| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 5, | |
| "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 derivative of the function $y=\\sqrt{\\frac{ x^5 \\cdot \\left(2 \\cdot x^6+3\\right) }{ \\sqrt[3]{1-2 \\cdot x} }}$ by taking the natural log of both sides of the equation.", | |
| "response": "Derivative: $y'=\\frac{-128\\cdot x^7+66\\cdot x^6-84\\cdot x+45}{-24\\cdot x^8+12\\cdot x^7-36\\cdot x^2+18\\cdot x}\\cdot\\sqrt{\\frac{x^5\\cdot\\left(2\\cdot x^6+3\\right)}{\\sqrt[3]{1-2\\cdot x}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "1. On a map, the scale is 1/2 inch : 25 miles. What is the actual distance between two cities that are 3 inches apart?\n\n2. What are the dimensions of a 15 foot by 10 foot room on a blueprint with a scale of 1.5 inches : 2 feet?\n\n3. The distance between Newark, NJ and San Francisco, CA is 2,888 miles. How far would that measure on a map with a scale of 3 cm to 1,000 miles?\n\n4. If the scale for a model train is 1 inch for every 10 feet of a real train, how big is the model for an 83-foot train?", | |
| "response": "1. $150$ miles\n\n2. $11.25$ feet by $7.5$ feet\n\n3. $8.664$ cm\n\n4. $8.3$ inches", | |
| "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": "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": "Determine the intervals of upward concavity and downward concavity for the function $f(x)=e^x \\cdot \\left(x^2-1\\right)$.", | |
| "response": "1. Interval(s) of upward concavity: $\\left(\\frac{2\\cdot\\sqrt{3}-4}{2},\\infty\\right), \\left(-\\infty,-\\frac{4+2\\cdot\\sqrt{3}}{2}\\right)$\n2. Interval(s) of downward concavity: $\\left(-\\frac{4+2\\cdot\\sqrt{3}}{2},\\frac{2\\cdot\\sqrt{3}-4}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Region $R$ is the region in the first quadrant bounded by the graphs of $y=2 \\cdot x$ and $y=x^2$.\n\n1. Write, but do not evaluate, an integral expression that gives the volume of the solid generated from revolving the region $R$ about the vertical line $x=3$.\n2. Write, but do not evaluate, an integral expression that gives the volume of the solid generated from revolving the region $R$ about the vertical line $x=-2$.", | |
| "response": "1. $V$ = $\\int_0^4\\pi\\cdot\\left(\\left(3-\\frac{1}{2}\\cdot y\\right)^2-\\left(3-\\sqrt{y}\\right)^2\\right)dy$\n2. $V$ = $\\int_0^4\\pi\\cdot\\left(\\left(-2-\\sqrt{y}\\right)^2-\\left(-2-\\frac{1}{2}\\cdot y\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve the integral: $\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$.", | |
| "response": "$\\int{\\left(\\frac{ x+4 }{ x-4 }\\right)^{\\frac{ 3 }{ 2 }} d x}$ =$C+\\sqrt{\\frac{x+4}{x-4}}\\cdot(x-20)-12\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x-4}-\\sqrt{x+4}}{\\sqrt{x-4}+\\sqrt{x+4}}\\right|\\right)$", | |
| "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": "Solve the following equations: 1. $13 x+6=6$ 2. $\\frac{ x }{ -4 }+11=5$ 3. $-4.5 x+12.3=-23.7$ 4. $\\frac{ x }{ 5 }+4=4.3$ 5. $-\\frac{ x }{ 3 }+(-7.2)=-2.1$ 6. $5.4 x-8.3=14.38$ 7. $\\frac{ x }{ 3 }-14=-8$", | |
| "response": "The solutions to the given equations are: 1. $x=0$\n2. $x=24$\n3. $x=8$\n4. $x=\\frac{ 3 }{ 2 }$\n5. $x=\\frac{ -153 }{ 10 }$\n6. $x=\\frac{ 21 }{ 5 }$\n7. $x=18$", | |
| "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": "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": "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": "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": "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": "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": "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": "$f(x)=\\frac{ 1 }{ 4 } \\cdot \\sqrt{x}+\\frac{ 1 }{ x }$, $x>0$. 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: $\\int{\\frac{ \\sqrt{x+10}+3 }{ (x+10)^2-\\sqrt{x+10} } d x}$.", | |
| "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": "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 the function $\\sqrt{x}$ with the center $a=9$.", | |
| "response": "$\\sqrt{x}$ =$\\sum_{n=0}^\\infty\\left(3^{1-2\\cdot n}\\cdot C_n^{\\frac{1}{2}}\\cdot(x-9)^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $f(x)=\\frac{ 1 }{ 15 } \\cdot \\left(\\cos(x)\\right)^3 \\cdot \\left(\\left(\\cos(x)\\right)^2-5\\right)$.", | |
| "response": "This is the final answer to the problem: $f'(x)=-\\frac{\\left(\\cos(x)\\right)^2}{15}\\cdot\\left(3\\cdot\\sin(x)\\cdot\\left(\\left(\\cos(x)\\right)^2-5\\right)+\\cos(x)\\cdot\\sin(2\\cdot x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ x^4 }{ 4 }-\\frac{ 11 }{ 3 } \\cdot x^3+15 \\cdot x^2+17$.", | |
| "response": "The point(s) where the function has a local minimum:$P(6,89), P(0,17)$ \nThe point(s) where the function has a local maximum:$P\\left(5,\\frac{1079}{12}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "A circle centered at $(-3,5)$ passes through the point $(5,-3)$. What is the equation of the circle in the general form?", | |
| "response": "This is the final answer to the problem: $x^2+y^2+6\\cdot x-10\\cdot y-94=0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Expand function $y=\\sin\\left(\\frac{ \\pi \\cdot x }{ 4 }\\right)$ in Taylor series at $x=2$, if $\\cos(x)=\\sum_{n=0}^\\infty\\left((-1)^n \\cdot \\frac{ x^{2 \\cdot n} }{ (2 \\cdot n)! }\\right)$.", | |
| "response": "This is the final answer to the problem: $\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{\\left(\\frac{\\pi}{4}\\cdot(x-2)\\right)^{2\\cdot n}}{(2\\cdot n)!}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Expand function $y=\\sin\\left(\\frac{ \\pi \\cdot x }{ 4 }\\right)$ in Taylor series at $x=2$, if $\\cos(x)=\\sum_{n=0}^\\infty\\left((-1)^n \\cdot \\frac{ x^{2 \\cdot n} }{ (2 \\cdot n)! }\\right)$.", | |
| "response": "This is the final answer to the problem: $\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{\\left(\\frac{\\pi}{4}\\cdot(x-2)\\right)^{2\\cdot n}}{(2\\cdot n)!}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Solve $\\left(\\sin(x)\\right)^2+\\left(\\cos(3 \\cdot x)\\right)^2=1$.", | |
| "response": "This is the final answer to the problem: $x=\\frac{n\\cdot\\pi}{4}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local extrema of the function $f(x)=2 \\cdot \\left(x^2\\right)^{\\frac{ 1 }{ 3 }}-x^2$ using the First Derivative Test.", | |
| "response": "Local maxima:$x=\\sqrt[4]{\\frac{8}{27}}, x=-\\sqrt[4]{\\frac{8}{27}}$\n\nLocal minima:$x=0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\left(\\cos(5 \\cdot x)\\right)^3 } d x}$ =$C+\\frac{\\sin(5\\cdot x)}{10\\cdot\\left(\\cos(5\\cdot x)\\right)^2}+\\frac{1}{10}\\cdot\\ln\\left(\\left|\\tan\\left(\\left(\\frac{5}{2}\\right)\\cdot x+\\frac{\\pi}{4}\\right)\\right|\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 5 \\cdot (x+2)^2 }$. \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: $y=\\frac{x}{5}-\\frac{4}{5}$\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$y=\\frac{ x^3 }{ 5 \\cdot (x+2)^2 }$. \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: $y=\\frac{x}{5}-\\frac{4}{5}$\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": "|$n$ \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": "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": "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": "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": "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": "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": "Evaluate the integral: $\\int_{0}^{\\frac{ 1 }{ 2 }}{\\sqrt[5]{1+x^3} d x}$ with explicitly guaranteed accuracy of $\\frac{ 1 }{ 100 }$ using power series expansion.", | |
| "response": "This is the final answer to the problem: $0.503$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral: $\\int_{0}^{\\frac{ 1 }{ 2 }}{\\sqrt[5]{1+x^3} d x}$ with explicitly guaranteed accuracy of $\\frac{ 1 }{ 100 }$ using power series expansion.", | |
| "response": "This is the final answer to the problem: $0.503$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function $y=\\frac{ 2 \\cdot \\csc(x)-7 \\cdot \\sin(x) }{ 4 \\cdot \\left(\\cos(x)\\right)^5 }-\\frac{ 3 }{ 5 } \\cdot \\cot(2 \\cdot x)$.", | |
| "response": "$y'$=$\\frac{6}{5\\cdot\\left(\\sin(2\\cdot x)\\right)^2}+\\frac{28\\cdot\\left(\\cos(x)\\right)^6-25\\cdot\\left(\\cos(x)\\right)^4-2\\cdot\\left(\\cos(x)\\right)^6\\cdot\\left(\\csc(x)\\right)^2}{4\\cdot\\left(\\cos(x)\\right)^{10}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate integral: $\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$. $\\left(M=4\\right)$, $\\left(N=5\\right)$, $\\left(p=2\\right)$, $\\left(q=9\\right)$, $\\left(m=2\\right)$.", | |
| "response": "$\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$ =$C+\\frac{x+1}{128+16\\cdot(x+1)^2}+\\frac{\\sqrt{2}}{64}\\cdot\\arctan\\left(\\frac{1}{2\\cdot\\sqrt{2}}\\cdot(x+1)\\right)-\\frac{2}{8+(x+1)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral using the Substitution Rule:\n\n$\\int{\\frac{ x^2+3 }{ \\sqrt{(2 \\cdot x-5)^3} } d x}$", | |
| "response": "This is the final answer to the problem: $C+\\frac{60\\cdot x+(2\\cdot x-5)^2-261}{12\\cdot\\sqrt{2\\cdot x-5}}$", | |
| "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 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": "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": "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 series expansion of the function $f(x)=\\frac{ x }{ 2 }$ with the period $4$ at interval $[-2,2]$.", | |
| "response": "The Fourier series is: $f(x)=\\sum_{n=1}^\\infty\\left(\\frac{(-1)^{n+1}\\cdot2}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of the function: $y=-4 \\cdot x^{\\sqrt{5 \\cdot x}}$.", | |
| "response": "$\\frac{ d y }{d x}$ =$-\\left(\\frac{4\\cdot\\sqrt{5}}{\\sqrt{x}}+\\frac{2\\cdot\\sqrt{5}\\cdot\\ln(x)}{\\sqrt{x}}\\right)\\cdot x^{\\sqrt{5}\\cdot\\sqrt{x}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Given two functions $f(x)=\\sqrt{x^2-1}$ and $g(x)=\\sqrt{3-x}$\n\n1. compute $f\\left(g(x)\\right)$\n2. compute $\\frac{ f(x) }{ g(x) }$ and find the domain of the new function.", | |
| "response": "1. the new function $f\\left(g(x)\\right)$ is$f\\left(g(x)\\right)=\\sqrt{2-x}$\n2. the function $\\frac{ f(x) }{ g(x) }$ is $\\frac{\\sqrt{x^2-1}}{\\sqrt{3-x}}$, and the domain for the function is $1\\le x<3 \\lor x\\le-1$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral $\\int{\\frac{ 6 \\cdot x^3-7 \\cdot x^2+3 \\cdot x-1 }{ 2 \\cdot x-3 \\cdot x^2 } d x}$.", | |
| "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": "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": "Compute the integral: $\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$.", | |
| "response": "$\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$ =$C+6\\cdot\\left(\\frac{1}{4}\\cdot\\sqrt[6]{x}^4+\\arctan\\left(\\sqrt[6]{x}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$.", | |
| "response": "$\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$ =$C+6\\cdot\\left(\\frac{1}{4}\\cdot\\sqrt[6]{x}^4+\\arctan\\left(\\sqrt[6]{x}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$.", | |
| "response": "$\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$ =$C+6\\cdot\\left(\\frac{1}{4}\\cdot\\sqrt[6]{x}^4+\\arctan\\left(\\sqrt[6]{x}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$.", | |
| "response": "$\\int{\\frac{ x+\\sqrt[3]{x^2}+\\sqrt[6]{x} }{ x \\cdot \\left(1+\\sqrt[3]{x}\\right) } d x}$ =$C+6\\cdot\\left(\\frac{1}{4}\\cdot\\sqrt[6]{x}^4+\\arctan\\left(\\sqrt[6]{x}\\right)\\right)$", | |
| "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": "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": "Let $R$ be the region in the first quadrant enclosed by the graph of $g(x)=\\frac{ 12 }{ 1+x^2 }-2$.\n\n1. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is revolved about the $x$-axis.\n2. Write, but do not evaluate, an integral expression that gives the volume of the solid generated when $R$ is revolved about the $y$-axis.", | |
| "response": "1. $\\int_0^{2.236}\\left(\\pi\\cdot\\left(\\frac{12}{1+x^2}-2\\right)^2\\right)dx$\n2. $\\int_0^{10}\\left(\\pi\\cdot\\left(\\frac{12}{y+2}-1\\right)\\right)dy$", | |
| "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": "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 function $s(t)=\\frac{ t }{ 1+t^2 }$ represents the position of a particle traveling along a horizontal line.\n\n1. Find the velocity and acceleration functions.\n2. Determine the time intervals when the object is slowing down or speeding up.", | |
| "response": "This is the final answer to the problem: 1. The velocity function $v(t)$ = $\\frac{1-t^2}{\\left(1+t^2\\right)^2}$ and acceleration function $a(t)$ = $\\frac{2\\cdot t\\cdot\\left(t^2-3\\right)}{\\left(1+t^2\\right)^3}$.\n2. The time intervals when the object speeds up $\\left(1,\\sqrt{3}\\right)$ and slows down $\\left(\\sqrt{3},\\infty\\right), (0,1)$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the antiderivative of $-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} }$.", | |
| "response": "$\\int{-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} } d x}$ =$C+\\ln\\left(\\frac{1+\\sqrt{1-x^2}}{|x|}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 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": "Evaluate the function at the indicated values $f(-3)$, $f(2)$, $f(-a)$,$-f(a)$, and $f(a+h)$. \n\n$f(x)=|x-1|-|x+1|$", | |
| "response": "This is the final answer to the problem: $f(-3)$=$2$\n\n$f(2)$=$-2$ \n\n$f(-a)$=$|a+1|-|-a+1|$ \n\n$-f(a)$= $-|a-1|+|a+1|$ \n\n$f(a+h)$=$|a+h-1|-|a+h+1|$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 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": "Compute the integral $\\int{\\left(2 \\cdot x^2+4 \\cdot x+5\\right) \\cdot \\sin(2 \\cdot x) d x}$.", | |
| "response": "$\\int{\\left(2 \\cdot x^2+4 \\cdot x+5\\right) \\cdot \\sin(2 \\cdot x) d x}$ =$C+x\\cdot\\sin(2\\cdot x)+\\frac{1}{2}\\cdot\\cos(2\\cdot x)+\\sin(2\\cdot x)-\\frac{1}{2}\\cdot\\left(2\\cdot x^2+4\\cdot x+5\\right)\\cdot\\cos(2\\cdot x)$", | |
| "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": "A sky diver jumps from a reasonable height above the ground. The air resistance she experiences is proportional to her velocity, and the constant of proportionality is $0.24$. It can be shown that the downward velocity of the sky diver at time $t$ is given by\n\n$v(t)=180 \\cdot \\left(1-e^{-0.24 \\cdot t}\\right)$ \n\nwhere $t$ is measured in seconds and $v(t)$ is measured in feet per second\n\n\n\n1. Find the initial velocity of the sky diver\n\n2. Find the velocity after $4$ seconds (round your answer to one decimal place)\n\n3. The maximum velocity of a falling object with wind resistance is called its terminal velocity. Find the terminal velocity of this sky diver. (round your answer to the nearest whole number)", | |
| "response": "1. $0$\n2. $111.1$\n3. $180$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Let $R$ be the region bounded by the graphs of $y=\\frac{ 1 }{ x+2 }$ and $y=-\\frac{ 1 }{ 2 } \\cdot x+3$.\n\nFind the volume of the solid generated when $R$ is rotated about the vertical line $x=-3$.", | |
| "response": "The volume of the solid is $292.097$ units³.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Let $Q$ be the region bounded by the graph of $x=\\frac{ 2 }{ 1-y }$, the line $x=-1$, and the line $y=\\frac{ 5 }{ 4 }$.\n\nWrite, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when $Q$ is revolved about the line $x=-1$.", | |
| "response": "$V$ = $\\int_{\\frac{5}{4}}^3\\left(\\pi\\cdot\\left(\\frac{2}{1-y}+1\\right)^2\\right)dy$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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": "Find the equations of the common tangent lines to the following ellipses:\n\n$\\frac{ x^2 }{ 6 }+y^2=1$ \n\n$\\frac{ x^2 }{ 4 }+\\frac{ y^2 }{ 9 }=1$", | |
| "response": "This is the final answer to the problem: $2\\cdot x+y-5=0 \\lor 2\\cdot x+y+5=0 \\lor 2\\cdot x-y-5=0 \\lor 2\\cdot x-y+5=0$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ 3 }{ 4 } \\cdot x^4-10 \\cdot x^3+24 \\cdot x^2-4$.", | |
| "response": "The point(s) where the function has a local minimum:$P(8,-516), P(0,-4)$ \nThe point(s) where the function has a local maximum:$P(2,24)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ 3 }{ 4 } \\cdot x^4-10 \\cdot x^3+24 \\cdot x^2-4$.", | |
| "response": "The point(s) where the function has a local minimum:$P(8,-516), P(0,-4)$ \nThe point(s) where the function has a local maximum:$P(2,24)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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": "For the function $y=\\frac{ 2 \\cdot x+3 }{ 4 \\cdot x+5 }$ find the derivative $y^{(n)}$ .", | |
| "response": "The General Form of the Derivative of $y=\\frac{ 2 \\cdot x+3 }{ 4 \\cdot x+5 }$: $y^{(n)}=\\frac{1}{2}\\cdot(-1)^n\\cdot\\left(n!\\right)\\cdot4^n\\cdot(4\\cdot x+5)^{-(n+1)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "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": "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": "Sketch the curve: \n\n$y=25 \\cdot x^2 \\cdot e^{\\frac{ 1 }{ 5 \\cdot x }}$ \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", | |
| "response": "1. The domain (in interval notation): $(-\\infty,0)\\cup(0,\\infty)$\n2. Vertical asymptotes: $x=0$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(\\frac{1}{10},\\infty\\right)$\n6. Intervals where the function is decreasing $\\left(0,\\frac{1}{10}\\right), (-\\infty,0)$\n7. Intervals where the function is concave up: $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down: None\n9. Points of inflection: None", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\arctan(x \\cdot y \\cdot z)$.", | |
| "response": "$f_{xx}(x,y,z)$=$\\frac{-2\\cdot x\\cdot y^3\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$\\frac{z-x^2\\cdot y^2\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yy}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$\\frac{x-x^3\\cdot y^2\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{zz}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y^3\\cdot z}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$\\frac{y-x^2\\cdot y^3\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "For the function: $f(x)=x^3+x^4$ determine\n\n1. intervals where $f$ is increasing or decreasing,\n2. local minima and maxima of $f$ ,\n3. intervals where $f$ is concave up and concave down, and\n4. the inflection points of $f$ .", | |
| "response": "1. Increasing over $\\left( -\\frac{3}{4}, 0 \\right) \\cup (0, \\infty)$ ; decreasing over $\\left(-\\infty,-\\frac{3}{4}\\right)$\n2. Local maxima at None ; local minima at $x=-\\frac{3}{4}$\n3. Concave up for $x>0, x<-\\frac{1}{2}$ ; concave down for $-\\frac{1}{2}<x<0$\n4. Inflection points at $P\\left(-\\frac{1}{2},-\\frac{1}{16}\\right); P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find 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 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": "" | |
| } | |
| ] | |
| } |