| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 0, | |
| "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": "Determine the intervals of upward concavity and downward concavity for the function $f(x)=e^x \\cdot \\left(x^2-1\\right)$.", | |
| "response": "1. Interval(s) of upward concavity: $\\left(\\frac{2\\cdot\\sqrt{3}-4}{2},\\infty\\right), \\left(-\\infty,-\\frac{4+2\\cdot\\sqrt{3}}{2}\\right)$\n2. Interval(s) of downward concavity: $\\left(-\\frac{4+2\\cdot\\sqrt{3}}{2},\\frac{2\\cdot\\sqrt{3}-4}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve $3 \\cdot \\sin(x)+4 \\cdot \\cos(x)=5$.", | |
| "response": "This is the final answer to the problem: $x=-\\arcsin\\left(\\frac{4}{5}\\right)+\\frac{\\pi}{2}+2\\cdot\\pi\\cdot n$ ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "$f(x)=x+\\sin(2 \\cdot x)$ over $x$= $\\left[-\\frac{ \\pi }{ 2 },\\frac{ \\pi }{ 2 }\\right]$. \n\nDetermine:\n1. intervals where $f$ is increasing\n2. intervals where $f$ is decreasing\n3. local minima of $f$\n4. local maxima of $f$\n5. intervals where $f$ is concave up\n6. intervals where $f$ is concave down\n7. the inflection points of $f$", | |
| "response": "1. intervals where $f$ is increasing : $\\left(-\\frac{\\pi}{3},\\frac{\\pi}{3}\\right)$\n2. intervals where $f$ is decreasing: $\\left(\\frac{\\pi}{3},\\frac{\\pi}{2}\\right), \\left(-\\frac{\\pi}{2},-\\frac{\\pi}{3}\\right)$\n3. local minima of $f$: $-\\frac{\\pi}{3}$\n4. local maxima of $f$: $\\frac{\\pi}{3}$\n5. intervals where $f$ is concave up : $\\left(-\\frac{\\pi}{2},0\\right)$\n6. intervals where $f$ is concave down: $\\left(0,\\frac{\\pi}{2}\\right)$\n7. the inflection points of $f$: $0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Determine the equation of the hyperbola using the information given: Endpoints of the conjugate axis located at $(3,2)$, $(3,4)$ and focus located at $(7,3)$.", | |
| "response": "The equation is: $\\frac{(x-3)^2}{15}-\\frac{(y-3)^2}{1}=1$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the derivative of the function $y=\\arcsin\\left(\\sqrt{1-9 \\cdot x^2}\\right)$.", | |
| "response": "$y'$= $-\\frac{3\\cdot x}{\\sqrt{1-9\\cdot x^2}\\cdot|x|}$", | |
| "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": "Let $R$ be the region bounded by the graphs of $y=\\ln(x)$ and $y=x-5$. Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated around the $y$-axis.", | |
| "response": "$V$ = $\\pi\\cdot\\int_b^d\\left((y+5)^2-\\left(e^y\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Consider the differential equation $\\frac{ d y }{d x}=\\frac{ 4+y }{ x }$. Find the particular solution $y=f(x)$ to the given differential equation with the initial condition $f(3)=-3$.", | |
| "response": "$y$ = $\\frac{|x|}{3}-4$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Determine a definite integral that represents the region enclosed by one petal of $r=\\cos\\left(3 \\cdot \\theta\\right)$ .", | |
| "response": "This is the final answer to the problem:$\\int_0^{\\frac{\\pi}{6}}\\cos\\left(3\\cdot\\theta\\right)^2d\\theta$ = $\\frac{\\pi}{12}$ ", | |
| "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": "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": "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": "Solve the following series via differentiation of the geomtric series \n\n$\\sum_{n=0}^\\infty\\left(x^n\\right)=\\frac{ 1 }{ 1-x }$,$|x|<1$\n\n1. $\\sum_{n=1}^\\infty\\left(\\frac{ n }{ 6^{n-1} }\\right)$\n2. $\\sum_{n=2}^\\infty\\left(\\frac{ n }{ 3^{n-1} }\\right)$\n3. $\\sum_{n=3}^\\infty\\left(\\frac{ n }{ 2^{n+1} }\\right)$\n4. $\\sum_{n=2}^\\infty\\left(\\frac{ n+1 }{ 5^n }\\right)$\n5. $\\sum_{n=2}^\\infty\\left(\\frac{ n \\cdot (n-1) }{ 4^n }\\right)$\n6. $\\sum_{n=2}^\\infty\\left(\\frac{ n^2 }{ 4^n }\\right)$", | |
| "response": "1. $\\frac{36}{25}$\n2. $\\frac{5}{4}$\n3. $\\frac{1}{2}$\n4. $\\frac{13}{80}$\n5. $\\frac{8}{27}$\n6. $\\frac{53}{108}$", | |
| "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": "Let $R$ be the region in the first quadrant bounded by the graph of $y=3 \\cdot \\arctan(x)$ and the lines $x=\\pi$ and $y=1$.\n\nFind the volume of the solid generated when $R$ is revolved about the line $x=\\pi$.", | |
| "response": "The volume of the solid is $36.736$ units³.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Make full curve sketching of $f(x)=\\frac{ 3 \\cdot x^3 }{ 3 \\cdot x^2-4 }$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptote(s)\n3. Horizontal asymptote(s)\n4. Slant asymptote(s)\n5. Interval(s) where the function is increasing\n6. Interval(s) where the function is decreasing\n7. Interval(s) where the function is concave up\n8. Interval(s) where the function is concave down\n9. Point(s) of inflection", | |
| "response": "This is the final answer to the problem:\n\n1. The domain (in interval notation) $\\left(-1\\cdot\\infty,-2\\cdot3^{-1\\cdot2^{-1}}\\right)\\cup\\left(-2\\cdot3^{-1\\cdot2^{-1}},2\\cdot3^{-1\\cdot2^{-1}}\\right)\\cup\\left(2\\cdot3^{-1\\cdot2^{-1}},\\infty\\right)$\n2. Vertical asymptote(s) $x=-\\frac{2}{\\sqrt{3}}, x=\\frac{2}{\\sqrt{3}}$\n3. Horizontal asymptote(s) None\n4. Slant asymptote(s) $y=x$\n5. Interval(s) where the function is increasing $(2,\\infty), (-\\infty,-2)$\n6. Interval(s) where the function is decreasing $\\left(0,\\frac{2}{\\sqrt{3}}\\right), \\left(-\\frac{2}{\\sqrt{3}},0\\right), \\left(-2,-\\frac{2}{\\sqrt{3}}\\right), \\left(\\frac{2}{\\sqrt{3}},2\\right)$\n7. Interval(s) where the function is concave up $\\left(\\frac{2}{\\sqrt{3}},\\infty\\right), \\left(-\\frac{2}{\\sqrt{3}},0\\right)$\n8. Interval(s) where the function is concave down $\\left(0,\\frac{2}{\\sqrt{3}}\\right), \\left(-\\infty,-\\frac{2}{\\sqrt{3}}\\right)$\n9. Point(s) of inflection $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative of the function: $y=(x+11)^5 \\cdot (3 \\cdot x-7)^4 \\cdot (x-12) \\cdot (x+4)$.", | |
| "response": "$y'$ =$\\left(\\frac{5}{x+11}+\\frac{12}{3\\cdot x-7}+\\frac{1}{x-12}+\\frac{1}{x+4}\\right)\\cdot(x+11)^5\\cdot(3\\cdot x-7)^4\\cdot(x-12)\\cdot(x+4)$", | |
| "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": "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": "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": "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": "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 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": "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": "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 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": "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": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 3 \\cdot (x+2)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", | |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-2)\\cup(-2,\\infty)$\n2. Vertical asymptotes: $x=-2$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{3}-\\frac{4}{3}$\n5. Intervals where the function is increasing: $(0,\\infty), (-2,0), (-\\infty,-6)$\n6. Intervals where the function is decreasing: $(-6,-2)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-2,0), (-\\infty,-2)$\n9. Points of inflection: $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 3 \\cdot (x+2)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", | |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-2)\\cup(-2,\\infty)$\n2. Vertical asymptotes: $x=-2$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{3}-\\frac{4}{3}$\n5. Intervals where the function is increasing: $(0,\\infty), (-2,0), (-\\infty,-6)$\n6. Intervals where the function is decreasing: $(-6,-2)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-2,0), (-\\infty,-2)$\n9. Points of inflection: $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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": "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 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": "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": "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": "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": "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": "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": "A town has an initial population of $80\\ 000$. It grows at a constant rate of $2200$ per year for $5$ years. The linear function that models the town’s population P as a function of the year is: $P(t)=80\\ 000+2200 \\cdot t$, where $t$ is the number of years since the model began. When will the population reach $120\\ 000$?", | |
| "response": "$t$: $\\frac{200}{11}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 6 \\cdot x^3-7 \\cdot x^2+3 \\cdot x-1 }{ 2 \\cdot x-3 \\cdot x^2 } d x}$.", | |
| "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": "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": "Find the Fourier series expansion for periodic (period $2$) $f(x)=x-1$ (at $(-1,1]$).", | |
| "response": "The Fourier series is: $f(x)\\approx\\sum_{n=1}^\\infty-\\frac{2\\cdot(-1)^n\\cdot\\sin(n\\cdot\\pi\\cdot x)}{\\pi\\cdot n}-1$", | |
| "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": "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": "Compute the integral: $\\int{\\frac{ 1 }{ (x-3) \\cdot \\sqrt{10 \\cdot x-24-x^2} } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ (x-3) \\cdot \\sqrt{10 \\cdot x-24-x^2} } d x}$ =$-\\frac{2}{\\sqrt{3}}\\cdot\\arctan\\left(\\frac{\\sqrt{-x^2+10\\cdot x-24}}{\\sqrt{3}\\cdot x-4\\cdot\\sqrt{3}}\\right)+C$", | |
| "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": "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": "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": "For which positive $p$ does the series $\\sum_{n=1}^\\infty\\left(\\frac{ p^{n^2} }{ 2^n }\\right)$ converge?", | |
| "response": "Converges for: $p\\le1$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For which positive $p$ does the series $\\sum_{n=1}^\\infty\\left(\\frac{ p^{n^2} }{ 2^n }\\right)$ converge?", | |
| "response": "Converges for: $p\\le1$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given $y=3 \\cdot x^5+10 \\cdot x^4-20$ find where the function is concave up, down, and point(s) of inflection.", | |
| "response": "Concave up:$(0,\\infty), (-2,0)$Concave down:$(-\\infty,-2)$Point(s) of Inflection:$P(-2,44)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $y=x \\cdot \\sin(x)+2 \\cdot x \\cdot \\cos(x)-2 \\cdot \\sin(x)+\\ln\\left(\\sin(x)\\right)+c^2$.", | |
| "response": "This is the final answer to the problem: $y'=\\frac{\\left(\\sin(x)\\right)^2+x\\cdot\\sin(x)\\cdot\\cos(x)-2\\cdot x\\cdot\\left(\\sin(x)\\right)^2+\\cos(x)}{\\sin(x)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $y=x \\cdot \\sin(x)+2 \\cdot x \\cdot \\cos(x)-2 \\cdot \\sin(x)+\\ln\\left(\\sin(x)\\right)+c^2$.", | |
| "response": "This is the final answer to the problem: $y'=\\frac{\\left(\\sin(x)\\right)^2+x\\cdot\\sin(x)\\cdot\\cos(x)-2\\cdot x\\cdot\\left(\\sin(x)\\right)^2+\\cos(x)}{\\sin(x)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the 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 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": "Differentiate $\\sqrt{x \\cdot y}-x=y^5$.", | |
| "response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^{\\frac{1}{2}}\\cdot y^{\\frac{1}{2}}-y}{x-10\\cdot x^{\\frac{1}{2}}\\cdot y^{\\frac{9}{2}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 2 \\cdot \\tan(x)+3 }{ \\sin(x)^2+2 \\cdot \\cos(x)^2 } d x}$.", | |
| "response": "$\\int{\\frac{ 2 \\cdot \\tan(x)+3 }{ \\sin(x)^2+2 \\cdot \\cos(x)^2 } d x}$ =$C+\\frac{3}{\\sqrt{2}}\\cdot\\arctan\\left(\\frac{1}{\\sqrt{2}}\\cdot\\tan(x)\\right)+\\ln\\left(2+\\left(\\tan(x)\\right)^2\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For the function $y=(4-x)^3 \\cdot (x+1)^2$ specify the points where local maxima and minima of $y$ occur. Submit as your final answer:\n\n1. The point(s) where local maxima occur\n2. The point(s) where local minima occur", | |
| "response": "This is the final answer to the problem: \n1. The point(s) where local maxima occur $P(1,108)$\n2. The point(s) where local minima occur $P(-1,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Identify the intervals where $f(x)=\\frac{ 1 }{ 4 } \\cdot x^4+\\frac{ 1 }{ 3 } \\cdot x^3-8 \\cdot x^2-16 \\cdot x$ is increasing and where decreasing on the interval $[-4,4]$. Then determine the relative extrema of $f(x)$. Lastly, determine the absolute extrema of $f(x)$.", | |
| "response": "$f(x)$ is increasing at the interval: $(-4,-1)$ and is decreasing at the interval: $(-1,4)$\n\nLocal Minimum: $P\\left(-4,-\\frac{64}{3}\\right)$\n\nLocal Maximum:$P\\left(-1,\\frac{95}{12}\\right)$\n\nAbsolute Minimum:$P\\left(4,-\\frac{320}{3}\\right)$\n\nAbsolute Maximum:$P\\left(-1,\\frac{95}{12}\\right)$", | |
| "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": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ deg with the horizontal. At what time is the maximum range of the projectile attained? ", | |
| "response": "$t$ = $88.37$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 5 \\cdot \\pi \\cdot x, & 0 \\le x \\le 4 \\\\ 0, & x>4 \\end{cases}$.", | |
| "response": "$q(t)$ = $\\int_0^\\infty\\frac{\\left(5\\cdot\\left(4\\cdot\\sin\\left(4\\cdot\\alpha\\right)\\cdot\\alpha+\\cos\\left(4\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot t\\right)+5\\cdot\\left(\\sin\\left(4\\cdot\\alpha\\right)-4\\cdot\\alpha\\cdot\\cos\\left(4\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot t\\right)\\right)}{\\alpha^2}d \\alpha$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Solve the integral: $\\int{\\frac{ 1 }{ \\sin(x)^5 } d x}$.", | |
| "response": "This is the final answer to the problem: $C+\\frac{1}{16}\\cdot\\left(2\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^2+6\\cdot\\ln\\left(\\left|\\tan\\left(\\frac{x}{2}\\right)\\right|\\right)+\\frac{1}{4}\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^4-\\frac{2}{\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^2}-\\frac{1}{4\\cdot\\left(\\tan\\left(\\frac{x}{2}\\right)\\right)^4}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative $y_{x}'$ of the function: $x=\\arcsin\\left(\\frac{ t }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $y=\\arccos\\left(\\frac{ 1 }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $t \\ge 0$.", | |
| "response": "$y_{x}'$ =$\\frac{t\\cdot\\sqrt{t^2+2}}{\\sqrt{2\\cdot t^2+1}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A profit is earned when revenue exceeds the cost. Suppose the profit function for a skateboard manufacturer is given by $P(x)=30 \\cdot x-0.3 \\cdot x^2-250$, where $x$ is the number of skateboards sold.\n\n1. Find the exact profit from the sale of the thirtieth skateboard.\n2. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.", | |
| "response": "This is the final answer to the problem:\n\n1. the exact profit from the sale of the thirtieth skateboard: $12.3$\n2. the marginal profit function and the sale of the thirtieth skateboard: $30-0.6\\cdot x, 12.6$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A profit is earned when revenue exceeds the cost. Suppose the profit function for a skateboard manufacturer is given by $P(x)=30 \\cdot x-0.3 \\cdot x^2-250$, where $x$ is the number of skateboards sold.\n\n1. Find the exact profit from the sale of the thirtieth skateboard.\n2. Find the marginal profit function and use it to estimate the profit from the sale of the thirtieth skateboard.", | |
| "response": "This is the final answer to the problem:\n\n1. the exact profit from the sale of the thirtieth skateboard: $12.3$\n2. the marginal profit function and the sale of the thirtieth skateboard: $30-0.6\\cdot x, 12.6$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For the curve $x=a\\left(t-\\sin(t)\\right)$, $y=a\\left(1-\\cos(t)\\right)$ determine the curvature. Use $a=12$.", | |
| "response": "The curvature is:$\\frac{1}{48\\cdot\\left|\\sin\\left(\\frac{t}{2}\\right)\\right|}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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 $\\frac{ d y }{d x}$ if $y=\\frac{ 5 \\cdot x^2-3 \\cdot x }{ \\left(3 \\cdot x^7+2 \\cdot x^6\\right)^4 }$.", | |
| "response": "$\\frac{ d y }{d x}$ = $\\frac{-390\\cdot x^2+23\\cdot x+138}{x^{24}\\cdot(3\\cdot x+2)^5}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Make full curve sketching of $f(x)=\\frac{ 3 \\cdot x^3 }{ 2 \\cdot x^2-3 }$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptote(s)\n3. Horizontal asymptote(s)\n4. Slant asymptote(s)\n5. Interval(s) where the function is increasing\n6. Interval(s) where the function is decreasing\n7. Interval(s) where the function is concave up\n8. Interval(s) where the function is concave down\n9. Point(s) of inflection", | |
| "response": "This is the final answer to the problem:\n\n1. The domain (in interval notation) $\\left(-1\\cdot\\infty,-1\\cdot3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}}\\right)\\cup\\left(-1\\cdot3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}},3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}}\\right)\\cup\\left(3^{2^{-1}}\\cdot2^{-1\\cdot2^{-1}},\\infty\\right)$\n2. Vertical asymptote(s) $x=\\frac{\\sqrt{3}}{\\sqrt{2}}, x=-\\frac{\\sqrt{3}}{\\sqrt{2}}$\n3. Horizontal asymptote(s) None\n4. Slant asymptote(s) $y=\\frac{3}{2}\\cdot x$\n5. Interval(s) where the function is increasing $\\left(\\frac{3}{\\sqrt{2}},\\infty\\right), \\left(-\\infty,-\\frac{3}{\\sqrt{2}}\\right)$\n6. Interval(s) where the function is decreasing $\\left(-\\frac{3}{\\sqrt{2}},-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\frac{3}{\\sqrt{2}}\\right), \\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right)$\n7. Interval(s) where the function is concave up $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n8. Interval(s) where the function is concave down $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n9. Point(s) of inflection $P(0,0)$", | |
| "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": "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": "Given $x^3-2 \\cdot x^2 \\cdot y^2+5 \\cdot x+y-5=0$, evaluate $\\frac{d ^2y}{ d x^2}$ at $x=1$.", | |
| "response": "This is the final answer to the problem: $\\frac{d^2y}{dx^2}=-\\frac{238}{27}$ or $\\frac{d^2y}{dx^2}=-\\frac{319}{27}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 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": "Compute the integral $\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$.", | |
| "response": "$\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$ =$\\frac{1}{16}\\cdot\\left(2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2+2\\cdot\\ln\\left(4\\cdot x^2+1\\right)+8\\cdot x^2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2-8\\cdot x\\cdot\\arctan(2\\cdot x)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$.", | |
| "response": "$\\int{x \\cdot \\arctan(2 \\cdot x)^2 d x}$ =$\\frac{1}{16}\\cdot\\left(2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2+2\\cdot\\ln\\left(4\\cdot x^2+1\\right)+8\\cdot x^2\\cdot\\left(\\arctan(2\\cdot x)\\right)^2-8\\cdot x\\cdot\\arctan(2\\cdot x)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\sin\\left(x+z^y\\right)$.", | |
| "response": "$f_{xx}(x,y,z)$=$-\\sin\\left(x+z^y\\right)$ \n\n$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$-z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)$ \n\n$f_{yy}(x,y,z)$=$-z^y\\cdot\\left(\\ln(z)\\right)^2\\cdot\\left(-\\cos\\left(x+z^y\\right)+z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$z^{-1+y}\\cdot\\left(\\cos\\left(x+z^y\\right)\\cdot\\left(1+y\\cdot\\ln(z)\\right)-y\\cdot z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{zz}(x,y,z)$=$y\\cdot z^{-2+y}\\cdot\\left((-1+y)\\cdot\\cos\\left(x+z^y\\right)-y\\cdot z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$-y\\cdot z^{-1+y}\\cdot\\sin\\left(x+z^y\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\sin\\left(x+z^y\\right)$.", | |
| "response": "$f_{xx}(x,y,z)$=$-\\sin\\left(x+z^y\\right)$ \n\n$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$-z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)$ \n\n$f_{yy}(x,y,z)$=$-z^y\\cdot\\left(\\ln(z)\\right)^2\\cdot\\left(-\\cos\\left(x+z^y\\right)+z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$z^{-1+y}\\cdot\\left(\\cos\\left(x+z^y\\right)\\cdot\\left(1+y\\cdot\\ln(z)\\right)-y\\cdot z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{zz}(x,y,z)$=$y\\cdot z^{-2+y}\\cdot\\left((-1+y)\\cdot\\cos\\left(x+z^y\\right)-y\\cdot z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$-y\\cdot z^{-1+y}\\cdot\\sin\\left(x+z^y\\right)$", | |
| "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": "" | |
| } | |
| ] | |
| } |