| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 6, | |
| "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": "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: $\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$.", | |
| "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": "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": "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": "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": "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": "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": "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": "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": "Find the directional derivative of $f(x,y,z)=x^2+y \\cdot z$ at $P(1,-3,2)$ in the direction of increasing $t$ along the path\n\n$\\vec{r}(t)=t^2 \\cdot \\vec{i}+3 \\cdot t \\cdot \\vec{j}+\\left(1-t^3\\right) \\cdot \\vec{k}$.", | |
| "response": "$f_{u}(P)$=$\\frac{11}{\\sqrt{22}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the method of Lagrange multipliers to maximize $U(x,y)=8 \\cdot x^{\\frac{ 4 }{ 5 }} \\cdot y^{\\frac{ 1 }{ 5 }}$; $4 \\cdot x+2 \\cdot y=12$.", | |
| "response": "Answer: maximum $16.715$ at $P(2.4,1.2)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find $\\frac{ d y }{d x}$ for $y=x \\cdot \\arccsc(x)$.", | |
| "response": "$\\frac{ d y }{d x}$= $-\\frac{x}{|x|\\cdot\\sqrt{x^2-1}}+\\arccsc(x)$", | |
| "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": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ \\pi \\cdot x, & 0 \\le x \\le 1 \\\\ 0, & x>1 \\end{cases}$.", | |
| "response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{\\left(\\alpha\\cdot\\sin\\left(\\alpha\\right)+\\cos\\left(\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+\\left(\\sin\\left(\\alpha\\right)-\\alpha\\cdot\\cos\\left(\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the initial value problem (find function $f$) for $f'(x)=\\frac{ 2 }{ x^2 }-\\frac{ x^2 }{ 2 }$, $f(1)=0$.", | |
| "response": "$f(x)=\\frac{13}{6}-\\frac{2}{x}-\\frac{1}{6}\\cdot x^3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the initial value problem (find function $f$) for $f'(x)=\\frac{ 2 }{ x^2 }-\\frac{ x^2 }{ 2 }$, $f(1)=0$.", | |
| "response": "$f(x)=\\frac{13}{6}-\\frac{2}{x}-\\frac{1}{6}\\cdot x^3$", | |
| "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": "Given $y=x^6-\\frac{ 9 }{ 2 } \\cdot x^5+\\frac{ 15 }{ 2 } \\cdot x^4-5 \\cdot x^3+10$ find where the function is concave up, down, and point(s) of inflection.", | |
| "response": "Concave up:$(1,\\infty), (-\\infty,0)$Concave down:$(0,1)$Point(s) of Inflection:$P(1,9), P(0,10)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $f(x)=\\frac{ 1 }{ 15 } \\cdot \\left(\\cos(x)\\right)^3 \\cdot \\left(\\left(\\cos(x)\\right)^2-5\\right)$.", | |
| "response": "This is the final answer to the problem: $f'(x)=-\\frac{\\left(\\cos(x)\\right)^2}{15}\\cdot\\left(3\\cdot\\sin(x)\\cdot\\left(\\left(\\cos(x)\\right)^2-5\\right)+\\cos(x)\\cdot\\sin(2\\cdot x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $f(x)=\\frac{ 1 }{ 15 } \\cdot \\left(\\cos(x)\\right)^3 \\cdot \\left(\\left(\\cos(x)\\right)^2-5\\right)$.", | |
| "response": "This is the final answer to the problem: $f'(x)=-\\frac{\\left(\\cos(x)\\right)^2}{15}\\cdot\\left(3\\cdot\\sin(x)\\cdot\\left(\\left(\\cos(x)\\right)^2-5\\right)+\\cos(x)\\cdot\\sin(2\\cdot x)\\right)$", | |
| "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": "Solve the integral: $\\int{\\tan(x)^4 d x}$.", | |
| "response": "$\\int{\\tan(x)^4 d x}$ = $C + x + 1/3 (sec^2(x) - 4) tan(x)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve $\\sin(x)+7 \\cdot \\cos(x)+7=0$.", | |
| "response": "This is the final answer to the problem: $x=2\\cdot\\pi\\cdot k-2\\cdot\\arctan(7) \\lor x=\\pi+2\\cdot\\pi\\cdot k$", | |
| "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": "The velocity of a bullet from a rifle can be approximated by $v(t)=6400 \\cdot t^2-6505 \\cdot t+2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half- second after the shot: $0 \\le t \\le 0.5$ What is the total distance the bullet travels in $0.5$ sec?", | |
| "response": "The total distance is:$796.54166667$", | |
| "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": "Find the Fourier series of the periodic function $f(x)=\\frac{ x^2 }{ 2 }$ in the interval $-2 \\cdot \\pi \\le x<2 \\cdot \\pi$ if $f(x)=f(x+4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $\\frac{2\\cdot\\pi^2}{3}+\\sum_{n=1}^\\infty\\left(\\frac{8\\cdot(-1)^n}{n^2}\\cdot\\cos\\left(\\frac{n\\cdot x}{2}\\right)\\right)$", | |
| "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": "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 $I=\\int{\\frac{ 1 }{ x^3+8 } d x}$.", | |
| "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": "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": "Calculate integral: $\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$. $\\left(M=4\\right)$, $\\left(N=5\\right)$, $\\left(p=2\\right)$, $\\left(q=9\\right)$, $\\left(m=2\\right)$.", | |
| "response": "$\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$ =$C+\\frac{x+1}{128+16\\cdot(x+1)^2}+\\frac{\\sqrt{2}}{64}\\cdot\\arctan\\left(\\frac{1}{2\\cdot\\sqrt{2}}\\cdot(x+1)\\right)-\\frac{2}{8+(x+1)^2}$", | |
| "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": "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": "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 $\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$ =$\\sqrt{x^2+1}+\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x^2+1}-1}{\\sqrt{x^2+1}+1}\\right|\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$.", | |
| "response": "$\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$ =$\\sqrt{x^2+1}+\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x^2+1}-1}{\\sqrt{x^2+1}+1}\\right|\\right)+C$", | |
| "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": "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 surface of a large cup is formed by revolving the graph of the function $y=0.25 \\cdot x^{1.6}$ from $x=0$ to $x=5$ about the $y$-axis (measured in centimeters). Find the curvature $\\kappa$ of the generating curve as a function of $x$.", | |
| "response": "$\\kappa$=$\\frac{30}{x^{\\frac{2}{5}}\\cdot\\left(25+4\\cdot x^{\\frac{6}{5}}\\right)^{\\frac{3}{2}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The surface of a large cup is formed by revolving the graph of the function $y=0.25 \\cdot x^{1.6}$ from $x=0$ to $x=5$ about the $y$-axis (measured in centimeters). Find the curvature $\\kappa$ of the generating curve as a function of $x$.", | |
| "response": "$\\kappa$=$\\frac{30}{x^{\\frac{2}{5}}\\cdot\\left(25+4\\cdot x^{\\frac{6}{5}}\\right)^{\\frac{3}{2}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x^2-x^4-2$. \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: $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{1}{\\sqrt{2}}\\right), \\left(\\frac{1}{\\sqrt{2}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right), P\\left(\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x^2-x^4-2$. \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: $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{1}{\\sqrt{2}}\\right), \\left(\\frac{1}{\\sqrt{2}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right), P\\left(\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right)$", | |
| "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": "Find a “reasonable” upper-bound on the error in approximating $f(x)=x \\cdot \\ln(x)$ by its 3rd order Taylor polynomial $P_{3}(x)$ at $a=1$ valid for all values of $x$ such that $|x-1| \\le 0.7$.", | |
| "response": "This is the final answer to the problem: $\\frac{2}{(0.3)^3}\\cdot\\frac{(0.7)^4}{4!}$", | |
| "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": "Compute the integral $\\int{\\frac{ x^3 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+5} } d x}$.", | |
| "response": "$\\int{\\frac{ x^3 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+5} } d x}$ =$\\left(\\frac{1}{12}\\cdot x^2-\\frac{5}{48}\\cdot x-\\frac{5}{96}\\right)\\cdot\\sqrt{4\\cdot x^2+4\\cdot x+5}+\\frac{5}{16}\\cdot\\ln\\left(x+\\frac{1}{2}+\\sqrt{1+\\left(x+\\frac{1}{2}\\right)^2}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The function $s(t)=\\frac{ t }{ 1+t^2 }$ 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 $v(t)$ = $\\frac{1-t^2}{\\left(1+t^2\\right)^2}$ and acceleration function $a(t)$ = $\\frac{2\\cdot t\\cdot\\left(t^2-3\\right)}{\\left(1+t^2\\right)^3}$.\n2. The time intervals when the object speeds up $\\left(1,\\sqrt{3}\\right)$ and slows down $\\left(\\sqrt{3},\\infty\\right), (0,1)$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the antiderivative of $-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} }$.", | |
| "response": "$\\int{-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} } d x}$ =$C+\\ln\\left(\\frac{1+\\sqrt{1-x^2}}{|x|}\\right)$", | |
| "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": "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": "Solve $\\sin(x)+\\cos(x)-2 \\cdot \\sqrt{2} \\cdot \\sin(x) \\cdot \\cos(x)=0$.", | |
| "response": "This is the final answer to the problem: $x=(-1)^{n+1}\\cdot\\frac{\\pi}{6}-\\frac{\\pi}{4}+\\pi\\cdot n, x=\\frac{\\pi}{4}+2\\cdot\\pi\\cdot k$", | |
| "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": "Compute the integral: $\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$.", | |
| "response": "$\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$ =$C-\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{1-\\sin(x)}{1+\\sin(x)}\\right|\\right)-\\frac{1}{3}\\cdot\\left(\\sin(x)\\right)^3-\\sin(x)$", | |
| "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": "Let $z=e^{1-x \\cdot y}$, $x=t^{\\frac{ 1 }{ 3 }}$, $y=t^3$. Find $\\frac{ d z }{d t}$.", | |
| "response": "$\\frac{ d z }{d t}$ =$\\frac{-(10\\cdot e)}{3}\\cdot e^{-t^3\\cdot\\sqrt[3]{t}}\\cdot t^2\\cdot\\sqrt[3]{t}$", | |
| "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": "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": "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 Taylor series of the given function $f(x)=\\cos(x)$ centered at the indicated point: $a=\\frac{ \\pi }{ 2 }$ .", | |
| "response": "$\\cos(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^{n+1}\\cdot\\frac{\\left(x-\\frac{\\pi}{2}\\right)^{2\\cdot n+1}}{(2\\cdot n+1)!}\\right)$", | |
| "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": "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": "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": "Find zeros of $f(x)=\\sin(x)+\\sin(2 \\cdot x)+2 \\cdot \\sin(x) \\cdot \\sin(2 \\cdot x)-2 \\cdot \\cos(x)-\\cos(2 \\cdot x)$.", | |
| "response": "This is the final answer to the problem: $x_1=-\\frac{\\pi}{2}+2\\cdot\\pi\\cdot n, x_2=-\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_3=\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_4=(-1)^n\\cdot\\frac{\\pi}{6}+\\pi\\cdot n$", | |
| "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": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{7}$ with accuracy 0.0001.", | |
| "response": "This is the final answer to the problem: $1.9129$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{7}$ with accuracy 0.0001.", | |
| "response": "This is the final answer to the problem: $1.9129$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series expansion of the function $f(x)=\\begin{cases} -x, & -\\pi<x \\le 0 \\\\ \\frac{ x^2 }{ \\pi }, & 0<x \\le \\pi \\end{cases}$ with the period $2 \\cdot \\pi$ at interval $[-\\pi,\\pi]$.", | |
| "response": "The Fourier series is: $f(x)=\\frac{5\\cdot\\pi}{12}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n\\cdot3-1}{\\pi\\cdot n^2}\\cdot\\cos(n\\cdot x)+\\left(\\frac{2}{\\pi^2\\cdot n^3}\\cdot\\left((-1)^n-1\\right)\\right)\\cdot\\sin(n\\cdot x)\\right)$", | |
| "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 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": "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": "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 the radius of convergence and sum of the series: $\\frac{ 3 }{ 2 }+\\frac{ 3 \\cdot x }{ 1 \\cdot 3 }+\\frac{ 3 \\cdot x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ 3 \\cdot 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{3}{x^2}+\\frac{3\\cdot x\\cdot e^x-3\\cdot e^x}{x^2},&x\\ne0\\\\\\frac{3}{2},&x=0\\end{cases}$", | |
| "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": "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": "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": "For the function $y=\\frac{ 2 \\cdot x+3 }{ 4 \\cdot x+5 }$ find the derivative $y^{(n)}$ .", | |
| "response": "The General Form of the Derivative of $y=\\frac{ 2 \\cdot x+3 }{ 4 \\cdot x+5 }$: $y^{(n)}=\\frac{1}{2}\\cdot(-1)^n\\cdot\\left(n!\\right)\\cdot4^n\\cdot(4\\cdot x+5)^{-(n+1)}$", | |
| "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": "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": "Find the area of a triangle bounded by the x-axis, the line $f(x)=12-\\frac{ 1 }{ 3 } \\cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.", | |
| "response": "The area of the triangle is $\\frac{972}{5}$", | |
| "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": "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": "Solve the integral: $\\int{22 \\cdot \\cot(-11 \\cdot x)^5 d x}$.", | |
| "response": "$\\int{22 \\cdot \\cot(-11 \\cdot x)^5 d x}$ =$C+\\frac{1}{2}\\cdot\\left(\\cot(11\\cdot x)\\right)^4+\\ln\\left(1+\\left(\\cot(11\\cdot x)\\right)^2\\right)-\\left(\\cot(11\\cdot x)\\right)^2$", | |
| "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": "Use integration by substitution and/or by parts to compute the integral \n $\\int{x \\cdot \\ln(5+x) d x}$.", | |
| "response": "This is the final answer to the problem:$D+5\\cdot(x+5)+\\left(\\frac{1}{2}\\cdot(x+5)^2-5\\cdot(x+5)\\right)\\cdot\\ln(x+5)-\\frac{1}{4}\\cdot(x+5)^2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use integration by substitution and/or by parts to compute the integral \n $\\int{x \\cdot \\ln(5+x) d x}$.", | |
| "response": "This is the final answer to the problem:$D+5\\cdot(x+5)+\\left(\\frac{1}{2}\\cdot(x+5)^2-5\\cdot(x+5)\\right)\\cdot\\ln(x+5)-\\frac{1}{4}\\cdot(x+5)^2$", | |
| "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": "" | |
| } | |
| ] | |
| } |