| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 3, | |
| "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": "Find the points at which the following polar curve $r=4 \\cdot \\cos\\left(\\theta\\right)$ has a horizontal or vertical tangent line.", | |
| "response": "This is the final answer to the problem:\n\nhorizontal tangents at: $\\boxed{(2, 2), (2, -2)}$\n\nvertical tangents at: $\\boxed{(4, 0), (0, 0)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the points at which the following polar curve $r=4 \\cdot \\cos\\left(\\theta\\right)$ has a horizontal or vertical tangent line.", | |
| "response": "This is the final answer to the problem:\n\nhorizontal tangents at: $\\boxed{(2, 2), (2, -2)}$\n\nvertical tangents at: $\\boxed{(4, 0), (0, 0)}$", | |
| "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": "Let $R$ be the region bounded by the graphs of $y=\\ln(x)$ and $y=x-5$. Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated around the $y$-axis.", | |
| "response": "$V$ = $\\pi\\cdot\\int_b^d\\left((y+5)^2-\\left(e^y\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Determine a definite integral that represents the region common to $r=3 \\cdot \\cos\\left(\\theta\\right)$ and $r=3 \\cdot \\sin\\left(\\theta\\right)$ .", | |
| "response": "This is the final answer to the problem:$9\\cdot\\int_0^{\\frac{\\pi}{4}}\\sin\\left(\\theta\\right)^2d\\theta$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Determine a definite integral that represents the region common to $r=3 \\cdot \\cos\\left(\\theta\\right)$ and $r=3 \\cdot \\sin\\left(\\theta\\right)$ .", | |
| "response": "This is the final answer to the problem:$9\\cdot\\int_0^{\\frac{\\pi}{4}}\\sin\\left(\\theta\\right)^2d\\theta$", | |
| "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": "Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):\n\nHalley’s Comet: length of major axis=$35.88$, eccentricity=$0.967$.", | |
| "response": "$r$ = $\\frac{1.16450334}{1+0.967\\cdot\\cos(t)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function $y=\\arcsin\\left(\\frac{ a }{ x }\\right)-\\arctan\\left(\\frac{ x }{ a }\\right)$.", | |
| "response": "This is the final answer to the problem: $-\\frac{a}{a^2+x^2}-\\frac{a}{|x|\\cdot\\sqrt{x^2-a^2}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.", | |
| "response": "$f'(1)$ =$-0.5244$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.", | |
| "response": "$f'(1)$ =$-0.5244$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours.", | |
| "response": "There will be $7426$ leaves after 4 hours.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$.", | |
| "response": "$\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$ =$C-\\frac{1}{9}\\cdot\\left(1+\\frac{3}{x^2}\\right)\\cdot\\sqrt{1+\\frac{3}{x^2}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "$f(x)=\\frac{ 1 }{ 4 } \\cdot \\sqrt{x}+\\frac{ 1 }{ x }$, $x>0$. 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": "Solve the integral: $\\int{\\cot(x)^4 d x}$.", | |
| "response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\cot(x)^4 d x}$.", | |
| "response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 4 \\cdot (x+3)^2 }$. \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: $y=\\frac{x}{4}-\\frac{3}{2}$\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": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{2 \\cdot x-x^2}$ with the center $a=1$.", | |
| "response": "$\\sqrt{2 \\cdot x-x^2}$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^{2\\cdot n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $u=\\left|\\sin(x)\\right|$ in the interval $[-\\pi,\\pi]$.", | |
| "response": "The Fourier series is: $\\frac{2}{\\pi}-\\frac{4}{\\pi}\\cdot\\left(\\frac{\\cos(2\\cdot x)}{1\\cdot3}+\\frac{\\cos(4\\cdot x)}{3\\cdot5}+\\frac{\\cos(6\\cdot x)}{5\\cdot7}+\\cdots\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 2 }{ 4 }+\\frac{ 2 \\cdot x }{ 1 \\cdot 5 }+\\frac{ 2 \\cdot (x)^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ 2 \\cdot (x)^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{12}{x^4}+\\frac{\\left(x\\cdot\\left(16\\cdot x\\cdot e^x-16\\cdot e^x\\right)-8\\cdot e^x\\cdot x^3\\right)\\cdot x^3+\\left(4\\cdot e^x+2\\cdot e^x\\cdot x^2+2\\cdot e^x\\cdot x^3-4\\cdot x\\cdot e^x\\right)\\cdot x^4}{x^8},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral: $I=\\int{\\left(x^3+3\\right) \\cdot \\cos(2 \\cdot x) d x}$.", | |
| "response": "This is the final answer to the problem: $\\frac{1}{256}\\cdot\\left(384\\cdot\\sin(2\\cdot x)+128\\cdot x^3\\cdot\\sin(2\\cdot x)+192\\cdot x^2\\cdot\\cos(2\\cdot x)-96\\cdot\\cos(2\\cdot x)-256\\cdot C-192\\cdot x\\cdot\\sin(2\\cdot x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the following system of equations:\n\n$\\sin(x) \\cdot \\sin(y)=\\frac{ \\sqrt{3} }{ 4 }$ \n\n$\\cos(x) \\cdot \\cos(y)=\\frac{ \\sqrt{3} }{ 4 }$", | |
| "response": "This is the final answer to the problem: $x=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$ or $x=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$, where $k$ and $n$ are integers ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the following system of equations:\n\n$\\sin(x) \\cdot \\sin(y)=\\frac{ \\sqrt{3} }{ 4 }$ \n\n$\\cos(x) \\cdot \\cos(y)=\\frac{ \\sqrt{3} }{ 4 }$", | |
| "response": "This is the final answer to the problem: $x=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$ or $x=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$, where $k$ and $n$ are integers ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use synthetic division to find the quotient and reminder. Ensure the equation is in the form required by synthetic division:\n\n$\\frac{ 4 \\cdot x^3-12 \\cdot x^2-5 \\cdot x-1 }{ 2 \\cdot x+1 }$ \n\nHint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.", | |
| "response": "Quotient: $2\\cdot x^2-7\\cdot x+1$Remainder: $-2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use synthetic division to find the quotient and reminder. Ensure the equation is in the form required by synthetic division:\n\n$\\frac{ 4 \\cdot x^3-12 \\cdot x^2-5 \\cdot x-1 }{ 2 \\cdot x+1 }$ \n\nHint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.", | |
| "response": "Quotient: $2\\cdot x^2-7\\cdot x+1$Remainder: $-2$", | |
| "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": "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": "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 Maclaurin series of $F(x)=\\arcsin(x)$ ( $F(x)=\\int_{0}^x{f(t) d t}$) by integrating the Maclaurin series of $f(t)=\\frac{ 1 }{ \\sqrt{1-t^2} }$ ( $f(t)=\\sum_{k=0}^\\infty\\left(\\frac{ 1 }{ 2^{2 \\cdot k-1} \\cdot k } \\cdot \\frac{ (2 \\cdot k-2)! }{ \\left((k-1)!\\right) \\cdot (k-1)! } \\cdot t^{2 \\cdot k}\\right)$) term by term. If is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.", | |
| "response": "$\\arcsin(x)$ =$\\sum_{k=0}^\\infty\\left(\\frac{1}{2^{2\\cdot k-1}\\cdot k}\\cdot\\frac{(2\\cdot k-2)!}{\\left((k-1)!\\right)\\cdot(k-1)!}\\cdot\\frac{x^{2\\cdot k+1}}{2\\cdot k+1}\\right)$", | |
| "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": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{29}$ with accuracy 0.0001.", | |
| "response": "This is the final answer to the problem: $3.0723$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Consider the function $f(x)=\\frac{ 1 }{ 2 } \\cdot x^5+2 \\cdot x$. Let $g$ denote the inverse of $f$. Find the derivative $g'(2.5)$ using the theorem $g'(c)=\\frac{ 1 }{ f'\\left(g(c)\\right) }$.", | |
| "response": "$g'(2.5)$ =$\\frac{2}{9}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the sum of the $\\sum_{n=0}^\\infty\\left(\\frac{ (-1)^n }{ (2 \\cdot n+1)! }\\right)$ with estimate error $0.01$.", | |
| "response": "This is the final answer to the problem: $\\frac{101}{120}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Consider points $P$$P(3,7,-2)$ and $Q$$P(1,1,-3)$. Determine the angle between vectors $\\vec{OP}$ and $\\vec{OQ}$. Express the answer in radians rounded to two decimal places.", | |
| "response": "$\\theta$ =$0.91$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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 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": "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": "Use the table of integrals to evaluate the integral $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$. \n\nUse this link to access the table of integrals: [Table of Integrals](https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals)", | |
| "response": "1. Submit the formula used: $\\int{\\left(\\cos(u)\\right)^3 d u}=\\frac{ 1 }{ 3 } \\cdot \\left(2+\\left(\\cos(u)\\right)^2\\right) \\cdot \\sin(u)+c, \\int{\\left(\\sin(u)\\right)^n \\cdot \\left(\\cos(u)\\right)^m d u}=-\\frac{ \\left(\\sin(u)\\right)^{n-1} \\cdot \\left(\\cos(u)\\right)^{m+1} }{ n+m }+\\frac{ n-1 }{ n+m } \\cdot \\int{\\left(\\sin(u)\\right)^{n-2} \\cdot \\left(\\cos(u)\\right)^m d u}$ (For example: to evaluate $\\int{(x+3)^2 d x}$ you would use and submit the formula $\\int{u^n d u}=\\frac{ u^{n+1} }{ n+1 }+C$).\n2. $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$=$-\\frac{\\sin(y)\\cdot\\left(\\cos(y)\\right)^4}{5}+\\frac{1}{5}\\cdot\\frac{1}{3}\\cdot\\left(2+\\left(\\cos(y)\\right)^2\\right)\\cdot\\sin(y)+c$", | |
| "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 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": "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 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": "A particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.", | |
| "response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.", | |
| "response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the derivative of the function: $f\\left(\\alpha\\right)=\\ln\\left(\\sqrt{\\frac{ 3-\\sin\\left(\\alpha\\right) }{ 4+2 \\cdot \\sin\\left(\\alpha\\right) }}\\right)$ at $\\alpha=\\frac{ \\pi }{ 4 }$.", | |
| "response": "$f'\\left(\\frac{ \\pi }{ 4 }\\right)$ =$-0.2848$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the Fourier series of the function $f(x)=\\frac{ -1 }{ 2 } \\cdot x$ in the interval $[-2,2]$.", | |
| "response": "The Fourier series is: $\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the derivative $y=\\arctan\\left(\\frac{ 4-\\cos\\left(\\frac{ x }{ 2 }\\right) }{ 1+4 \\cdot \\cos\\left(\\frac{ x }{ 2 }\\right) }\\right)$.", | |
| "response": "$y'$= $\\frac{\\sin\\left(\\frac{x}{2}\\right)}{2+2\\cdot\\left(\\cos\\left(\\frac{x}{2}\\right)\\right)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 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 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 $\\psi(x)=2 \\cdot e^{-2 \\cdot x}$ in the interval $(-\\pi,\\pi)$.", | |
| "response": "The Fourier series is: $2\\cdot e^{-2\\cdot x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{2}+\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n\\cdot\\left(2\\cdot\\cos(n\\cdot x)+n\\cdot\\sin(n\\cdot x)\\right)}{4+n^2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate $\\int\\int\\int_{E}{\\left(x^3+y^3+z^3\\right) d V}$, where $E$=$\\left\\{(x,y,z)|0 \\le x \\le 2,0 \\le y \\le 2 \\cdot x,0 \\le z \\le 4-x-y\\right\\}$.", | |
| "response": "$I$ = $\\frac{112}{5}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x^2+2}$ with the center $a=0$.", | |
| "response": "$\\sqrt{x^2+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^{2\\cdot n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$ =$C+2\\cdot\\cot(6\\cdot x)+\\frac{2}{5}\\cdot\\left(\\cot(6\\cdot x)\\right)^5+\\frac{4}{3}\\cdot\\left(\\cot(6\\cdot x)\\right)^3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Taylor series for $f(x)=\\ln\\left(\\sqrt{3} \\cdot x+\\sqrt{1+3 \\cdot x^2}\\right)$, centered at $x=0$. Write out the sum of the first four non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $\\sqrt{3} x - (\\sqrt{3} x^3)/2 + (27 \\sqrt{3} x^5)/40 - (135 \\sqrt{3} x^7)/112 + O(x^9)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Taylor series for $f(x)=\\ln\\left(\\sqrt{3} \\cdot x+\\sqrt{1+3 \\cdot x^2}\\right)$, centered at $x=0$. Write out the sum of the first four non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $\\sqrt{3} x - (\\sqrt{3} x^3)/2 + (27 \\sqrt{3} x^5)/40 - (135 \\sqrt{3} x^7)/112 + O(x^9)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given $g(x)=\\frac{ 1 }{ 3 } \\cdot (a+b) \\cdot x^3+\\frac{ 1 }{ 2 } \\cdot (a+b+c) \\cdot x^2-(a+b+c+d) \\cdot x+a \\cdot b \\cdot c \\cdot d$, simplify the derivative of $g(x)$ if $x^2+x=a+b$.", | |
| "response": "This is the final answer to the problem: $g'(x)=(a+b)^2+c\\cdot x-(a+b+c+d)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of the function $x^{\\frac{ 1 }{ 3 }}$ with the center $a=27$.", | |
| "response": "$x^{\\frac{ 1 }{ 3 }}$ =$\\sum_{n=0}^\\infty\\left(3^{1-3\\cdot n}\\cdot C_n^{\\frac{1}{3}}\\cdot(x-27)^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$.", | |
| "response": "$\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$ =$C+\\frac{1}{21}\\cdot\\left(\\frac{1}{2}\\cdot\\left(\\cot(10\\cdot x)\\right)^4+\\frac{1}{3}\\cdot\\left(\\cot(10\\cdot x)\\right)^6\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 equations of the common tangent lines to the following ellipses:\n\n$\\frac{ x^2 }{ 6 }+y^2=1$ \n\n$\\frac{ x^2 }{ 4 }+\\frac{ y^2 }{ 9 }=1$", | |
| "response": "This is the final answer to the problem: $2\\cdot x+y-5=0 \\lor 2\\cdot x+y+5=0 \\lor 2\\cdot x-y-5=0 \\lor 2\\cdot x-y+5=0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A box is to be made with the following properties:\n\nThe length of the base, $l$, is twice the length of a width $w$.\n\nThe cost of material to be used for the lateral faces and the top of the box is three times as the cost of the material to be used for the lower base.\n\nFind the dimensions of the box in terms of its fixed volume $V$ such that the cost of the used material is the minimum.", | |
| "response": "This is the final answer to the problem: $y=\\frac{2}{3}\\cdot\\sqrt[3]{\\frac{4\\cdot V}{3}}, w=\\frac{1}{2}\\cdot\\sqrt[3]{\\frac{9\\cdot V}{2}}, l=\\sqrt[3]{\\frac{9\\cdot V}{2}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 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": "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 tangential and normal components of acceleration for $\\vec{r}(t)=\\left\\langle a \\cdot \\cos\\left(\\omega \\cdot t\\right),b \\cdot \\sin\\left(\\omega \\cdot t\\right) \\right\\rangle$ at $t=0$ with positive coefficients.", | |
| "response": "This is the final answer to the problem:\n\n$a_{t}$: $0$ \n\n$a_{N}$: $\\omega^2\\cdot|a|$", | |
| "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": "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": "Compute the integral $\\int{\\frac{ -2 }{ \\left(4+\\sqrt{x}\\right) \\cdot \\sqrt{x-x^2} } d x}$.", | |
| "response": "$\\int{\\frac{ -2 }{ \\left(4+\\sqrt{x}\\right) \\cdot \\sqrt{x-x^2} } d x}$ = $C-\\frac{8}{\\sqrt{15}}\\cdot\\arctan\\left(\\frac{1}{\\sqrt{15}}\\cdot\\left(1+4\\cdot\\tan\\left(\\frac{1}{2}\\cdot\\arcsin\\left(\\sqrt{x}\\right)\\right)\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the second derivative $\\frac{d ^2y}{ d x^2}$ of the function $x=\\left(4 \\cdot \\sin(t)\\right)^3$, $y=2 \\cdot \\sin(2 \\cdot t)$.", | |
| "response": "$\\frac{d ^2y}{ d x^2}$=$\\frac{\\left(2304\\cdot\\left(\\sin(t)\\right)^3-1536\\cdot\\sin(t)\\right)\\cdot\\cos(2\\cdot t)-1536\\cdot\\left(\\sin(t)\\right)^2\\cdot\\cos(t)\\cdot\\sin(2\\cdot t)}{7077888\\cdot\\left(\\cos(t)\\right)^3\\cdot\\left(\\sin(t)\\right)^6}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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": "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 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": "Find the dimensions of the box whose length is twice as long as the width, whose height is $2$ inches greater than the width, and whose volume is $192$ cubic inches.", | |
| "response": "The dimensions of the box are $6, 4, 8$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the second derivative test to identify any critical points of the function $f(x,y)=x^2 \\cdot y^2$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these.", | |
| "response": "Maximum:NoneMinimum: $P(0,0)$ \n\nSaddle point: None\n\nThe second derivative test is inconclusive at: None", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find 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": "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 the extrema of a function $y=\\frac{ x^4 }{ 4 }-\\frac{ 2 \\cdot x^3 }{ 3 }-\\frac{ x^2 }{ 2 }+2$. 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: $P\\left(\\frac{2-2\\cdot\\sqrt{2}}{2},1.969\\right), P(0,2), P\\left(\\frac{2+2\\cdot\\sqrt{2}}{2},-1.8023\\right)$\n2. The largest value: $\\frac{46}{3}$\n3. The smallest value: $-1.8023$", | |
| "system": "" | |
| } | |
| ] | |
| } |