\int_0^4\pi\cdot\left(\left(3-\frac{1}{2}\cdot y\right)^2-\left(3-\sqrt{y}\right)^2\right)dy$
2. $V$ = $\int_0^4\pi\cdot\left(\left(-2-\sqrt{y}\right)^2-\left(-2-\frac{1}{2}\cdot y\right)^2\right)dy$ |
07418180-04cf-49a2-b949-e280e3a05ff4 | precalculus_review | false | null | Find the equivalent of $\sin\left(\arctan(x)\right)$ in terms of an algebraic expression. | The final answer: $\frac{x}{\sqrt{1+x^2}}$ |
076885c3-f6a8-4790-934e-58926a0cab6b | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYgAAACOCAIAAACkDwjYAAAQSklEQVR4nO2dO3KzPhfGD9/8lwLzToosQOzAnhSpXLoUrV2kTJUyBbSmTOnKRQZ2YBaQwpOR9sJXcDHG4EvM5QDPr0psk0jH0oN0JPQYcRwTAABw4n99FwAAAMpAmAAA7Pgv/8kwjB7LAQCYOMW00n91b4BbMAxjIkGbTk1bBWGsozQwwlQOAMAOCBMAgB0QJgAAO3gKk/Zs4xTb05XvOWGvBQUAtAFPYTJX+1i5IvtVuGq/Mo/vBZJIyE... | Points $P(1.5,0)$ and $Q(\varphi,y)$ are on the graph of the function $f(\varphi) = \cos(\pi \cdot \varphi)$. Complete the following table with the appropriate values: $y$-coordinate of $Q$, the point $Q(x,y)$, and the slope of the secant line passing through points $P$ and $Q$. Round your answer to eight significant d... | 1. $-0.30901699$
2. $-0.031410759$
3. $-0.0031415875$
4. $-0.00031415926$
5. $P(1.4,-0.30901699)$
6. $P(1.49,-0.031410759)$
7. $P(1.499,-0.0031415875)$
8. $P(1.4999,-0.00031415926)$
9. $3.0901699$
10. $3.1410759$
11. $3.1415875$
12. $3.1415926$ |
084e9296-4724-4e37-97bf-b5fcfa6cf40d | multivariable_calculus | false | null | Differentiate $y = \sqrt{\frac{ (x-1) \cdot (x-2) }{ (x-3) \cdot (x-4) }}$. | $\frac{ d y }{d x}$ = $-\left(\frac{2\cdot x^2-10\cdot x+11}{(x-1)^{\frac{1}{2}}\cdot(x-2)^{\frac{1}{2}}\cdot(x-3)^{\frac{3}{2}}\cdot(x-4)^{\frac{3}{2}}}\right)$ |
0873b3d9-9d96-4990-a73c-56df41352356 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQoAAAA1CAIAAAA28jJSAAARFElEQVR4nO1da5Ac1XX+zrm3e2ZXWj2QQBEEJCQhEMXTCDAGEQRYeIUX5DIQi7KDY9nIZbBdSUglP+KkUpBAsJ0KlKGMESlISKkcpKTMQwiBhAlOKIOFEHqAhAJYFCAvkhZpV7sz0/eekx+3u3f2ySOIndmdr6ZW09O9rXt7znfe9y6pKhpooIHBwCM9gAYaqF006NFAA0OiQY8GGhgSDXo00MCQaNCjgQaGRIMeDTQwJBr0aKCBIdGgxy... | Write the set of numbers represented on the number line in interval notation. | The interval is: $(-2,-1]$ |
087ae21d-53fb-497e-b533-5cfa609dbaba | differential_calc | false | null | For the function $y = 3 \cdot x \cdot \sqrt[3]{x} - 6 \cdot \sqrt[3]{x^2} + 24 \cdot \sqrt[3]{x} - 8 \cdot x$, specify the points where local maxima and minima of $y$ occur.
1. The point(s) where local maxima occur
2. The point(s) where local minima occur | 1. The point(s) where local maxima occur: $P(1,13)$
2. The point(s) where local minima occur: $P(-1,-19)$, $P(8,8)$ |
08c72d46-1abd-49e1-9c9c-ce509902be6e | integral_calc | false | null | Solve the integral:
$$
\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} \, dx
$$ | $\int \left(\frac{ x+4 }{ x-4 } \right)^{\frac{ 3 }{ 2 }} \, dx$ = $C+\sqrt{\frac{x+4}{x-4}}\cdot(x-20)-12\cdot\ln\left(\left|\frac{\sqrt{x-4}-\sqrt{x+4}}{\sqrt{x-4}+\sqrt{x+4}}\right|\right)$ |
0960b413-513e-4f04-874f-3bd377eb5f75 | precalculus_review | false | null | Solve the differential equation:
$$
\frac{ d y }{d \theta} = \sin\left(\pi \cdot \theta\right)^4
$$ | $y(\theta)$ = $\frac{\sin\left(4\cdot\pi\cdot\theta\right)}{32\cdot\pi}+\frac{3\cdot\theta}{8}-\frac{\sin\left(2\cdot\pi\cdot\theta\right)}{4\cdot\pi}+C$ |
09720053-8865-4ffe-955e-ef9daebd51b2 | algebra | false | null | A doctor injects a patient with $13$ milligrams of radioactive dye that decays exponentially. After $12$ minutes, there are $4.75$ milligrams of dye remaining in the patient's system. Which is an appropriate model for this situation? | Model formula: $f(t)=13\cdot e^{\frac{\ln\left(\frac{4.75}{13}\right)}{12}\cdot t}$ |
09e81d01-5fa2-46cc-a6ba-9e61ccb1e6f9 | multivariable_calculus | false | null | Find the first derivative $y_{x}'$ of the function:
$$
x = \arcsin\left(\frac{ 2 \cdot t }{ \sqrt{4+4 \cdot t^2} }\right), \quad y = \arccos\left(\frac{ 2 }{ \sqrt{4+4 \cdot t^2} }\right), \quad t \ge 0
$$ | $y_{x}'$ = $1$ |
09f0d305-d8b8-4a6f-a593-518b403c6cb4 | algebra | false | null | Find all real solutions of the following equation:
$$
\frac{ x }{ 2 x+6 }-\frac{ 2 }{ 5 x+5 }=\frac{ 5 x^2-3 x-7 }{ 10 x^2+40 x+30 }
$$ | The real solution(s) to the given equation are: $x=\frac{5}{4}$ |
09f686b5-fa1b-4bbd-bdc5-71030c3b61e5 | algebra | false | null | Solve the following equations:
1. $8.6 = 6j + 4j$
2. $12z - (4z + 6) = 82$
3. $5.4d - 2.3d + 3(d - 4) = 16.67$
4. $2.6f - 1.3(3f - 4) = 6.5$
5. $-5.3m + (-3.9m) - 17 = -94.28$
6. $6(3.5y + 4.2) - 2.75y = 134.7$ | The solutions to the given equations are:
1. $j=0.86$
2. $z=11$
3. $d=\frac{ 47 }{ 10 }$
4. $f=-1$
5. $m=\frac{ 42 }{ 5 }$
6. $y=6$ |
0a64232b-0119-4b4b-9f1d-eea854c90bd4 | multivariable_calculus | false | null | Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function $x = 2 \cdot e^{3 \cdot t} \cdot \cos(t)$, $y = e^{3 \cdot t} \cdot \sin(t)$. | $\frac{d ^2y}{ d x^2}$ = $\frac{5}{2\cdot e^{3\cdot t}\cdot\left(3\cdot\cos(t)-\sin(t)\right)^3}$ |
0ad9f006-5f19-45a1-8138-0c9b37517662 | multivariable_calculus | false | null | Find the volume of the prism with vertices $A(0,0,0)$, $B(4,0,0)$, $C(4,6,0)$, $D(0,6,0)$, $E(0,0,1)$, and $F(4,0,1)$. | $V$ = $12$ |
0b26afe2-981d-494f-9b16-bc1fa0664b39 | multivariable_calculus | false | null | Find the point of intersection of the following surfaces:
1. $\frac{ 1 }{ 2 } \cdot y = x - z - 5$
2. $4 \cdot x^2 + 9 \cdot y^2 - 72 \cdot z = 0$ | The final answer: $x=9$, $y=-2$, $z=5$ |
0b68a76c-626f-4fef-8e94-5fe256e29883 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARcAAAEdCAIAAACpKrTlAAAztUlEQVR4nO2deXyUVZb3z3meSq2pqqSSkH2FLOxhCYgIyCLaYiuo0y50tygfe7Ff29HpVoeeaZdedLobe95xpF8Xxq3VbnVcwEalgzYgCEJYAmHLQsy+71tVqp7z/nGrHooEQqoqoSrmfNVYy1P3uc/ye8695557LhIRMAwTAJpgV4A5h/pEQ8Tg1oTxCSnYFRinCMGosiEiIkIPA75lQhxWURAQgvEWiRCP91vvjS9r5RjfQb5IwUJoye... | Use the graph of the function $y = h(x)$ shown here to find $\lim_{x \to 0^{-}}\left(h(x)\right)$ if possible. Estimate when necessary. | $\lim_{x \to 0^{-}}\left(h(x)\right)$ = $0$ |
0b9ad9f5-dc17-4d95-bc5c-a2fb158e69b6 | sequences_series | false | null | Consider the power series $\sum_{k=0}^\infty \frac{ (2 \cdot x-1)^k }{ 3^k \cdot \sqrt{k+1} }$.
1. Find the center of convergence of the series.
2. Find the radius of convergence of the series. | 1. The center of convergence is $\frac{1}{2}$.
2. The radius of convergence is $\frac{3}{2}$. |
0be6193e-1388-4738-9378-d2052c687bd6 | multivariable_calculus | false | null | Find the points at which the following polar curve $r = 4 \cdot \cos(\theta)$ has a horizontal or vertical tangent line. | The final answer:
1. Horizontal tangents at: $P\left(2\cdot\sqrt{2},\frac{\pi}{4}\right)$, $P\left(-2\cdot\sqrt{2},\frac{3\cdot\pi}{4}\right)$
2. Vertical tangents at: $P\left(0,\frac{\pi}{2}\right)$, $P(4,0)$ |
0c0ba3db-1470-4c36-975c-91ff5f51986f | integral_calc | false | null | Compute the integral:
$$
\int \sin(x)^4 \cdot \cos(x)^6 \, dx
$$ | $\int \sin(x)^4 \cdot \cos(x)^6 \, dx$ = $C+\frac{1}{320}\cdot\left(\sin(2\cdot x)\right)^5+\frac{1}{128}\cdot\left(\frac{3\cdot x}{2}-\frac{\sin(4\cdot x)}{2}+\frac{\sin(8\cdot x)}{16}\right)$ |
0c1585dc-3062-4fc7-9964-8ec9308d478b | precalculus_review | false | null | A biologist observes that a certain bacterial colony triples every 4 hours. Approximately how many hours does it take for it to double in size? (Give your answer to two decimal places.) | The final answer: $2.52$ |
0c3d74eb-2924-4ec0-8f74-6bf110e5daa4 | multivariable_calculus | false | null | Compute the second order derivative $\frac{d ^2y}{ d x^2}$ for the parametrically defined function
$$
x = 4 \cdot e^t \cdot \cos(3 \cdot t), \quad y = e^t \cdot \sin(3 \cdot t)
$$ | $\frac{d ^2y}{ d x^2}$ = $\frac{15}{8\cdot e^t\cdot\left(\cos(3\cdot t)-3\cdot\sin(3\cdot t)\right)^3}$ |
0cdec09c-5655-41df-856e-2a1537553741 | integral_calc | false | null | Compute the volume of the solid formed by rotating about the x-axis the area bounded by the axes and the parabola $x^{\frac{ 1 }{ 2 }}+y^{\frac{ 1 }{ 2 }}=6^{\frac{ 1 }{ 2 }}$. | Volume = $\pi\cdot\frac{72}{5}$ |
0d047527-42dc-4b9d-86ab-a98315d470d5 | precalculus_review | false | null | Solve $3 \cdot \sin(x) + 4 \cdot \cos(x) = 5$. | The final answer: $x=-\arcsin\left(\frac{4}{5}\right)+\frac{\pi}{2}+2\cdot\pi\cdot n$, $x=-\arccos\left(\frac{3}{5}\right)+\frac{\pi}{2}+2\cdot\pi\cdot n$ |
0d1189b8-7da9-4918-b7c1-eca7aeebe695 | integral_calc | false | null | Calculate the integral:
$$
\int_{-\sqrt{5}}^{\sqrt{5}} \frac{ 4 \cdot x^7+6 \cdot x^6-12 \cdot x^5-14 \cdot x^3-24 \cdot x^2+2 \cdot x+4 }{ x^2+2 } \, dx
$$ | $\int_{-\sqrt{5}}^{\sqrt{5}} \frac{ 4 \cdot x^7+6 \cdot x^6-12 \cdot x^5-14 \cdot x^3-24 \cdot x^2+2 \cdot x+4 }{ x^2+2 } \, dx$ = $20\cdot\sqrt{5}+4\cdot\sqrt{2}\cdot\arctan\left(\frac{\sqrt{5}}{\sqrt{2}}\right)$ |
0d292144-e47b-4b2a-8006-c3d316be019c | multivariable_calculus | false | null | Find a rectangular equation which is equivalent to the following parametric equations:
1. $x^2 = t^3 - 3 \cdot t^2 + 3 \cdot t - 1$
2. $y^2 = t^3 + 6 \cdot t^2 + 12 \cdot t + 8$ | The final answer: $\sqrt[3]{x^2}-\sqrt[3]{y^2}=-3$ |
0d3ad896-eab6-4f0c-9265-20ae824c572c | algebra | false | null | Solve the following equations:
1. Five less than twice a number $x$ is nineteen. What is the number $x$?
2. Four times the sum of a number $x$ and −3 is 28.
3. Three times twenty-one more than a number $x$ is twelve. What is the number?
4. Find the width $w$ of a rectangle if its length is 5 more than the width and it... | The solutions to the given equations are:
1. $x=12$
2. $x=10$
3. $x=-17$
4. $w=20$ |
0d736a08-bfaf-4a90-8c3c-0ef02fd113b7 | differential_calc | false | null | $f(x) = x + \sin(2 \cdot x)$ over $x = \left[-\frac{ \pi }{ 2 },\frac{ \pi }{ 2 }\right]$. Determine:
1. intervals where $f$ is increasing
2. intervals where $f$ is decreasing
3. local minima of $f$
4. local maxima of $f$
5. intervals where $f$ is concave up
6. intervals where $f$ is concave down
7. the inflection poin... | 1. intervals where $f$ is increasing: $\left(-\frac{\pi}{3},\frac{\pi}{3}\right)$
2. intervals where $f$ is decreasing: $\left(-\frac{\pi}{2},-\frac{\pi}{3}\right)$, $\left(\frac{\pi}{3},\frac{\pi}{2}\right)$
3. local minima of $f$: $-\frac{\pi}{3}$
4. local maxima of $f$: $\frac{\pi}{3}$
5. intervals where $f$ is conc... |
0deb486d-fc94-4e66-a267-22f7bc60d49e | algebra | false | null | Solve the following equations:
1. $13x + 6 = 6$
2. $\frac{x}{-4} + 11 = 5$
3. $-4.5x + 12.3 = -23.7$
4. $\frac{x}{5} + 4 = 4.3$
5. $-\frac{x}{3} + (-7.2) = -2.1$
6. $5.4x - 8.3 = 14.38$
7. $\frac{x}{3} - 14 = -8$ | The solutions to the given equations are:
1. $x=0$
2. $x=24$
3. $x=8$
4. $x=\frac{ 3 }{ 2 }$
5. $x=\frac{ -153 }{ 10 }$
6. $x=\frac{ 21 }{ 5 }$
7. $x=18$ |
0e30e42e-91dc-4962-9046-59dc3f6fcbd1 | integral_calc | false | null | Compute the integral:
$$
-10 \cdot \int \frac{ \cos(5 \cdot x)^4 }{ \sin(5 \cdot x)^3 } \, dx
$$ | $-10 \cdot \int \frac{ \cos(5 \cdot x)^4 }{ \sin(5 \cdot x)^3 } \, dx$ = $C+3\cdot\cos(5\cdot x)+\frac{\left(\cos(5\cdot x)\right)^3}{1-\left(\cos(5\cdot x)\right)^2}-\frac{3}{2}\cdot\ln\left(\frac{1+\cos(5\cdot x)}{1-\cos(5\cdot x)}\right)$ |
0e5a9971-ba63-4edd-be97-016995f50ab3 | algebra | false | null | Divide the rational expressions:
$$
\frac{ q^2-36 }{ q^2+12 \cdot q+36 } \div \frac{ q^2-4 \cdot q-12 }{ q^2+4 \cdot q-12 }
$$ | The final answer: $\frac{q-2}{q+2}$ |
0e9df16f-3fab-4eae-9a02-430147ee240a | precalculus_review | false | null | Find all values of $t$ that satisfy the following equation:
$$
\log_{a}(t) + \log_{2 \cdot a}(t) = \frac{ \ln\left(2 \cdot a^2\right) }{ \ln(2 \cdot a) }
$$
for $a > 0$ and $a \ne 1$. | $t$ = $a$ |
0eb75475-d51c-4bc8-9d9b-da721151197d | differential_calc | false | null | Find the maximum and minimum values of the function $y = 4 \cdot \sin(x) + \sin(4 \cdot x)$ in the closed interval $[0, \pi]$. | Maximum Value: $\frac{5\cdot\sqrt{2}\cdot\sqrt{5+\sqrt{5}}}{4}$
Minimum Value: $0$ |
0f24f9a7-cb9d-4a3b-9793-2aaa6802d316 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZkAAAGXCAIAAADuzbrpAADPDklEQVR4nOz913ccSZoniH6fuXtojQhoLUlQk0mRmZWVVVmVmZXTUy12tntnxTnzch/vPuzz3b9g3/aeed0zO3PPnt7u071T01XTVZ1aJ5NaggChgYCOQARChwv77oO5OxyKBJlQBON3KCI8XJibm//s04ZEBDW8InA+LETczc7P3a2GGo4HsMZlRxm75KMabdVQQ43Ljj9qTFfD6wD5sBtQw26hqurs7OzU1FQymSyXy/X19T//+c/j8b... | Use the graph of the function to estimate the intervals on which the function is increasing or decreasing.
A function is increasing/decreasing on an interval when its values increase/decrease as $x$ values increase (moving to the right on the graph). | The final answer:
1. Interval(s) of increase: $(1,\infty)$
2. Interval(s) of decrease: $(-\infty,1)$ |
0f8cbe1f-3952-4c8d-91b5-2f2ee3dbedbe | algebra | false | null | Divide the rational expressions:
$$
\frac{ q^2-16 }{ q^2+8 \cdot q+16 } \div \frac{ q^2+q-20 }{ q^2-q-20 }
$$ | The final answer: $\frac{q-5}{q+5}$ |
0fa16123-96ba-47d9-bb59-3e8955e2acdb | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbUAAAGcCAIAAADRa6l0AAEAAElEQVR4nOz9Z7AdRZonjD9PVtXx59xzvZH3QqIxAuFBIKwA0dA03dNuut+Zd2ZiZ3YiNjZiZz9txH7YiI1YE/NG/GOmp2ene2e6Z5qGdnghEAJkQEgICSQQyCJ/vTveVD7/D2kqq06dK0FLIF3u0+pLnaqsrMysyl/+HpOZSEQwI18BeeWVVz788MNKpWKeXLly5c0339zR0XFBHnHw4MFdu3adPHnSPNnV1XX77bcvWbLkgjziggsRIe... | The solid $E$ bounded by $z=1-x^2$ and situated in the first octant is given in the following figure:
Find the volume of the solid. | $V$ = $\frac{10}{3}$ |
0faf776f-2bc5-4452-8cc1-ac74f4ba286b | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATMAAAEwCAIAAABUrgGnAAA5lElEQVR4nO2deXxTZfb/z5M2adOkS7oXutFCW8raQin7IrKDCCqbfhUdvjOOM+j3Jzpu428c5+vMy2XUGeUno6iIKIygwJSyF4qUxdLSBVq6722a7s2+3vP746axtqUtTdrkps/79RLT3OTk3Nx87rOd5xyCiEChUBwMnr0doFAofUCVSaE4IlSZHIaORJwYqkwOQwgBqk8nhSqTM+Tl5aWmpsrlcgCoq6s7ceJEVVUVABBCEFGpVMpkMl... | Use the graph of the function $y = f(x)$ shown here to find $\lim_{x \to 2}\left(f(x)\right)$, if possible. Estimate when necessary. | $\lim_{x \to 2}\left(f(x)\right)$ = $0$ |
0fec4721-ec02-4a72-aa1a-2ed7b0f8276d | differential_calc | false | null | Sketch the curve:
$y = \sqrt{\frac{ 27 - x^3 }{ 2 \cdot x }}$.
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 the fun... | 1. The domain (in interval notation): $(0,3]$
2. Vertical asymptotes: $x=0$
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: None
6. Intervals where the function is decreasing: $(0,3]$
7. Intervals where the function is concave up: $\left(0,\frac{3}{\sqrt[3]{4}}\ri... |
0ffdf602-5652-4443-8918-b7a0da6d9e63 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ \cos(x)^3 }{ \sin(x)^9 } \, dx
$$ | $\int \frac{ \cos(x)^3 }{ \sin(x)^9 } \, dx$ = $C-\left(\frac{1}{3}\cdot\left(\cot(x)\right)^6+\frac{1}{4}\cdot\left(\cot(x)\right)^4+\frac{1}{8}\cdot\left(\cot(x)\right)^8\right)$ |
10185b4b-6b20-42fa-a5bc-a50510a49033 | algebra | false | null | An epidemiological study of the spread of a certain influenza strain that hit a small school population found that the total number of students, $P$, who contracted the flu $t$ days after it broke out is given by the model $P = t^2 - 14t + 200$, where $1 \leq t \leq 6$. Find the day that 160 students had the flu. Recal... | The final answer: $4$ |
10885317-f542-4f96-9f87-2fa854f4e4e5 | algebra | false | null | Using the Rational Zero Theorem, list all possible rational zeros of the following polynomial:
$p(x) = 2 \cdot x^3 + 3 \cdot x^2 - 8 \cdot x + 5$ | Possible rational zeros are $1$, $-1$, $5$, $-5$, $\frac{1}{2}$, $-\frac{1}{2}$, $\frac{5}{2}$, $-\frac{5}{2}$ |
108e1278-c12b-42ee-aefe-f6eec1047374 | algebra | false | null | You sold 4 more than three times as many newspapers this week as last week. If you sold 112 newspapers altogether, how many did you sell this week? | The number of newspapers sold this week is: $85$ |
10c7d30d-0451-4438-91b6-e8c0dea5d378 | sequences_series | false | null | Find the radius of convergence and sum of the series:
$$
\frac{ 1 }{ 2 }+\frac{ x }{ 1 \cdot 3 }+\frac{ x^2 }{ 1 \cdot 2 \cdot 4 }+\cdots+\frac{ x^n }{ \left(n!\right) \cdot (n+2) }+\cdots
$$ | 1. Radius of convergence: $R=\infty$
2. Sum: $f(x)=\begin{cases}\frac{1}{x^2}+\frac{x\cdot e^x-e^x}{x^2},&x\ne0\\\frac{1}{2},&x=0\end{cases}$ |
10dd6f8a-e223-4f9c-b3c6-310d5cc5f159 | algebra | false | null | Identify all points of removable discontinuity (singularity) of the function $f(x) = \frac{ x^2 - 16 }{ x - 4 }$. | $f(x)$ has removable discontinuities at $x=4$ |
11009eac-e3ab-4462-9b27-919429198672 | algebra | false | null | Find the equations of the following objects:
1. The sphere centered at $P(1,3,5)$ through $P(0,1,7)$. Also find its radius.
2. The points equidistant from $P(0,0,0)$ and $P(0,1,3)$.
3. The cylinder with radius $10$ and central axis line $y=3$, $z=5$. | 1. Equation: $(x-1)^2+(y-3)^2+(z-5)^2=9$ Radius: $3$
2. Equation: $y+3\cdot z=5$
3. Equation: $(y-3)^2+(z-5)^2=10^2$ |
110a226d-8231-48ac-9b5e-d4d78cd4f045 | multivariable_calculus | false | null | Find the volume of the solid whose boundaries are given in the rectangular coordinates:
$$
\sqrt{x^2+y^2} \le z \le \sqrt{16-x^2-y^2}, \quad x \ge 0, \quad y \ge 0
$$ | Volume: $\frac{64\cdot\pi-32\cdot\pi\cdot\sqrt{2}}{6}$ |
1115e7e1-487a-4437-bcec-4456804e484e | multivariable_calculus | false | null | Evaluate $L=\lim_{(x,y) \to (2,2)}\left(\frac{ 2 \cdot x^2+2 \cdot x^2 \cdot y+2 \cdot x-2 \cdot x \cdot y-2 \cdot x \cdot y^2-2 \cdot y }{ 3 \cdot x^2-6 \cdot x \cdot y-3 \cdot x^2 \cdot y+3 \cdot x \cdot y^2+3 \cdot y^2 }\right)$ | The final answer: $L=-\frac{7}{6}$ |
112457f8-271f-4e2a-a30b-0de94ab3ec1a | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXEAAAEBCAIAAAABxwIyAAB4hUlEQVR4nO2deSBU39/Hz9gmS8geUtlLKaRNhbRKkkr7virapD2UUtpXpZ20SEWlUiqUtNkiO5F9N3ZmzNznj/Pt/O4zM7INxnRff81Z7sxZ7px77jmf83mTMAwDAKSkpBw8ePDevXuA4A+enp4AgPXr13d3QThPZGTk1atXr1692t0F4SJOnTrVt2/fRYsWdXdBeiT9+vXLycmBn/m6tygEBAQ8BjGmEBAQcBKBVub7+PEjnU5HQS0trb... | The data in the following table are used to estimate the average power output produced by Peter Sagan for each $15$-min interval of Stage $1$ of the $2012$ Tour de France.
Average Power Output:
Estimate the net energy used in kilojoules, noting that $1 \cdot W = 1 \cdot \frac{ J }{ s }$. | $3820.5$ kJ |
11812666-0007-42b5-bad6-f6f023717b4b | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABV5UlEQVR4nO3deZTV9X34/9cMy7CJgMq+qqCg4oK4BPckbtG4ppqkSYhLF23TNCa2yS9pmuXbLK0xNZsxxqRqTFzaGKlGj1utEsQouLAISAQRZEBkG2BgGOb3B53LDLMw272fz+fex+McDnfuvTPzmhw9mafv9/vzKaurq6sLSImbbropbrrppoiIuOGGG+KGG25IeCIAgOy46aab4nvf+16zr11wwQXx05/+tMATNVWe9AAAAEDnZSE+Ig... | Let $R$ be the shaded region bounded by the graphs of $y = \ln(x)$ and $y = x - 5$, as shown in the figure above. Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated around the $y$-axis. | $V$ = $\pi\cdot\int_b^d\left((y+5)^2-\left(e^y\right)^2\right)dy$ |
11dd8f03-d770-455c-817d-ada6446f1ef1 | differential_calc | false | null | Consider the differential equation $\frac{ d y }{d x} = \frac{ 4+y }{ x }$. Find the particular solution $y=f(x)$ to the given differential equation with the initial condition $f(3) = -3$. | $y$ = $\frac{|x|}{3}-4$ |
11e1113c-9eed-4cb1-bd73-92ec080a13df | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdoAAAGxCAIAAABKteBhAAC2k0lEQVR4nOz9d2Acd53/j79eM9u7VnXVi2VJttxbnLglcRLiNEIgIRwQ2l0OOOA4Pnzu4McX7gIHfIDjjgOOy8FBgBBII6Q323Fsx72rWb13rVbb+7x+f7xnRmu5Sbak3ZXmcZwj7Y523jM785zX+/V+FSQiUFBIW4gIEdkPAMB+nv5fKSikDqjIsYKCgkIqoEr2ABQUrh0i2r179549e2pra3fu3JmXlwcAzc3Nb775ptfr3bp167Zt22... | For the following graph:
1. Determine for which values of $x=a$ the $\lim_{x \to a}\left(f(x)\right)$ exists but $f$ is not continuous at $x=a$.
2. Determine for which values of $x=a$ the function is continuous but not differentiable at $x=a$. | The limit exists but the function is not continuous at: $x=4$
The function is continuous but not differentiable at: None |
11e12386-d3f0-475a-a199-05ef0c74ca0f | algebra | false | null | Given the rational function $f(x) = \frac{ 3 \cdot x - 4 }{ x^3 - 16 \cdot x }$, 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 $(-\infty,-4)\cup(-4,0)\cup(0,4)\cup(4,\infty)$
2. Vertical asymptotes of $f(x)$: $x=4$, $x=0$, $x=-4$
3. Horizontal asymptotes of $f(x)$: $y=0$ |
12202748-6302-4b55-9526-cba280dafb55 | multivariable_calculus | false | null | Determine a definite integral that represents the region enclosed by one petal of $r = \cos(3 \cdot \theta)$. | The final answer: $\int_0^{\frac{\pi}{6}}\cos\left(3\cdot\theta\right)^2d\theta$ |
126c4165-b3d5-4470-8412-08e79d9821cf | integral_calc | false | null | Calculate the integral:
$$
\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } \, dx
$$ | $\int \frac{ \sqrt[5]{x}+\sqrt[5]{x^4}+x \cdot \sqrt[5]{x} }{ x \cdot \left(1+\sqrt[5]{x^2}\right) } \, dx$ = $C+5\cdot\arctan\left(\sqrt[5]{x}\right)+\frac{5}{4}\cdot\sqrt[5]{x}^4$ |
12b8f445-4743-4251-93f9-9904096422b2 | multivariable_calculus | false | null | A small appliances company makes toaster ovens and pizza cookers, and have noticed their customer base's purchasing habits give them the price-demand equations given below, where $p$ is the price and $x$ is the quantity of toaster ovens, $q$ is the price and $y$ is the quantity of pizza cookers. The Cost function is $C... | 1. $R(x,y)$ = $R(x,y)=50\cdot x+90\cdot y+2\cdot x\cdot y-3\cdot y^2-5\cdot x^2$
2. $R(6,10)$ = $840$
3. $P(x,y)$ = $P(x,y)=40\cdot x+74\cdot y+2\cdot x\cdot y-800-3\cdot y^2-5\cdot x^2$
4. $P(6,10)$ = $-180$ |
12dcabd5-54c4-4577-8754-c8d5c69abbb0 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAawAAAGzCAIAAAD8KrS1AACCMElEQVR4nO29Z1zUzPf/PcvSO9gQARFQQOyCBUEUEBWwoKKCl72BotjFCnZQUS+wi72LvVyiFBtYwIqAoIL03nvZcj+Y+5t/ftllpWw2W+b9wFcySWY+WZPDZObMOTQ2mw0QCARCUpGiWgACgUBQiTTqCSIQCEmGhowgAoGQZNDnMAKBkGiQEUQgEBINMoIIBEKiQUYQgUBINMgIIhAIiQYZQQQCIdEgP0EEAiHRID9BBAIh0aDPYQQCId... | Find the area of the surface obtained by rotating the closed loop formed by the curves $y=x^2$ and $x=y^2$ about the x-axis. | The final answer: $\frac{1072\cdot\pi\cdot\sqrt{5}-128\cdot\pi-24\cdot\pi\cdot\ln\left(2+\sqrt{5}\right)}{768}$ |
13021374-3ba0-4635-a682-b456cc9bd8c6 | integral_calc | false | null | Compute the integral:
$$
\int_{0}^3 \frac{ 1 }{ x^2 + 2 \cdot x - 8 } \, dx
$$ | $\int_{0}^3 \frac{ 1 }{ x^2 + 2 \cdot x - 8 } \, dx$ = $\infty$ |
1304f734-bec9-42b7-a6d4-863d7557902f | sequences_series | false | null | Find the Taylor series of $f'(x)$ about $a=0$ if $f(x) = \frac{ x - \ln(1 + x) }{ x^2 }$. Use sigma notation in the final answer. | The final answer: $\sum_{k=0}^\infty\left((-1)^{k+1}\cdot\frac{(k+1)}{(k+3)}\cdot x^k\right)$ |
1343ccc0-bf47-410c-aa9c-66daab82714c | differential_calc | false | null | Find the maximum and minimum values of the function $r = 3 \cdot \sin(x) + \sin(3 \cdot x)$ in the closed interval $\left[0, \frac{ 3 }{ 2 } \cdot \pi\right]$. | Maximum Value: $2\cdot\sqrt{2}$
Minimum Value: $-2\cdot\sqrt{2}$ |
1365e400-f4a0-48b2-8a9e-aa2648d70ec5 | multivariable_calculus | false | null | Find $L=\lim_{(x,y) \to (2,3)}\left(\frac{ x^2-y^2+10 \cdot y-25 }{ x^2-y^2-10 \cdot x+25 }\right)$. | The final answer: $L=-\frac{2}{3}$ |
13af41aa-a2d3-4154-8c2f-cc9fb3ecf5ad | multivariable_calculus | false | null | Find the mass of the solid $Q=\left\{(x,y,z) | 1 \le x^2+z^2 \le 25, y \le 1-x^2-z^2 \right\}$ whose density is $\rho(x,y,z) = k$, where $k > 0$. | $m$ = $288\cdot\pi\cdot k$ |
13bd7aaa-4a81-4021-90a2-28b642c80cf7 | multivariable_calculus | false | null | Find the directional derivative using the limit definition only:
$$
f(x,y) = y^2 \cdot \cos(2 \cdot x)
$$
at point $P\left(\frac{ \pi }{ 3 },2\right)$ in the direction of $\vec{u}=\left\langle \cos\left(\frac{ \pi }{ 4 }\right),\sin\left(\frac{ \pi }{ 4 }\right) \right\rangle$. | Directional derivative: $-\frac{2+4\cdot\sqrt{3}}{\sqrt{2}}$ |
147944c5-b782-48c5-a664-d66deb92d9a7 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} } \, dx
$$ | $\int \frac{ 4 \cdot x+\sqrt{4 \cdot x-5} }{ 5 \cdot \sqrt[4]{4 \cdot x-5}+\sqrt[4]{(4 \cdot x-5)^3} } \, dx$ = $C+25\cdot\sqrt[4]{4\cdot x-5}+\frac{1}{5}\cdot\sqrt[4]{4\cdot x-5}^5-\frac{4}{3}\cdot\sqrt[4]{4\cdot x-5}^3-\frac{125}{\sqrt{5}}\cdot\arctan\left(\frac{1}{\sqrt{5}}\cdot\sqrt[4]{4\cdot x-5}\right)$ |
14a384bf-b5f1-4423-b276-6a3e3054cae8 | algebra | false | null | A basic cellular package costs $\$30$/mo. for $60$ min of calling, with an additional charge of $\$0.4$/min beyond that time. The cost formula would be: $C = 30 + 0.4 \cdot (x - 60)$. If you have to keep your bill lower than $\$50$, what is the maximum calling minutes you can use? | Maximum calling minutes: $110$ |
14f3d6f9-4a4c-4b5c-ba5b-18578deec3e8 | multivariable_calculus | false | null | Evaluate the triple integral $\int \int \int f(x,y,z) \, dx \, dy \, dz$ over the solid $f(x,y,z) = e^{\sqrt{x^2+y^2}}$, $1 \le x^2+y^2 \le 4$, $y \le 0$, $x \le y \cdot \sqrt{3}$, $2 \le z \le 3$. | $\int \int \int f(x,y,z) \, dx \, dy \, dz$ = $\frac{\pi\cdot e^2}{6}$ |
153fb930-0922-4e19-a9e3-14b1d2b3cd2a | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{4 \cdot \pi} \left(\cos\left(\frac{ x }{ 2 }\right) \cdot \sin\left(\frac{ x }{ 2 }\right)\right) \, dx
$$ | $\int_{0}^{4 \cdot \pi} \left(\cos\left(\frac{ x }{ 2 }\right) \cdot \sin\left(\frac{ x }{ 2 }\right)\right) \, dx$ = $0$ |
15780608-0149-4c65-a0bd-fdae388be774 | multivariable_calculus | false | null | Find the volume of the solid that lies under the double cone $z^2 = 4 \cdot x^2 + 4 \cdot y^2$, inside the cylinder $x^2 + y^2 = x$, and above the plane $z = 0$. | The volume is $\frac{8}{9}$ |
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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.