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 |
|---|---|---|---|---|---|
592ed5f2-ba58-4094-ae40-57d78056e804 | differential_calc | false | null | Sketch the curve:
$$
y = \frac{ x^3 }{ 5 \cdot (x+2)^2 }
$$
Provide the following:
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): $(-1\cdot\infty,-2)\cup(-2,\infty)$
2. Vertical asymptotes: $x=-2$
3. Horizontal asymptotes: None
4. Slant asymptotes: $y=\frac{x}{5}-\frac{4}{5}$
5. Intervals where the function is increasing: $(-\infty,-6)$, $(-2,0)$, $(0,\infty)$
6. Intervals where the function is decreasing: $(... |
59685710-67d1-4c31-ae2d-1177a32f16bd | differential_calc | false | null | Sketch the curve:
$y = 5 \cdot x^2 - 2 \cdot x^4 - 3$
Submit as your final answer:
1. The domain (in interval notation)
2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)
3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)
4. Slant asymptotes (Leave blank if there a... | 1. The domain (in interval notation): $(-1\cdot\infty,\infty)$
2. Vertical asymptotes: None
3. Horizontal asymptotes: None
4. Slant asymptotes: None
5. Intervals where the function is increasing: $\left(0,\frac{\sqrt{5}}{2}\right)$, $\left(-\infty,-\frac{\sqrt{5}}{2}\right)$
6. Intervals where the function is decreasin... |
59a56679-d33b-4d05-8931-d23bf94bac14 | integral_calc | false | null | The velocity of a bullet from a rifle can be approximated by $v(t) = 6400 \cdot t^2 - 6505 \cdot t + 2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half-second after the shot: $0 \le t \le 0.5$. What is the total distan... | The total distance is: $796.54166667$ |
59b0822d-ad82-45a7-a9c0-aaf87e74534c | integral_calc | false | null | Solve the integral:
$$
\int -16 \cdot \sin(-3 \cdot x)^4 \cdot \cos(-3 \cdot x)^2 \, dx
$$ | $\int -16 \cdot \sin(-3 \cdot x)^4 \cdot \cos(-3 \cdot x)^2 \, dx$ = $-x+\frac{\sin(12\cdot x)}{12}+\frac{\sin(6\cdot x)}{12}-\frac{\sin(18\cdot x)}{36}+C$ |
59b526a5-7bc0-4b9f-ba17-4c66b6214b30 | precalculus_review | false | null | Which equations have a common zero?
1. $x^2-11 \cdot x+28=0$
2. $x^2-5 \cdot x-84=0$
3. $x^2-11 \cdot x+30=0$
4. $x^2-14 \cdot x+49=0$ | The final answer: $x^2-11\cdot x+28=0 \land x^2-14\cdot x+49=0$ |
59fa34d0-c00a-4a33-8290-f19aee1c458c | multivariable_calculus | false | null | Find the gradient of the function $f(x,y) = \frac{ \sqrt{x} + y^2 }{ x \cdot y }$. | $\nabla f(x,y)$ = $\left\langle\frac{1}{2\cdot x\cdot y\cdot\sqrt{x}}-\frac{\sqrt{x}+y^2}{y\cdot x^2},\frac{2}{x}-\frac{\sqrt{x}+y^2}{x\cdot y^2}\right\rangle$ |
5a0dece2-d699-4427-bab6-500d06750213 | sequences_series | false | null | Find the Fourier series of the periodic function $f(x) = \frac{ x^2 }{ 2 }$ in the interval $-2 \cdot \pi \le x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$. | The Fourier series is: $\frac{2\cdot\pi^2}{3}+\sum_{n=1}^\infty\left(\frac{8\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$ |
5a655b20-28b1-46be-9cdf-2c37ea9ff252 | integral_calc | false | null | Solve the integral:
$$
\int \sqrt{\frac{ 4 \cdot \sin(4 \cdot x) }{ 9 \cdot \cos(4 \cdot x)^9 }} \, dx
$$ | $\int \sqrt{\frac{ 4 \cdot \sin(4 \cdot x) }{ 9 \cdot \cos(4 \cdot x)^9 }} \, dx$ = $C+\frac{1}{9}\cdot\left(\tan(4\cdot x)\right)^{\frac{3}{2}}+\frac{1}{21}\cdot\left(\tan(4\cdot x)\right)^{\frac{7}{2}}$ |
5a8bb6a1-8496-4d5d-8662-e751b6df6339 | multivariable_calculus | false | null | Use the method of Lagrange multipliers to find the maximum and minimum values of the function $f(x,y,z) = x^2 + y^2 + z^2$ subject to the constraint $x \cdot y \cdot z = 4$. | Minimum: $6\cdot\sqrt[3]{2}$
Maximum: None |
5a9aa7d4-53f9-4e57-a13a-78001e1c3908 | sequences_series | false | null | Find the Fourier series of the function $\varphi(x) = 2 \cdot x$ in the interval $(0, 4 \cdot \pi)$. | The Fourier series is: $4\cdot\pi-8\cdot\sum_{n=1}^\infty\left(\frac{\sin\left(\frac{n\cdot x}{2}\right)}{n}\right)$ |
5ae51755-5e9f-4084-a568-044ca413b4e8 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAbUAAAG2CAIAAAB3XI8kAAAKMWlDQ1BJQ0MgUHJvZmlsZQAAeJydlndUU9kWh8+9N71QkhCKlNBraFICSA29SJEuKjEJEErAkAAiNkRUcERRkaYIMijggKNDkbEiioUBUbHrBBlE1HFwFBuWSWStGd+8ee/Nm98f935rn73P3Wfvfda6AJD8gwXCTFgJgAyhWBTh58WIjYtnYAcBDPAAA2wA4HCzs0IW+EYCmQJ82IxsmRP4F726DiD5+yrTP4zBAP+flLlZIjEAUJiM5/... | Find the slope of the line graphed: | The final answer: $-\frac{5}{4}$ |
5ba7658c-7761-4d8c-8b36-495732a2732d | integral_calc | false | null | Solve the integral:
$$
\int 2 \cdot \cot(14 \cdot x)^6 \, dx
$$ | $\int 2 \cdot \cot(14 \cdot x)^6 \, dx$ = $C-\frac{1}{7}\cdot\left(\frac{1}{5}\cdot\left(\cot(14\cdot x)\right)^5+\cot(14\cdot x)-\frac{1}{3}\cdot\left(\cot(14\cdot x)\right)^3-\arctan\left(\cot(14\cdot x)\right)\right)$ |
5bd25e22-ae72-4e7d-b526-af88e4214542 | integral_calc | false | null | Compute the integral:
$$
\int \frac{ 1 }{ 2 \cdot \sin\left(\frac{ x }{ 2 }\right)^6 } \, dx
$$ | $\int \frac{ 1 }{ 2 \cdot \sin\left(\frac{ x }{ 2 }\right)^6 } \, dx$ = $C-\frac{1}{5}\cdot\left(\cot\left(\frac{x}{2}\right)\right)^5-\frac{2}{3}\cdot\left(\cot\left(\frac{x}{2}\right)\right)^3-\cot\left(\frac{x}{2}\right)$ |
5be9b46e-ff5c-4541-8760-070e946c1791 | algebra | false | null | Divide rational expressions and simplify it:
$$
\frac{ x^2-1 }{ 2 } \div \frac{ x-1 }{ 5 }
$$ | The final answer: $\frac{5}{2}\cdot x+\frac{5}{2}$ |
5c0972a3-bf3f-4283-858b-291d0ec05d29 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACHd0lEQVR4nO3dd3jO5//+8VdIzIi9qlRtrb2KVo0qqlTRqqoaVaWoKiWVSEQkkYTYu/asvalVqmjVrFFaK7Ei9g4SyfePfvjmdr0Tme/rHs/HcfyOgzPcOfv5/o7W6T0up9jY2FgBHNSaNWuke/fuSl69enVZuXKl+YXsVEhIiISEhIiISL9+/aRfv36aG8FafPfdd7Jo0SKLbOTIkdK2bVtNjQAAaS2d7gKATs2bN5euXbsq+d69eyU4OF... | Let $R$ and $S$ be the regions in the first quadrant shown in the figure above. The region $R$ is bounded above by the $y$-axis and the graphs of $y=\tan(x)$ and $y=3-x$. The region $S$ is bounded by the $x$-axis and the graphs of $y=\tan(x)$ and $y=3-x$.
1. Write, but do not evaluate, an integral expression that give... | 1. $V$ = $\pi\cdot\int_0^a\left((3-x)^2-\left(\tan(x)\right)^2\right)dx$
2. $V$ = $\pi\cdot\int_0^b\left((3-y)^2-\left(\arctan(y)\right)^2\right)dy$ |
5c71f23e-54e6-42e7-9885-fa95f24ba31b | precalculus_review | false | null | Using the table above, estimate the logarithm.
1. $\ln(16)$
2. $\ln\left(3^4\right)$
3. $\ln(2.5)$
4. $\ln\left(\sqrt{630}\right)$
5. $\ln(0.4)$
Estimate the values using the table provided. | 1. $\ln(16)$≈ $2.78$
2. $\ln\left(3^4\right)$≈ $4.4$
3. $\ln(2.5)$≈ $0.92$
4. $\ln\left(\sqrt{630}\right)$≈ $3.225$
5. $\ln(0.4)$≈ $-0.92$ |
5cb50bf1-4fba-4f9e-addb-670f57499bba | precalculus_review | false | null | Which expression is equivalent to $\left(\frac{ 1 }{ \sin(x) }+\frac{ 1 }{ \tan(x) }\right)^2$? | The final answer: $T=\frac{1+\cos(x)}{1-\cos(x)}$ |
5d17f8a8-9af0-4b8e-bd06-745728009986 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXcAAAF9CAYAAAD/WyiYAACJH0lEQVR4nO2dd5wdVfn/38+ZuwmEJJtGSUgjZZPQCSmAIiotoagUEVTgJwSpQlBBQEFpUvwqiNKLqCAQsVGSkFBEBdJDC+mQ3ggkm4QA2Tvn+f1xZuaWvVuSbXd3z/tF2N25c2fmzp35zHOe8xRRVcXjaWaoKiJSaRlQabnH0xoxTX0AHs/2UpWwi0idhT3b1vF2j6c548Xd0yzIFtpsAc+31qsT5Pi1/J/Zr2dv248APM0Z8W4ZT7FTyF... | Use the graph to write the formula for a polynomial function of least degree. | The final answer: $f(x)=\frac{1}{2}\cdot(x+2)\cdot(x-1)\cdot(x-3)$ |
5d216ea8-1451-4963-a04a-80e5f673fd43 | integral_calc | false | null | Compute the integral:
$$
\int \sin\left(\frac{ x }{ 2 }\right)^6 \cdot \cos\left(\frac{ x }{ 2 }\right)^2 \, dx
$$ | $\int \sin\left(\frac{ x }{ 2 }\right)^6 \cdot \cos\left(\frac{ x }{ 2 }\right)^2 \, dx$ = $\frac{\sin\left(\frac{x}{2}\right)^7\cdot\cos\left(\frac{x}{2}\right)}{4}+\frac{1}{8}\cdot\left(-\frac{1}{3}\cdot\sin\left(\frac{x}{2}\right)^5\cdot\cos\left(\frac{x}{2}\right)-\frac{5}{12}\cdot\sin\left(\frac{x}{2}\right)^3\cdo... |
5d2dd0b4-0c6e-4b86-bedf-5136e6541392 | precalculus_review | false | null | Find all values of $t$ that satisfy the following equation:
$$
3^{t^2} = \frac{ 1 }{ 3^t } \cdot 27^{-(1+t)}
$$ | $t$ = $-3$, $-1$ |
5db683f0-f16c-40eb-9d74-5b4eeb8da1f0 | algebra | false | null | Use synthetic division to find the quotient and remainder. Ensure the equation is in the form required by synthetic division:
$$
\frac{ x^3-21 \cdot x^2+147 \cdot x-343 }{ x-7 }
$$
Hint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient fo... | Quotient: $x^2-14\cdot x+49$ Remainder: $0$ |
5e0fca6c-a20d-4a5c-bab2-e2ee9121e589 | integral_calc | false | null | Compute the integral:
$$
3 \cdot \int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x) } \, dx
$$ | $3 \cdot \int \frac{ \cos(2 \cdot x)^4 }{ \sin(2 \cdot x) } \, dx$ = $\frac{3}{2}\cdot\left(C+\frac{1}{3}\cdot\left(\cos(2\cdot x)\right)^3+\cos(2\cdot x)-\frac{1}{2}\cdot\ln\left(\frac{\left|1+\cos(2\cdot x)\right|}{\left|\cos(2\cdot x)-1\right|}\right)\right)$ |
5e2062d8-aedb-4907-86a4-72ba608bb5eb | precalculus_review | false | null | Calculate the sum $S=\cos(a)+\cos(2 \cdot a)+\cos(3 \cdot a) + \ldots + \cos(n \cdot a)$, where $a \ne 2 \cdot \pi \cdot k$ for integers $k$. | The final answer: $S=\frac{\sin\left(\frac{n\cdot a}{2}\right)\cdot\cos\left(\frac{(n+1)\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}$ |
5e3ca8d7-3d5a-4a86-a6ba-0fae9301f584 | differential_calc | false | null | Find $\left(\left(f(a)\right)^{-1}\right)'$ for the function $f(x) = x + \sqrt{x}$ at $a = 2$. | $\left(\left(f(a)\right)^{-1}\right)'$ = $\frac{2}{3}$ |
5eccb470-c198-4287-b80a-0ec45388c1d3 | sequences_series | false | null | Find the 3rd order Taylor polynomial $P_{3}(x)$ of $f(x) = \sqrt[3]{2 \cdot x + 1}$ about $a = 0$. | The final answer: $1+\frac{2}{3}\cdot x-\frac{4}{9}\cdot x^2+\frac{40}{81}\cdot x^3$ |
5edcfb39-ae07-4781-b43b-7c648518a3ee | differential_calc | false | null | Find asymptotes of $y = \frac{ 3 \cdot x }{ 2 } \cdot \ln\left(e-\frac{ 1 }{ 3 \cdot x }\right)$. Submit as your final answer:
1. the equation(s) of horizontal asymptotes
2. the equation(s) of vertical asymptotes
3. the equation(s) of slant asymptotes | 1. Horizontal Asymptote(s): None
2. Vertical Asymptote(s): $x=\frac{1}{3\cdot e}$
3. Slant Asymptote(s): $y=\frac{3\cdot x}{2}-\frac{1}{2\cdot e}$ |
5eebe4a7-75d3-4e52-913a-54badf5c3c51 | sequences_series | false | null | Using the series expansion for the function $(1+x)^m$, calculate approximately $\sqrt[3]{29}$ with an accuracy of 0.0001. | The final answer: $3.0722$ |
5ef63adb-67ec-433f-9448-1ea9de25ceeb | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUIAAAE1CAIAAAAd6H6yAABpgUlEQVR4nO2dd1xUV9rHn+fcmaEM1d4VNFKUIkpTERR77ym70ZiyUbO7aZvNZtMTNWVTdrPvZtN7M5pqwYIlKvYuAjZEbKAISJthZu553j/OvcOAZIPKwAzc72c3Mu3OmZn73HPO83sKlpeXg0YLgogQsblHodGk6BhjzT0GjRtHGG1DTFcz7xaMTvtp3Rphww25Fms/dAtGm41bIL82RWsTcktFM2M35sqVKxcvXvT19e3ZsycAWK3WM2... | Use the transformation $y-x=u$, $x+y=v$ to evaluate the integral on the square $R$ determined by the lines $y=x$, $y=-x+2$, $y=x+2$, and $y=-x$ shown in the following figure:
$$
\int\int_{R} \sin(x-y) \, dA
$$ | $I$ = $\cos(2)-1$ |
5f3e685b-fb16-4ed7-92b3-0cc904da61c6 | sequences_series | false | null | Find the Taylor series for $f(x) = \frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots. | The final answer: $x\cdot\left(1-3\cdot(x+1)+6\cdot(x+1)^2-10\cdot(x+1)^3+\cdots\right)$ |
5f9fc356-2813-445a-9081-10ae9c1969c3 | sequences_series | false | null | Write the Taylor series for the function $f(x) = x \cdot \cos(2 \cdot x)$ at the point $x = \frac{ \pi }{ 2 }$ up to the third term (zero or non-zero). | The final answer: $-\frac{\pi}{2}-\left(x-\frac{\pi}{2}\right)+\pi\cdot\left(x-\frac{\pi}{2}\right)^2$ |
5ff7c384-fb2a-4566-98b7-4976d8fcea39 | sequences_series | false | null | Find the sum of the series $\sum_{n=1}^\infty \frac{ x^{\frac{ n }{ 2 }+1} }{ \frac{ n }{ 2 }+1 }$. (Use differentiation of the series) | The sum of the series is $-x-2\cdot\sqrt{x}-2\cdot\ln\left(\left|\sqrt{x}-1\right|\right)$ |
60291c17-4868-4b90-9588-a7554af40ee2 | precalculus_review | false | null | Consider the function $f(x) = \frac{ 1 }{ 2 } \cdot x^5 + 2 \cdot x$. Let $g$ denote the inverse of $f$. Find the derivative $g'(2.5)$ using the theorem $g'(c) = \frac{ 1 }{ f'\left(g(c)\right) }$. | $g'(2.5)$ = $\frac{2}{9}$ |
60a8d80f-c925-47de-9ef0-b9fed3fc658a | sequences_series | false | null | Compute $\int_{0}^{\frac{ 1 }{ 3 }} e^{-\frac{ x^2 }{ 3 }} \, dx$ with accuracy $0.00001$. | The final answer: $0.32926$ |
61355bbe-39e3-4766-afc8-de6800a5c7a3 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAVoAAAFKCAIAAADT5QC6AADPxElEQVR4nOxdd3wU1RY+525v6b0DofcOIr13UFQEAQFBBez6LNgQC3ax94YgIgKiYEN67x1CCSmQRvruZuvMeX/cmc0CASHZJBvl+70XQ7K5c2fm3nNP+c45SERwHf8lEBEiVseHr6Oug9X2BK6jpoGIV3MG8M9clwX/KVwXB/8tZGZm/vzzz4cOHeL/zMvLW7lyZUpKiiiKns/k5+cXFhZyQXBdefxP4bo4+G/h9OnT06ZNmz9/Pv/n/P... | The graph of the polar rectangular region $D$ is given. Express the region $D$ in polar coordinates: | 1. The interval of $r$ is $[3,5]$
2. The interval of $\theta$ is $\left[\frac{3}{4}\cdot\pi,\frac{5}{4}\cdot\pi\right]$ |
617b2571-094d-4b7b-9de5-6e1c7d7eaddc | integral_calc | false | null | Compute the integral:
$$
\int \frac{ \sin\left(\frac{ x }{ 2 }\right)^4 }{ \cos\left(\frac{ x }{ 2 }\right)^2 } \, dx
$$ | $\int \frac{ \sin\left(\frac{ x }{ 2 }\right)^4 }{ \cos\left(\frac{ x }{ 2 }\right)^2 } \, dx$ = $\frac{2\cdot\sin\left(\frac{x}{2}\right)^3}{\cos\left(\frac{x}{2}\right)}-\frac{3}{2}\cdot x+\frac{3}{2}\cdot\sin(x)+C$ |
618fb3aa-74d9-419a-bea7-0d8bafa34035 | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{\left(y \cdot \ln(x)+z\right) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 1 \le x \le e, 0 \le y \le \ln(x), 0 \le z \le 1\right\}
$$ | $I$ = $\frac{7}{2}-e$ |
619170f6-53b2-46db-9a48-36aacde4a799 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASkAAAEYCAIAAACP3ZfdAAEAAElEQVR4nOy9d7xlWVUnvtba++Sb732pXtWrqq7qqs6JppsmNEhSxoQJVDDOOIMoiukHIsGEKM6oozADKjqoYCKYR4IEydB007G6unJ68eZ04l7r98c599XrhCgwgLL/qM+9t+4975y999orfdd3oYjAV8cXbwgAgogg4mf9Tr4KxXeEhUGARUCYOV8jEWHOTp48vXr+AioiwEkUHjtxsrO5qQgFAAGFAAVExHXdK664olKp5D8vlUrXXn... | Washington, D.C. is located at $39$ deg N and $77$ deg W (see following figure). Assume the radius of Earth is $4000$ mi. Express the location of Washington, D.C. in spherical coordinates (use radians). | $P\left(r,\theta,\varphi\right)$ = $P(4000,1.34,0.89)$ |
620f7f37-a726-4e19-ab07-35d802396575 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAtsAAADmCAYAAAAEC0nyAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Using the given graphs of $f(x)$ and $g(x)$, find $f\left(g(5)\right)$. | $f\left(g(5)\right)$ = $3$ |
62310ba5-128c-4f73-915f-448d1af9b46b | sequences_series | false | null | Write the Taylor series for the function $f(x) = x \cdot \cos(x)$ at the point $x = \frac{ \pi }{ 2 }$ up to the third term (zero or non-zero). | The final answer: $0-\frac{\pi}{2}\cdot\left(x-\frac{\pi}{2}\right)-\left(x-\frac{\pi}{2}\right)^2$ |
623c522f-4d1f-4262-9642-f854d9f41ab7 | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAekAAAFsCAIAAABn0darAAC7q0lEQVR4nOy9V4wkWXYefCMjM9Lb8tXee2+mfU/PzM7OzDpyDSmKECFBhChBfKBAAQIEPVACIUDQoyA9SAAJ6JcAieRSS3J2d3pcm+lqb6q6uqu9LW/SZ4bNiPs/fMqjmCzTVdXl634PhaqsyLA3zj33nO98R+Kcs+UKy7LK5XIikchms8lkkjFmmqZhGOFw2OPxlMvlcDjMGHMchzHm8Xjm+XQFBAQEqpCWs+2ePDjnkiTN91kICAgI/F... | Find the moment of inertia of an isosceles triangle $I_{x}$ relative to its hypotenuse, if at each of its points the surface density is proportional to its distance to the hypotenuse. | $I_{x}$ = $\frac{k}{10}\cdot a^5$ |
624efa84-cb80-467d-a3d8-d8fec7ab3510 | differential_calc | false | null | Make full curve sketching of $y = \ln\left(\left|\frac{ 3 \cdot x-2 }{ 3 \cdot x+2 }\right|\right)$. 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 dec... | 1. The domain (in interval notation) $\left(-\infty,-\frac{2}{3}\right)\cup\left(-\frac{2}{3},1\right)\cup\left(\frac{2}{3},\infty\right)$
2. Vertical asymptotes $x=-\frac{2}{3}$, $x=\frac{2}{3}$
3. Horizontal asymptotes $y=0$
4. Slant asymptotes None
5. Intervals where the function is increasing $\left(-\infty,-\frac{... |
6257ca0f-7795-41a8-99ec-5a8183d30978 | precalculus_review | false | null | If $x = \sin(\theta)$, find $\cot(\arcsin(x))$. | $\cot(\arcsin(x))$ = $\frac{\sqrt{1-x^2}}{x}$ (Enter your solution as an expression using only the variable $x$.) |
62adcd38-fed2-403f-9857-9f9c49504578 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkcAAAI6CAYAAAAt/+zoAAEAAElEQVR4nOz9Z3MkSZaegb6hIzJSIaFVadW6e7p7ZmdWkcPZWXKXtvxAGu3yC41/jb+Aa8YljaQZry3vzs7sTPW2ri4toYHUmaGF3w9ZHhUJoASqPAqNxHnMYCUAuKer468fP+4uMcYYCIIgCIIgCACAfNwfgCAIgiAI4scEiSOCIAiCIIgcJI4IgiAIgiBykDgiCIIgCILIQeKIIAiCIAgiB4kjgiAIgiCIHCSOCIIgCIIgcpA4IgiCIA... | Use the graph to answer the following questions:
1. State the intervals of increase:
2. State the intervals of decrease:
3. State the local minimum in point form
4. State the local maximum in point form
5. For what values of $x$ is $f(x) > 0$ (express your answer in interval notation) | 1. Intervals of increase: $(-4,0)$
2. Intervals of decrease: $(-\infty,-4) \cup (0,\infty)$
3. Local minimum: $P(-4,-32)$
4. Local maximum: $P(0,0)$
5. Values of $x$ for which $f(x) > 0$: $(-\infty,-6)$ |
62b5b762-36d4-4d83-a134-33f1b7699526 | multivariable_calculus | false | null | The force of gravity $\vec{F}$ acting on an object is given by $\vec{F} = m \cdot \vec{g}$, where $m$ is the mass of the object (expressed in kilograms) and $\vec{g}$ is acceleration resulting from gravity, with $\left\lVert\vec{g}\right\rVert = 9.8$ N/kg. A $2$-kg disco ball hangs by a chain from the ceiling of a room... | 1. $\vec{F} = $-19.6\cdot\vec{k}$$; $\left\lVert\vec{F}\right\rVert = $19.6$$
2. $\vec{T} = $-19.6\cdot\vec{k}$$; $\left\lVert\vec{T}\right\rVert = $19.6$$ |
62c22c75-1dc7-4a32-a21b-70f80a9bee23 | sequences_series | false | null | Evaluate the integral:
$$
\int_{0}^{\frac{ 1 }{ 2 }} \sqrt[5]{1+x^3} \, dx
$$
with accuracy $\frac{ 1 }{ 100 }$ using power series expansion. | The final answer: $0.503$ |
62fa87bc-5d7d-4724-a87a-fa6c86ee2cb7 | differential_calc | false | null | Compute the limit:
$$
\lim_{x \to 0} \left( \frac{ x-1 }{ 2 \cdot x^2 } + \frac{ 1 }{ x \cdot \left( e^{2 \cdot x} - 1 \right) } \right)
$$ | The final answer to the problem is: $\frac{1}{6}$ |
630ccb25-22c1-49b5-acca-5ee84f2d1f1e | algebra | false | null | For what value of $m$ does the inequality $m \cdot x + 4 \le 2 \cdot x + 5 \cdot (x - 1)$ have no real solutions? | $m$ = $7$ |
634ff1c9-b9cc-4e54-a01f-7a5a3af01865 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAlYAAAD5CAIAAABqEuxaAAEAAElEQVR4nOz96Zokx7ElCB4RUTOPJRcAJO9S1beqZ3rm/Z9ifs0LzDfT33R1zSUvL4klMyPCF1MVOfNDVM3NIzITAEkAmUAIwUhfzM3MzU31qIgcOSIk8WzP9mzP9mzP9tsz/aVP4Nme7dme7dme7ZexZwh8tmd7tmd7tt+olV/6BJ7t2Z7t12wEHMh0S664rb8cyCwMBVj/uzQBZAEWIhrEocQkmAJCIIAAuPmw9EO4IAQA1GH5Yr5rgI... | A lampshade is constructed by rotating $y=\frac{ 1 }{ x }$ around the $x$-axis from $y=1$ to $y=2$ as seen here. Determine how much material you would need to construct this lampshade - that is, the surface area - accurate to four decimal places. | Surface Area = $10.5017$ |
6350245d-a020-4f20-b18d-8420582c7b2e | sequences_series | false | null | Compute $\lim_{x \to 0}\left(\frac{ \cos(x)+2 }{ 3 \cdot x^3 \cdot \sin(x) }-\frac{ 3 }{ 3 \cdot x^4 }\right)$. Use the expansion of the function in the Taylor series. | The final answer: $\frac{1}{180}$ |
6359d8d6-dd56-4fb5-8e66-e731b72e2ea1 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAc0AAAIKCAIAAADOIoPbAACZyklEQVR4nO39f3QT553vgX/YixVbOGa8MUQ4uATLxtiJDgkTc4K7jl3Clps9JknJ3dBLyo3zY+0lNG2+X5ptyOGG65RvfNPC3bRdkkXbpmRZfErPIW1jTnPoJdQqXZe1M1COWsABK3ENtoLJWlWN7Ijcw/ePj/To0fzSo9FIGo2f10mpNBrNjH+99Zn38/kx58aNG8DhxJEkafPmzRcuXCguLvZ4PGT7unXryOPW1lby+O677y4rK8vlFX... | Find the volume of the shape created when rotating the curve from $x=1$ to $x=2$ around the $x$-axis, as pictured here: | $V$ = $2\cdot\pi\cdot\left(\ln(2)-1\right)^2$ |
6384f2d3-a49e-4db1-a04d-6cb03a14813a | algebra | false | null | Rewrite the quadratic expression $3 + 2 \cdot z - 5 \cdot z^2$ by completing the square. | $3 + 2 \cdot z - 5 \cdot z^2$ = $-5\cdot\left(z-\frac{1}{5}\right)^2+\frac{16}{5}$ |
6393eea4-766f-400b-a497-def1c92321c4 | differential_calc | false | null | Sketch the curve:
$$
y = \sqrt{\frac{ 216-x^3 }{ 4 \cdot x }}
$$
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 where the fun... | 1. The domain (in interval notation): $(0,6]$
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: $(0,6]$
7. Intervals where the function is concave up: $\left(0,3\cdot\sqrt[3]{2}\right)... |
63cc78bd-ed0e-4995-b5d7-112269bc35b9 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABopUlEQVR4nO3dd1yV5eP/8TeKE8G9Z5Yrc2eouHNjzly5c5QrB1rOsnJUilq4V/lxpaZo7rSMSg034dYM1MS9GC70/P7wKz/1HBQMrpvxej4ePB5yXfc55+0Az5v7uu7byWaz2YR45+3tLW9vb0mSl5eXvLy8LE4EAAAAmJfC6gAAAAAAkg8KCAAAAABjKCAAAAAAjKGAAAAAADCGAgIAAADAGAoIAAAAAGMoIAAAAACMoYAAAAAAMIYCAg... | A function $f$ has a derivative $f'$ whose graph is shown below. Use the graph of $f'$ to answer questions about $f$.
1. For what values of $x$ is $f$ increasing? (Enter your answer using interval notation).
2. For what values of $x$ is $f$ decreasing? (Enter your answer using interval notation).
3. For what values ... | 1. The function $f$ is increasing on the interval(s): $\left(0,\ 1\right)$, $\left(3,\ 5\right)$
2. The function $f$ is decreasing on the interval(s): $\left(1,\ 3\right)$
3. The function $f$ has a local maximum or local minimum at the value(s) $x$ = $1$, $3$ |
64067e9f-bbfb-48b1-b763-734ea00fe8e4 | integral_calc | false | null | Find the integral:
$$
\int \frac{ \arcsin(4 \cdot x) }{ \sqrt{4 \cdot x+1} } \, dx
$$ | Answer is: $\frac{1}{2}\cdot\sqrt{4\cdot x+1}\cdot\arcsin(4\cdot x)-\left(C-\sqrt{1-4\cdot x}\right)$ |
6438919f-d8f6-4e72-8d71-6f562f93a51e | precalculus_review | false | null | $P = \left(x,-\frac{ \sqrt{15} }{ 4 }\right)$, $x > 0$ is a point on the unit circle.
1. Find the (exact) missing coordinate value of the point.
2. Find the values of the six trigonometric functions for the angle $\theta$ with a terminal side that passes through point $P$.
Rationalize denominators. | 1. The (exact) missing coordinate value of the point is: $\frac{1}{4}$
2. The values of the six trigonometric functions are:
* $\sin\left(\theta\right)$ = $-\frac{\sqrt{15}}{4}$
* $\cos\left(\theta\right)$ = $\frac{1}{4}$
* $\tan\left(\theta\right)$ = $-\sqrt{15}$
* $\csc\left(\theta\right)$ = $-\frac{4... |
64927e69-37fa-45c5-b927-f586521421b3 | 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 $\lim_{x \to 0^{-}}\left(f(x)\right)$, if possible. Estimate when necessary. | $\lim_{x \to 0^{-}}\left(f(x)\right)$ = $-2$ |
6501a0ec-185e-403b-94d4-328cdcb034f7 | differential_calc | false | null | Find all values of the constant $c$ such that the limit
$$\lim_{x \to -\infty} \left(\frac{ 3^{c \cdot x}+4 }{ 3^{4 \cdot x}+4 }\right)$$
exists. Enter the range for the constant $c$ as an interval of the real line. | The final answer: $(-\infty,\infty)$ |
65049e00-213a-4574-82d7-5808de990ee2 | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{2 \cdot \pi} \left(\sin(x) \cdot \sin(2 \cdot x) \cdot \sin(3 \cdot x)\right) \, dx
$$ | $\int_{0}^{2 \cdot \pi} \left(\sin(x) \cdot \sin(2 \cdot x) \cdot \sin(3 \cdot x)\right) \, dx$ = $0$ |
65adfaa3-961d-42d7-8748-108e8865c96c | multivariable_calculus | false | null | $E$ is located inside the sphere $x^2+y^2+z^2=1$, above the xy-plane, and inside the circular cone $z=\sqrt{x^2+y^2}$. Find the volume of $E$. | Volume = $\frac{2\cdot\pi-\pi\cdot\sqrt{2}}{3}$ |
65b511d6-b6cc-419c-94dc-4d5f8aaddae4 | precalculus_review | false | null | Find expressions for $\cosh(x) + \sinh(x)$ and $\cosh(x) - \sinh(x)$. | The final answer:
$\cosh(x) + \sinh(x)$: $e^x$
$\cosh(x) - \sinh(x)$: $e^{-x}$ |
65c1a270-1f76-4be3-9b46-402724ce0b86 | precalculus_review | false | null | Find points on a coordinate plane that satisfy the following equation:
$$
x^4 + y^4 + 2 \cdot x^2 \cdot y^2 - 9 \cdot x^2 - 10 \cdot y^2 - 4 \cdot x + 29 = 0
$$ | The final answer: $(2,-1)$, $(2,1)$ |
65c1e512-3dd9-46b1-9033-d6804ee5d9de | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQIAAAC4CAIAAACHC3WgAACHmElEQVR4nO19dXwUx/vw7ElycXd3JwkRLAQSLLgUdy0uxaXQIsUKhRYpVtyh0OKUAsE1aGjQQEgIIZ5c7O52d94/nt8973CBtF+attDy/JHPZW9vdnZmHjeOUko+wnsJlFKO4wghgiBIpVK4yPO8TCajlFJKJRIJXBRFET//OwDf/XcvVgtwH9Hg/QQ42YIgSCQSnb1n0QM+/MtwgAVYB/gLyA+vXL348K9dvg8d4GQDDgiCcPv2bUIIpV... | Find the area of the figure enclosed between the curves $y = 4 \cdot x^2$, $y = \frac{ x^2 }{ 9 }$, and $y = 2$: | Area: $\frac{20\cdot\sqrt{2}}{3}$ |
66acec6b-4d9c-4bd8-b009-32ee36aefe18 | algebra | false | null | Divide the rational expressions:
$$
\frac{ q^2-9 }{ q^2+6 \cdot q+9 } \div \frac{ q^2-2 \cdot q-3 }{ q^2+2 \cdot q-3 }
$$ | The final answer: $\frac{q-1}{q+1}$ |
66af2cdd-b16b-4737-a140-2fd39c485799 | sequences_series | false | null | Find a “reasonable” upper-bound on the error in approximating $f(x) = (x-1) \cdot \ln(x-1)$ by its 3rd order Taylor polynomial $P_{3}(x)$ about $a=2$ valid for all values of $x$ such that $|x-2| \le 0.3$. | The final answer: $\frac{2}{(0.7)^3}\cdot\frac{(0.3)^4}{4!}$ |
66e32cbc-860b-49a6-9cc6-2f5e65c8cb51 | precalculus_review | false | null | Evaluate the sum $S = \sin(a) + \sin(2 \cdot a) + \sin(3 \cdot a) + \ldots + \sin(n \cdot a)$, where $a \ne \pi \cdot k$. | The final answer: $S=\frac{\sin\left(\frac{(n+1)\cdot a}{2}\right)\cdot\sin\left(\frac{n\cdot a}{2}\right)}{\sin\left(\frac{a}{2}\right)}$ |
671f80e1-5201-4313-9f84-f34e02bc1cfa | differential_calc | false | null | Find the derivative of the function $y = \frac{ 2 \cdot \csc(x) - 7 \cdot \sin(x) }{ 4 \cdot \left(\cos(x)\right)^5 } - \frac{ 3 }{ 5 } \cdot \cot(2 \cdot x)$. | $y'$ = $\frac{6}{5\cdot\left(\sin(2\cdot x)\right)^2}+\frac{28\cdot\left(\cos(x)\right)^6-25\cdot\left(\cos(x)\right)^4-2\cdot\left(\cos(x)\right)^6\cdot\left(\csc(x)\right)^2}{4\cdot\left(\cos(x)\right)^{10}}$ |
672bf8b7-70c8-4296-96a2-3427f9e31551 | differential_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAACO7UlEQVR4nOzdd3iUVcLG4SeVkmToNYk0aYk0g0AiIC5KEXsBKwoKuzYU46rrZ0HWrgiIBQ3i2g0WLCggqIAQCEtAkBiKICGhhM6EhPT5/mAzMswEkkx5p/zu69prfU8mycMwJHly3nNOkMVisUjS22+/LUkaP368vFFGRoYkKSEhweAkjpGvdqZMmaIpU6ZIkpKTk5WcnGxwIse89fmrRD7nBNLXv40bN2rIkCE2Y2effbaWLVtW64/p7X... | Begin with box (A). To advance to the next box, look for the answer located above each box. Continue working in this manner until you complete the circuit. When you are done, submit the answer to the last limit statement completed.
START HERE
| A$\lim_{x \to 10}\left(-|1-x|\right)$ |
| --- |
| Previous a... | The result of evaluating the final limit is: $2$ |
67313a83-62b3-4200-a3c7-a19ec49a8683 | precalculus_review | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAkUAAAF9CAYAAAAKvxycAAA+8ElEQVR4nO3de3hU5bn//88kkxOQhARBKBhOFREoVAE1UAMWS4JitYqyEUGkUvdvszfHzc9DWwjY7tpaUGjpLli1HjhIESyKgt1ADIJYIFBFTkowAQMIORBISEiG+f6BiRlymklmzVqz5v26rlyXTGbmftY8ruTOuu/nWY6cnBy3DLZixQqNGTPG0Bh5eXn6zne+U/3vjIwMTZs2TYWFhR7PS0lJ0eLFixUXF9fsGEYIRAzJnDkhRsM6d+... | Find formulas for
$f(x)$ and
$g(x)$, the perpendicular lines whose graphs are shown below (note that
$f$ is the black line and
$g$ is the red line): | The final answer: $f(x)=-\frac{3}{5}\cdot x+3$, $g(x)=\frac{5}{3}\cdot x$ |
674563c1-43e2-47d9-be85-2566027df8cc | integral_calc | false | null | Compute the integral:
$$
4 \cdot \int \cos(4 \cdot x)^4 \, dx
$$ | $4 \cdot \int \cos(4 \cdot x)^4 \, dx$ = $\frac{\sin(4\cdot x)\cdot\cos(4\cdot x)^3}{4}+3\cdot\left(\frac{1}{8}\cdot\sin(4\cdot x)\cdot\cos(4\cdot x)+\frac{x}{2}\right)+C$ |
674ce6c5-cc42-4940-87a7-f55788a5f6c9 | integral_calc | false | null | Solve the integral:
$$
\int 2 \cdot \tan(-10 \cdot x)^4 \, dx
$$ | $\int 2 \cdot \tan(-10 \cdot x)^4 \, dx$ = $C+\frac{1}{5}\cdot\left(\frac{1}{3}\cdot\left(\tan(10\cdot x)\right)^3+\arctan\left(\tan(10\cdot x)\right)-\tan(10\cdot x)\right)$ |
675e419a-20df-4103-9bf4-92cf40c77234 | algebra | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQ4AAAEwCAYAAABGwZ94AAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum... | Write the equation for the graphed function. | The final answer: $-2\cdot(x-2)^2+7$ |
6773e6e9-681b-483f-8c75-43833b81fd8a | algebra | false | null | A falling object travels a distance given by the formula $d = 5 \cdot t + 16 \cdot t^2$ (ft), where $t$ is measured in seconds. How long will it take for the object to travel $74$ ft? | The final answer: $t=2$ |
67ca31aa-5290-45c5-9922-f42e135847f7 | algebra | false | null | A town's population has been decreasing at a constant rate. In 2010, the population was $5900$. By 2012, the population had dropped to $4700$. Assuming this trend continues, predict the population in 2016. | The population in 2016 is predicted to be $2300$ |
685d1667-9fd9-4d85-b338-8ad9eed93d40 | algebra | false | null | Find the formula for an exponential function that passes through the two points: $P(-2,6)$ and $P(3,1)$. | $f(x)$ = $6^{\frac{3}{5}}\cdot6^{-\frac{x}{5}}$ |
68996589-17fd-459a-946d-71f87e57b7a3 | integral_calc | false | null | Find $I=\int \frac{ 5 }{ 1+\sqrt{(x+1)^2+1} } \, dx$. | The final answer: $I=5\cdot\ln\left(x+1+\sqrt{x^2+2\cdot x+2}\right)+\frac{10}{x+2+\sqrt{x^2+2\cdot x+2}}+C$ |
68edc24a-5486-4fd2-b684-187893550518 | 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 $t=15$. | The final answer: $-\frac{3}{5}$ |
6908fd1a-de7a-4133-99a9-9c5bf4932f3f | multivariable_calculus | false | null | Evaluate $\int\int\int_{E}{(x \cdot z+1) d V}$, where $E$ is the region defined by:
$$
E = \left\{(x,y,z) | 0 \le x \le \sqrt{y}, 0 \le y \le 2, 0 \le z \le 1-x^2-y^2\right\}
$$ | $I$ = $\frac{73}{30}-\frac{52\cdot\sqrt{2}}{35}$ |
69536a3f-0fe2-40fc-8f40-9642f76f67b1 | integral_calc | false | null | Calculate the integral:
$$
\int \frac{ M \cdot x + N }{ \left( x^2 + p \cdot x + q \right)^m } \, dx
$$
where $M = 4$, $N = 5$, $p = 2$, $q = 9$, and $m = 2$. | $\int \frac{ M \cdot x + N }{ \left( x^2 + p \cdot x + q \right)^m } \, dx$ = $C+\frac{x+1}{128+16\cdot(x+1)^2}+\frac{\sqrt{2}}{64}\cdot\arctan\left(\frac{1}{2\cdot\sqrt{2}}\cdot(x+1)\right)-\frac{2}{8+(x+1)^2}$ |
695892c9-6ed2-4332-a5fb-5be48d39a53d | differential_calc | false | null | Make full curve sketching of $y = \sqrt[3]{5 \cdot x^2 - \frac{ x^3 }{ 2 }}$. 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 wh... | 1. The domain (in interval notation) $(-1\cdot\infty,\infty)$
2. Vertical asymptotes None
3. Horizontal asymptotes None
4. Slant asymptotes $y=-\frac{1}{\sqrt[3]{2}}\cdot x+\frac{10}{3\cdot\sqrt[3]{2}}$
5. Intervals where the function is increasing $\left(0,\frac{20}{3}\right)$
6. Intervals where the function is decrea... |
69980591-c61d-4569-9d4a-dc1f584a059b | integral_calc | false | null | Find the mass of an oversized hockey puck of radius 2 in. with density function $\rho(x) = x^3 - 2 \cdot x + 5$ that is centered at the origin. | $m$ = $\frac{332\cdot\pi}{15}$ |
6a01b887-afea-45a4-a721-2b93a32d3659 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcIAAADvCAIAAADuP0NOAAEAAElEQVR4nOy9d5xuVXU3vtba+5SnTJ+5cwtc2kVABKQpSECaQRQFSxJsFGvwNcauscZfYt6Y1ySWxBgLdn1jUFNeE7sgihCRJkiVdrlt+szTTtl7rd8f6zlnzsy9FJEWctfn3vk888yp+5z93at811ooIrBbHpPinLPWAoCIMLMxRkQQMc/zIAjyPCciADDGlFvulv9hwsVPCwC7mskMAAgEACAg2P/KrNiXAACAyt0RGIBBAFAAUP+605... | The base of a lamp is constructed by revolving a quarter circle $y = \sqrt{2 \cdot x - x^2}$ around the $y$-axis from $x = 1$ to $x = 2$ as seen here:
Create an integral for the surface area of this curve and compute it. | Surface Area = $2\cdot\pi+\pi^2$ |
6a3b6a6e-8d7d-4bc2-ada7-1ff894479ae8 | multivariable_calculus | false | null | The position function for an object in three dimensions is given by the equation $\vec{r}(t) = t \cdot \cos(t) \cdot \vec{i} + t \cdot \sin(t) \cdot \vec{j} + 3 \cdot t \cdot \vec{k}$. Find the tangential and normal components of acceleration when $t = 1.5$. Round your answer to two decimal digits. | $a_{T}(1.5)$ = $0.43$ meters per second squared
$a_{N}(1.5)$ = $2.46$ meters per second squared |
6a71b57b-416e-48e5-ad32-d4511d34e88b | sequences_series | false | null | Find the Fourier series of the function $f(x) = \frac{ 1 }{ 3 } \cdot x$ in the interval $[-4,4]$. | The Fourier series is: $\sum_{n=1}^\infty\left(\frac{\frac{8}{3}\cdot(-1)^{n+1}}{\pi\cdot n}\cdot\sin\left(\frac{\pi\cdot n\cdot x}{4}\right)\right)$ |
6b3c2531-9f19-4e7b-9266-5b329b94ce58 | integral_calc | false | null | Solve the integral:
$$
\int \frac{ 7 }{ \sin(-2 \cdot x)^3 \cdot \cos(2 \cdot x)^2 } \, dx
$$ | $\int \frac{ 7 }{ \sin(-2 \cdot x)^3 \cdot \cos(2 \cdot x)^2 } \, dx$ = $-7\cdot\left(C+\frac{3}{8}\cdot\ln\left(\left|\cos(2\cdot x)-1\right|\right)+\frac{1}{2\cdot\cos(2\cdot x)}+\frac{1}{8\cdot\left(1+\cos(2\cdot x)\right)}+\frac{1}{8\cdot\left(\cos(2\cdot x)-1\right)}-\frac{3}{8}\cdot\ln\left(\left|1+\cos(2\cdot x)... |
6b42b93e-7542-4939-9fac-3c233632843b | sequences_series | false | null | Expand the function $f(x) = \ln(1 + 4 \cdot x)$ 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 estimate of the absolu... | 1. $4\cdot x-\frac{(4\cdot x)^2}{2}+\frac{(4\cdot x)^3}{3}-\frac{(4\cdot x)^4}{4}+\frac{(4\cdot x)^5}{5}-\frac{(4\cdot x)^6}{6}+\frac{(4\cdot x)^7}{7}-\frac{(4\cdot x)^8}{8}+\frac{(4\cdot x)^9}{9}$
2. $\left|R_{10}(x)\right|<\frac{4^{10}}{10}$ |
6b629d16-c2b1-4623-82b2-02dbc4e7f2e4 | integral_calc | false | null | Compute the integral using the Substitution Rule:
$$
\int \frac{ x^2+3 }{ \sqrt{(2 \cdot x-5)^3} } \, dx
$$ | The final answer: $C+\frac{60\cdot x+(2\cdot x-5)^2-261}{12\cdot\sqrt{2\cdot x-5}}$ |
6b9b82b0-0daf-4868-b36b-698a539d3df2 | sequences_series | false | null | Use partial fractions to find the power series of the function $\frac{ 5 }{ \left(x^2+4\right) \cdot \left(x^2-1\right) }$. | $\frac{ 5 }{ \left(x^2+4\right) \cdot \left(x^2-1\right) }$ = $\sum_{n=0}^\infty\left(\left(-1+(-1)^{n+1}\cdot\frac{1}{2^{2\cdot n+2}}\right)\cdot x^{2\cdot n}\right)$ |
6bb32e56-be8b-434d-bd37-b11b1d2b9bba | precalculus_review | false | null | Evaluate the definite integral. Express answer in exact form whenever possible:
$$
\int_{0}^{\frac{ \pi }{ 2 }} \sqrt{1-\cos(2 \cdot x)} \, dx
$$ | $\int_{0}^{\frac{ \pi }{ 2 }} \sqrt{1-\cos(2 \cdot x)} \, dx$ = $\sqrt{2}$ |
6bd3ca7c-5934-4620-8202-14e34d1af387 | integral_calc | false | null | Compute the integral:
$$
-\int \frac{ \cos\left(\frac{ x }{ 2 }\right)^4 }{ \sin\left(\frac{ x }{ 2 }\right)^3 } \, dx
$$ | $-\int \frac{ \cos\left(\frac{ x }{ 2 }\right)^4 }{ \sin\left(\frac{ x }{ 2 }\right)^3 } \, dx$ = $C+3\cdot\cos\left(\frac{1}{2}\cdot x\right)+\frac{\left(\cos\left(\frac{1}{2}\cdot x\right)\right)^3}{1-\left(\cos\left(\frac{1}{2}\cdot x\right)\right)^2}-\frac{3}{2}\cdot\ln\left(\frac{1+\cos\left(\frac{1}{2}\cdot x\rig... |
6bdd2bef-2357-4e25-9fb4-6fe1f7d87733 | precalculus_review | false | null | Find the solutions of $(x+2)^4 + (x+5)^4 = 17$. | The final answer: $x_1=-4$, $x_2=-3$ |
6be04ab0-da51-4839-a9ba-166317b288ab | multivariable_calculus | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWcAAAFeCAIAAADWr0m/AADFVElEQVR4nOy9dUAU3/f4Pdu7sCyNdJcgINKKIKmCBSI2FnY3xluxu7sTW1FQESRElBCUkO7uju3def64v888+11C7FXn9RfcvXPnzOzOmXvPPYGBYRjqBzAMYzCY/vT89aP9cERcvB/LP3WxKD8E7O8WAAUF5Q8D1RooKChfB6o1UFBQvg5Ua6CgoHwdqNZAQUH5OlCtgYKC8nWgWgMFBeXrQLUGCgrK14FqjT+DAwcOjBs3bsyYMevXr+... | Find the surface area bounded by the curves $\left(x^2+y^2\right)^2 = 2 \cdot a^2 \cdot x \cdot y$. | $S$ = $a^2$ |
6c4086cf-b0c4-42a0-917f-e052b8133ee4 | sequences_series | false | null | Find the sum of the series $\sum_{n=0}^\infty \left(\frac{ (-1)^n }{ (2 \cdot n+1)! }\right)$ with an estimate error of $0.01$. | The final answer: $\frac{101}{120}$ |
6c95222e-64ae-4653-9815-9cfd31df92ac | algebra | false | null | Use the vertex $P(h,k) = P(-3,-2)$ and a point $P(x,y) = P(-1,3)$ on the graph of $f(x)$ to find the general form of the quadratic function. | The final answer: $f(x)=\frac{1}{4}\cdot\left(5\cdot x^2+30\cdot x+37\right)$ |
6c9b4f86-ec25-4c37-935e-acc9c8d1e528 | sequences_series | false | null | Compute the integral $\int_{0}^1{e^{-x^2} \, dx}$ with an estimated error of $0.0001$ using its series expansion. | The final answer: $0.74683603$ |
6cd14f24-ead8-41f0-bb5d-090488789a6d | differential_calc | false | null | For the function $f(x) = x^{11} - 6 \cdot x^{10}$, determine:
1. Intervals where:
1. $f$ is increasing
2. $f$ is decreasing
3. $f$ is concave up
4. $f$ is concave down
2. find:
1. local minima
2. local maxima
3. the inflection points of $f$ | 1. Intervals where:
1. $f$ is increasing: $(-\infty,0)$, $\left(\frac{60}{11},\infty\right)$
2. $f$ is decreasing: $\left(0,\frac{60}{11}\right)$
3. $f$ is concave up: $\left(\frac{54}{11},\infty\right)$
4. $f$ is concave down: $(-\infty,0)$, $\left(0,\frac{54}{11}\right)$
2. find:
1. local minima:... |
6d9acd8c-731d-41f3-b2a6-e84ea4e9f1ac | multivariable_calculus | false | null | Consider points $P(3,7,-2)$ and $Q(1,1,-3)$. Determine the angle between vectors $\vec{OP}$ and $\vec{OQ}$. Express the answer in radians, rounded to two decimal places. | $\theta$ = $0.91$ |
6da4aebe-04a0-4537-b2cc-214c8d3346dd | multivariable_calculus | false | null | Determine the equation of the parabola with a focus at $(-1,4)$ and a directrix of $x=5$. | The equation is: $(y-4)^2=-12\cdot(x-2)$ |
6ded3bb5-48f9-4882-bab5-147dbdbbf8e1 | algebra | false | null | Add and subtract the rational expressions, and then simplify:
$$
\frac{ x-1 }{ x+1 } - \frac{ 2 \cdot x + 3 }{ 2 \cdot x + 1 }
$$ | The final answer: $\frac{-6\cdot x-4}{(x+1)\cdot(2\cdot x+1)}$ |
6df6cc08-95e3-4b69-a68b-eb839c0fea7b | multivariable_calculus | false | null | Find the distance from $\left\langle 2,3,1 \right\rangle$ to the line through the origin parallel to $\left\langle 5,1,4 \right\rangle$. | The final answer: $\sqrt{\frac{299}{42}}$ |
6e10ffbf-706f-4aca-8017-ca016f7ab943 | integral_calc | true | data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAAB25UlEQVR4nO3deXiV9Z3//9dJQkJIDktYA0pUhAAaAgEMyGYCUhzEalt1OjPoz2ppZWoppdpOWzvOWK0LzNQNFZcZrdb26uXXUUR2guyRNWwB2UQkCVuArCQ5yf37g3LkeO6Ek5Bz3/c59/NxXVyXvDnJeef2ZHnl874/H49hGIYsMG/ePEnStGnTrHi6iDVnzhzNmTNHkjRr1izNmjXL5o6ciddT07g+oeE6hYbr1DSuT2i4TqHhOjWN6x... | Graph of $D'(t)$
The rate, in miles per hour, at which a delivery truck driver drives the first five minutes of their delivery route can be modeled by the function $D'(t)$, where $t$ measures the number of minutes since the driver began their route. The graph of $D'(t)$ is shown above. What is the average rate at whic... | The average rate at which the driver drives during the first five minutes of their route is $23$ miles per hour. |
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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.