| { |
| "dataset_name": "Mu-Math", |
| "group_index": 4, |
| "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": "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": "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": "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": "Compute the integral $-2 \\cdot \\int{x^{-4} \\cdot \\left(4+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$.", |
| "response": "$-2 \\cdot \\int{x^{-4} \\cdot \\left(4+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$ =$C+\\frac{1}{6}\\cdot\\left(\\frac{4}{x^2}+1\\right)\\cdot\\sqrt{\\frac{4}{x^2}+1}$", |
| "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 following equations: 1. $8.6=6 j+4 j$\n2. $12 z-(4 z+6)=82$\n3. $5.4 d-2.3 d+3 (d-4)=16.67$\n4. $2.6 f-1.3 (3 f-4)=6.5$\n5. $-5.3 m+(-3.9 m)-17=-94.28$\n6. $6 (3.5 y+4.2)-2.75 y=134.7$", |
| "response": "The solutions to the given equations are: 1. $j=0.86$\n2. $z=11$\n3. $d=\\frac{ 47 }{ 10 }$\n4. $f=-1$\n5. $m=\\frac{ 42 }{ 5 }$\n6. $y=6$", |
| "system": "" |
| }, |
| { |
| "prompt": "$f(x)=x+\\sin(2 \\cdot x)$ 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": "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": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.", |
| "response": "$f'(1)$ =$-0.5244$", |
| "system": "" |
| }, |
| { |
| "prompt": "Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours.", |
| "response": "There will be $7426$ leaves after 4 hours.", |
| "system": "" |
| }, |
| { |
| "prompt": "Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours.", |
| "response": "There will be $7426$ leaves after 4 hours.", |
| "system": "" |
| }, |
| { |
| "prompt": "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: $\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$.", |
| "response": "$\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$ =$\\frac{\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)^3}{2}+\\frac{3}{4}\\cdot\\left(\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)+\\frac{x}{2}\\right)+C$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the integral: $\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$.", |
| "response": "$\\int{\\cos\\left(\\frac{ x }{ 2 }\\right)^4 d x}$ =$\\frac{\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)^3}{2}+\\frac{3}{4}\\cdot\\left(\\sin\\left(\\frac{x}{2}\\right)\\cdot\\cos\\left(\\frac{x}{2}\\right)+\\frac{x}{2}\\right)+C$", |
| "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": "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": "Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-x-6 \\cdot y+z=3 \\cdot \\cos(-x-6 \\cdot y+z)$.\n\nSubmit as your final answer:\n\na. $\\frac{\\partial z}{\\partial x}$;\n\nb. $\\frac{\\partial z}{\\partial y}$.", |
| "response": "This is the final answer to the problem: \na. $1$;\n\nb. $6$.", |
| "system": "" |
| }, |
| { |
| "prompt": "Solve the initial value problem (find function $f$) for $f'(x)=\\frac{ 2 }{ x^2 }-\\frac{ x^2 }{ 2 }$, $f(1)=0$.", |
| "response": "$f(x)=\\frac{13}{6}-\\frac{2}{x}-\\frac{1}{6}\\cdot x^3$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the partial derivatives of the implicit function $z(x,y)$, given by the equation $-10 \\cdot x-9 \\cdot y+8 \\cdot z=4 \\cdot \\cos(-10 \\cdot x-9 \\cdot y+8 \\cdot z)$.\n\nSubmit as your final answer:\n\na. $\\frac{\\partial z}{\\partial x}$;\n\nb. $\\frac{\\partial z}{\\partial y}$.", |
| "response": "This is the final answer to the problem: \na. $\\frac{5}{4}$;\n\nb. $\\frac{9}{8}$.", |
| "system": "" |
| }, |
| { |
| "prompt": "Given $y=x^6-\\frac{ 9 }{ 2 } \\cdot x^5+\\frac{ 15 }{ 2 } \\cdot x^4-5 \\cdot x^3+10$ find where the function is concave up, down, and point(s) of inflection.", |
| "response": "Concave up:$(1,\\infty), (-\\infty,0)$Concave down:$(0,1)$Point(s) of Inflection:$P(1,9), P(0,10)$", |
| "system": "" |
| }, |
| { |
| "prompt": "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": "Find the derivative of $f(x)=\\frac{ 1 }{ 15 } \\cdot \\left(\\cos(x)\\right)^3 \\cdot \\left(\\left(\\cos(x)\\right)^2-5\\right)$.", |
| "response": "This is the final answer to the problem: $f'(x)=-\\frac{\\left(\\cos(x)\\right)^2}{15}\\cdot\\left(3\\cdot\\sin(x)\\cdot\\left(\\left(\\cos(x)\\right)^2-5\\right)+\\cos(x)\\cdot\\sin(2\\cdot x)\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the integral: $\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$.", |
| "response": "$\\int{\\sin\\left(\\frac{ x }{ 2 }\\right)^5 d x}$ =$-\\frac{2\\cdot\\sin\\left(\\frac{x}{2}\\right)^4\\cdot\\cos\\left(\\frac{x}{2}\\right)}{5}+\\frac{4}{5}\\cdot\\left(-\\frac{2}{3}\\cdot\\sin\\left(\\frac{x}{2}\\right)^2\\cdot\\cos\\left(\\frac{x}{2}\\right)-\\frac{4}{3}\\cdot\\cos\\left(\\frac{x}{2}\\right)\\right)+C$", |
| "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 $\\sqrt{2 \\cdot x-x^2}$ with the center $a=1$.", |
| "response": "$\\sqrt{2 \\cdot x-x^2}$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^{2\\cdot n}\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Solve the integral: $\\int{\\tan(x)^4 d x}$.", |
| "response": "$\\int{\\tan(x)^4 d x}$ = $C + x + 1/3 (sec^2(x) - 4) tan(x)$", |
| "system": "" |
| }, |
| { |
| "prompt": "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 $\\cos(2 \\cdot t)-5 \\cdot \\sin(t)-3=0$.", |
| "response": "This is the final answer to the problem: $t=(-1)^{n+1}\\cdot\\frac{\\pi}{6}+n\\cdot\\pi$", |
| "system": "" |
| }, |
| { |
| "prompt": "Solve $\\cos(2 \\cdot t)-5 \\cdot \\sin(t)-3=0$.", |
| "response": "This is the final answer to the problem: $t=(-1)^{n+1}\\cdot\\frac{\\pi}{6}+n\\cdot\\pi$", |
| "system": "" |
| }, |
| { |
| "prompt": "Solve the following equations: 1. $-10 c=-80$\n2. $n-(-6)=12$\n3. $-82+x=-20$\n4. $-\\frac{ r }{ 2 }=5$\n5. $r-3.4=7.1$\n6. $\\frac{ g }{ 2.5 }=1.8$\n7. $4.8 m=43.2$\n8. $\\frac{ 3 }{ 4 } t=\\frac{ 9 }{ 20 }$\n9. $3\\frac{2}{3}+m=5\\frac{1}{6}$", |
| "response": "The solutions to the given equations are: \n1. $c=8$\n2. $n=6$\n3. $x=62$\n4. $r=-10$\n5. $r=10.5$\n6. $g=\\frac{ 9 }{ 2 }$\n7. $m=9$\n8. $t=\\frac{3}{5}$\n9. $m=\\frac{3}{2}$", |
| "system": "" |
| }, |
| { |
| "prompt": "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": "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": "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": "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": "Find the derivative of the function $y=\\frac{ 3 \\cdot \\csc(x)-4 \\cdot \\sin(x) }{ 8 \\cdot \\left(\\cos(x)\\right)^5 }-\\frac{ 76 }{ 5 } \\cdot \\cot(3 \\cdot x)$.", |
| "response": "$y'$=$\\frac{228}{5\\cdot\\left(\\sin(3\\cdot x)\\right)^2}+\\frac{16\\cdot\\left(\\cos(x)\\right)^6-5\\cdot\\left(\\cos(x)\\right)^4-3\\cdot\\left(\\cos(x)\\right)^6\\cdot\\left(\\csc(x)\\right)^2}{8\\cdot\\left(\\cos(x)\\right)^{10}}$", |
| "system": "" |
| }, |
| { |
| "prompt": "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": "What are the points of inflection of the graph of $f(x)=\\frac{ x+1 }{ x^2+1 }$?", |
| "response": "This is the final answer to the problem: $x_1=1 \\land x_2=\\sqrt{3}-2 \\land x_3=-2-\\sqrt{3}$", |
| "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": "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": "The force of gravity $\\vec{F}$ acting on an object is given by $\\vec{F}=m \\cdot \\vec{g}$, where $m$ is the mass of the object (expressed in kilograms) and $\\vec{g}$ is acceleration resulting from gravity, with $\\left\\lVert\\vec{g}\\right\\rVert=9.8$ N/kg. A $2$-kg disco ball hangs by a chain from the ceiling of a room. \n\n1. Find the force of gravity $\\vec{F}$ acting on a disco ball and find its magnitude.\n2. Find the force of tension $\\vec{T}$ in the chain and its magnitude.\n\nExpress answers using standard unit vectors.", |
| "response": "1. $\\vec{F}$=$-19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{F}\\right\\rVert$=$19.6$\n2. $\\vec{T}$=$19.6\\cdot\\vec{k}$; $\\left\\lVert\\vec{T}\\right\\rVert$=$19.6$", |
| "system": "" |
| }, |
| { |
| "prompt": "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": "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": "Find the derivative of the function $y=\\arcsin\\left(\\sqrt{1-x^2}\\right)$.", |
| "response": "$y'$= $-\\frac{x}{|x|\\cdot\\sqrt{1-x^2}}$", |
| "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": "Calculate $\\sqrt[3]{30}$ with estimate error $0.001$, using series expansion.", |
| "response": "This is the final answer to the problem: $\\frac{755}{243}$", |
| "system": "" |
| }, |
| { |
| "prompt": "The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has $$84,000$$ subscribers at a quarterly charge of $$30\\space \\text{USD}$$. Market research has suggested that if the owners raise the price to $$34\\space \\text{USD}$$, they would lose $$9,000$$ subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?", |
| "response": "This is the final answer to the problem: $\\frac{101}{3}$", |
| "system": "" |
| }, |
| { |
| "prompt": "Make full curve sketching of $y=2 \\cdot \\arcsin\\left(\\frac{ 1-7 \\cdot x^2 }{ 1+7 \\cdot x^2 }\\right)$. Submit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation) $(-1\\cdot\\infty,\\infty)$\n2. Vertical asymptotes $\\text{None}$\n3. Horizontal asymptotes $y=-\\pi$\n4. Slant asymptotes $\\text{None}$\n5. Intervals where the function is increasing $(-\\infty,0)$\n6. Intervals where the function is decreasing $(0,\\infty)$\n7. Intervals where the function is concave up $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down $\\text{None}$\n9. Points of inflection $\\text{None}$", |
| "system": "" |
| }, |
| { |
| "prompt": "Use the table of integrals to evaluate the integral $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$. \n\nUse this link to access the table of integrals: [Table of Integrals](https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals)", |
| "response": "1. Submit the formula used: $\\int{\\left(\\cos(u)\\right)^3 d u}=\\frac{ 1 }{ 3 } \\cdot \\left(2+\\left(\\cos(u)\\right)^2\\right) \\cdot \\sin(u)+c, \\int{\\left(\\sin(u)\\right)^n \\cdot \\left(\\cos(u)\\right)^m d u}=-\\frac{ \\left(\\sin(u)\\right)^{n-1} \\cdot \\left(\\cos(u)\\right)^{m+1} }{ n+m }+\\frac{ n-1 }{ n+m } \\cdot \\int{\\left(\\sin(u)\\right)^{n-2} \\cdot \\left(\\cos(u)\\right)^m d u}$ (For example: to evaluate $\\int{(x+3)^2 d x}$ you would use and submit the formula $\\int{u^n d u}=\\frac{ u^{n+1} }{ n+1 }+C$).\n2. $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$=$-\\frac{\\sin(y)\\cdot\\left(\\cos(y)\\right)^4}{5}+\\frac{1}{5}\\cdot\\frac{1}{3}\\cdot\\left(2+\\left(\\cos(y)\\right)^2\\right)\\cdot\\sin(y)+c$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the integral $\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$.", |
| "response": "$\\int{\\frac{ \\sqrt{4+x^2} }{ x } d x}$ =$C+\\sqrt{4+x^2}+\\ln\\left(\\left|\\frac{\\sqrt{4+x^2}-2}{2+\\sqrt{4+x^2}}\\right|\\right)$", |
| "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": "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 particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.", |
| "response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$", |
| "system": "" |
| }, |
| { |
| "prompt": "A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t)=6-\\frac{ 5 }{ t^2+1 }$, where $t$ is time measured in hours since a circle of a $1$-cm radius of the bacterium was put into the culture.\n\n1. Express the area of the bacteria, $A(t)$, as a function of time.\n2. Find the exact and approximate area of the bacterial culture in $3$ hours.\n3. Express the circumference of the bacteria, $C(t)$, as a function of time.\n4. Find the exact and approximate circumference of the bacteria in $3$ hours.", |
| "response": "This is the final answer to the problem:\n\n1. $A(t)$ = $\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)^2$square centimeters\n2. The exact area of the bacterial culture in $3$ hours = $\\frac{121}{4}\\cdot\\pi$ square centimeters. The approximate area of the bacterial culture in $3$ hours = $95.03317777$ square centimeters.\n3. $C(t)$ = $2\\cdot\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)$centimeters\n4. The exact circumference of the bacteria in $3$ hours = $11\\cdot\\pi$ centimeters. The approximate circumference of the bacteria in $3$ hours = $34.55751919$ centimeters.", |
| "system": "" |
| }, |
| { |
| "prompt": "The function $s(t)=\\frac{ t }{ 1+t^2 }$ represents the position of a particle traveling along a horizontal line.\n\n1. Find the velocity and acceleration functions.\n2. Determine the time intervals when the object is slowing down or speeding up.", |
| "response": "This is the final answer to the problem: 1. The velocity function $v(t)$ = $\\frac{1-t^2}{\\left(1+t^2\\right)^2}$ and acceleration function $a(t)$ = $\\frac{2\\cdot t\\cdot\\left(t^2-3\\right)}{\\left(1+t^2\\right)^3}$.\n2. The time intervals when the object speeds up $\\left(1,\\sqrt{3}\\right)$ and slows down $\\left(\\sqrt{3},\\infty\\right), (0,1)$.", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the antiderivative of $-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} }$.", |
| "response": "$\\int{-\\frac{ 1 }{ x \\cdot \\sqrt{1-x^2} } d x}$ =$C+\\ln\\left(\\frac{1+\\sqrt{1-x^2}}{|x|}\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "The electrical resistance $R$ produced by wiring resistors $R_{1}$ and $R_{2}$ in parallel can be calculated from the formula $\\frac{ 1 }{ R }=\\frac{ 1 }{ R_{1} }+\\frac{ 1 }{ R_{2} }$. If $R_{1}$ and $R_{2}$ are measured to be $7$ ohm and $6$ ohm respectively, and if these measurements are accurate to within $0.05$ ohm, estimate the maximum possible error in computing $R$.", |
| "response": "Maximum possible error:$0.02514793$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute $\\sqrt[3]{130}$ with accuracy $0.0001$.", |
| "response": "This is the final answer to the problem: $5.0658$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute $\\sqrt[3]{130}$ with accuracy $0.0001$.", |
| "response": "This is the final answer to the problem: $5.0658$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the derivative of the function: $f\\left(\\alpha\\right)=\\ln\\left(\\sqrt{\\frac{ 3-\\sin\\left(\\alpha\\right) }{ 4+2 \\cdot \\sin\\left(\\alpha\\right) }}\\right)$ at $\\alpha=\\frac{ \\pi }{ 4 }$.", |
| "response": "$f'\\left(\\frac{ \\pi }{ 4 }\\right)$ =$-0.2848$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the derivative of the function: $f\\left(\\alpha\\right)=\\ln\\left(\\sqrt{\\frac{ 3-\\sin\\left(\\alpha\\right) }{ 4+2 \\cdot \\sin\\left(\\alpha\\right) }}\\right)$ at $\\alpha=\\frac{ \\pi }{ 4 }$.", |
| "response": "$f'\\left(\\frac{ \\pi }{ 4 }\\right)$ =$-0.2848$", |
| "system": "" |
| }, |
| { |
| "prompt": "An airplane’s Mach number $M$ is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\\mu=2 \\cdot \\arcsin\\left(\\frac{ 1 }{ M }\\right)$. Find the Mach angles for the following Mach numbers.\n\n1. $M=1.4$\n2. $M=2.8$\n3. $M=4.3$", |
| "response": "This is the final answer to the problem: \n\n1. $1.59120591$\n2. $0.73041444$\n3. $0.46941423$", |
| "system": "" |
| }, |
| { |
| "prompt": "An airplane’s Mach number $M$ is the ratio of its speed to the speed of sound. When a plane is flying at a constant altitude, then its Mach angle is given by $\\mu=2 \\cdot \\arcsin\\left(\\frac{ 1 }{ M }\\right)$. Find the Mach angles for the following Mach numbers.\n\n1. $M=1.4$\n2. $M=2.8$\n3. $M=4.3$", |
| "response": "This is the final answer to the problem: \n\n1. $1.59120591$\n2. $0.73041444$\n3. $0.46941423$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the moment of inertia of one arch of the cycloid $x=3 \\cdot a \\cdot \\left(\\frac{ t }{ 2 }-\\sin\\left(\\frac{ t }{ 2 }\\right)\\right)$, $y=3 \\cdot a \\cdot \\left(1-\\cos\\left(\\frac{ t }{ 2 }\\right)\\right)$ relative to the x-axis.", |
| "response": "Moment of Inertia: $\\frac{1152}{5}\\cdot a^3$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 4 }+\\frac{ x }{ 1 \\cdot 5 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .", |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{6 - 6 \\cdot e^x + 6 \\cdot e^x \\cdot x - 3 \\cdot e^x \\cdot x^2 + e^x \\cdot x^3}{x^4},&x\\ne0\\\\\\frac{1}{4},&x=0\\end{cases}$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the Fourier series expansion of the function $f(x)=\\begin{cases} x, & -\\pi \\le x<0 \\\\ \\pi, & 0 \\le x \\le \\pi \\end{cases}$ with the period $2 \\cdot \\pi$ at interval $[-\\pi,\\pi]$.", |
| "response": "The Fourier series is: $f(x)=\\frac{\\pi}{4}+\\sum_{n=1}^\\infty\\left(\\frac{1+(-1)^{n+1}}{\\pi\\cdot n^2}\\cdot\\cos(n\\cdot x)+\\frac{(-1)^{n+1}\\cdot2+1}{n}\\cdot\\sin(n\\cdot x)\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find zeros of $f(x)=\\sin(x)+\\sin(2 \\cdot x)+2 \\cdot \\sin(x) \\cdot \\sin(2 \\cdot x)-2 \\cdot \\cos(x)-\\cos(2 \\cdot x)$.", |
| "response": "This is the final answer to the problem: $x_1=-\\frac{\\pi}{2}+2\\cdot\\pi\\cdot n, x_2=-\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_3=\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_4=(-1)^n\\cdot\\frac{\\pi}{6}+\\pi\\cdot n$", |
| "system": "" |
| }, |
| { |
| "prompt": "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": "Find the Fourier series of the function $\\psi(x)=e^{-x}$ in the interval $(-2 \\cdot \\pi,\\pi \\cdot 2)$.", |
| "response": "The Fourier series is: $e^{-x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{4}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n}{4+n^2}\\cdot\\left(2\\cdot\\cos\\left(\\frac{n}{2}\\cdot x\\right)+n\\cdot\\sin\\left(\\frac{n}{2}\\cdot x\\right)\\right)\\right)\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the Fourier series 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": "Consider the differential equation $\\frac{ d y }{d x}=e^y \\cdot (5 \\cdot x-1)$. Find $y=g(x)$, the particular solution to the differential equation for $-0.819 \\le x \\le 1.219$ that passes through the point $P(1,0)$.", |
| "response": "$y$ = $-\\ln\\left(-\\frac{5}{2}\\cdot x^2+x+\\frac{5}{2}\\right)$", |
| "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": "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": "Given $g(x)=\\frac{ 1 }{ 3 } \\cdot (a+b) \\cdot x^3+\\frac{ 1 }{ 2 } \\cdot (a+b+c) \\cdot x^2-(a+b+c+d) \\cdot x+a \\cdot b \\cdot c \\cdot d$, simplify the derivative of $g(x)$ if $x^2+x=a+b$.", |
| "response": "This is the final answer to the problem: $g'(x)=(a+b)^2+c\\cdot x-(a+b+c+d)$", |
| "system": "" |
| }, |
| { |
| "prompt": "For the function $\\varphi(x)=(10-x) \\cdot \\sqrt{x^2+8}$ specify the points where local maxima and minima of $\\varphi(x)$ 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: \n\n1. The point(s) where local maxima occur $P\\left(4,12\\cdot\\sqrt{6}\\right)$\n2. The point(s) where local minima occur $P(1,27)$", |
| "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": "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": "Compute the integral: $\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$.", |
| "response": "$\\int{\\frac{ \\sqrt{25-x^2} }{ 4 \\cdot x^2 } d x}$ =$C+\\frac{1}{4}\\arccos\\left(\\frac{x}{5}\\right)-\\frac{1}{4}\\tan\\left(\\arccos\\left(\\frac{x}{5}\\right)\\right)$", |
| "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": "Compute the derivative of the complex function $p=u^v$ given $u=3 \\cdot \\ln(x-2 \\cdot y)$ and $v=e^{\\frac{ x }{ y }}$.", |
| "response": "$\\frac{\\partial p}{\\partial x} = (3 \\ln(x - 2y))^{e^{\\frac{x}{y}}} \\left[ \\frac{e^{\\frac{x}{y}}}{y} \\ln(3 \\ln(x - 2y)) + \\frac{e^{\\frac{x}{y}}}{(x - 2y) \\ln(x - 2y)} \\right]$\n\n$\\frac{\\partial p}{\\partial y} = (3 \\ln(x - 2y))^{e^{\\frac{x}{y}}} \\left[ -\\frac{x e^{\\frac{x}{y}}}{y^2} \\ln(3 \\ln(x - 2y)) - \\frac{2e^{\\frac{x}{y}}}{(x - 2y) \\ln(x - 2y)} \\right]$\n", |
| "system": "" |
| }, |
| { |
| "prompt": "Find points on a coordinate plane that satisfy the following equation:\n\n$10 \\cdot x^2+29 \\cdot y^2+34 \\cdot x \\cdot y+8 \\cdot x+14 \\cdot y+2=0$", |
| "response": "This is the final answer to the problem: $(3,-2)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 3 }{ 2 }+\\frac{ 3 \\cdot x }{ 1 \\cdot 3 }+\\frac{ 3 \\cdot x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ 3 \\cdot x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .", |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{3}{x^2}+\\frac{3\\cdot x\\cdot e^x-3\\cdot e^x}{x^2},&x\\ne0\\\\\\frac{3}{2},&x=0\\end{cases}$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the radius of convergence and sum of the series: $\\frac{ 3 }{ 2 }+\\frac{ 3 \\cdot x }{ 1 \\cdot 3 }+\\frac{ 3 \\cdot x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ 3 \\cdot x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .", |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{3}{x^2}+\\frac{3\\cdot x\\cdot e^x-3\\cdot e^x}{x^2},&x\\ne0\\\\\\frac{3}{2},&x=0\\end{cases}$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the tangential and normal components of acceleration for $\\vec{r}(t)=\\left\\langle a \\cdot \\cos\\left(\\omega \\cdot t\\right),b \\cdot \\sin\\left(\\omega \\cdot t\\right) \\right\\rangle$ at $t=0$ with positive coefficients.", |
| "response": "This is the final answer to the problem:\n\n$a_{t}$: $0$ \n\n$a_{N}$: $\\omega^2\\cdot|a|$", |
| "system": "" |
| }, |
| { |
| "prompt": "Find the tangential and normal components of acceleration for $\\vec{r}(t)=\\left\\langle a \\cdot \\cos\\left(\\omega \\cdot t\\right),b \\cdot \\sin\\left(\\omega \\cdot t\\right) \\right\\rangle$ at $t=0$ with positive coefficients.", |
| "response": "This is the final answer to the problem:\n\n$a_{t}$: $0$ \n\n$a_{N}$: $\\omega^2\\cdot|a|$", |
| "system": "" |
| }, |
| { |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x \\cdot \\sqrt{2-x^2}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation): $\\left[-1\\cdot2^{2^{-1}},2^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $(-1,1)$\n6. Intervals where the function is decreasing: $\\left(-2^{2^{-1}},-1\\right), \\left(1,2^{2^{-1}}\\right)$\n7. Intervals where the function is concave up: $\\left(-2^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,2^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Determine the Taylor series for $f(x)=\\frac{ 2 \\cdot x-1 }{ x^2-3 \\cdot x+2 }$, centered at $x_{0}=4$. Write out the sum of the first four non-zero terms, followed by dots.", |
| "response": "This is the final answer to the problem: $\\frac{7}{6}+\\left(\\frac{1}{3^2}-\\frac{3}{2^2}\\right)\\cdot(x-4)-\\left(\\frac{1}{3^3}-\\frac{3}{2^3}\\right)\\cdot(x-4)^2+\\left(\\frac{1}{3^4}-\\frac{3}{2^4}\\right)\\cdot(x-4)^3+\\cdots$", |
| "system": "" |
| }, |
| { |
| "prompt": "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 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": "Compute the integral: $\\int{\\frac{ 1 }{ \\sin(x)^6 } d x}$.", |
| "response": "$\\int{\\frac{ 1 }{ \\sin(x)^6 } d x}$ =$-\\frac{\\cos(x)}{5\\cdot\\sin(x)^5}+\\frac{4}{5}\\cdot\\left(-\\frac{\\cos(x)}{3\\cdot\\sin(x)^3}-\\frac{2}{3}\\cdot\\cot(x)\\right)$", |
| "system": "" |
| }, |
| { |
| "prompt": "Determine a definite integral that represents the region enclosed by the inner loop of $r=3-4 \\cdot \\cos\\left(\\theta\\right)$ .", |
| "response": "This is the final answer to the problem:$\\int_0^{\\arccos\\left(\\frac{3}{4}\\right)}\\left(3-4\\cdot\\cos\\left(\\theta\\right)\\right)^2d\\theta$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$.", |
| "response": "$\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$ =$\\frac{1}{2}\\cdot\\ln\\left(\\left|2\\cdot x+1+\\sqrt{4\\cdot x^2+4\\cdot x+3}\\right|\\right)+C$", |
| "system": "" |
| }, |
| { |
| "prompt": "Compute the integral $\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$.", |
| "response": "$\\int{\\frac{ 1 }{ \\sqrt{4 \\cdot x^2+4 \\cdot x+3} } d x}$ =$\\frac{1}{2}\\cdot\\ln\\left(\\left|2\\cdot x+1+\\sqrt{4\\cdot x^2+4\\cdot x+3}\\right|\\right)+C$", |
| "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": "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": "Use integration by substitution and/or by parts to compute the integral \n $\\int{x \\cdot \\ln(5+x) d x}$.", |
| "response": "This is the final answer to the problem:$D+5\\cdot(x+5)+\\left(\\frac{1}{2}\\cdot(x+5)^2-5\\cdot(x+5)\\right)\\cdot\\ln(x+5)-\\frac{1}{4}\\cdot(x+5)^2$", |
| "system": "" |
| }, |
| { |
| "prompt": "Use the second derivative test to identify any critical points of the function $f(x,y)=x^2 \\cdot y^2$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these.", |
| "response": "Maximum:NoneMinimum: $P(0,0)$ \n\nSaddle point: None\n\nThe second derivative test is inconclusive at: None", |
| "system": "" |
| }, |
| { |
| "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": "" |
| } |
| ] |
| } |
