| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 8, | |
| "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": "Evaluate the integral by choosing the order of integration: $\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ y }{ x+y^2 }\\right) d y} d x}$.", | |
| "response": "$\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ y }{ x+y^2 }\\right) d y} d x}$ =$\\ln\\left(\\frac{25\\cdot\\sqrt{5}}{32}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given $f(x)=x^2+\\frac{ 16 }{ x^2 }$, find the intervals where $f$ increases and the intervals where $f$ is decreasing.", | |
| "response": "$f$ is increasing on the intervals: $(2,\\infty), (-2,0)$\n\n$f$ is decreasing on the intervals: $(-\\infty,-2), (0,2)$ ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given $f(x)=x^2+\\frac{ 16 }{ x^2 }$, find the intervals where $f$ increases and the intervals where $f$ is decreasing.", | |
| "response": "$f$ is increasing on the intervals: $(2,\\infty), (-2,0)$\n\n$f$ is decreasing on the intervals: $(-\\infty,-2), (0,2)$ ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the periodic function $f(x)=x^2$ in the interval $-2 \\cdot \\pi \\le x<2 \\cdot \\pi$ if $f(x)=f(x+4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $\\frac{4\\cdot\\pi^2}{3}+\\sum_{n=1}^\\infty\\left(\\frac{16\\cdot(-1)^n}{n^2}\\cdot\\cos\\left(\\frac{n\\cdot x}{2}\\right)\\right)$", | |
| "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": "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": "Find a rectangular equation which is equivalent to the following parametric equations:\n\n$x^2=t^3-3 \\cdot t^2+3 \\cdot t-1$ \n\n$y^2=t^3+6 \\cdot t^2+12 \\cdot t+8$", | |
| "response": "This is the final answer to the problem: $\\sqrt[3]{x^2}-\\sqrt[3]{y^2}=-3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find a rectangular equation which is equivalent to the following parametric equations:\n\n$x^2=t^3-3 \\cdot t^2+3 \\cdot t-1$ \n\n$y^2=t^3+6 \\cdot t^2+12 \\cdot t+8$", | |
| "response": "This is the final answer to the problem: $\\sqrt[3]{x^2}-\\sqrt[3]{y^2}=-3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "$f(x)=x+\\sin(2 \\cdot x)$ over $x$= $\\left[-\\frac{ \\pi }{ 2 },\\frac{ \\pi }{ 2 }\\right]$. \n\nDetermine:\n1. intervals where $f$ is increasing\n2. intervals where $f$ is decreasing\n3. local minima of $f$\n4. local maxima of $f$\n5. intervals where $f$ is concave up\n6. intervals where $f$ is concave down\n7. the inflection points of $f$", | |
| "response": "1. intervals where $f$ is increasing : $\\left(-\\frac{\\pi}{3},\\frac{\\pi}{3}\\right)$\n2. intervals where $f$ is decreasing: $\\left(\\frac{\\pi}{3},\\frac{\\pi}{2}\\right), \\left(-\\frac{\\pi}{2},-\\frac{\\pi}{3}\\right)$\n3. local minima of $f$: $-\\frac{\\pi}{3}$\n4. local maxima of $f$: $\\frac{\\pi}{3}$\n5. intervals where $f$ is concave up : $\\left(-\\frac{\\pi}{2},0\\right)$\n6. intervals where $f$ is concave down: $\\left(0,\\frac{\\pi}{2}\\right)$\n7. the inflection points of $f$: $0$", | |
| "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": "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": "Let $R$ be the region bounded by the graphs of $y=\\ln(x)$ and $y=x-5$. Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated around the $y$-axis.", | |
| "response": "$V$ = $\\pi\\cdot\\int_b^d\\left((y+5)^2-\\left(e^y\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 $f(3)=-3$.", | |
| "response": "$y$ = $\\frac{|x|}{3}-4$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $u=\\left|\\sin\\left(\\frac{ x }{ 2 }\\right)\\right|$ in the interval $[-2 \\cdot \\pi,2 \\cdot \\pi]$.", | |
| "response": "The Fourier series is: $\\frac{2}{\\pi}+\\frac{4}{\\pi}\\cdot\\sum_{k=1}^\\infty\\left(\\frac{1}{\\left(1-4\\cdot k^2\\right)}\\cdot\\cos(k\\cdot x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate integral: $\\int_{\\frac{ 1 }{ 2 }}^{\\frac{ \\sqrt{3} }{ 2 }}{\\frac{ 1 }{ x \\cdot \\sqrt{9-9 \\cdot x^2} } d x}$.", | |
| "response": "$$\\int_{\\frac{ 1 }{ 2 }}^{\\frac{ \\sqrt{3} }{ 2 }}{\\frac{ 1 }{ x \\cdot \\sqrt{9-9 \\cdot x^2} } d x}=\\frac{1}{6}\\ln\\left(\\frac{7}{3}+\\frac{4}{\\sqrt{3}}\\right)$$", | |
| "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": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.", | |
| "response": "$f'(1)$ =$-0.5244$", | |
| "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: $\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$.", | |
| "response": "$\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$ =$\\frac{\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)^3}{2}+\\frac{3}{4}\\cdot\\left(\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)+\\frac{x}{2}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\cot(x)^4 d x}$.", | |
| "response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of the function $\\sqrt{x}$ with the center $a=9$.", | |
| "response": "$\\sqrt{x}$ =$\\sum_{n=0}^\\infty\\left(3^{1-2\\cdot n}\\cdot C_n^{\\frac{1}{2}}\\cdot(x-9)^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute $\\sqrt[4]{90}$ with accuracy $0.0001$.", | |
| "response": "This is the final answer to the problem: $3.0801$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ x^4 }{ 4 }-\\frac{ 11 }{ 3 } \\cdot x^3+15 \\cdot x^2+17$.", | |
| "response": "The point(s) where the function has a local minimum:$P(6,89), P(0,17)$ \nThe point(s) where the function has a local maximum:$P\\left(5,\\frac{1079}{12}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{2 \\cdot x-x^2}$ with the center $a=1$.", | |
| "response": "$\\sqrt{2 \\cdot x-x^2}$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^{2\\cdot n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $u=\\left|\\sin(x)\\right|$ in the interval $[-\\pi,\\pi]$.", | |
| "response": "The Fourier series is: $\\frac{2}{\\pi}-\\frac{4}{\\pi}\\cdot\\left(\\frac{\\cos(2\\cdot x)}{1\\cdot3}+\\frac{\\cos(4\\cdot x)}{3\\cdot5}+\\frac{\\cos(6\\cdot x)}{5\\cdot7}+\\cdots\\right)$", | |
| "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": "Expand function $y=\\sin\\left(\\frac{ \\pi \\cdot x }{ 4 }\\right)$ in Taylor series at $x=2$, if $\\cos(x)=\\sum_{n=0}^\\infty\\left((-1)^n \\cdot \\frac{ x^{2 \\cdot n} }{ (2 \\cdot n)! }\\right)$.", | |
| "response": "This is the final answer to the problem: $\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{\\left(\\frac{\\pi}{4}\\cdot(x-2)\\right)^{2\\cdot n}}{(2\\cdot n)!}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function $y=\\frac{ 3 \\cdot \\csc(x)-4 \\cdot \\sin(x) }{ 8 \\cdot \\left(\\cos(x)\\right)^5 }-\\frac{ 76 }{ 5 } \\cdot \\cot(3 \\cdot x)$.", | |
| "response": "$y'$=$\\frac{228}{5\\cdot\\left(\\sin(3\\cdot x)\\right)^2}+\\frac{16\\cdot\\left(\\cos(x)\\right)^6-5\\cdot\\left(\\cos(x)\\right)^4-3\\cdot\\left(\\cos(x)\\right)^6\\cdot\\left(\\csc(x)\\right)^2}{8\\cdot\\left(\\cos(x)\\right)^{10}}$", | |
| "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": "Sketch the curve: \n\n$y=5 \\cdot x^2-2 \\cdot x^4-3$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)\n3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)\n4. Slant asymptotes (Leave blank if there are no slant asymptotes)\n5. Intervals where the function is increasing (Leave blank if there are no such intervals)\n6. Intervals where the function is decreasing (Leave blank if there are no such intervals)\n7. Intervals where the function is concave up (Leave blank if there are no such intervals)\n8. Intervals where the function is concave down (Leave blank if there are no such intervals)\n9. Points of inflection (Leave blank if there are no 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: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(0,\\frac{\\sqrt{5}}{2}\\right), \\left(-\\infty,-\\frac{\\sqrt{5}}{2}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{5}}{2},0\\right), \\left(\\frac{\\sqrt{5}}{2},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right), \\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right), P\\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the gradient: $f(x,y)=\\frac{ \\sqrt{x}+y^2 }{ x \\cdot y }$.", | |
| "response": "$\\nabla f(x,y)$ =$\\left\\langle\\frac{1}{2\\cdot x\\cdot y\\cdot\\sqrt{x}}-\\frac{\\sqrt{x}+y^2}{y\\cdot x^2},\\frac{2}{x}-\\frac{\\sqrt{x}+y^2}{x\\cdot y^2}\\right\\rangle$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 constraints $x \\cdot y \\cdot z=4$.", | |
| "response": "Minimum: $6\\cdot\\sqrt[3]{2}$\n\nMaximum: None", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\varphi(x)=2 \\cdot x$ in the interval $(0,4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $4\\cdot\\pi-8\\cdot\\sum_{n=1}^\\infty\\left(\\frac{\\sin\\left(\\frac{n\\cdot x}{2}\\right)}{n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\varphi(x)=2 \\cdot x$ in the interval $(0,4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $4\\cdot\\pi-8\\cdot\\sum_{n=1}^\\infty\\left(\\frac{\\sin\\left(\\frac{n\\cdot x}{2}\\right)}{n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ 1 }{ 2 \\cdot \\sin\\left(\\frac{ x }{ 2 }\\right)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ 2 \\cdot \\sin\\left(\\frac{ x }{ 2 }\\right)^6 } d x}$ =$C-\\frac{1}{5}\\cdot\\left(\\cot\\left(\\frac{x}{2}\\right)\\right)^5-\\frac{2}{3}\\cdot\\left(\\cot\\left(\\frac{x}{2}\\right)\\right)^3-\\cot\\left(\\frac{x}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 $g'(c)=\\frac{ 1 }{ f'\\left(g(c)\\right) }$.", | |
| "response": "$g'(2.5)$ =$\\frac{2}{9}$", | |
| "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": "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.", | |
| "response": "$I_{x}$ = $\\frac{k}{10}\\cdot a^5$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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. \n\n1. Find the force of gravity $\\vec{F}$ acting on a disco ball and find its magnitude.\n2. Find the force of tension $\\vec{T}$ in the chain and its magnitude.\n\nExpress answers using standard unit vectors.", | |
| "response": "1. $\\vec{F}$=$-19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{F}\\right\\rVert$=$19.6$\n2. $\\vec{T}$=$19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{T}\\right\\rVert$=$19.6$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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. \n\n1. Find the force of gravity $\\vec{F}$ acting on a disco ball and find its magnitude.\n2. Find the force of tension $\\vec{T}$ in the chain and its magnitude.\n\nExpress answers using standard unit vectors.", | |
| "response": "1. $\\vec{F}$=$-19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{F}\\right\\rVert$=$19.6$\n2. $\\vec{T}$=$19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{T}\\right\\rVert$=$19.6$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function $y=\\frac{ 2 \\cdot \\csc(x)-7 \\cdot \\sin(x) }{ 4 \\cdot \\left(\\cos(x)\\right)^5 }-\\frac{ 3 }{ 5 } \\cdot \\cot(2 \\cdot x)$.", | |
| "response": "$y'$=$\\frac{6}{5\\cdot\\left(\\sin(2\\cdot x)\\right)^2}+\\frac{28\\cdot\\left(\\cos(x)\\right)^6-25\\cdot\\left(\\cos(x)\\right)^4-2\\cdot\\left(\\cos(x)\\right)^6\\cdot\\left(\\csc(x)\\right)^2}{4\\cdot\\left(\\cos(x)\\right)^{10}}$", | |
| "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": "Find the sum of the $\\sum_{n=0}^\\infty\\left(\\frac{ (-1)^n }{ (2 \\cdot n+1)! }\\right)$ with estimate error $0.01$.", | |
| "response": "This is the final answer to the problem: $\\frac{101}{120}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Consider points $P$$P(3,7,-2)$ and $Q$$P(1,1,-3)$. Determine the angle between vectors $\\vec{OP}$ and $\\vec{OQ}$. Express the answer in radians rounded to two decimal places.", | |
| "response": "$\\theta$ =$0.91$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the second derivative $\\frac{d ^2y}{ d x^2}$ for the parametrically defined function $x=2 \\cdot \\cos(3 \\cdot t)$, $y=\\sin(2 \\cdot t)$.", | |
| "response": "$\\frac{d ^2y}{ d x^2}$=$-\\frac{24\\cdot\\sin(2\\cdot t)\\cdot\\sin(3\\cdot t)+36\\cdot\\cos(2\\cdot t)\\cdot\\cos(3\\cdot t)}{216\\cdot\\left(\\sin(3\\cdot t)\\right)^3}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function $y=\\arcsin\\left(\\sqrt{1-x^2}\\right)$.", | |
| "response": "$y'$= $-\\frac{x}{|x|\\cdot\\sqrt{1-x^2}}$", | |
| "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": "Compute the integral: $\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$.", | |
| "response": "$\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$ =$\\frac{1}{25}\\cdot\\sqrt{6+10\\cdot x+25\\cdot x^2}+\\frac{9}{25}\\cdot\\ln\\left(1+5\\cdot x+\\sqrt{1+(5\\cdot x+1)^2}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$.", | |
| "response": "$\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$ =$\\frac{1}{25}\\cdot\\sqrt{6+10\\cdot x+25\\cdot x^2}+\\frac{9}{25}\\cdot\\ln\\left(1+5\\cdot x+\\sqrt{1+(5\\cdot x+1)^2}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$.", | |
| "response": "$\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$ =$\\frac{1}{25}\\cdot\\sqrt{6+10\\cdot x+25\\cdot x^2}+\\frac{9}{25}\\cdot\\ln\\left(1+5\\cdot x+\\sqrt{1+(5\\cdot x+1)^2}\\right)+C$", | |
| "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": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x+2}$ with the center $a=1$.", | |
| "response": "$\\sqrt{x+2}$ =$\\sum_{n=0}^\\infty\\left(3^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x+2}$ with the center $a=1$.", | |
| "response": "$\\sqrt{x+2}$ =$\\sum_{n=0}^\\infty\\left(3^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^n\\right)$", | |
| "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": "Compute the integral $\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$ =$C+\\sqrt{4+x^2}+\\ln\\left(\\left|\\frac{\\sqrt{4+x^2}-2}{2+\\sqrt{4+x^2}}\\right|\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$ =$C+\\sqrt{4+x^2}+\\ln\\left(\\left|\\frac{\\sqrt{4+x^2}-2}{2+\\sqrt{4+x^2}}\\right|\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ with the horizontal. What is the maximum range? Round your answer to one decimal digit.", | |
| "response": "Answer: $22092.5$ m", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The electrical resistance $R$ produced by wiring resistors $R_{1}$ and $R_{2}$ in parallel can be calculated from the formula $\\frac{ 1 }{ R }=\\frac{ 1 }{ R_{1} }+\\frac{ 1 }{ R_{2} }$. If $R_{1}$ and $R_{2}$ are measured to be $7$ ohm and $6$ ohm respectively, and if these measurements are accurate to within $0.05$ ohm, estimate the maximum possible error in computing $R$.", | |
| "response": "Maximum possible error:$0.02514793$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute $\\sqrt[3]{130}$ with accuracy $0.0001$.", | |
| "response": "This is the final answer to the problem: $5.0658$", | |
| "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": "Find the derivative of the function: $f\\left(\\alpha\\right)=\\ln\\left(\\sqrt{\\frac{ 3-\\sin\\left(\\alpha\\right) }{ 4+2 \\cdot \\sin\\left(\\alpha\\right) }}\\right)$ at $\\alpha=\\frac{ \\pi }{ 4 }$.", | |
| "response": "$f'\\left(\\frac{ \\pi }{ 4 }\\right)$ =$-0.2848$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Write the Taylor series for the function $f(x)=-2 \\cdot x \\cdot \\sin(x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).", | |
| "response": "This is the final answer to the problem: $2\\cdot\\pi\\cdot(x-\\pi)+2\\cdot(x-\\pi)^2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the moment of inertia of one arch of the cycloid $x=3 \\cdot a \\cdot \\left(\\frac{ t }{ 2 }-\\sin\\left(\\frac{ t }{ 2 }\\right)\\right)$, $y=3 \\cdot a \\cdot \\left(1-\\cos\\left(\\frac{ t }{ 2 }\\right)\\right)$ relative to the x-axis.", | |
| "response": "Moment of Inertia: $\\frac{1152}{5}\\cdot a^3$", | |
| "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": "Find the tangential and normal components of acceleration if $\\vec{r}(t)=\\left\\langle 6 \\cdot t,3 \\cdot t^2,2 \\cdot t^3 \\right\\rangle$", | |
| "response": "$a_{T}$ =$\\frac{12\\cdot t^3+6\\cdot t}{\\sqrt{t^4+t^2+1}}$ ; $a_{N}$ = $\\frac{6\\cdot\\sqrt{t^4+4\\cdot t^2+1}}{\\sqrt{t^4+t^2+1}}$", | |
| "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": "Compute the integral $\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$.", | |
| "response": "$\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$ =$C-\\frac{1}{4}\\cdot\\left(\\frac{3}{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}+3\\cdot\\ln\\left(\\frac{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}-1}{1+e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $y=x \\cdot \\sin(x)+2 \\cdot x \\cdot \\cos(x)-2 \\cdot \\sin(x)+\\ln\\left(\\sin(x)\\right)+c^2$.", | |
| "response": "This is the final answer to the problem: $y'=\\frac{\\left(\\sin(x)\\right)^2+x\\cdot\\sin(x)\\cdot\\cos(x)-2\\cdot x\\cdot\\left(\\sin(x)\\right)^2+\\cos(x)}{\\sin(x)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function: $y=-3 \\cdot x^{\\sqrt[3]{2 \\cdot x}}$.", | |
| "response": "$\\frac{ d y }{d x}$ =$-\\left(\\frac{3\\cdot\\sqrt[3]{2}}{x^{\\frac{2}{3}}}+\\frac{\\sqrt[3]{2}\\cdot\\ln(x)}{x^{\\frac{2}{3}}}\\right)\\cdot x^{\\sqrt[3]{2}\\cdot\\sqrt[3]{x}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\left(\\cos(2 \\cdot x)\\right)^3 } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\left(\\cos(2 \\cdot x)\\right)^3 } d x}$ =$C+\\frac{\\sin(2\\cdot x)}{4\\cdot\\left(\\cos(2\\cdot x)\\right)^2}+\\frac{1}{4}\\cdot\\ln\\left(\\left|\\tan\\left(\\frac{1}{2}\\cdot\\left(2\\cdot x+\\frac{\\pi}{2}\\right)\\right)\\right|\\right)$", | |
| "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": "Find the Fourier series of the function $\\psi(x)=2 \\cdot e^{-2 \\cdot x}$ in the interval $(-\\pi,\\pi)$.", | |
| "response": "The Fourier series is: $2\\cdot e^{-2\\cdot x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{2}+\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n\\cdot\\left(2\\cdot\\cos(n\\cdot x)+n\\cdot\\sin(n\\cdot x)\\right)}{4+n^2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x^2+2}$ with the center $a=0$.", | |
| "response": "$\\sqrt{x^2+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^{2\\cdot n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$ =$C+2\\cdot\\cot(6\\cdot x)+\\frac{2}{5}\\cdot\\left(\\cot(6\\cdot x)\\right)^5+\\frac{4}{3}\\cdot\\left(\\cot(6\\cdot x)\\right)^3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ deg with the horizontal. At what time is the maximum range of the projectile attained? ", | |
| "response": "$t$ = $88.37$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For the function $\\varphi(x)=(10-x) \\cdot \\sqrt{x^2+8}$ specify the points where local maxima and minima of $\\varphi(x)$ 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: \n\n1. The point(s) where local maxima occur $P\\left(4,12\\cdot\\sqrt{6}\\right)$\n2. The point(s) where local minima occur $P(1,27)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of the function $x^{\\frac{ 1 }{ 3 }}$ with the center $a=27$.", | |
| "response": "$x^{\\frac{ 1 }{ 3 }}$ =$\\sum_{n=0}^\\infty\\left(3^{1-3\\cdot n}\\cdot C_n^{\\frac{1}{3}}\\cdot(x-27)^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$.", | |
| "response": "$\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$ =$C+\\frac{1}{21}\\cdot\\left(\\frac{1}{2}\\cdot\\left(\\cot(10\\cdot x)\\right)^4+\\frac{1}{3}\\cdot\\left(\\cot(10\\cdot x)\\right)^6\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$.", | |
| "response": "$\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$ =$C+\\frac{1}{21}\\cdot\\left(\\frac{1}{2}\\cdot\\left(\\cot(10\\cdot x)\\right)^4+\\frac{1}{3}\\cdot\\left(\\cot(10\\cdot x)\\right)^6\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the equations of the common tangent lines to the following ellipses:\n\n$\\frac{ x^2 }{ 6 }+y^2=1$ \n\n$\\frac{ x^2 }{ 4 }+\\frac{ y^2 }{ 9 }=1$", | |
| "response": "This is the final answer to the problem: $2\\cdot x+y-5=0 \\lor 2\\cdot x+y+5=0 \\lor 2\\cdot x-y-5=0 \\lor 2\\cdot x-y+5=0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A box is to be made with the following properties:\n\nThe length of the base, $l$, is twice the length of a width $w$.\n\nThe cost of material to be used for the lateral faces and the top of the box is three times as the cost of the material to be used for the lower base.\n\nFind the dimensions of the box in terms of its fixed volume $V$ such that the cost of the used material is the minimum.", | |
| "response": "This is the final answer to the problem: $y=\\frac{2}{3}\\cdot\\sqrt[3]{\\frac{4\\cdot V}{3}}, w=\\frac{1}{2}\\cdot\\sqrt[3]{\\frac{9\\cdot V}{2}}, l=\\sqrt[3]{\\frac{9\\cdot V}{2}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$ =$C+\\frac{1}{4}\\arccos\\left(\\frac{x}{5}\\right)-\\frac{1}{4}\\tan\\left(\\arccos\\left(\\frac{x}{5}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$ =$C+\\frac{1}{4}\\arccos\\left(\\frac{x}{5}\\right)-\\frac{1}{4}\\tan\\left(\\arccos\\left(\\frac{x}{5}\\right)\\right)$", | |
| "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 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 integral $\\int{\\frac{ 1 }{ \\sqrt[3]{\\left(\\sin(x)\\right)^{11} \\cdot \\cos(x)} } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\sqrt[3]{\\left(\\sin(x)\\right)^{11} \\cdot \\cos(x)} } d x}$ =$-\\frac{3\\cdot\\left(1+4\\cdot\\left(\\tan(x)\\right)^2\\right)}{8\\cdot\\left(\\tan(x)\\right)^2\\cdot\\sqrt[3]{\\left(\\tan(x)\\right)^2}}+C$", | |
| "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": "Sketch the curve: \n\n$y=25 \\cdot x^2 \\cdot e^{\\frac{ 1 }{ 5 \\cdot x }}$ \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": "1. The domain (in interval notation): $(-\\infty,0)\\cup(0,\\infty)$\n2. Vertical asymptotes: $x=0$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(\\frac{1}{10},\\infty\\right)$\n6. Intervals where the function is decreasing $\\left(0,\\frac{1}{10}\\right), (-\\infty,0)$\n7. Intervals where the function is concave up: $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down: None\n9. Points of inflection: None", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\arctan(x \\cdot y \\cdot z)$.", | |
| "response": "$f_{xx}(x,y,z)$=$\\frac{-2\\cdot x\\cdot y^3\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$\\frac{z-x^2\\cdot y^2\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yy}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$\\frac{x-x^3\\cdot y^2\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{zz}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y^3\\cdot z}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$\\frac{y-x^2\\cdot y^3\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For the function: $f(x)=x^3+x^4$ determine\n\n1. intervals where $f$ is increasing or decreasing,\n2. local minima and maxima of $f$ ,\n3. intervals where $f$ is concave up and concave down, and\n4. the inflection points of $f$ .", | |
| "response": "1. Increasing over $\\left( -\\frac{3}{4}, 0 \\right) \\cup (0, \\infty)$ ; decreasing over $\\left(-\\infty,-\\frac{3}{4}\\right)$\n2. Local maxima at None ; local minima at $x=-\\frac{3}{4}$\n3. Concave up for $x>0, x<-\\frac{1}{2}$ ; concave down for $-\\frac{1}{2}<x<0$\n4. Inflection points at $P\\left(-\\frac{1}{2},-\\frac{1}{16}\\right); P(0,0)$", | |
| "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": "Find the derivative of $y=\\sin(2 \\cdot x) \\cdot \\cos(3 \\cdot x)-\\frac{ \\ln(x-1) }{ \\ln(x+1) }+c$", | |
| "response": "This is the final answer to the problem: $y'=2\\cdot\\cos(5\\cdot x)-\\sin(3\\cdot x)\\cdot\\sin(2\\cdot x)-\\frac{(x+1)\\cdot\\ln(x+1)-(x-1)\\cdot\\ln(x-1)}{(x-1)\\cdot(x+1)\\cdot\\left(\\ln(x+1)\\right)^2}$", | |
| "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": "Consider points $A$$P(3,-1,2)$, $B$$P(2,1,5)$, and $C$$P(1,-2,-2)$.\n\n1. Find the area of parallelogram ABCD with adjacent sides $\\vec{AB}$ and $\\vec{AC}$.\n2. Find the area of triangle ABC.\n3. Find the distance from point $A$ to line BC.", | |
| "response": "1. $A$=$5\\cdot\\sqrt{6}$\n2. $A$=$\\frac{5\\cdot\\sqrt{6}}{2}$\n3. $d$=$\\frac{5\\cdot\\sqrt{6}}{\\sqrt{59}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the generalized center of mass between $y=b \\cdot \\sin(a \\cdot x)$, $x=0$, and $x=\\frac{ \\pi }{ a }$ . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.", | |
| "response": "$(x,y)$ = $P\\left(\\frac{\\pi}{2\\cdot a},\\frac{\\pi\\cdot b}{8}\\right)$ \n\n$V$ = $\\frac{2\\cdot\\pi^2\\cdot b}{a^2}$", | |
| "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": "" | |
| } | |
| ] | |
| } |