| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 1, | |
| "source_file": "/hkfs/work/workspace/scratch/tum_fmp0582-dndworkspace/自己训练lora/prepare/data/math/Mu-Math.json", | |
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| "total_records": 1084, | |
| "sample_count": 100, | |
| "generated_at": "2025-11-05T12:58:28Z", | |
| "seed": 4171135963, | |
| "samples": [ | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$.", | |
| "response": "$\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$ =$C+128\\cdot\\left(\\frac{1}{3\\cdot\\left(\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)\\right)^3}-\\frac{1}{\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=5 \\cdot x \\cdot \\sqrt{4-x^2}$. \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): $[-2,2]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(-\\sqrt{2},\\sqrt{2}\\right)$\n6. Intervals where the function is decreasing: $\\left(-2,-\\sqrt{2}\\right), \\left(\\sqrt{2},2\\right)$\n7. Intervals where the function is concave up: $(-2,0)$\n8. Intervals where the function is concave down: $(0,2)$\n9. Points of inflection: $P(0,0)$", | |
| "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": "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": "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": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 2 }+\\frac{ x }{ 1 \\cdot 3 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{1}{x^2}+\\frac{x\\cdot e^x-e^x}{x^2},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ 6 }{ \\sin(3 \\cdot x)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ 6 }{ \\sin(3 \\cdot x)^6 } d x}$ =$-\\frac{2\\cdot\\cos(3\\cdot x)}{5\\cdot\\sin(3\\cdot x)^5}+\\frac{24}{5}\\cdot\\left(-\\frac{\\cos(3\\cdot x)}{9\\cdot\\sin(3\\cdot x)^3}-\\frac{2}{9}\\cdot\\cot(3\\cdot x)\\right)+C$", | |
| "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": "Find a normal vector and a tangent vector for $2 \\cdot x^3-x^2 \\cdot y^2=3 \\cdot x-y-7$ at point $P$ : $(1,-2)$", | |
| "response": "Normal vector:$\\vec{N}=\\vec{i}-\\vec{j}$Tangent vector: $\\vec{T}=\\vec{i}+\\vec{j}$", | |
| "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": "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": "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": "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": "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": "An alternating current for outlets in a home has voltage given by the function $V(t)=150 \\cdot \\cos(368 \\cdot t)$, where $V$ is the voltage in volts at time $t$ in seconds.\n\n1. Find the period of the function.\n2. Determine the number of periods that occur when $1$ sec. has passed.", | |
| "response": "This is the final answer to the problem: \n\n1. the period of the function: $\\frac{\\pi}{184}$\n2. the number of periods: $58.56901906$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute $\\sqrt[4]{90}$ with accuracy $0.0001$.", | |
| "response": "This is the final answer to the problem: $3.0801$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve 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 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 $\\cos(2 \\cdot t)-5 \\cdot \\sin(t)-3=0$.", | |
| "response": "This is the final answer to the problem: $t=(-1)^{n+1}\\cdot\\frac{\\pi}{6}+n\\cdot\\pi$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the following equations: 1. $-10 c=-80$\n2. $n-(-6)=12$\n3. $-82+x=-20$\n4. $-\\frac{ r }{ 2 }=5$\n5. $r-3.4=7.1$\n6. $\\frac{ g }{ 2.5 }=1.8$\n7. $4.8 m=43.2$\n8. $\\frac{ 3 }{ 4 } t=\\frac{ 9 }{ 20 }$\n9. $3\\frac{2}{3}+m=5\\frac{1}{6}$", | |
| "response": "The solutions to the given equations are: \n1. $c=8$\n2. $n=6$\n3. $x=62$\n4. $r=-10$\n5. $r=10.5$\n6. $g=\\frac{ 9 }{ 2 }$\n7. $m=9$\n8. $t=\\frac{3}{5}$\n9. $m=\\frac{3}{2}$", | |
| "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": "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": "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": "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": "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 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": "|$n$ \n\n$\\ln(n)$ \n\n| |\n| 1 | 0.00 |\n| 2 | 0.69 |\n| 3 | 1.10 |\n| 4 | 1.39 |\n| 5 | 1.61 |\n| 6 | 1.79 |\n| 7 | 1.95 |\n| 8 | 2.08 |\n| 9 | 2.20 |\n| 10 | 2.30 |\n\n \n \nUsing the table above, estimate the logarithm.1. $\\ln(16)$\n2. $\\ln\\left(3^4\\right)$\n3. $\\ln(2.5)$\n4. $\\ln\\left(\\sqrt{630}\\right)$\n5. $\\ln(0.4)$", | |
| "response": "1. $\\ln(16)$≈$2.78$\n2. $\\ln\\left(3^4\\right)$≈$4.4$\n3. $\\ln(2.5)$≈$0.92$\n4. $\\ln\\left(\\sqrt{630}\\right)$≈$3.2228$\n5. $\\ln(0.4)$≈$-0.92$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the following problems by integration of the geometric series:\n\n$\\sum_{n=0}^\\infty\\left(x^n\\right)=\\frac{ 1 }{ 1-x }$, $|x|<1$\n\n1. $\\sum_{n=0}^\\infty\\left(\\frac{ 1 }{ (n+1) \\cdot 2^{n+1} }\\right)$\n2. $\\sum_{n=2}^\\infty\\left(\\frac{ 1 }{ n \\cdot 5^{n+1} }\\right)$\n3. $\\sum_{n=1}^\\infty\\left(\\frac{ 1 }{ n \\cdot 6^{n+3} }\\right)$\n4. $\\sum_{n=0}^\\infty\\left(\\frac{ 1 }{ (n+1) \\cdot (n+2) \\cdot 4^{n+2} }\\right)$\n5. $\\sum_{n=3}^\\infty\\left(\\frac{ 1 }{ n \\cdot (n+1) \\cdot 4^{n+3} }\\right)$", | |
| "response": "1. $\\ln(2)$\n2. $\\frac{1}{5}\\cdot\\ln\\left(\\frac{5}{4}\\right)-\\frac{1}{25}$\n3. $\\frac{1}{216}\\cdot\\ln\\left(\\frac{6}{5}\\right)$\n4. $\\frac{ 3 }{ 4 } \\cdot \\ln\\left(\\frac{ 3 }{ 4 }\\right)+\\frac{ 1 }{ 4 }$\n5. $\\frac{3}{64}\\cdot\\ln\\left(\\frac{3}{4}\\right)+\\frac{83}{6144}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Taylor series for $f(x)=\\frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $x\\cdot\\left(1-3\\cdot(x+1)+6\\cdot(x+1)^2-10\\cdot(x+1)^3+\\cdots\\right)$=\r\n= $-1 + 4 (x + 1) - 9 (x + 1)^2 + 16 (x + 1)^3$\r\n", | |
| "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": "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": "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": "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": "Find the derivative of the function $y=\\arcsin\\left(\\frac{ 2 \\cdot x }{ 1+x^2 }\\right)$.", | |
| "response": "$y'$=$\\frac{2\\cdot\\left(1-x^2\\right)}{\\left|1-x^2\\right|\\cdot\\left(1+x^2\\right)}$", | |
| "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": "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": "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": "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": "Find the Fourier expansion of this function: $f(x)=x^2$ at $(-\\pi,\\pi)$.", | |
| "response": "The Fourier series is: $f(x)=\\frac{\\pi^2}{3}+4\\cdot\\sum_{n=1}^\\infty\\left(\\frac{(-1)^{n}\\cdot\\cos(n\\cdot x)}{n^2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier expansion of this function: $f(x)=x^2$ at $(-\\pi,\\pi)$.", | |
| "response": "The Fourier series is: $f(x)=\\frac{\\pi^2}{3}+4\\cdot\\sum_{n=1}^\\infty\\left(\\frac{(-1)^{n}\\cdot\\cos(n\\cdot x)}{n^2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Write the Taylor series for the function $f(x)=x \\cdot \\sin(2 \\cdot x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).", | |
| "response": "This is the final answer to the problem: $0+2\\cdot\\pi\\cdot(x-\\pi)+\\frac{4}{2}\\cdot(x-\\pi)^2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Write the Taylor series for the function $f(x)=x \\cdot \\sin(2 \\cdot x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).", | |
| "response": "This is the final answer to the problem: $0+2\\cdot\\pi\\cdot(x-\\pi)+\\frac{4}{2}\\cdot(x-\\pi)^2$", | |
| "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": "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": "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": "Solve the following equations: 1. $-t+(5 t-7)=-5$\n2. $21-3 (2-w)=-12$\n3. $9=8 b-(2 b-3)$\n4. $4.5 r-2 r+3 (r-1)=10.75$\n5. $1.2 (x-8)+2.4 (x+1)=7.2$\n6. $4.9 m+(-3.2 m)-13=-2.63$\n7. $4 (2.25 w+3.1)-2.75 w=44.9$", | |
| "response": "The solutions to the given equations are: 1. $t=\\frac{ 1 }{ 2 }$\n2. $w=-9$\n3. $b=1$\n4. $r=\\frac{ 5 }{ 2 }$\n5. $x=4$\n6. $m=\\frac{ 61 }{ 10 }$\n7. $w=\\frac{ 26 }{ 5 }$", | |
| "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": "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": "Given that $\\frac{ 1 }{ 1-x }=\\sum_{n=0}^\\infty x^n$ , use term-by-term differentiation or integration to find power series for function $f(x)=\\ln(x)$ centered at $x=1$ .", | |
| "response": "$\\ln(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{(x-1)^{n+1}}{n+1}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ \\tan(x) }{ \\sqrt{\\sin(x)^4+\\cos(x)^4} } d x}$.", | |
| "response": "$\\int{\\frac{ \\tan(x) }{ \\sqrt{\\sin(x)^4+\\cos(x)^4} } d x}$ =$\\frac{1}{2}\\cdot\\ln\\left(\\tan(x)^2+\\sqrt{\\tan(x)^4+1}\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative of the function: $y=\\left(3 \\cdot a^2-2 \\cdot a \\cdot b \\cdot x+\\frac{ 5 }{ 3 } \\cdot b^2 \\cdot x^2\\right) \\cdot \\sqrt[3]{\\left(\\frac{ a }{ 3 }+\\frac{ b }{ 3 } \\cdot x\\right)^2}$.", | |
| "response": "The first derivative is:$\\frac{40\\cdot b^3\\cdot x^2}{9\\cdot3^{\\frac{2}{3}}\\cdot\\sqrt[3]{a+b\\cdot x}}$", | |
| "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": "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": "For the curve $x=a\\left(t-\\sin(t)\\right)$, $y=a\\left(1-\\cos(t)\\right)$ determine the curvature. Use $a=10$.", | |
| "response": "The curvature is:$\\frac{1}{40\\cdot\\left|\\sin\\left(\\frac{t}{2}\\right)\\right|}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute $\\sqrt[3]{130}$ with accuracy $0.0001$.", | |
| "response": "This is the final answer to the problem: $5.0658$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x }{ \\left(x^2-4 \\cdot x+8\\right)^2 } d x}$.", | |
| "response": "$\\int{\\frac{ x }{ \\left(x^2-4 \\cdot x+8\\right)^2 } d x}$ =$C+\\frac{x-2}{2\\cdot\\left(8+2\\cdot(x-2)^2\\right)}+\\frac{1}{8}\\cdot\\arctan\\left(\\frac{1}{2}\\cdot(x-2)\\right)-\\frac{1}{2\\cdot\\left(x^2-4\\cdot x+8\\right)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Determine the Taylor series for $y=\\left(\\sin(x)\\right)^2$, centered at $x_{0}=\\frac{ \\pi }{ 2 }$. Write out the sum of the first three non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $1-\\frac{2}{2!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^2+\\frac{2^3}{4!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^4+\\cdots$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the Fourier series of the function $f(x)=\\frac{ -1 }{ 2 } \\cdot x$ in the interval $[-2,2]$.", | |
| "response": "The Fourier series is: $\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Find $\\frac{ d y }{d x}$, given $y=\\tan(2 \\cdot v)$ and $v=\\arctan(2 \\cdot x-1)$.", | |
| "response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^2-2\\cdot x+1}{2\\cdot\\left(x-x^2\\right)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find $\\frac{ d y }{d x}$, given $y=\\tan(2 \\cdot v)$ and $v=\\arctan(2 \\cdot x-1)$.", | |
| "response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^2-2\\cdot x+1}{2\\cdot\\left(x-x^2\\right)^2}$", | |
| "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 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": "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": "Find the Fourier series of the function $\\psi(x)=e^{-x}$ in the interval $(-2 \\cdot \\pi,\\pi \\cdot 2)$.", | |
| "response": "The Fourier series is: $e^{-x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{4}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n}{4+n^2}\\cdot\\left(2\\cdot\\cos\\left(\\frac{n}{2}\\cdot x\\right)+n\\cdot\\sin\\left(\\frac{n}{2}\\cdot x\\right)\\right)\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\psi(x)=e^{-x}$ in the interval $(-2 \\cdot \\pi,\\pi \\cdot 2)$.", | |
| "response": "The Fourier series is: $e^{-x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{4}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n}{4+n^2}\\cdot\\left(2\\cdot\\cos\\left(\\frac{n}{2}\\cdot x\\right)+n\\cdot\\sin\\left(\\frac{n}{2}\\cdot x\\right)\\right)\\right)\\right)$", | |
| "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": "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": "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": "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": "Evaluate the integral: $I=\\int{3 \\cdot x \\cdot \\ln\\left(4+\\frac{ 1 }{ x }\\right) d x}$.", | |
| "response": "This is the final answer to the problem: $\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(4\\cdot x+1)-\\frac{3\\cdot x^2}{4}+\\frac{3\\cdot x}{8}-\\frac{3}{32}\\cdot\\ln\\left(x+\\frac{1}{4}\\right)\\right)-\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(x)-\\left(C+\\frac{3}{4}\\cdot x^2\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\psi(x)=e^{-x}$ in the interval $(-\\pi,\\pi)$.", | |
| "response": "The Fourier series is: $e^{-x}=\\frac{e^\\pi-e^{-\\pi}}{2\\cdot\\pi}\\cdot\\left(\\frac{1}{2}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n}{1+n^2}\\cdot\\left(\\cos(n\\cdot x)+n\\cdot\\sin(n\\cdot x)\\right)\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x \\cdot \\sqrt{2-x^2}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", | |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation): $\\left[-1\\cdot2^{2^{-1}},2^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $(-1,1)$\n6. Intervals where the function is decreasing: $\\left(-2^{2^{-1}},-1\\right), \\left(1,2^{2^{-1}}\\right)$\n7. Intervals where the function is concave up: $\\left(-2^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,2^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 the 3rd order Taylor polynomial $P_{3}(x)$ for the function $f(x)=\\arctan(x)$ in powers of $x-1$ and give the Lagrange form of the remainder.", | |
| "response": "$P_{3}(x)$=$\\frac{ \\pi }{ 4 }+\\frac{ 1 }{ 2 } \\cdot (x-1)-\\frac{ 1 }{ 4 } \\cdot (x-1)^2+\\frac{ 1 }{ 12 } \\cdot (x-1)^3$ \n\n$R_{3}(x)$=$\\frac{ -\\frac{ 48 \\cdot c^3 }{ \\left(1+c^2\\right)^4 }+\\frac{ 24 \\cdot c }{ \\left(1+c^2\\right)^3 } }{ 4! } \\cdot (x-1)^4$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the area of the surface formed by rotating the arc of the circle $x^2+y^2=1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x+y=1$.", | |
| "response": "This is the final answer to the problem: $\\frac{4\\cdot\\pi-\\pi^2}{\\sqrt{2}}$", | |
| "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": "Apply the gradient descent algorithm to the function $g(x,y)=\\left(x^2-1\\right) \\cdot \\left(x^2-3 \\cdot x+1\\right)+y^2$ with step size $\\frac{ 1 }{ 5 }$ and initial guess $p_{0}$=$\\left\\langle 0,0 \\right\\rangle$ for three steps (so steps $p_{1}$, $p_{2}$, and $p_{3}$).", | |
| "response": "| $i$ | $1$ | $2$ | $3$ |\n| --- | --- | --- | --- |\n| $p_{i}$ | $\\left\\langle-\\frac{3}{5},0\\right\\rangle$ | $\\left\\langle-\\frac{237}{625},0\\right\\rangle$ | $\\left\\langle-\\frac{826\\ 113\\ 663}{1\\ 220\\ 703\\ 125},0\\right\\rangle$ |\n| $g\\left(p_{i}\\right)$ | $-\\frac{1264}{625}$ | $-\\frac{99667587}{1220703125}$ |$-\\frac{2760602760604515522296126283436630289590864}{2220446049250313080847263336181640625}$ |", | |
| "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": "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": "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": "Find the equation of the tangent line to the curve: $r=3+\\cos(2 \\cdot t)$, $t=\\frac{ 3 \\cdot \\pi }{ 4 }$.", | |
| "response": "$y$ = $\\frac{1}{5}\\cdot\\left(x+\\frac{3}{\\sqrt{2}}\\right)+\\frac{3}{\\sqrt{2}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find and classify all critical points of the function $f(x,y)=x \\cdot y \\cdot (1-7 \\cdot x-9 \\cdot y)$.", | |
| "response": "Points of local minima: None.\n\nPoints of local maxima: $P\\left(\\frac{1}{21},\\frac{1}{27}\\right)$.\n\nSaddle points: $P\\left(0,\\frac{1}{9}\\right), P\\left(\\frac{1}{7},0\\right), P(0,0)$.", | |
| "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": "" | |
| }, | |
| { | |
| "prompt": "Find $\\frac{d ^3}{ d x^3}f(x)$, given $f(x)=\\ln\\left(\\frac{ x+7 }{ x-7 }\\right)$.", | |
| "response": "This is the final answer to the problem: $\\frac{d^3}{dx^3}f(x)=-\\frac{84\\cdot x^2+1372}{\\left(x^2-49\\right)^3}$", | |
| "system": "" | |
| } | |
| ] | |
| } |