| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 2, | |
| "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": "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 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": "Determine the intervals of upward concavity and downward concavity for the function $f(x)=e^x \\cdot \\left(x^2-1\\right)$.", | |
| "response": "1. Interval(s) of upward concavity: $\\left(\\frac{2\\cdot\\sqrt{3}-4}{2},\\infty\\right), \\left(-\\infty,-\\frac{4+2\\cdot\\sqrt{3}}{2}\\right)$\n2. Interval(s) of downward concavity: $\\left(-\\frac{4+2\\cdot\\sqrt{3}}{2},\\frac{2\\cdot\\sqrt{3}-4}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Region $R$ is the region in the first quadrant bounded by the graphs of $y=2 \\cdot x$ and $y=x^2$.\n\n1. Write, but do not evaluate, an integral expression that gives the volume of the solid generated from revolving the region $R$ about the vertical line $x=3$.\n2. Write, but do not evaluate, an integral expression that gives the volume of the solid generated from revolving the region $R$ about the vertical line $x=-2$.", | |
| "response": "1. $V$ = $\\int_0^4\\pi\\cdot\\left(\\left(3-\\frac{1}{2}\\cdot y\\right)^2-\\left(3-\\sqrt{y}\\right)^2\\right)dy$\n2. $V$ = $\\int_0^4\\pi\\cdot\\left(\\left(-2-\\sqrt{y}\\right)^2-\\left(-2-\\frac{1}{2}\\cdot y\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the following equations: 1. $8.6=6 j+4 j$\n2. $12 z-(4 z+6)=82$\n3. $5.4 d-2.3 d+3 (d-4)=16.67$\n4. $2.6 f-1.3 (3 f-4)=6.5$\n5. $-5.3 m+(-3.9 m)-17=-94.28$\n6. $6 (3.5 y+4.2)-2.75 y=134.7$", | |
| "response": "The solutions to the given equations are: 1. $j=0.86$\n2. $z=11$\n3. $d=\\frac{ 47 }{ 10 }$\n4. $f=-1$\n5. $m=\\frac{ 42 }{ 5 }$\n6. $y=6$", | |
| "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": "Solve the integral: $\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$.", | |
| "response": "$\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$ =$C-\\left(\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^6+\\frac{1}{4}\\cdot\\left(\\cot(x)\\right)^4+\\frac{1}{8}\\cdot\\left(\\cot(x)\\right)^8\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$.", | |
| "response": "$\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$ =$C-\\left(\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^6+\\frac{1}{4}\\cdot\\left(\\cot(x)\\right)^4+\\frac{1}{8}\\cdot\\left(\\cot(x)\\right)^8\\right)$", | |
| "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": "Calculate integral: $I=\\int{4 \\cdot \\cos\\left(3 \\cdot \\ln(2 \\cdot x)\\right) d x}$.", | |
| "response": "This is the final answer to the problem: $\\frac{1}{10}\\cdot\\left(C+4\\cdot x\\cdot\\cos\\left(3\\cdot\\ln(2\\cdot x)\\right)+12\\cdot x\\cdot\\sin\\left(3\\cdot\\ln(2\\cdot x)\\right)\\right)$", | |
| "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": "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": "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": "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": "$f(x)=\\frac{ 1 }{ 4 } \\cdot \\sqrt{x}+\\frac{ 1 }{ x }$, $x>0$. Determine:1. 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 : $(4,\\infty)$\n2. intervals where $f$ is decreasing: $(0,4)$\n3. local minima of $f$: $4$\n4. local maxima of $f$: None\n5. intervals where $f$ is concave up : $\\left(0,8\\cdot\\sqrt[3]{2}\\right)$\n6. intervals where $f$ is concave down: $\\left(8\\cdot\\sqrt[3]{2},\\infty\\right)$\n7. the inflection points of $f$: $8\\cdot\\sqrt[3]{2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the extrema of a function $y=\\frac{ 2 \\cdot x^4 }{ 4 }-\\frac{ x^3 }{ 3 }-\\frac{ 3 \\cdot x^2 }{ 2 }+2$. Then determine the largest and smallest value of the function when $-2 \\le x \\le 4$.", | |
| "response": "This is the final answer to the problem: \n\n1. Extrema points: $P\\left(\\frac{3}{2},\\frac{1}{32}\\right), P\\left(-1,\\frac{4}{3}\\right), P(0,2)$\n2. The largest value: $\\frac{254}{3}$\n3. The smallest value: $\\frac{1}{32}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-x-6 \\cdot y+z=3 \\cdot \\cos(-x-6 \\cdot y+z)$.\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. $1$;\n\nb. $6$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\frac{ \\sqrt{x+10}+3 }{ (x+10)^2-\\sqrt{x+10} } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{x+10}+3 }{ (x+10)^2-\\sqrt{x+10} } d x}$ =$C+\\frac{8}{3}\\cdot\\ln\\left(\\left|\\sqrt{x+10}-1\\right|\\right)-\\frac{4}{3}\\cdot\\ln\\left(11+\\sqrt{x+10}+x\\right)-\\frac{4}{\\sqrt{3}}\\cdot\\arctan\\left(\\frac{1}{\\sqrt{3}}\\cdot\\left(1+2\\cdot\\sqrt{x+10}\\right)\\right)$", | |
| "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 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": "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": "Compute the integral: $\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$.", | |
| "response": "$\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$ =$-\\frac{2\\cdot\\sin\\left(\\frac{x}{2}\\right)^4\\cdot\\cos\\left(\\frac{x}{2}\\right)}{5}+\\frac{4}{5}\\cdot\\left(-\\frac{2}{3}\\cdot\\sin\\left(\\frac{x}{2}\\right)^2\\cdot\\cos\\left(\\frac{x}{2}\\right)-\\frac{4}{3}\\cdot\\cos\\left(\\frac{x}{2}\\right)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$.", | |
| "response": "$\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$ =$-\\frac{2\\cdot\\sin\\left(\\frac{x}{2}\\right)^4\\cdot\\cos\\left(\\frac{x}{2}\\right)}{5}+\\frac{4}{5}\\cdot\\left(-\\frac{2}{3}\\cdot\\sin\\left(\\frac{x}{2}\\right)^2\\cdot\\cos\\left(\\frac{x}{2}\\right)-\\frac{4}{3}\\cdot\\cos\\left(\\frac{x}{2}\\right)\\right)+C$", | |
| "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": "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 $y'$ and $y''$ for $x^2+6 \\cdot x \\cdot y-2 \\cdot y^2=3$.", | |
| "response": "$y'$= $\\frac{x+3\\cdot y}{2\\cdot y-3\\cdot x}$; \n\n$y''$= $\\frac{11\\cdot\\left(x^2+6\\cdot x\\cdot y-2\\cdot y^2\\right)}{(3\\cdot x-2\\cdot y)^3}$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve $\\left(\\sin(x)\\right)^2+\\left(\\cos(3 \\cdot x)\\right)^2=1$.", | |
| "response": "This is the final answer to the problem: $x=\\frac{n\\cdot\\pi}{4}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve $\\left(\\sin(x)\\right)^2+\\left(\\cos(3 \\cdot x)\\right)^2=1$.", | |
| "response": "This is the final answer to the problem: $x=\\frac{n\\cdot\\pi}{4}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local extrema of the function $f(x)=2 \\cdot \\left(x^2\\right)^{\\frac{ 1 }{ 3 }}-x^2$ using the First Derivative Test.", | |
| "response": "Local maxima:$x=\\sqrt[4]{\\frac{8}{27}}, x=-\\sqrt[4]{\\frac{8}{27}}$\n\nLocal minima:$x=0$", | |
| "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": "Compute the integral $\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$ =$C+\\frac{\\sin(5\\cdot x)}{10\\cdot\\left(\\cos(5\\cdot x)\\right)^2}+\\frac{1}{10}\\cdot\\ln\\left(\\left|\\tan\\left(\\left(\\frac{5}{2}\\right)\\cdot x+\\frac{\\pi}{4}\\right)\\right|\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$ =$C+\\frac{\\sin(5\\cdot x)}{10\\cdot\\left(\\cos(5\\cdot x)\\right)^2}+\\frac{1}{10}\\cdot\\ln\\left(\\left|\\tan\\left(\\left(\\frac{5}{2}\\right)\\cdot x+\\frac{\\pi}{4}\\right)\\right|\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "What are the points of inflection of the graph of $f(x)=\\frac{ x+1 }{ x^2+1 }$?", | |
| "response": "This is the final answer to the problem: $x_1=1 \\land x_2=\\sqrt{3}-2 \\land x_3=-2-\\sqrt{3}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the second derivative $\\frac{d ^2y}{ d x^2}$ of the function $x=5 \\cdot \\left(\\cos(t)\\right)^3$, $y=6 \\cdot \\left(\\sin(3 \\cdot t)\\right)^3$.", | |
| "response": "$\\frac{d ^2y}{ d x^2}$=$\\frac{-15\\cdot\\left(\\cos(t)\\right)^2\\cdot\\sin(t)\\cdot\\left(324\\cdot\\sin(3\\cdot t)-486\\cdot\\left(\\sin(3\\cdot t)\\right)^3\\right)-\\left(30\\cdot\\cos(t)-45\\cdot\\left(\\cos(t)\\right)^3\\right)\\cdot54\\cdot\\left(\\sin(3\\cdot t)\\right)^2\\cdot\\cos(3\\cdot t)}{\\left(-15\\cdot\\left(\\cos(t)\\right)^2\\cdot\\sin(t)\\right)^3}$", | |
| "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": "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": "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": "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": "Given functions $p(x)=\\frac{ 1 }{ \\sqrt{x} }$ and $m(x)=x^2-4$,\nstate the domain of each of the following functions\nusing interval notation:\n\n1. $\\frac{ p(x) }{ m(x) }$\n2. $p\\left(m(x)\\right)$\n3. $m\\left(p(x)\\right)$", | |
| "response": "1. Domain of $\\frac{ p(x) }{ m(x) }$: $(0,2)\\cup(2,\\infty)$\n2. Domain of $p\\left(m(x)\\right)$: $(-\\infty,-2)\\cup(2,\\infty)$\n3. Domain of $m\\left(p(x)\\right)$: $(0,\\infty)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given functions $p(x)=\\frac{ 1 }{ \\sqrt{x} }$ and $m(x)=x^2-4$,\nstate the domain of each of the following functions\nusing interval notation:\n\n1. $\\frac{ p(x) }{ m(x) }$\n2. $p\\left(m(x)\\right)$\n3. $m\\left(p(x)\\right)$", | |
| "response": "1. Domain of $\\frac{ p(x) }{ m(x) }$: $(0,2)\\cup(2,\\infty)$\n2. Domain of $p\\left(m(x)\\right)$: $(-\\infty,-2)\\cup(2,\\infty)$\n3. Domain of $m\\left(p(x)\\right)$: $(0,\\infty)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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).", | |
| "response": "This is the final answer to the problem: $-\\frac{\\pi}{2}-\\left(x-\\frac{\\pi}{2}\\right)+\\pi\\cdot\\left(x-\\frac{\\pi}{2}\\right)^2$", | |
| "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": "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:\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:\n\n1. The domain (in interval notation) $\\left(-\\infty,-\\frac{2}{3}\\right)\\cup\\left(-\\frac{2}{3},\\frac{2}{3}\\right)\\cup\\left(\\frac{2}{3},\\infty\\right)$\n2. Vertical asymptotes $x=\\frac{2}{3}, x=-\\frac{2}{3}$\n3. Horizontal asymptotes $y=0$\n4. Slant asymptotes None\n5. Intervals where the function is increasing $\\left(\\frac{2}{3},\\infty\\right), \\left(-\\infty,-\\frac{2}{3}\\right)$\n6. Intervals where the function is decreasing $\\left(-\\frac{2}{3},\\frac{2}{3}\\right)$\n7. Intervals where the function is concave up $\\left(-\\frac{2}{3},0\\right), \\left(-\\infty,-\\frac{2}{3}\\right)$\n8. Intervals where the function is concave down $\\left(0,\\frac{2}{3}\\right) \\cup \\left(\\frac{2}{3}, \\infty\\right)$\n9. Points of inflection $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral: $\\int_{0}^{\\frac{ 1 }{ 2 }}{\\sqrt[5]{1+x^3} d x}$ with explicitly guaranteed accuracy of $\\frac{ 1 }{ 100 }$ using power series expansion.", | |
| "response": "This is the final answer to the problem: $0.503$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate $I=\\int{\\frac{ 1 }{ x^3+8 } d x}$.", | |
| "response": "This is the final answer to the problem: $I=\\frac{\\sqrt{3}}{12}\\cdot\\arctan\\left(\\frac{x-1}{\\sqrt{3}}\\right)+\\frac{1}{24}\\cdot\\ln\\left(\\frac{(x+2)^2}{x^2-2\\cdot x+4}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate $I=\\int{\\frac{ 1 }{ x^3+8 } d x}$.", | |
| "response": "This is the final answer to the problem: $I=\\frac{\\sqrt{3}}{12}\\cdot\\arctan\\left(\\frac{x-1}{\\sqrt{3}}\\right)+\\frac{1}{24}\\cdot\\ln\\left(\\frac{(x+2)^2}{x^2-2\\cdot x+4}\\right)+C$", | |
| "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": "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": "For the function $f(x)=x^{11}-6 \\cdot x^{10}$, determine:\n\n1. Intervals where:\n1. $f$ is increasing\n2. $f$ is decreasing\n3. $f$ is concave up\n4. $f$ is concave down\n\n3. find:\n1. local minima\n2. local maxima\n3. the inflection points of $f$", | |
| "response": "This is the final answer to the problem:1. Intervals where:\n1. $f$ is increasing: $\\left(\\frac{60}{11},\\infty\\right), (-\\infty,0)$\n2. $f$ is decreasing: $\\left(0,\\frac{60}{11}\\right)$\n3. $f$ is concave up: $\\left(\\frac{54}{11},\\infty\\right)$\n4. $f$ is concave down: $\\left(0,\\frac{54}{11}\\right), (-\\infty,0)$\n\n3. find:\n1. local minima: $\\frac{60}{11}$\n2. local maxima: $0$\n3. the inflection points of $f$: $P\\left(\\frac{54}{11},-\\frac{2529990231179046912}{285311670611}\\right)$", | |
| "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 Fourier series expansion of the function $f(x)=\\frac{ x }{ 2 }$ with the period $4$ at interval $[-2,2]$.", | |
| "response": "The Fourier series is: $f(x)=\\sum_{n=1}^\\infty\\left(\\frac{(-1)^{n+1}\\cdot2}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 6 \\cdot x^3-7 \\cdot x^2+3 \\cdot x-1 }{ 2 \\cdot x-3 \\cdot x^2 } d x}$.", | |
| "response": "Answer is:$-x^2+x-\\frac{1}{3}\\cdot\\ln\\left(\\left|x-\\frac{2}{3}\\right|\\right)+\\frac{1}{2}\\cdot\\ln\\left(\\left|1-\\frac{2}{3\\cdot x}\\right|\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate $\\sqrt[3]{30}$ with estimate error $0.001$, using series expansion.", | |
| "response": "This is the final answer to the problem: $\\frac{755}{243}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has $$84,000$$ subscribers at a quarterly charge of $$30\\space \\text{USD}$$. Market research has suggested that if the owners raise the price to $$34\\space \\text{USD}$$, they would lose $$9,000$$ subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?", | |
| "response": "This is the final answer to the problem: $\\frac{101}{3}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has $$84,000$$ subscribers at a quarterly charge of $$30\\space \\text{USD}$$. Market research has suggested that if the owners raise the price to $$34\\space \\text{USD}$$, they would lose $$9,000$$ subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?", | |
| "response": "This is the final answer to the problem: $\\frac{101}{3}$", | |
| "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 certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t)=6-\\frac{ 5 }{ t^2+1 }$, where $t$ is time measured in hours since a circle of a $1$-cm radius of the bacterium was put into the culture.\n\n1. Express the area of the bacteria, $A(t)$, as a function of time.\n2. Find the exact and approximate area of the bacterial culture in $3$ hours.\n3. Express the circumference of the bacteria, $C(t)$, as a function of time.\n4. Find the exact and approximate circumference of the bacteria in $3$ hours.", | |
| "response": "This is the final answer to the problem:\n\n1. $A(t)$ = $\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)^2$square centimeters\n2. The exact area of the bacterial culture in $3$ hours = $\\frac{121}{4}\\cdot\\pi$ square centimeters. The approximate area of the bacterial culture in $3$ hours = $95.03317777$ square centimeters.\n3. $C(t)$ = $2\\cdot\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)$centimeters\n4. The exact circumference of the bacteria in $3$ hours = $11\\cdot\\pi$ centimeters. The approximate circumference of the bacteria in $3$ hours = $34.55751919$ centimeters.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=2 \\cdot x \\cdot \\sqrt{3-x^2}$. \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): $\\left[-1\\cdot3^{2^{-1}},3^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(-\\sqrt{\\frac{3}{2}},\\sqrt{\\frac{3}{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(\\sqrt{\\frac{3}{2}},3^{2^{-1}}\\right), \\left(-3^{2^{-1}},-\\sqrt{\\frac{3}{2}}\\right)$\n7. Intervals where the function is concave up: $\\left(-3^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,3^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "An airplane’s Mach number $M$ is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\\mu=2 \\cdot \\arcsin\\left(\\frac{ 1 }{ M }\\right)$. Find the Mach angles for the following Mach numbers.\n\n1. $M=1.4$\n2. $M=2.8$\n3. $M=4.3$", | |
| "response": "This is the final answer to the problem: \n\n1. $1.59120591$\n2. $0.73041444$\n3. $0.46941423$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "An airplane’s Mach number $M$ is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\\mu=2 \\cdot \\arcsin\\left(\\frac{ 1 }{ M }\\right)$. Find the Mach angles for the following Mach numbers.\n\n1. $M=1.4$\n2. $M=2.8$\n3. $M=4.3$", | |
| "response": "This is the final answer to the problem: \n\n1. $1.59120591$\n2. $0.73041444$\n3. $0.46941423$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 4 }+\\frac{ x }{ 1 \\cdot 5 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{6 - 6 \\cdot e^x + 6 \\cdot e^x \\cdot x - 3 \\cdot e^x \\cdot x^2 + e^x \\cdot x^3}{x^4},&x\\ne0\\\\\\frac{1}{4},&x=0\\end{cases}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 4 }+\\frac{ x }{ 1 \\cdot 5 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{6 - 6 \\cdot e^x + 6 \\cdot e^x \\cdot x - 3 \\cdot e^x \\cdot x^2 + e^x \\cdot x^3}{x^4},&x\\ne0\\\\\\frac{1}{4},&x=0\\end{cases}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 4 }+\\frac{ x }{ 1 \\cdot 5 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{6 - 6 \\cdot e^x + 6 \\cdot e^x \\cdot x - 3 \\cdot e^x \\cdot x^2 + e^x \\cdot x^3}{x^4},&x\\ne0\\\\\\frac{1}{4},&x=0\\end{cases}$", | |
| "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": "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": "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": "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": "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": "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": "Let $R$ be the region bounded by the graphs of $y=\\frac{ 1 }{ x+2 }$ and $y=-\\frac{ 1 }{ 2 } \\cdot x+3$.\n\nFind the volume of the solid generated when $R$ is rotated about the vertical line $x=-3$.", | |
| "response": "The volume of the solid is $292.097$ units³.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Let $Q$ be the region bounded by the graph of $x=\\frac{ 2 }{ 1-y }$, the line $x=-1$, and the line $y=\\frac{ 5 }{ 4 }$.\n\nWrite, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $Q$ is revolved about the line $x=-1$.", | |
| "response": "$V$ = $\\int_{\\frac{5}{4}}^3\\left(\\pi\\cdot\\left(\\frac{2}{1-y}+1\\right)^2\\right)dy$", | |
| "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": "Consider the differential equation $\\frac{ d y }{d x}=e^y \\cdot (5 \\cdot x-1)$. Find $y=g(x)$, the particular solution to the differential equation for $-0.819 \\le x \\le 1.219$ that passes through the point $P(1,0)$.", | |
| "response": "$y$ = $-\\ln\\left(-\\frac{5}{2}\\cdot x^2+x+\\frac{5}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ deg with the horizontal. At what time is the maximum range of the projectile attained? ", | |
| "response": "$t$ = $88.37$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 5 \\cdot \\pi \\cdot x, & 0 \\le x \\le 4 \\\\ 0, & x>4 \\end{cases}$.", | |
| "response": "$q(t)$ = $\\int_0^\\infty\\frac{\\left(5\\cdot\\left(4\\cdot\\sin\\left(4\\cdot\\alpha\\right)\\cdot\\alpha+\\cos\\left(4\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot t\\right)+5\\cdot\\left(\\sin\\left(4\\cdot\\alpha\\right)-4\\cdot\\alpha\\cdot\\cos\\left(4\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot t\\right)\\right)}{\\alpha^2}d \\alpha$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 5 \\cdot \\pi \\cdot x, & 0 \\le x \\le 4 \\\\ 0, & x>4 \\end{cases}$.", | |
| "response": "$q(t)$ = $\\int_0^\\infty\\frac{\\left(5\\cdot\\left(4\\cdot\\sin\\left(4\\cdot\\alpha\\right)\\cdot\\alpha+\\cos\\left(4\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot t\\right)+5\\cdot\\left(\\sin\\left(4\\cdot\\alpha\\right)-4\\cdot\\alpha\\cdot\\cos\\left(4\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot t\\right)\\right)}{\\alpha^2}d \\alpha$", | |
| "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": "Find the centers of symmetry of the curve of $f(x)=\\left(\\sin(x)\\right)^2$.", | |
| "response": "This is the final answer to the problem: $\\left(k\\cdot\\pi+\\frac{\\pi}{4},\\frac{1}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ 3 }{ 4 } \\cdot x^4-10 \\cdot x^3+24 \\cdot x^2-4$.", | |
| "response": "The point(s) where the function has a local minimum:$P(8,-516), P(0,-4)$ \nThe point(s) where the function has a local maximum:$P(2,24)$", | |
| "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 $a=0$.", | |
| "response": "$\\sqrt{x+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative $y_{x}'$ of the function: $x=\\arcsin\\left(\\frac{ t }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $y=\\arccos\\left(\\frac{ 1 }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $t \\ge 0$.", | |
| "response": "$y_{x}'$ =$\\frac{t\\cdot\\sqrt{t^2+2}}{\\sqrt{2\\cdot t^2+1}}$", | |
| "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": "Find points on a coordinate plane that satisfy the following equation:\n\n$10 \\cdot x^2+29 \\cdot y^2+34 \\cdot x \\cdot y+8 \\cdot x+14 \\cdot y+2=0$", | |
| "response": "This is the final answer to the problem: $(3,-2)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find points on a coordinate plane that satisfy the following equation:\n\n$10 \\cdot x^2+29 \\cdot y^2+34 \\cdot x \\cdot y+8 \\cdot x+14 \\cdot y+2=0$", | |
| "response": "This is the final answer to the problem: $(3,-2)$", | |
| "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": "Determine the Taylor series for $f(x)=\\frac{ 2 \\cdot x-1 }{ x^2-3 \\cdot x+2 }$, centered at $x_{0}=4$. Write out the sum of the first four non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $\\frac{7}{6}+\\left(\\frac{1}{3^2}-\\frac{3}{2^2}\\right)\\cdot(x-4)-\\left(\\frac{1}{3^3}-\\frac{3}{2^3}\\right)\\cdot(x-4)^2+\\left(\\frac{1}{3^4}-\\frac{3}{2^4}\\right)\\cdot(x-4)^3+\\cdots$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find $\\frac{ d y }{d x}$ if $y=\\frac{ 5 \\cdot x^2-3 \\cdot x }{ \\left(3 \\cdot x^7+2 \\cdot x^6\\right)^4 }$.", | |
| "response": "$\\frac{ d y }{d x}$ = $\\frac{-390\\cdot x^2+23\\cdot x+138}{x^{24}\\cdot(3\\cdot x+2)^5}$", | |
| "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": "Expand the function: $y=\\ln\\left(x+\\sqrt{1+x^2}\\right)$ in a power series.", | |
| "response": "This is the final answer to the problem: $x-\\frac{1}{2}\\cdot\\frac{x^3}{3}+\\frac{1\\cdot3}{4\\cdot2}\\cdot\\frac{x^5}{5}-\\frac{1\\cdot3\\cdot5}{2\\cdot4\\cdot6}\\cdot\\frac{x^7}{7}+\\cdots+\\frac{(2\\cdot n-1)!!}{(2\\cdot n)!!}\\cdot\\frac{x^{2\\cdot n+1}}{2\\cdot n+1}+\\cdots$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$ =$\\frac{1}{2}\\cdot\\ln\\left(\\left|2\\cdot x+1+\\sqrt{4\\cdot x^2+4\\cdot x+3}\\right|\\right)+C$", | |
| "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": "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": "Differentiate the function \n $f(x)=\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }$.", | |
| "response": "$\\frac{ d }{d x}\\left(\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }\\right)$=$\\frac{18\\cdot x\\cdot\\sqrt[6]{x}+75\\cdot x^3\\cdot\\sqrt[6]{x}+\\sqrt{x}-6\\cdot\\sqrt[6]{x}}{6\\cdot x\\cdot x^{\\frac{2}{3}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Differentiate the function \n $f(x)=\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }$.", | |
| "response": "$\\frac{ d }{d x}\\left(\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }\\right)$=$\\frac{18\\cdot x\\cdot\\sqrt[6]{x}+75\\cdot x^3\\cdot\\sqrt[6]{x}+\\sqrt{x}-6\\cdot\\sqrt[6]{x}}{6\\cdot x\\cdot x^{\\frac{2}{3}}}$", | |
| "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": "Compute the integral $\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$.", | |
| "response": "$\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$ =$\\frac{1}{16}\\cdot\\left(2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2+2\\cdot\\ln\\left(4\\cdot x^2+1\\right)+8\\cdot x^2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2-8\\cdot x\\cdot\\arctan(2\\cdot x)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "When hired at a new job selling electronics, you are given two pay options:\n\nOption A: Base salary of $20\\ 000$ USD a year with a commission of $12$ percent of your sales.\n\nOption B: Base salary of $26\\ 000$ USD a year with a commission of $3$ percent of your sales.\n\nHow much electronics would you need to sell for Option A to produce a larger income? Give your answer either exactly or rounded to two decimal places.", | |
| "response": "This is the final answer to the problem: $66666.67$", | |
| "system": "" | |
| } | |
| ] | |
| } |