| { | |
| "dataset_name": "Mu-Math", | |
| "group_index": 7, | |
| "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": "Given $f(x)=x^2+\\frac{ 16 }{ x^2 }$, find the intervals where $f$ increases and the intervals where $f$ is decreasing.", | |
| "response": "$f$ is increasing on the intervals: $(2,\\infty), (-2,0)$\n\n$f$ is decreasing on the intervals: $(-\\infty,-2), (0,2)$ ", | |
| "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": "Find the points at which the following polar curve $r=4 \\cdot \\cos\\left(\\theta\\right)$ has a horizontal or vertical tangent line.", | |
| "response": "This is the final answer to the problem:\n\nhorizontal tangents at: $\\boxed{(2, 2), (2, -2)}$\n\nvertical tangents at: $\\boxed{(4, 0), (0, 0)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the radius of convergence and sum of the series: $\\frac{ 1 }{ 2 }+\\frac{ x }{ 1 \\cdot 3 }+\\frac{ x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .", | |
| "response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{1}{x^2}+\\frac{x\\cdot e^x-e^x}{x^2},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the 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": "Determine a definite integral that represents the region enclosed by one petal of $r=\\cos\\left(3 \\cdot \\theta\\right)$ .", | |
| "response": "This is the final answer to the problem:$\\int_0^{\\frac{\\pi}{6}}\\cos\\left(3\\cdot\\theta\\right)^2d\\theta$ = $\\frac{\\pi}{12}$ ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $u=\\left|\\sin\\left(\\frac{ x }{ 2 }\\right)\\right|$ in the interval $[-2 \\cdot \\pi,2 \\cdot \\pi]$.", | |
| "response": "The Fourier series is: $\\frac{2}{\\pi}+\\frac{4}{\\pi}\\cdot\\sum_{k=1}^\\infty\\left(\\frac{1}{\\left(1-4\\cdot k^2\\right)}\\cdot\\cos(k\\cdot x)\\right)$", | |
| "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": "Let $f(x)=\\ln\\left(x^2+1\\right)$.\n\n1. Find the 4th order Taylor polynomial $P_{4}(x)$ of $f(x)$ about $a=0$.\n2. Compute $\\left|f(1)-P_{4}(1)\\right|$.\n3. Compute $\\left|f(0.1)-P_{4}(0.1)\\right|$.", | |
| "response": "This is the final answer to the problem: \n\n1. $x^2-\\frac{x^4}{2}$\n2. $0.1931$\n3. $0.0003\\cdot10^{-3}$", | |
| "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": "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": "Solve the integral: $\\int{\\cot(x)^4 d x}$.", | |
| "response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the derivative of $f(x)=\\frac{ \\left(\\left(\\tan(x)\\right)^2-1\\right) \\cdot \\left(\\left(\\tan(x)\\right)^4+10 \\cdot \\left(\\tan(x)\\right)^2+1\\right) }{ 3 \\cdot \\left(\\tan(x)\\right)^3 }$.", | |
| "response": "This is the final answer to the problem: $f'(x)=\\left(\\tan(x)\\right)^4+4\\cdot\\left(\\tan(x)\\right)^2+\\frac{4}{\\left(\\tan(x)\\right)^2}+\\frac{1}{\\left(\\tan(x)\\right)^4}+6$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ \\pi \\cdot x, & 0 \\le x \\le 1 \\\\ 0, & x>1 \\end{cases}$.", | |
| "response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{\\left(\\alpha\\cdot\\sin\\left(\\alpha\\right)+\\cos\\left(\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+\\left(\\sin\\left(\\alpha\\right)-\\alpha\\cdot\\cos\\left(\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Solve the following equations: 1. $-10 c=-80$\n2. $n-(-6)=12$\n3. $-82+x=-20$\n4. $-\\frac{ r }{ 2 }=5$\n5. $r-3.4=7.1$\n6. $\\frac{ g }{ 2.5 }=1.8$\n7. $4.8 m=43.2$\n8. $\\frac{ 3 }{ 4 } t=\\frac{ 9 }{ 20 }$\n9. $3\\frac{2}{3}+m=5\\frac{1}{6}$", | |
| "response": "The solutions to the given equations are: \n1. $c=8$\n2. $n=6$\n3. $x=62$\n4. $r=-10$\n5. $r=10.5$\n6. $g=\\frac{ 9 }{ 2 }$\n7. $m=9$\n8. $t=\\frac{3}{5}$\n9. $m=\\frac{3}{2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The function $s(t)=2 \\cdot t^3-3 \\cdot t^2-12 \\cdot t+8$ 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 $v(t)$ = $6\\cdot t^2-6\\cdot t-12$ and acceleration function $a(t)$ = $12\\cdot t-6$.\n\n2. The time intervals when the object speeds up $\\left(2,\\infty\\right), \\left(0,\\frac{1}{2}\\right)$ and slows down $\\left(\\frac{1}{2},2\\right)$.", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "The function $s(t)=2 \\cdot t^3-3 \\cdot t^2-12 \\cdot t+8$ 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 $v(t)$ = $6\\cdot t^2-6\\cdot t-12$ and acceleration function $a(t)$ = $12\\cdot t-6$.\n\n2. The time intervals when the object speeds up $\\left(2,\\infty\\right), \\left(0,\\frac{1}{2}\\right)$ and slows down $\\left(\\frac{1}{2},2\\right)$.", | |
| "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": "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": "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": "Solve the following system of equations:\n\n$\\sin(x) \\cdot \\sin(y)=\\frac{ \\sqrt{3} }{ 4 }$ \n\n$\\cos(x) \\cdot \\cos(y)=\\frac{ \\sqrt{3} }{ 4 }$", | |
| "response": "This is the final answer to the problem: $x=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$ or $x=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$, where $k$ and $n$ are integers ", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Sketch the curve: \n\n$y=5 \\cdot x^2-2 \\cdot x^4-3$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)\n3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)\n4. Slant asymptotes (Leave blank if there are no slant asymptotes)\n5. Intervals where the function is increasing (Leave blank if there are no such intervals)\n6. Intervals where the function is decreasing (Leave blank if there are no such intervals)\n7. Intervals where the function is concave up (Leave blank if there are no such intervals)\n8. Intervals where the function is concave down (Leave blank if there are no such intervals)\n9. Points of inflection (Leave blank if there are no 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: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(0,\\frac{\\sqrt{5}}{2}\\right), \\left(-\\infty,-\\frac{\\sqrt{5}}{2}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{5}}{2},0\\right), \\left(\\frac{\\sqrt{5}}{2},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right), \\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right), P\\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=5 \\cdot x^2-2 \\cdot x^4-3$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)\n3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)\n4. Slant asymptotes (Leave blank if there are no slant asymptotes)\n5. Intervals where the function is increasing (Leave blank if there are no such intervals)\n6. Intervals where the function is decreasing (Leave blank if there are no such intervals)\n7. Intervals where the function is concave up (Leave blank if there are no such intervals)\n8. Intervals where the function is concave down (Leave blank if there are no such intervals)\n9. Points of inflection (Leave blank if there are no 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: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(0,\\frac{\\sqrt{5}}{2}\\right), \\left(-\\infty,-\\frac{\\sqrt{5}}{2}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{5}}{2},0\\right), \\left(\\frac{\\sqrt{5}}{2},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}}\\right), \\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right), P\\left(\\frac{\\sqrt{5}}{2\\cdot\\sqrt{3}},-\\frac{91}{72}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\varphi(x)=2 \\cdot x$ in the interval $(0,4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $4\\cdot\\pi-8\\cdot\\sum_{n=1}^\\infty\\left(\\frac{\\sin\\left(\\frac{n\\cdot x}{2}\\right)}{n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $\\varphi(x)=2 \\cdot x$ in the interval $(0,4 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $4\\cdot\\pi-8\\cdot\\sum_{n=1}^\\infty\\left(\\frac{\\sin\\left(\\frac{n\\cdot x}{2}\\right)}{n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{29}$ with accuracy 0.0001.", | |
| "response": "This is the final answer to the problem: $3.0723$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Taylor series for $f(x)=\\frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $x\\cdot\\left(1-3\\cdot(x+1)+6\\cdot(x+1)^2-10\\cdot(x+1)^3+\\cdots\\right)$=\r\n= $-1 + 4 (x + 1) - 9 (x + 1)^2 + 16 (x + 1)^3$\r\n", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Write the Taylor series for the function $f(x)=x \\cdot \\cos(2 \\cdot x)$ at the point $x=\\frac{ \\pi }{ 2 }$ up to the third term (zero or non-zero).", | |
| "response": "This is the final answer to the problem: $-\\frac{\\pi}{2}-\\left(x-\\frac{\\pi}{2}\\right)+\\pi\\cdot\\left(x-\\frac{\\pi}{2}\\right)^2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the 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": "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 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 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": "Write the Taylor series for the function $f(x)=x \\cdot \\sin(2 \\cdot x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).", | |
| "response": "This is the final answer to the problem: $0+2\\cdot\\pi\\cdot(x-\\pi)+\\frac{4}{2}\\cdot(x-\\pi)^2$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the arc length of the curve $x=\\frac{ 5 \\cdot y^2 }{ 6 }-\\frac{ \\ln(5 \\cdot y) }{ 2 }$ enclosed between $y=3$ and $y=5$.", | |
| "response": "Arc Length: $-\\frac{\\operatorname{arsinh}\\left(\\frac{200y^{2} - 24}{12 \\sqrt{21}}\\right)}{10} + \\frac{\\sqrt{100y^{4} - 24y^{2} + 9}}{12} - \\frac{\\operatorname{arsinh}\\left(\\frac{3}{2 \\sqrt{21} \\, y^{2}} - \\frac{2}{\\sqrt{21}}\\right)}{4} \\approx \\boxed{13.23342845}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x^2-x^4-2$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)\n3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)\n4. Slant asymptotes (Leave blank if there are no slant asymptotes)\n5. Intervals where the function is increasing (Leave blank if there are no such intervals)\n6. Intervals where the function is decreasing (Leave blank if there are no such intervals)\n7. Intervals where the function is concave up (Leave blank if there are no such intervals)\n8. Intervals where the function is concave down (Leave blank if there are no such intervals)\n9. Points of inflection (Leave blank if there are no 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: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{1}{\\sqrt{2}}\\right), \\left(\\frac{1}{\\sqrt{2}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right), P\\left(\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Sketch the curve: \n\n$y=3 \\cdot x^2-x^4-2$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes (Leave blank if there are no vertical asymptotes)\n3. Horizontal asymptotes (Leave blank if there are no horizontal asymptotes)\n4. Slant asymptotes (Leave blank if there are no slant asymptotes)\n5. Intervals where the function is increasing (Leave blank if there are no such intervals)\n6. Intervals where the function is decreasing (Leave blank if there are no such intervals)\n7. Intervals where the function is concave up (Leave blank if there are no such intervals)\n8. Intervals where the function is concave down (Leave blank if there are no such intervals)\n9. Points of inflection (Leave blank if there are no 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: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(0,\\frac{\\sqrt{3}}{\\sqrt{2}}\\right), \\left(-\\infty,-\\frac{\\sqrt{3}}{\\sqrt{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(-\\frac{\\sqrt{3}}{\\sqrt{2}},0\\right), \\left(\\frac{\\sqrt{3}}{\\sqrt{2}},\\infty\\right)$\n7. Intervals where the function is concave up: $\\left(-\\frac{1}{\\sqrt{2}},\\frac{1}{\\sqrt{2}}\\right)$\n8. Intervals where the function is concave down: $\\left(-\\infty,-\\frac{1}{\\sqrt{2}}\\right), \\left(\\frac{1}{\\sqrt{2}},\\infty\\right)$\n9. Points of inflection: $P\\left(-\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right), P\\left(\\frac{1}{\\sqrt{2}},-\\frac{3}{4}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find a “reasonable” upper-bound on the error in approximating $f(x)=x \\cdot \\ln(x)$ by its 3rd order Taylor polynomial $P_{3}(x)$ at $a=1$ valid for all values of $x$ such that $|x-1| \\le 0.7$.", | |
| "response": "This is the final answer to the problem: $\\frac{2}{(0.3)^3}\\cdot\\frac{(0.7)^4}{4!}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "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": "Sketch the curve: \n\n$y=2 \\cdot x \\cdot \\sqrt{3-x^2}$. \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection", | |
| "response": "This is the final answer to the problem: \n1. The domain (in interval notation): $\\left[-1\\cdot3^{2^{-1}},3^{2^{-1}}\\right]$\n2. Vertical asymptotes: None\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(-\\sqrt{\\frac{3}{2}},\\sqrt{\\frac{3}{2}}\\right)$\n6. Intervals where the function is decreasing: $\\left(\\sqrt{\\frac{3}{2}},3^{2^{-1}}\\right), \\left(-3^{2^{-1}},-\\sqrt{\\frac{3}{2}}\\right)$\n7. Intervals where the function is concave up: $\\left(-3^{2^{-1}},0\\right)$\n8. Intervals where the function is concave down: $\\left(0,3^{2^{-1}}\\right)$\n9. Points of inflection: $P(0,0)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "On a cylinder 6 cm in diameter, a channel is cut out along the surface, having an equilateral triangle with a side of 1.5 cm in cross section. Compute the volume of the cut out material.", | |
| "response": "$V$ =$\\frac{108\\cdot\\sqrt{3}-27}{32}\\cdot\\pi$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Determine the Taylor series for $y=\\left(\\sin(x)\\right)^2$, centered at $x_{0}=\\frac{ \\pi }{ 2 }$. Write out the sum of the first three non-zero terms, followed by dots.", | |
| "response": "This is the final answer to the problem: $1-\\frac{2}{2!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^2+\\frac{2^3}{4!}\\cdot\\left(x-\\frac{\\pi}{2}\\right)^4+\\cdots$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$.", | |
| "response": "$\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$ =$C-\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{1-\\sin(x)}{1+\\sin(x)}\\right|\\right)-\\frac{1}{3}\\cdot\\left(\\sin(x)\\right)^3-\\sin(x)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the function $f(x)=\\frac{ -1 }{ 2 } \\cdot x$ in the interval $[-2,2]$.", | |
| "response": "The Fourier series is: $\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find $\\frac{ d y }{d x}$, given $y=\\tan(2 \\cdot v)$ and $v=\\arctan(2 \\cdot x-1)$.", | |
| "response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^2-2\\cdot x+1}{2\\cdot\\left(x-x^2\\right)^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the derivative of $y=x \\cdot \\sin(x)+2 \\cdot x \\cdot \\cos(x)-2 \\cdot \\sin(x)+\\ln\\left(\\sin(x)\\right)+c^2$.", | |
| "response": "This is the final answer to the problem: $y'=\\frac{\\left(\\sin(x)\\right)^2+x\\cdot\\sin(x)\\cdot\\cos(x)-2\\cdot x\\cdot\\left(\\sin(x)\\right)^2+\\cos(x)}{\\sin(x)}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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 $Q$ 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": "Evaluate $\\int\\int\\int_{E}{\\left(x^3+y^3+z^3\\right) d V}$, where $E$=$\\left\\{(x,y,z)|0 \\le x \\le 2,0 \\le y \\le 2 \\cdot x,0 \\le z \\le 4-x-y\\right\\}$.", | |
| "response": "$I$ = $\\frac{112}{5}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x^2+2}$ with the center $a=0$.", | |
| "response": "$\\sqrt{x^2+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^{2\\cdot n}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$ =$C+2\\cdot\\cot(6\\cdot x)+\\frac{2}{5}\\cdot\\left(\\cot(6\\cdot x)\\right)^5+\\frac{4}{3}\\cdot\\left(\\cot(6\\cdot x)\\right)^3$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the 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": "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": "Solve the integral: $\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$.", | |
| "response": "$\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$ =$C+\\frac{1}{21}\\cdot\\left(\\frac{1}{2}\\cdot\\left(\\cot(10\\cdot x)\\right)^4+\\frac{1}{3}\\cdot\\left(\\cot(10\\cdot x)\\right)^6\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the centers of symmetry of the curve of $f(x)=\\left(\\sin(x)\\right)^2$.", | |
| "response": "This is the final answer to the problem: $\\left(k\\cdot\\pi+\\frac{\\pi}{4},\\frac{1}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the centers of symmetry of the curve of $f(x)=\\left(\\sin(x)\\right)^2$.", | |
| "response": "This is the final answer to the problem: $\\left(k\\cdot\\pi+\\frac{\\pi}{4},\\frac{1}{2}\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral: $I=\\int{3 \\cdot x \\cdot \\ln\\left(4+\\frac{ 1 }{ x }\\right) d x}$.", | |
| "response": "This is the final answer to the problem: $\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(4\\cdot x+1)-\\frac{3\\cdot x^2}{4}+\\frac{3\\cdot x}{8}-\\frac{3}{32}\\cdot\\ln\\left(x+\\frac{1}{4}\\right)\\right)-\\left(\\frac{3}{2}\\cdot x^2\\cdot\\ln(x)-\\left(C+\\frac{3}{4}\\cdot x^2\\right)\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x+2}$ with the center $a=0$.", | |
| "response": "$\\sqrt{x+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^n\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$ =$\\frac{\\cos\\left(\\frac{x}{3}\\right)}{5\\cdot\\sin\\left(\\frac{x}{3}\\right)^5}-\\frac{4}{15}\\cdot\\left(-\\frac{\\cos\\left(\\frac{x}{3}\\right)}{\\sin\\left(\\frac{x}{3}\\right)^3}-2\\cdot\\cot\\left(\\frac{x}{3}\\right)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Compute the integral: $\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$.", | |
| "response": "$\\int{\\frac{ -1 }{ 3 \\cdot \\sin\\left(\\frac{ x }{ 3 }\\right)^6 } d x}$ =$\\frac{\\cos\\left(\\frac{x}{3}\\right)}{5\\cdot\\sin\\left(\\frac{x}{3}\\right)^5}-\\frac{4}{15}\\cdot\\left(-\\frac{\\cos\\left(\\frac{x}{3}\\right)}{\\sin\\left(\\frac{x}{3}\\right)^3}-2\\cdot\\cot\\left(\\frac{x}{3}\\right)\\right)+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative $y_{x}'$ of the function: $x=\\arcsin\\left(\\frac{ t }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $y=\\arccos\\left(\\frac{ 1 }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $t \\ge 0$.", | |
| "response": "$y_{x}'$ =$\\frac{t\\cdot\\sqrt{t^2+2}}{\\sqrt{2\\cdot t^2+1}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the first derivative $y_{x}'$ of the function: $x=\\arcsin\\left(\\frac{ t }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $y=\\arccos\\left(\\frac{ 1 }{ \\sqrt{2+2 \\cdot t^2} }\\right)$, $t \\ge 0$.", | |
| "response": "$y_{x}'$ =$\\frac{t\\cdot\\sqrt{t^2+2}}{\\sqrt{2\\cdot t^2+1}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the Fourier series of the periodic function $f(x)=x^2$ in the interval $-\\pi \\le x<\\pi$ if $f(x)=f(x+2 \\cdot \\pi)$.", | |
| "response": "The Fourier series is: $x^2=\\frac{\\pi^2}{3}-4\\cdot\\left(\\frac{\\cos(x)}{1^2}-\\frac{\\cos(2\\cdot x)}{2^2}+\\frac{\\cos(3\\cdot x)}{3^2}-\\cdots\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the integral $\\int{\\frac{ 1 }{ \\sqrt[3]{\\left(\\sin(x)\\right)^{11} \\cdot \\cos(x)} } d x}$.", | |
| "response": "$\\int{\\frac{ 1 }{ \\sqrt[3]{\\left(\\sin(x)\\right)^{11} \\cdot \\cos(x)} } d x}$ =$-\\frac{3\\cdot\\left(1+4\\cdot\\left(\\tan(x)\\right)^2\\right)}{8\\cdot\\left(\\tan(x)\\right)^2\\cdot\\sqrt[3]{\\left(\\tan(x)\\right)^2}}+C$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the 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 $\\frac{ d y }{d x}$ 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": "Evaluate the integral by choosing the order of integration: $\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$.", | |
| "response": "$\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$ =$\\frac{2\\cdot\\pi+8\\cdot\\ln\\left(\\frac{5}{4}\\right)-4\\cdot\\ln(2)-8\\cdot\\arctan\\left(\\frac{1}{2}\\right)}{8}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral by choosing the order of integration: $\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$.", | |
| "response": "$\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ x }{ x^2+y^2 }\\right) d y} d x}$ =$\\frac{2\\cdot\\pi+8\\cdot\\ln\\left(\\frac{5}{4}\\right)-4\\cdot\\ln(2)-8\\cdot\\arctan\\left(\\frac{1}{2}\\right)}{8}$", | |
| "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": "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": "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": "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": "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": "Evaluate the integral by interchanging the order of integration: $\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$.", | |
| "response": "$\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$ =$39\\cdot\\sqrt[3]{2}+\\frac{81}{2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral by interchanging the order of integration: $\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$.", | |
| "response": "$\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$ =$39\\cdot\\sqrt[3]{2}+\\frac{81}{2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Evaluate the integral by interchanging the order of integration: $\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$.", | |
| "response": "$\\int_{1}^{27}{\\int_{1}^2{\\left(\\sqrt[3]{x}+\\sqrt[3]{y}\\right) d y} d x}$ =$39\\cdot\\sqrt[3]{2}+\\frac{81}{2}$", | |
| "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": "Find the area of a triangle bounded by the x-axis, the line $f(x)=12-\\frac{ 1 }{ 3 } \\cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.", | |
| "response": "The area of the triangle is $\\frac{972}{5}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the area of a triangle bounded by the x-axis, the line $f(x)=12-\\frac{ 1 }{ 3 } \\cdot x$, and the line perpendicular to $f(x)$ that passes through the origin.", | |
| "response": "The area of the triangle is $\\frac{972}{5}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "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": "Find the generalized center of mass between $y=b \\cdot \\sin(a \\cdot x)$, $x=0$, and $x=\\frac{ \\pi }{ a }$ . Then, use the Pappus theorem to find the volume of the solid generated when revolving around the $y$-axis.", | |
| "response": "$(x,y)$ = $P\\left(\\frac{\\pi}{2\\cdot a},\\frac{\\pi\\cdot b}{8}\\right)$ \n\n$V$ = $\\frac{2\\cdot\\pi^2\\cdot b}{a^2}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Differentiate the function \n $f(x)=\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }$.", | |
| "response": "$\\frac{ d }{d x}\\left(\\frac{ 5 \\cdot x^3-\\sqrt[3]{x}+6 \\cdot x+2 }{ \\sqrt{x} }\\right)$=$\\frac{18\\cdot x\\cdot\\sqrt[6]{x}+75\\cdot x^3\\cdot\\sqrt[6]{x}+\\sqrt{x}-6\\cdot\\sqrt[6]{x}}{6\\cdot x\\cdot x^{\\frac{2}{3}}}$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\sin\\left(x+z^y\\right)$.", | |
| "response": "$f_{xx}(x,y,z)$=$-\\sin\\left(x+z^y\\right)$ \n\n$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$-z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)$ \n\n$f_{yy}(x,y,z)$=$-z^y\\cdot\\left(\\ln(z)\\right)^2\\cdot\\left(-\\cos\\left(x+z^y\\right)+z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$z^{-1+y}\\cdot\\left(\\cos\\left(x+z^y\\right)\\cdot\\left(1+y\\cdot\\ln(z)\\right)-y\\cdot z^y\\cdot\\ln(z)\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{zz}(x,y,z)$=$y\\cdot z^{-2+y}\\cdot\\left((-1+y)\\cdot\\cos\\left(x+z^y\\right)-y\\cdot z^y\\cdot\\sin\\left(x+z^y\\right)\\right)$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$-y\\cdot z^{-1+y}\\cdot\\sin\\left(x+z^y\\right)$", | |
| "system": "" | |
| }, | |
| { | |
| "prompt": "Find the extrema of a function $y=\\frac{ x^4 }{ 4 }-\\frac{ 2 \\cdot x^3 }{ 3 }-\\frac{ x^2 }{ 2 }+2$. Then determine the largest and smallest value of the function when $-2 \\le x \\le 4$.", | |
| "response": "This is the final answer to the problem: \n\n1. Extrema points: $P\\left(\\frac{2-2\\cdot\\sqrt{2}}{2},1.969\\right), P(0,2), P\\left(\\frac{2+2\\cdot\\sqrt{2}}{2},-1.8023\\right)$\n2. The largest value: $\\frac{46}{3}$\n3. The smallest value: $-1.8023$", | |
| "system": "" | |
| } | |
| ] | |
| } |