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{
"dataset_name": "Mu-Math",
"group_index": 9,
"source_file": "/hkfs/work/workspace/scratch/tum_fmp0582-dndworkspace/自己训练lora/prepare/data/math/Mu-Math.json",
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"total_records": 1084,
"sample_count": 100,
"generated_at": "2025-11-05T12:58:28Z",
"seed": 4171135963,
"samples": [
{
"prompt": "Compute the integral: $\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$.",
"response": "$\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$ =$C+128\\cdot\\left(\\frac{1}{3\\cdot\\left(\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)\\right)^3}-\\frac{1}{\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)}\\right)$",
"system": ""
},
{
"prompt": "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.",
"response": "Derivative: $y'=\\frac{-128\\cdot x^7+66\\cdot x^6-84\\cdot x+45}{-24\\cdot x^8+12\\cdot x^7-36\\cdot x^2+18\\cdot x}\\cdot\\sqrt{\\frac{x^5\\cdot\\left(2\\cdot x^6+3\\right)}{\\sqrt[3]{1-2\\cdot x}}}$",
"system": ""
},
{
"prompt": "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.",
"response": "Derivative: $y'=\\frac{-128\\cdot x^7+66\\cdot x^6-84\\cdot x+45}{-24\\cdot x^8+12\\cdot x^7-36\\cdot x^2+18\\cdot x}\\cdot\\sqrt{\\frac{x^5\\cdot\\left(2\\cdot x^6+3\\right)}{\\sqrt[3]{1-2\\cdot x}}}$",
"system": ""
},
{
"prompt": "Compute the integral $-2 \\cdot \\int{x^{-4} \\cdot \\left(4+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$.",
"response": "$-2 \\cdot \\int{x^{-4} \\cdot \\left(4+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$ =$C+\\frac{1}{6}\\cdot\\left(\\frac{4}{x^2}+1\\right)\\cdot\\sqrt{\\frac{4}{x^2}+1}$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$.",
"response": "$\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$ =$C+\\sqrt{\\frac{x+4}{x-4}}\\cdot(x-20)-12\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x-4}-\\sqrt{x+4}}{\\sqrt{x-4}+\\sqrt{x+4}}\\right|\\right)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$.",
"response": "$\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$ =$C+\\sqrt{\\frac{x+4}{x-4}}\\cdot(x-20)-12\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x-4}-\\sqrt{x+4}}{\\sqrt{x-4}+\\sqrt{x+4}}\\right|\\right)$",
"system": ""
},
{
"prompt": "Solve $3 \\cdot \\sin(x)+4 \\cdot \\cos(x)=5$.",
"response": "This is the final answer to the problem: $x=-\\arcsin\\left(\\frac{4}{5}\\right)+\\frac{\\pi}{2}+2\\cdot\\pi\\cdot n$ ",
"system": ""
},
{
"prompt": "Solve the following equations: 1. $13 x+6=6$ 2. $\\frac{ x }{ -4 }+11=5$ 3. $-4.5 x+12.3=-23.7$ 4. $\\frac{ x }{ 5 }+4=4.3$ 5. $-\\frac{ x }{ 3 }+(-7.2)=-2.1$ 6. $5.4 x-8.3=14.38$ 7. $\\frac{ x }{ 3 }-14=-8$",
"response": "The solutions to the given equations are: 1. $x=0$\n2. $x=24$\n3. $x=8$\n4. $x=\\frac{ 3 }{ 2 }$\n5. $x=\\frac{ -153 }{ 10 }$\n6. $x=\\frac{ 21 }{ 5 }$\n7. $x=18$",
"system": ""
},
{
"prompt": "Solve the following equations: 1. $13 x+6=6$ 2. $\\frac{ x }{ -4 }+11=5$ 3. $-4.5 x+12.3=-23.7$ 4. $\\frac{ x }{ 5 }+4=4.3$ 5. $-\\frac{ x }{ 3 }+(-7.2)=-2.1$ 6. $5.4 x-8.3=14.38$ 7. $\\frac{ x }{ 3 }-14=-8$",
"response": "The solutions to the given equations are: 1. $x=0$\n2. $x=24$\n3. $x=8$\n4. $x=\\frac{ 3 }{ 2 }$\n5. $x=\\frac{ -153 }{ 10 }$\n6. $x=\\frac{ 21 }{ 5 }$\n7. $x=18$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=\\sqrt{\\frac{ 27-x^3 }{ 2 \\cdot x }}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(0,3]$\n2. Vertical asymptotes: $x=0$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: None\n6. Intervals where the function is decreasing: $(0,3]$\n7. Intervals where the function is concave up: $\\left(0,\\frac{3}{\\sqrt[3]{4}}\\right)$\n8. Intervals where the function is concave down: $\\left(\\frac{3}{\\sqrt[3]{4}},3\\right)$\n9. Points of inflection: $P\\left(\\frac{3}{\\sqrt[3]{4}},2.3146\\right)$",
"system": ""
},
{
"prompt": "Determine the equation of the hyperbola using the information given: Endpoints of the conjugate axis located at $(3,2)$, $(3,4)$ and focus located at $(7,3)$.",
"response": "The equation is: $\\frac{(x-3)^2}{15}-\\frac{(y-3)^2}{1}=1$",
"system": ""
},
{
"prompt": "Determine the equation of the hyperbola using the information given: Endpoints of the conjugate axis located at $(3,2)$, $(3,4)$ and focus located at $(7,3)$.",
"response": "The equation is: $\\frac{(x-3)^2}{15}-\\frac{(y-3)^2}{1}=1$",
"system": ""
},
{
"prompt": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 2 }+\\frac{ x }{ 1 \\cdot 3 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .",
"response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{1}{x^2}+\\frac{x\\cdot e^x-e^x}{x^2},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$",
"system": ""
},
{
"prompt": "Compute the derivative of the function $y=\\arcsin\\left(\\sqrt{1-9 \\cdot x^2}\\right)$.",
"response": "$y'$= $-\\frac{3\\cdot x}{\\sqrt{1-9\\cdot x^2}\\cdot|x|}$",
"system": ""
},
{
"prompt": "Determine a definite integral that represents the region enclosed by one petal of $r=\\cos\\left(3 \\cdot \\theta\\right)$ .",
"response": "This is the final answer to the problem:$\\int_0^{\\frac{\\pi}{6}}\\cos\\left(3\\cdot\\theta\\right)^2d\\theta$ = $\\frac{\\pi}{12}$ ",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ -9 \\cdot \\sqrt[3]{x} }{ 9 \\cdot \\sqrt[3]{x^2}+3 \\cdot \\sqrt{x} } d x}$.",
"response": "$\\int{\\frac{ -9 \\cdot \\sqrt[3]{x} }{ 9 \\cdot \\sqrt[3]{x^2}+3 \\cdot \\sqrt{x} } d x}$ =$-\\left(C+\\frac{1}{3}\\cdot\\sqrt[6]{x}^2+\\frac{2}{27}\\cdot\\ln\\left(\\frac{1}{3}\\cdot\\left|1+3\\cdot\\sqrt[6]{x}\\right|\\right)+\\frac{3}{2}\\cdot\\sqrt[6]{x}^4-\\frac{2}{3}\\cdot\\sqrt[6]{x}^3-\\frac{2}{9}\\cdot\\sqrt[6]{x}\\right)$",
"system": ""
},
{
"prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 2 \\cdot \\pi \\cdot x, & 0 \\le x \\le 1 \\\\ 0, & x>1 \\end{cases}$.",
"response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{2\\cdot\\left(\\alpha\\cdot\\sin\\left(\\alpha\\right)+\\cos\\left(\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+2\\cdot\\left(\\sin\\left(\\alpha\\right)-\\alpha\\cdot\\cos\\left(\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$",
"system": ""
},
{
"prompt": "Let $f(x)=\\ln\\left(x^2+1\\right)$.\n\n1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.\n2. Compute $\\left|f(1)-P_{4}(1)\\right|$.\n3. Compute $\\left|f(0.1)-P_{4}(0.1)\\right|$.",
"response": "This is the final answer to the problem: \n\n1. $x^2-\\frac{x^4}{2}$\n2. $0.1931$\n3. $0.0003\\cdot10^{-3}$",
"system": ""
},
{
"prompt": "Let $f(x)=\\ln\\left(x^2+1\\right)$.\n\n1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.\n2. Compute $\\left|f(1)-P_{4}(1)\\right|$.\n3. Compute $\\left|f(0.1)-P_{4}(0.1)\\right|$.",
"response": "This is the final answer to the problem: \n\n1. $x^2-\\frac{x^4}{2}$\n2. $0.1931$\n3. $0.0003\\cdot10^{-3}$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ 10 \\cdot x-8 }{ \\sqrt{2 \\cdot x^2+4 \\cdot x+10} } d x}$.",
"response": "$\\int{\\frac{ 10 \\cdot x-8 }{ \\sqrt{2 \\cdot x^2+4 \\cdot x+10} } d x}$ =$\\frac{1}{\\sqrt{2}}\\cdot\\left(10\\cdot\\sqrt{x^2+2\\cdot x+5}-18\\cdot\\ln\\left(\\left|2+2\\cdot x+2\\cdot\\sqrt{x^2+2\\cdot x+5}\\right|\\right)\\right)+C$",
"system": ""
},
{
"prompt": "Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):\n\nHalley’s Comet: length of major axis=$35.88$, eccentricity=$0.967$.",
"response": "$r$ = $\\frac{1.16450334}{1+0.967\\cdot\\cos(t)}$",
"system": ""
},
{
"prompt": "Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):\n\nHalley’s Comet: length of major axis=$35.88$, eccentricity=$0.967$.",
"response": "$r$ = $\\frac{1.16450334}{1+0.967\\cdot\\cos(t)}$",
"system": ""
},
{
"prompt": "Let $R$ be the region in the first quadrant bounded by the graph of $y=3 \\cdot \\arctan(x)$ and the lines $x=\\pi$ and $y=1$.\n\nFind the volume of the solid generated when $R$ is revolved about the line $x=\\pi$.",
"response": "The volume of the solid is $36.736$ units³.",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ 1 }{ \\sin(8 \\cdot x)^5 } d x}$.",
"response": "This is the final answer to the problem: $C+\\frac{1}{128}\\cdot\\left(2\\cdot\\left(\\tan(4\\cdot x)\\right)^2+6\\cdot\\ln\\left(\\left|\\tan(4\\cdot x)\\right|\\right)+\\frac{1}{4}\\cdot\\left(\\tan(4\\cdot x)\\right)^4-\\frac{2}{\\left(\\tan(4\\cdot x)\\right)^2}-\\frac{1}{4\\cdot\\left(\\tan(4\\cdot x)\\right)^4}\\right)$",
"system": ""
},
{
"prompt": "What is the general solution to the differential equation $\\frac{ d y }{d x}=-3 \\cdot y+12$ for $y>4$?",
"response": "$y$ = $C\\cdot e^{-3\\cdot x}+4$",
"system": ""
},
{
"prompt": "Make full curve sketching of $f(x)=\\frac{ 3 \\cdot x^3 }{ 3 \\cdot x^2-4 }$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptote(s)\n3. Horizontal asymptote(s)\n4. Slant asymptote(s)\n5. Interval(s) where the function is increasing\n6. Interval(s) where the function is decreasing\n7. Interval(s) where the function is concave up\n8. Interval(s) where the function is concave down\n9. Point(s) of inflection",
"response": "This is the final answer to the problem:\n\n1. The domain (in interval notation) $\\left(-1\\cdot\\infty,-2\\cdot3^{-1\\cdot2^{-1}}\\right)\\cup\\left(-2\\cdot3^{-1\\cdot2^{-1}},2\\cdot3^{-1\\cdot2^{-1}}\\right)\\cup\\left(2\\cdot3^{-1\\cdot2^{-1}},\\infty\\right)$\n2. Vertical asymptote(s) $x=-\\frac{2}{\\sqrt{3}}, x=\\frac{2}{\\sqrt{3}}$\n3. Horizontal asymptote(s) None\n4. Slant asymptote(s) $y=x$\n5. Interval(s) where the function is increasing $(2,\\infty), (-\\infty,-2)$\n6. Interval(s) where the function is decreasing $\\left(0,\\frac{2}{\\sqrt{3}}\\right), \\left(-\\frac{2}{\\sqrt{3}},0\\right), \\left(-2,-\\frac{2}{\\sqrt{3}}\\right), \\left(\\frac{2}{\\sqrt{3}},2\\right)$\n7. Interval(s) where the function is concave up $\\left(\\frac{2}{\\sqrt{3}},\\infty\\right), \\left(-\\frac{2}{\\sqrt{3}},0\\right)$\n8. Interval(s) where the function is concave down $\\left(0,\\frac{2}{\\sqrt{3}}\\right), \\left(-\\infty,-\\frac{2}{\\sqrt{3}}\\right)$\n9. Point(s) of inflection $P(0,0)$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$.",
"response": "$\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$ =$C-\\frac{1}{9}\\cdot\\left(1+\\frac{3}{x^2}\\right)\\cdot\\sqrt{1+\\frac{3}{x^2}}$",
"system": ""
},
{
"prompt": "Find the derivative of $f(x)=\\frac{ \\left(\\left(\\tan(x)\\right)^2-1\\right) \\cdot \\left(\\left(\\tan(x)\\right)^4+10 \\cdot \\left(\\tan(x)\\right)^2+1\\right) }{ 3 \\cdot \\left(\\tan(x)\\right)^3 }$.",
"response": "This is the final answer to the problem: $f'(x)=\\left(\\tan(x)\\right)^4+4\\cdot\\left(\\tan(x)\\right)^2+\\frac{4}{\\left(\\tan(x)\\right)^2}+\\frac{1}{\\left(\\tan(x)\\right)^4}+6$",
"system": ""
},
{
"prompt": "Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-10 \\cdot x-9 \\cdot y+8 \\cdot z=4 \\cdot \\cos(-10 \\cdot x-9 \\cdot y+8 \\cdot z)$.\n\nSubmit as your final answer:\n\na. $\\frac{\\partial z}{\\partial x}$;\n\nb. $\\frac{\\partial z}{\\partial y}$.",
"response": "This is the final answer to the problem: \na. $\\frac{5}{4}$;\n\nb. $\\frac{9}{8}$.",
"system": ""
},
{
"prompt": "Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-10 \\cdot x-9 \\cdot y+8 \\cdot z=4 \\cdot \\cos(-10 \\cdot x-9 \\cdot y+8 \\cdot z)$.\n\nSubmit as your final answer:\n\na. $\\frac{\\partial z}{\\partial x}$;\n\nb. $\\frac{\\partial z}{\\partial y}$.",
"response": "This is the final answer to the problem: \na. $\\frac{5}{4}$;\n\nb. $\\frac{9}{8}$.",
"system": ""
},
{
"prompt": "An alternating current for outlets in a home has voltage given by the function $V(t)=150 \\cdot \\cos(368 \\cdot t)$, where $V$ is the voltage in volts at time $t$ in seconds.\n\n1. Find the period of the function.\n2. Determine the number of periods that occur when $1$ sec. has passed.",
"response": "This is the final answer to the problem: \n\n1. the period of the function: $\\frac{\\pi}{184}$\n2. the number of periods: $58.56901906$",
"system": ""
},
{
"prompt": "Compute $\\sqrt[4]{90}$ with accuracy $0.0001$.",
"response": "This is the final answer to the problem: $3.0801$",
"system": ""
},
{
"prompt": "Solve $\\cos(2 \\cdot t)-5 \\cdot \\sin(t)-3=0$.",
"response": "This is the final answer to the problem: $t=(-1)^{n+1}\\cdot\\frac{\\pi}{6}+n\\cdot\\pi$",
"system": ""
},
{
"prompt": "The region bounded by the arc of the curve $y=\\sqrt{2} \\cdot \\sin(2 \\cdot x)$, $0 \\le x \\le \\frac{ \\pi }{ 2 }$, is revolved around the X-axis. Compute the surface area of this solid of revolution.",
"response": "Surface Area: $\\frac{\\pi}{4}\\cdot\\left(12\\cdot\\sqrt{2}+\\ln\\left(17+12\\cdot\\sqrt{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Find the radius of convergence and sum of the series: $\\frac{ 2 }{ 4 }+\\frac{ 2 \\cdot x }{ 1 \\cdot 5 }+\\frac{ 2 \\cdot (x)^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ 2 \\cdot (x)^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .",
"response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{12}{x^4}+\\frac{\\left(x\\cdot\\left(16\\cdot x\\cdot e^x-16\\cdot e^x\\right)-8\\cdot e^x\\cdot x^3\\right)\\cdot x^3+\\left(4\\cdot e^x+2\\cdot e^x\\cdot x^2+2\\cdot e^x\\cdot x^3-4\\cdot x\\cdot e^x\\right)\\cdot x^4}{x^8},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$",
"system": ""
},
{
"prompt": "The function $s(t)=2 \\cdot t^3-3 \\cdot t^2-12 \\cdot t+8$ represents the position of a particle traveling along a horizontal line.\n\n1. Find the velocity and acceleration functions.\n2. Determine the time intervals when the object is slowing down or speeding up.",
"response": "This is the final answer to the problem: 1. The velocity function $v(t)$ = $6\\cdot t^2-6\\cdot t-12$ and acceleration function $a(t)$ = $12\\cdot t-6$.\n\n2. The time intervals when the object speeds up $\\left(2,\\infty\\right), \\left(0,\\frac{1}{2}\\right)$ and slows down $\\left(\\frac{1}{2},2\\right)$.",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 5 \\cdot (x+2)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-2)\\cup(-2,\\infty)$\n2. Vertical asymptotes: $x=-2$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{5}-\\frac{4}{5}$\n5. Intervals where the function is increasing: $(0,\\infty), (-2,0), (-\\infty,-6)$\n6. Intervals where the function is decreasing: $(-6,-2)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-2,0), (-\\infty,-2)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "|$n$ \n\n$\\ln(n)$ \n\n| |\n| 1 | 0.00 |\n| 2 | 0.69 |\n| 3 | 1.10 |\n| 4 | 1.39 |\n| 5 | 1.61 |\n| 6 | 1.79 |\n| 7 | 1.95 |\n| 8 | 2.08 |\n| 9 | 2.20 |\n| 10 | 2.30 |\n\n \n \nUsing the table above, estimate the logarithm.1. $\\ln(16)$\n2. $\\ln\\left(3^4\\right)$\n3. $\\ln(2.5)$\n4. $\\ln\\left(\\sqrt{630}\\right)$\n5. $\\ln(0.4)$",
"response": "1. $\\ln(16)$≈$2.78$\n2. $\\ln\\left(3^4\\right)$≈$4.4$\n3. $\\ln(2.5)$≈$0.92$\n4. $\\ln\\left(\\sqrt{630}\\right)$≈$3.2228$\n5. $\\ln(0.4)$≈$-0.92$",
"system": ""
},
{
"prompt": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{29}$ with accuracy 0.0001.",
"response": "This is the final answer to the problem: $3.0723$",
"system": ""
},
{
"prompt": "Compute $\\int_{0}^{\\frac{ 1 }{ 3 }}{e^{-\\frac{ x^2 }{ 3 }} d x}$ with accuracy $0.00001$.",
"response": "This is the final answer to the problem: $0.32926$",
"system": ""
},
{
"prompt": "Washington, D.C. is located at $39$ deg N and $77$ deg W. Assume the radius of Earth is $4000$ mi. Express the location of Washington, D.C. in spherical coordinates (use radians).",
"response": "$P\\left(r,\\theta,\\varphi\\right)$ = $P(4000,-1.34,0.89)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{2 \\cdot \\tan(-10 \\cdot x)^4 d x}$.",
"response": "$\\int{2 \\cdot \\tan(-10 \\cdot x)^4 d x}$ =$C+\\frac{1}{5}\\cdot\\left(\\frac{1}{3}\\cdot\\left(\\tan(10\\cdot x)\\right)^3+\\arctan\\left(\\tan(10\\cdot x)\\right)-\\tan(10\\cdot x)\\right)$",
"system": ""
},
{
"prompt": "Compute the integral using the Substitution Rule:\n\n$\\int{\\frac{ x^2+3 }{ \\sqrt{(2 \\cdot x-5)^3} } d x}$",
"response": "This is the final answer to the problem: $C+\\frac{60\\cdot x+(2\\cdot x-5)^2-261}{12\\cdot\\sqrt{2\\cdot x-5}}$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 6 \\cdot (x+3)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-3)\\cup(-3,\\infty)$\n2. Vertical asymptotes: $x=-3$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{6}-1$\n5. Intervals where the function is increasing: $(-3,0), (0,\\infty), (-\\infty,-9)$\n6. Intervals where the function is decreasing: $(-9,-3)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-3,0), (-\\infty,-3)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "Find the derivative of the function: $y=-4 \\cdot x^{\\sqrt{5 \\cdot x}}$.",
"response": "$\\frac{ d y }{d x}$ =$-\\left(\\frac{4\\cdot\\sqrt{5}}{\\sqrt{x}}+\\frac{2\\cdot\\sqrt{5}\\cdot\\ln(x)}{\\sqrt{x}}\\right)\\cdot x^{\\sqrt{5}\\cdot\\sqrt{x}}$",
"system": ""
},
{
"prompt": "Given two functions $f(x)=\\sqrt{x^2-1}$ and $g(x)=\\sqrt{3-x}$\n\n1. compute $f\\left(g(x)\\right)$\n2. compute $\\frac{ f(x) }{ g(x) }$ and find the domain of the new function.",
"response": "1. the new function $f\\left(g(x)\\right)$ is$f\\left(g(x)\\right)=\\sqrt{2-x}$\n2. the function $\\frac{ f(x) }{ g(x) }$ is $\\frac{\\sqrt{x^2-1}}{\\sqrt{3-x}}$, and the domain for the function is $1\\le x<3 \\lor x\\le-1$",
"system": ""
},
{
"prompt": "A town has an initial population of $80\\ 000$. It grows at a constant rate of $2200$ per year for $5$ years. The linear function that models the town’s population P as a function of the year is: $P(t)=80\\ 000+2200 \\cdot t$, where $t$ is the number of years since the model began. When will the population reach $120\\ 000$?",
"response": "$t$: $\\frac{200}{11}$",
"system": ""
},
{
"prompt": "Find the arc length of the curve $x=\\frac{ 5 \\cdot y^2 }{ 6 }-\\frac{ \\ln(5 \\cdot y) }{ 2 }$ enclosed between $y=3$ and $y=5$.",
"response": "Arc Length: $-\\frac{\\operatorname{arsinh}\\left(\\frac{200y^{2} - 24}{12 \\sqrt{21}}\\right)}{10} + \\frac{\\sqrt{100y^{4} - 24y^{2} + 9}}{12} - \\frac{\\operatorname{arsinh}\\left(\\frac{3}{2 \\sqrt{21} \\, y^{2}} - \\frac{2}{\\sqrt{21}}\\right)}{4} \\approx \\boxed{13.23342845}$",
"system": ""
},
{
"prompt": "Find the Fourier series expansion for periodic (period $2$) $f(x)=x-1$ (at $(-1,1]$).",
"response": "The Fourier series is: $f(x)\\approx\\sum_{n=1}^\\infty-\\frac{2\\cdot(-1)^n\\cdot\\sin(n\\cdot\\pi\\cdot x)}{\\pi\\cdot n}-1$",
"system": ""
},
{
"prompt": "Make full curve sketching of $y=2 \\cdot \\arcsin\\left(\\frac{ 1-7 \\cdot x^2 }{ 1+7 \\cdot x^2 }\\right)$. Submit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation) $(-1\\cdot\\infty,\\infty)$\n2. Vertical asymptotes $\\text{None}$\n3. Horizontal asymptotes $y=-\\pi$\n4. Slant asymptotes $\\text{None}$\n5. Intervals where the function is increasing $(-\\infty,0)$\n6. Intervals where the function is decreasing $(0,\\infty)$\n7. Intervals where the function is concave up $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down $\\text{None}$\n9. Points of inflection $\\text{None}$",
"system": ""
},
{
"prompt": "Given that $\\frac{ 1 }{ 1-x }=\\sum_{n=0}^\\infty x^n$ , use term-by-term differentiation or integration to find power series for function $f(x)=\\ln(x)$ centered at $x=1$ .",
"response": "$\\ln(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{(x-1)^{n+1}}{n+1}\\right)$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ \\tan(x) }{ \\sqrt{\\sin(x)^4+\\cos(x)^4} } d x}$.",
"response": "$\\int{\\frac{ \\tan(x) }{ \\sqrt{\\sin(x)^4+\\cos(x)^4} } d x}$ =$\\frac{1}{2}\\cdot\\ln\\left(\\tan(x)^2+\\sqrt{\\tan(x)^4+1}\\right)+C$",
"system": ""
},
{
"prompt": "Solve the following system of equations:\n\n$x+y=\\frac{ 2 \\cdot \\pi }{ 3 }$ \n\n$\\frac{ \\sin(x) }{ \\sin(y) }=2$",
"response": "This is the final answer to the problem: $x=\\frac{\\pi}{2}+\\pi\\cdot k, y=\\frac{\\pi}{6}-\\pi\\cdot k$",
"system": ""
},
{
"prompt": "Solve the following system of equations:\n\n$x+y=\\frac{ 2 \\cdot \\pi }{ 3 }$ \n\n$\\frac{ \\sin(x) }{ \\sin(y) }=2$",
"response": "This is the final answer to the problem: $x=\\frac{\\pi}{2}+\\pi\\cdot k, y=\\frac{\\pi}{6}-\\pi\\cdot k$",
"system": ""
},
{
"prompt": "A particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.",
"response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 1 }{ (x-3) \\cdot \\sqrt{10 \\cdot x-24-x^2} } d x}$.",
"response": "$\\int{\\frac{ 1 }{ (x-3) \\cdot \\sqrt{10 \\cdot x-24-x^2} } d x}$ =$-\\frac{2}{\\sqrt{3}}\\cdot\\arctan\\left(\\frac{\\sqrt{-x^2+10\\cdot x-24}}{\\sqrt{3}\\cdot x-4\\cdot\\sqrt{3}}\\right)+C$",
"system": ""
},
{
"prompt": "For the curve $x=a\\left(t-\\sin(t)\\right)$, $y=a\\left(1-\\cos(t)\\right)$ determine the curvature. Use $a=10$.",
"response": "The curvature is:$\\frac{1}{40\\cdot\\left|\\sin\\left(\\frac{t}{2}\\right)\\right|}$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ x }{ \\left(x^2-4 \\cdot x+8\\right)^2 } d x}$.",
"response": "$\\int{\\frac{ x }{ \\left(x^2-4 \\cdot x+8\\right)^2 } d x}$ =$C+\\frac{x-2}{2\\cdot\\left(8+2\\cdot(x-2)^2\\right)}+\\frac{1}{8}\\cdot\\arctan\\left(\\frac{1}{2}\\cdot(x-2)\\right)-\\frac{1}{2\\cdot\\left(x^2-4\\cdot x+8\\right)}$",
"system": ""
},
{
"prompt": "On a cylinder 6 cm in diameter, a channel is cut out along the surface, having an equilateral triangle with a side of 1.5 cm in cross section. Compute the volume of the cut out material.",
"response": "$V$ =$\\frac{108\\cdot\\sqrt{3}-27}{32}\\cdot\\pi$",
"system": ""
},
{
"prompt": "Determine the Taylor series for $y=\\left(\\sin(x)\\right)^2$, centered at $x_{0}=\\frac{ \\pi }{ 2 }$. Write out the sum of the first three non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $1-\\frac{2}{2!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^2+\\frac{2^3}{4!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^4+\\cdots$",
"system": ""
},
{
"prompt": "Evaluate the function at the indicated values $f(-3)$, $f(2)$, $f(-a)$,$-f(a)$, and $f(a+h)$. \n\n$f(x)=|x-1|-|x+1|$",
"response": "This is the final answer to the problem: $f(-3)$=$2$\n\n$f(2)$=$-2$ \n\n$f(-a)$=$|a+1|-|-a+1|$ \n\n$-f(a)$= $-|a-1|+|a+1|$ \n\n$f(a+h)$=$|a+h-1|-|a+h+1|$",
"system": ""
},
{
"prompt": "Find the Fourier series of the function $f(x)=\\frac{ -1 }{ 2 } \\cdot x$ in the interval $[-2,2]$.",
"response": "The Fourier series is: $\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Compute the derivative $y=\\arctan\\left(\\frac{ 4-\\cos\\left(\\frac{ x }{ 2 }\\right) }{ 1+4 \\cdot \\cos\\left(\\frac{ x }{ 2 }\\right) }\\right)$.",
"response": "$y'$= $\\frac{\\sin\\left(\\frac{x}{2}\\right)}{2+2\\cdot\\left(\\cos\\left(\\frac{x}{2}\\right)\\right)^2}$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\left(2 \\cdot x^2+4 \\cdot x+5\\right) \\cdot \\sin(2 \\cdot x) d x}$.",
"response": "$\\int{\\left(2 \\cdot x^2+4 \\cdot x+5\\right) \\cdot \\sin(2 \\cdot x) d x}$ =$C+x\\cdot\\sin(2\\cdot x)+\\frac{1}{2}\\cdot\\cos(2\\cdot x)+\\sin(2\\cdot x)-\\frac{1}{2}\\cdot\\left(2\\cdot x^2+4\\cdot x+5\\right)\\cdot\\cos(2\\cdot x)$",
"system": ""
},
{
"prompt": "A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is $0.24$. It can be shown that the downward velocity of the sky diver at time $t$ is given by\n\n$v(t)=180 \\cdot \\left(1-e^{-0.24 \\cdot t}\\right)$ \n\nwhere $t$ is measured in seconds and $v(t)$ is measured in feet per second\n\n\n\n1. Find the initial velocity of the sky diver\n\n2. Find the velocity after $4$ seconds (round your answer to one decimal place)\n\n3. The maximum velocity of a falling object with wind resistance is called its terminal velocity. Find the terminal velocity of this sky diver. (round your answer to the nearest whole number)",
"response": "1. $0$\n2. $111.1$\n3. $180$",
"system": ""
},
{
"prompt": "Find the Fourier series expansion of the function $f(x)=\\begin{cases} x, & -\\pi \\le x<0 \\\\ \\pi, & 0 \\le x \\le \\pi \\end{cases}$ with the period $2 \\cdot \\pi$ at interval $[-\\pi,\\pi]$.",
"response": "The Fourier series is: $f(x)=\\frac{\\pi}{4}+\\sum_{n=1}^\\infty\\left(\\frac{1+(-1)^{n+1}}{\\pi\\cdot n^2}\\cdot\\cos(n\\cdot x)+\\frac{(-1)^{n+1}\\cdot2+1}{n}\\cdot\\sin(n\\cdot x)\\right)$",
"system": ""
},
{
"prompt": "Given $y=3 \\cdot x^5+10 \\cdot x^4-20$ find where the function is concave up, down, and point(s) of inflection.",
"response": "Concave up:$(0,\\infty), (-2,0)$Concave down:$(-\\infty,-2)$Point(s) of Inflection:$P(-2,44)$",
"system": ""
},
{
"prompt": "Differentiate $\\sqrt{x \\cdot y}-x=y^5$.",
"response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^{\\frac{1}{2}}\\cdot y^{\\frac{1}{2}}-y}{x-10\\cdot x^{\\frac{1}{2}}\\cdot y^{\\frac{9}{2}}}$",
"system": ""
},
{
"prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y)=\\frac{ A \\cdot x+B \\cdot y }{ C \\cdot x+D \\cdot y }$.",
"response": "$f_{xx}(x,y)$=$\\frac{2\\cdot C\\cdot(B\\cdot C-A\\cdot D)\\cdot y}{(C\\cdot x+D\\cdot y)^3}$$f_{xy}(x,y)$=$f_{yx}(x,y)$=$\\frac{-(B\\cdot C-A\\cdot D)\\cdot(C\\cdot x-D\\cdot y)}{(C\\cdot x+D\\cdot y)^3}$ \n\n$f_{yy}(x,y)$=$-2\\cdot D\\cdot\\frac{B\\cdot C\\cdot x-A\\cdot D\\cdot x}{(C\\cdot x+D\\cdot y)^3}$",
"system": ""
},
{
"prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y)=\\frac{ A \\cdot x+B \\cdot y }{ C \\cdot x+D \\cdot y }$.",
"response": "$f_{xx}(x,y)$=$\\frac{2\\cdot C\\cdot(B\\cdot C-A\\cdot D)\\cdot y}{(C\\cdot x+D\\cdot y)^3}$$f_{xy}(x,y)$=$f_{yx}(x,y)$=$\\frac{-(B\\cdot C-A\\cdot D)\\cdot(C\\cdot x-D\\cdot y)}{(C\\cdot x+D\\cdot y)^3}$ \n\n$f_{yy}(x,y)$=$-2\\cdot D\\cdot\\frac{B\\cdot C\\cdot x-A\\cdot D\\cdot x}{(C\\cdot x+D\\cdot y)^3}$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ 2 \\cdot \\tan(x)+3 }{ \\sin(x)^2+2 \\cdot \\cos(x)^2 } d x}$.",
"response": "$\\int{\\frac{ 2 \\cdot \\tan(x)+3 }{ \\sin(x)^2+2 \\cdot \\cos(x)^2 } d x}$ =$C+\\frac{3}{\\sqrt{2}}\\cdot\\arctan\\left(\\frac{1}{\\sqrt{2}}\\cdot\\tan(x)\\right)+\\ln\\left(2+\\left(\\tan(x)\\right)^2\\right)$",
"system": ""
},
{
"prompt": "For the function $y=(4-x)^3 \\cdot (x+1)^2$ specify the points where local maxima and minima of $y$ occur. Submit as your final answer:\n\n1. The point(s) where local maxima occur\n2. The point(s) where local minima occur",
"response": "This is the final answer to the problem: \n1. The point(s) where local maxima occur $P(1,108)$\n2. The point(s) where local minima occur $P(-1,0)$",
"system": ""
},
{
"prompt": "Given $y=3 \\cdot x^5+20 \\cdot x^4+40 \\cdot x^3+100$ find where the function is concave up, down, and point(s) of inflection.",
"response": "Concave up:$(0,\\infty)$Concave down:$(-2,0), (-\\infty,-2)$Point(s) of Inflection:$P(0,100)$",
"system": ""
},
{
"prompt": "Find the Taylor series for $f(x)=\\ln\\left(\\sqrt{3} \\cdot x+\\sqrt{1+3 \\cdot x^2}\\right)$, centered at $x=0$. Write out the sum of the first four non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $\\sqrt{3} x - (\\sqrt{3} x^3)/2 + (27 \\sqrt{3} x^5)/40 - (135 \\sqrt{3} x^7)/112 + O(x^9)$",
"system": ""
},
{
"prompt": "Given $g(x)=\\frac{ 1 }{ 3 } \\cdot (a+b) \\cdot x^3+\\frac{ 1 }{ 2 } \\cdot (a+b+c) \\cdot x^2-(a+b+c+d) \\cdot x+a \\cdot b \\cdot c \\cdot d$, simplify the derivative of $g(x)$ if $x^2+x=a+b$.",
"response": "This is the final answer to the problem: $g'(x)=(a+b)^2+c\\cdot x-(a+b+c+d)$",
"system": ""
},
{
"prompt": "Evaluate the integral: $I=\\int{3 \\cdot x \\cdot \\ln\\left(4+\\frac{ 1 }{ x }\\right) d x}$.",
"response": "This is the final answer to the problem: $\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(4\\cdot x+1)-\\frac{3\\cdot x^2}{4}+\\frac{3\\cdot x}{8}-\\frac{3}{32}\\cdot\\ln\\left(x+\\frac{1}{4}\\right)\\right)-\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(x)-\\left(C+\\frac{3}{4}\\cdot x^2\\right)\\right)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ 1 }{ \\sin(x)^5 } d x}$.",
"response": "This is the final answer to the problem: $C+\\frac{1}{16}\\cdot\\left(2\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^2+6\\cdot\\ln\\left(\\left|\\tan\\left(\\frac{x}{2}\\right)\\right|\\right)+\\frac{1}{4}\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^4-\\frac{2}{\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^2}-\\frac{1}{4\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^4}\\right)$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$.",
"response": "$\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$ =$\\frac{\\cos\\left(\\frac{x}{3}\\right)}{5\\cdot\\sin\\left(\\frac{x}{3}\\right)^5}-\\frac{4}{15}\\cdot\\left(-\\frac{\\cos\\left(\\frac{x}{3}\\right)}{\\sin\\left(\\frac{x}{3}\\right)^3}-2\\cdot\\cot\\left(\\frac{x}{3}\\right)\\right)+C$",
"system": ""
},
{
"prompt": "Compute the derivative of the complex function $p=u^v$ given $u=3 \\cdot \\ln(x-2 \\cdot y)$ and $v=e^{\\frac{ x }{ y }}$.",
"response": "$\\frac{\\partial p}{\\partial x} = (3 \\ln(x - 2y))^{e^{\\frac{x}{y}}} \\left[ \\frac{e^{\\frac{x}{y}}}{y} \\ln(3 \\ln(x - 2y)) + \\frac{e^{\\frac{x}{y}}}{(x - 2y) \\ln(x - 2y)} \\right]$\n\n$\\frac{\\partial p}{\\partial y} = (3 \\ln(x - 2y))^{e^{\\frac{x}{y}}} \\left[ -\\frac{x e^{\\frac{x}{y}}}{y^2} \\ln(3 \\ln(x - 2y)) - \\frac{2e^{\\frac{x}{y}}}{(x - 2y) \\ln(x - 2y)} \\right]$\n",
"system": ""
},
{
"prompt": "For the curve $x=a\\left(t-\\sin(t)\\right)$, $y=a\\left(1-\\cos(t)\\right)$ determine the curvature. Use $a=12$.",
"response": "The curvature is:$\\frac{1}{48\\cdot\\left|\\sin\\left(\\frac{t}{2}\\right)\\right|}$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$.",
"response": "$\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$ =$C-\\frac{1}{2}\\cdot\\left(\\cot(4\\cdot x)\\right)^5-\\frac{5}{2}\\cdot\\cot(4\\cdot x)-\\frac{5}{3}\\cdot\\left(\\cot(4\\cdot x)\\right)^3$",
"system": ""
},
{
"prompt": "Find the Fourier series of the periodic function $f(x)=x^2$ in the interval $-\\pi \\le x<\\pi$ if $f(x)=f(x+2 \\cdot \\pi)$.",
"response": "The Fourier series is: $x^2=\\frac{\\pi^2}{3}-4\\cdot\\left(\\frac{\\cos(x)}{1^2}-\\frac{\\cos(2\\cdot x)}{2^2}+\\frac{\\cos(3\\cdot x)}{3^2}-\\cdots\\right)$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=3 \\cdot x \\cdot \\sqrt{2-x^2}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $\\left[-1\\cdot2^{2^{-1}},2^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $(-1,1)$\n6. Intervals where the function is decreasing: $\\left(-2^{2^{-1}},-1\\right), \\left(1,2^{2^{-1}}\\right)$\n7. Intervals where the function is concave up: $\\left(-2^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,2^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=3 \\cdot x \\cdot \\sqrt{2-x^2}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $\\left[-1\\cdot2^{2^{-1}},2^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $(-1,1)$\n6. Intervals where the function is decreasing: $\\left(-2^{2^{-1}},-1\\right), \\left(1,2^{2^{-1}}\\right)$\n7. Intervals where the function is concave up: $\\left(-2^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,2^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "Find the 3rd order Taylor polynomial $P_{3}(x)$ for the function $f(x)=\\arctan(x)$ in powers of $x-1$ and give the Lagrange form of the remainder.",
"response": "$P_{3}(x)$=$\\frac{ \\pi }{ 4 }+\\frac{ 1 }{ 2 } \\cdot (x-1)-\\frac{ 1 }{ 4 } \\cdot (x-1)^2+\\frac{ 1 }{ 12 } \\cdot (x-1)^3$ \n\n$R_{3}(x)$=$\\frac{ -\\frac{ 48 \\cdot c^3 }{ \\left(1+c^2\\right)^4 }+\\frac{ 24 \\cdot c }{ \\left(1+c^2\\right)^3 } }{ 4! } \\cdot (x-1)^4$",
"system": ""
},
{
"prompt": "Make full curve sketching of $f(x)=\\frac{ 3 \\cdot x^3 }{ 2 \\cdot x^2-3 }$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptote(s)\n3. Horizontal asymptote(s)\n4. Slant asymptote(s)\n5. Interval(s) where the function is increasing\n6. Interval(s) where the function is decreasing\n7. Interval(s) where the function is concave up\n8. Interval(s) where the function is concave down\n9. Point(s) of inflection",
"response": "This is the final answer to the problem:\n\n1. The domain (in interval notation) $\\left(-1\\cdot\\infty,-1\\cdot3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}}\\right)\\cup\\left(-1\\cdot3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}},3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}}\\right)\\cup\\left(3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}},\\infty\\right)$\n2. Vertical asymptote(s) $x=\\frac{\\sqrt{3}}{\\sqrt{2}}, x=-\\frac{\\sqrt{3}}{\\sqrt{2}}$\n3. Horizontal asymptote(s) None\n4. Slant asymptote(s) $y=\\frac{3}{2}\\cdot x$\n5. Interval(s) where the function is increasing $\\left(\\frac{3}{\\sqrt{2}},\\infty\\right), \\left(-\\infty,-\\frac{3}{\\sqrt{2}}\\right)$\n6. Interval(s) where the function is decreasing $\\left(-\\frac{3}{\\sqrt{2}},-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\frac{3}{\\sqrt{2}}\\right), \\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right)$\n7. Interval(s) where the function is concave up $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n8. Interval(s) where the function is concave down $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n9. Point(s) of inflection $P(0,0)$",
"system": ""
},
{
"prompt": "Evaluate the integral by choosing the order of integration: $\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$.",
"response": "$\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$ =$\\frac{2\\cdot\\pi+8\\cdot\\ln\\left(\\frac{5}{4}\\right)-4\\cdot\\ln(2)-8\\cdot\\arctan\\left(\\frac{1}{2}\\right)}{8}$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 1 }{ \\sin(x)^6 } d x}$.",
"response": "$\\int{\\frac{ 1 }{ \\sin(x)^6 } d x}$ =$-\\frac{\\cos(x)}{5\\cdot\\sin(x)^5}+\\frac{4}{5}\\cdot\\left(-\\frac{\\cos(x)}{3\\cdot\\sin(x)^3}-\\frac{2}{3}\\cdot\\cot(x)\\right)$",
"system": ""
},
{
"prompt": "Determine a definite integral that represents the region enclosed by the inner loop of $r=3-4 \\cdot \\cos\\left(\\theta\\right)$ .",
"response": "This is the final answer to the problem:$\\int_0^{\\arccos\\left(\\frac{3}{4}\\right)}\\left(3-4\\cdot\\cos\\left(\\theta\\right)\\right)^2d\\theta$",
"system": ""
},
{
"prompt": "Find the second derivative $\\frac{d ^2y}{ d x^2}$ of the function $x=\\left(4 \\cdot \\sin(t)\\right)^3$, $y=2 \\cdot \\sin(2 \\cdot t)$.",
"response": "$\\frac{d ^2y}{ d x^2}$=$\\frac{\\left(2304\\cdot\\left(\\sin(t)\\right)^3-1536\\cdot\\sin(t)\\right)\\cdot\\cos(2\\cdot t)-1536\\cdot\\left(\\sin(t)\\right)^2\\cdot\\cos(t)\\cdot\\sin(2\\cdot t)}{7077888\\cdot\\left(\\cos(t)\\right)^3\\cdot\\left(\\sin(t)\\right)^6}$",
"system": ""
},
{
"prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ \\pi \\cdot x, & 0 \\le x \\le 2 \\\\ 0, & x>2 \\end{cases}$.",
"response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{\\left(2\\cdot\\alpha\\cdot\\sin\\left(2\\cdot\\alpha\\right)+\\cos\\left(2\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+\\left(\\sin\\left(2\\cdot\\alpha\\right)-2\\cdot\\alpha\\cdot\\cos\\left(2\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$",
"system": ""
},
{
"prompt": "Given $x^3-2 \\cdot x^2 \\cdot y^2+5 \\cdot x+y-5=0$, evaluate $\\frac{d ^2y}{ d x^2}$ at $x=1$.",
"response": "This is the final answer to the problem: $\\frac{d^2y}{dx^2}=-\\frac{238}{27}$ or $\\frac{d^2y}{dx^2}=-\\frac{319}{27}$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 1 }{ (x+4) \\cdot \\sqrt{x^2+2 \\cdot x+5} } d x}$.",
"response": "$\\int{\\frac{ 1 }{ (x+4) \\cdot \\sqrt{x^2+2 \\cdot x+5} } d x}$ =$C+\\frac{1}{\\sqrt{13}}\\cdot\\ln\\left(\\sqrt{13}-4-x-\\sqrt{x^2+2\\cdot x+5}\\right)-\\frac{1}{\\sqrt{13}}\\cdot\\ln\\left(4+\\sqrt{13}+x+\\sqrt{x^2+2\\cdot x+5}\\right)$",
"system": ""
},
{
"prompt": "Determine the interval(s) on which $f(x)=x^3 \\cdot e^{-x}$ is decreasing.",
"response": "The function is decreasing on the interval(s) $(3,\\infty)$.",
"system": ""
},
{
"prompt": "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.",
"response": "The dimensions of the box are $6, 4, 8$",
"system": ""
},
{
"prompt": "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.",
"response": "Maximum:NoneMinimum: $P(0,0)$ \n\nSaddle point: None\n\nThe second derivative test is inconclusive at: None",
"system": ""
},
{
"prompt": "Find the equation of the tangent line to the curve: $r=3+\\cos(2 \\cdot t)$, $t=\\frac{ 3 \\cdot \\pi }{ 4 }$.",
"response": "$y$ = $\\frac{1}{5}\\cdot\\left(x+\\frac{3}{\\sqrt{2}}\\right)+\\frac{3}{\\sqrt{2}}$",
"system": ""
},
{
"prompt": "Find $\\frac{d ^3}{ d x^3}f(x)$, given $f(x)=\\ln\\left(\\frac{ x+7 }{ x-7 }\\right)$.",
"response": "This is the final answer to the problem: $\\frac{d^3}{dx^3}f(x)=-\\frac{84\\cdot x^2+1372}{\\left(x^2-49\\right)^3}$",
"system": ""
},
{
"prompt": "Find $\\frac{d ^3}{ d x^3}f(x)$, given $f(x)=\\ln\\left(\\frac{ x+7 }{ x-7 }\\right)$.",
"response": "This is the final answer to the problem: $\\frac{d^3}{dx^3}f(x)=-\\frac{84\\cdot x^2+1372}{\\left(x^2-49\\right)^3}$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ -\\sqrt[3]{2 \\cdot x} }{ \\sqrt[3]{(2 \\cdot x)^2}-\\sqrt{2 \\cdot x} } d x}$.",
"response": "$\\int{\\frac{ -\\sqrt[3]{2 \\cdot x} }{ \\sqrt[3]{(2 \\cdot x)^2}-\\sqrt{2 \\cdot x} } d x}$ =$C-3\\cdot\\left(\\frac{1}{2}\\cdot\\sqrt[6]{2\\cdot x}^2+\\frac{1}{3}\\cdot\\sqrt[6]{2\\cdot x}^3+\\frac{1}{4}\\cdot\\sqrt[6]{2\\cdot x}^4+\\sqrt[6]{2\\cdot x}+\\ln\\left(\\left|\\sqrt[6]{2\\cdot x}-1\\right|\\right)\\right)$",
"system": ""
}
]
} |