File size: 43,901 Bytes
0529935 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 |
{
"dataset_name": "Mu-Math",
"group_index": 3,
"source_file": "/hkfs/work/workspace/scratch/tum_fmp0582-dndworkspace/自己训练lora/prepare/data/math/Mu-Math.json",
"selected_indices": [
1,
9,
14,
49,
51,
59,
76,
91,
111,
122,
123,
125,
140,
145,
153,
155,
167,
186,
194,
201,
203,
209,
237,
246,
270,
277,
299,
310,
328,
329,
341,
342,
350,
365,
366,
392,
401,
406,
409,
411,
418,
432,
470,
479,
486,
498,
530,
545,
565,
579,
581,
582,
596,
603,
606,
607,
608,
660,
668,
678,
682,
689,
708,
717,
736,
763,
769,
789,
794,
798,
803,
817,
819,
822,
827,
839,
843,
846,
858,
865,
876,
896,
912,
920,
948,
950,
971,
974,
987,
1012,
1022,
1024,
1030,
1046,
1047,
1051,
1057,
1059,
1065,
1080
],
"total_records": 1084,
"sample_count": 100,
"generated_at": "2025-11-05T12:58:28Z",
"seed": 4171135963,
"samples": [
{
"prompt": "Evaluate the integral by choosing the order of integration: $\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ y }{ x+y^2 }\\right) d y} d x}$.",
"response": "$\\int_{0}^1{\\int_{1}^2{\\left(\\frac{ y }{ x+y^2 }\\right) d y} d x}$ =$\\ln\\left(\\frac{25\\cdot\\sqrt{5}}{32}\\right)$",
"system": ""
},
{
"prompt": "Find the Fourier series of the periodic function $f(x)=x^2$ in the interval $-2 \\cdot \\pi \\le x<2 \\cdot \\pi$ if $f(x)=f(x+4 \\cdot \\pi)$.",
"response": "The Fourier series is: $\\frac{4\\cdot\\pi^2}{3}+\\sum_{n=1}^\\infty\\left(\\frac{16\\cdot(-1)^n}{n^2}\\cdot\\cos\\left(\\frac{n\\cdot x}{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$.",
"response": "$\\int{\\frac{ 2 \\cdot x^3 }{ \\sqrt{x^2+16} } d x}$ =$C+128\\cdot\\left(\\frac{1}{3\\cdot\\left(\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)\\right)^3}-\\frac{1}{\\cos\\left(\\arctan\\left(\\frac{x}{4}\\right)\\right)}\\right)$",
"system": ""
},
{
"prompt": "Find the points at which the following polar curve $r=4 \\cdot \\cos\\left(\\theta\\right)$ has a horizontal or vertical tangent line.",
"response": "This is the final answer to the problem:\n\nhorizontal tangents at: $\\boxed{(2, 2), (2, -2)}$\n\nvertical tangents at: $\\boxed{(4, 0), (0, 0)}$",
"system": ""
},
{
"prompt": "Find the points at which the following polar curve $r=4 \\cdot \\cos\\left(\\theta\\right)$ has a horizontal or vertical tangent line.",
"response": "This is the final answer to the problem:\n\nhorizontal tangents at: $\\boxed{(2, 2), (2, -2)}$\n\nvertical tangents at: $\\boxed{(4, 0), (0, 0)}$",
"system": ""
},
{
"prompt": "Find a rectangular equation which is equivalent to the following parametric equations:\n\n$x^2=t^3-3 \\cdot t^2+3 \\cdot t-1$ \n\n$y^2=t^3+6 \\cdot t^2+12 \\cdot t+8$",
"response": "This is the final answer to the problem: $\\sqrt[3]{x^2}-\\sqrt[3]{y^2}=-3$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$.",
"response": "$\\int{\\frac{ \\cos(x)^3 }{ \\sin(x)^9 } d x}$ =$C-\\left(\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^6+\\frac{1}{4}\\cdot\\left(\\cot(x)\\right)^4+\\frac{1}{8}\\cdot\\left(\\cot(x)\\right)^8\\right)$",
"system": ""
},
{
"prompt": "Let $R$ be the region bounded by the graphs of $y=\\ln(x)$ and $y=x-5$. Write, but do not evaluate, an integral expression for the volume of the solid generated when $R$ is rotated around the $y$-axis.",
"response": "$V$ = $\\pi\\cdot\\int_b^d\\left((y+5)^2-\\left(e^y\\right)^2\\right)dy$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 6 }{ \\sin(3 \\cdot x)^6 } d x}$.",
"response": "$\\int{\\frac{ 6 }{ \\sin(3 \\cdot x)^6 } d x}$ =$-\\frac{2\\cdot\\cos(3\\cdot x)}{5\\cdot\\sin(3\\cdot x)^5}+\\frac{24}{5}\\cdot\\left(-\\frac{\\cos(3\\cdot x)}{9\\cdot\\sin(3\\cdot x)^3}-\\frac{2}{9}\\cdot\\cot(3\\cdot x)\\right)+C$",
"system": ""
},
{
"prompt": "Determine a definite integral that represents the region common to $r=3 \\cdot \\cos\\left(\\theta\\right)$ and $r=3 \\cdot \\sin\\left(\\theta\\right)$ .",
"response": "This is the final answer to the problem:$9\\cdot\\int_0^{\\frac{\\pi}{4}}\\sin\\left(\\theta\\right)^2d\\theta$",
"system": ""
},
{
"prompt": "Determine a definite integral that represents the region common to $r=3 \\cdot \\cos\\left(\\theta\\right)$ and $r=3 \\cdot \\sin\\left(\\theta\\right)$ .",
"response": "This is the final answer to the problem:$9\\cdot\\int_0^{\\frac{\\pi}{4}}\\sin\\left(\\theta\\right)^2d\\theta$",
"system": ""
},
{
"prompt": "Calculate integral: $\\int_{\\frac{ 1 }{ 2 }}^{\\frac{ \\sqrt{3} }{ 2 }}{\\frac{ 1 }{ x \\cdot \\sqrt{9-9 \\cdot x^2} } d x}$.",
"response": "$$\\int_{\\frac{ 1 }{ 2 }}^{\\frac{ \\sqrt{3} }{ 2 }}{\\frac{ 1 }{ x \\cdot \\sqrt{9-9 \\cdot x^2} } d x}=\\frac{1}{6}\\ln\\left(\\frac{7}{3}+\\frac{4}{\\sqrt{3}}\\right)$$",
"system": ""
},
{
"prompt": "Determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbit of the comet or planet. Distance is given in astronomical units (AU):\n\nHalley’s Comet: length of major axis=$35.88$, eccentricity=$0.967$.",
"response": "$r$ = $\\frac{1.16450334}{1+0.967\\cdot\\cos(t)}$",
"system": ""
},
{
"prompt": "Find the derivative of the function $y=\\arcsin\\left(\\frac{ a }{ x }\\right)-\\arctan\\left(\\frac{ x }{ a }\\right)$.",
"response": "This is the final answer to the problem: $-\\frac{a}{a^2+x^2}-\\frac{a}{|x|\\cdot\\sqrt{x^2-a^2}}$",
"system": ""
},
{
"prompt": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.",
"response": "$f'(1)$ =$-0.5244$",
"system": ""
},
{
"prompt": "Compute the derivative of the function $f(x)=\\sqrt[x]{3}+\\frac{ 1 }{ 6^{5 \\cdot x} }+4^{\\sqrt{x}}$ at $x=1$.",
"response": "$f'(1)$ =$-0.5244$",
"system": ""
},
{
"prompt": "Leaves of deciduous trees fall in an exponential rate during autumn. A large maple has 10,000 leaves and the wind starts blowing. If there are 8,000 leaves left after 3 hours, how many will be left after 4 hours.",
"response": "There will be $7426$ leaves after 4 hours.",
"system": ""
},
{
"prompt": "Compute the integral $\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$.",
"response": "$\\int{x^{-4} \\cdot \\left(3+x^2\\right)^{\\frac{ 1 }{ 2 }} d x}$ =$C-\\frac{1}{9}\\cdot\\left(1+\\frac{3}{x^2}\\right)\\cdot\\sqrt{1+\\frac{3}{x^2}}$",
"system": ""
},
{
"prompt": "$f(x)=\\frac{ 1 }{ 4 } \\cdot \\sqrt{x}+\\frac{ 1 }{ x }$, $x>0$. Determine:1. intervals where $f$ is increasing\n2. intervals where $f$ is decreasing\n3. local minima of $f$\n4. local maxima of $f$\n5. intervals where $f$ is concave up\n6. intervals where $f$ is concave down\n7. the inflection points of $f$",
"response": "1. intervals where $f$ is increasing : $(4,\\infty)$\n2. intervals where $f$ is decreasing: $(0,4)$\n3. local minima of $f$: $4$\n4. local maxima of $f$: None\n5. intervals where $f$ is concave up : $\\left(0,8\\cdot\\sqrt[3]{2}\\right)$\n6. intervals where $f$ is concave down: $\\left(8\\cdot\\sqrt[3]{2},\\infty\\right)$\n7. the inflection points of $f$: $8\\cdot\\sqrt[3]{2}$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\cot(x)^4 d x}$.",
"response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\cot(x)^4 d x}$.",
"response": "$\\int{\\cot(x)^4 d x}$ =$C+\\cot(x)-\\frac{1}{3}\\cdot\\left(\\cot(x)\\right)^3-\\arctan\\left(\\cot(x)\\right)$",
"system": ""
},
{
"prompt": "Find $\\frac{ d y }{d x}$ for $y=x \\cdot \\arccsc(x)$.",
"response": "$\\frac{ d y }{d x}$= $-\\frac{x}{|x|\\cdot\\sqrt{x^2-1}}+\\arccsc(x)$",
"system": ""
},
{
"prompt": "An alternating current for outlets in a home has voltage given by the function $V(t)=150 \\cdot \\cos(368 \\cdot t)$, where $V$ is the voltage in volts at time $t$ in seconds.\n\n1. Find the period of the function.\n2. Determine the number of periods that occur when $1$ sec. has passed.",
"response": "This is the final answer to the problem: \n\n1. the period of the function: $\\frac{\\pi}{184}$\n2. the number of periods: $58.56901906$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 4 \\cdot (x+3)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-3)\\cup(-3,\\infty)$\n2. Vertical asymptotes: $x=-3$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{4}-\\frac{3}{2}$\n5. Intervals where the function is increasing: $(-3,0), (0,\\infty), (-\\infty,-9)$\n6. Intervals where the function is decreasing: $(-9,-3)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-3,0), (-\\infty,-3)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{2 \\cdot x-x^2}$ with the center $a=1$.",
"response": "$\\sqrt{2 \\cdot x-x^2}$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot C_{\\frac{1}{2}}^n\\cdot(x-1)^{2\\cdot n}\\right)$",
"system": ""
},
{
"prompt": "Find the Fourier series of the function $u=\\left|\\sin(x)\\right|$ in the interval $[-\\pi,\\pi]$.",
"response": "The Fourier series is: $\\frac{2}{\\pi}-\\frac{4}{\\pi}\\cdot\\left(\\frac{\\cos(2\\cdot x)}{1\\cdot3}+\\frac{\\cos(4\\cdot x)}{3\\cdot5}+\\frac{\\cos(6\\cdot x)}{5\\cdot7}+\\cdots\\right)$",
"system": ""
},
{
"prompt": "Find the radius of convergence and sum of the series: $\\frac{ 2 }{ 4 }+\\frac{ 2 \\cdot x }{ 1 \\cdot 5 }+\\frac{ 2 \\cdot (x)^2 }{ 1 \\cdot 2 \\cdot 6 }+\\cdots+\\frac{ 2 \\cdot (x)^n }{ \\left(n!\\right) \\cdot (n+4) }+\\cdots$ .",
"response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{12}{x^4}+\\frac{\\left(x\\cdot\\left(16\\cdot x\\cdot e^x-16\\cdot e^x\\right)-8\\cdot e^x\\cdot x^3\\right)\\cdot x^3+\\left(4\\cdot e^x+2\\cdot e^x\\cdot x^2+2\\cdot e^x\\cdot x^3-4\\cdot x\\cdot e^x\\right)\\cdot x^4}{x^8},&x\\ne0\\\\\\frac{1}{2},&x=0\\end{cases}$",
"system": ""
},
{
"prompt": "Evaluate the integral: $I=\\int{\\left(x^3+3\\right) \\cdot \\cos(2 \\cdot x) d x}$.",
"response": "This is the final answer to the problem: $\\frac{1}{256}\\cdot\\left(384\\cdot\\sin(2\\cdot x)+128\\cdot x^3\\cdot\\sin(2\\cdot x)+192\\cdot x^2\\cdot\\cos(2\\cdot x)-96\\cdot\\cos(2\\cdot x)-256\\cdot C-192\\cdot x\\cdot\\sin(2\\cdot x)\\right)$",
"system": ""
},
{
"prompt": "Solve the following system of equations:\n\n$\\sin(x) \\cdot \\sin(y)=\\frac{ \\sqrt{3} }{ 4 }$ \n\n$\\cos(x) \\cdot \\cos(y)=\\frac{ \\sqrt{3} }{ 4 }$",
"response": "This is the final answer to the problem: $x=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$ or $x=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$, where $k$ and $n$ are integers ",
"system": ""
},
{
"prompt": "Solve the following system of equations:\n\n$\\sin(x) \\cdot \\sin(y)=\\frac{ \\sqrt{3} }{ 4 }$ \n\n$\\cos(x) \\cdot \\cos(y)=\\frac{ \\sqrt{3} }{ 4 }$",
"response": "This is the final answer to the problem: $x=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$ or $x=\\frac{\\pi}{3}+\\frac{\\pi}{2}\\cdot(2\\cdot n+k)$ and $y=\\frac{\\pi}{6}+\\frac{\\pi}{2}\\cdot(k-2\\cdot n)$, where $k$ and $n$ are integers ",
"system": ""
},
{
"prompt": "Use synthetic division to find the quotient and reminder. Ensure the equation is in the form required by synthetic division:\n\n$\\frac{ 4 \\cdot x^3-12 \\cdot x^2-5 \\cdot x-1 }{ 2 \\cdot x+1 }$ \n\nHint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.",
"response": "Quotient: $2\\cdot x^2-7\\cdot x+1$Remainder: $-2$",
"system": ""
},
{
"prompt": "Use synthetic division to find the quotient and reminder. Ensure the equation is in the form required by synthetic division:\n\n$\\frac{ 4 \\cdot x^3-12 \\cdot x^2-5 \\cdot x-1 }{ 2 \\cdot x+1 }$ \n\nHint: divide the dividend and divisor by the coefficient of the linear term in the divisor. Solve on a paper, if it is more convenient for you.",
"response": "Quotient: $2\\cdot x^2-7\\cdot x+1$Remainder: $-2$",
"system": ""
},
{
"prompt": "Find the second derivative $\\frac{d ^2y}{ d x^2}$ of the function $x=5 \\cdot \\left(\\cos(t)\\right)^3$, $y=6 \\cdot \\left(\\sin(3 \\cdot t)\\right)^3$.",
"response": "$\\frac{d ^2y}{ d x^2}$=$\\frac{-15\\cdot\\left(\\cos(t)\\right)^2\\cdot\\sin(t)\\cdot\\left(324\\cdot\\sin(3\\cdot t)-486\\cdot\\left(\\sin(3\\cdot t)\\right)^3\\right)-\\left(30\\cdot\\cos(t)-45\\cdot\\left(\\cos(t)\\right)^3\\right)\\cdot54\\cdot\\left(\\sin(3\\cdot t)\\right)^2\\cdot\\cos(3\\cdot t)}{\\left(-15\\cdot\\left(\\cos(t)\\right)^2\\cdot\\sin(t)\\right)^3}$",
"system": ""
},
{
"prompt": "The velocity of a bullet from a rifle can be approximated by $v(t)=6400 \\cdot t^2-6505 \\cdot t+2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half- second after the shot: $0 \\le t \\le 0.5$ What is the total distance the bullet travels in $0.5$ sec?",
"response": "The total distance is:$796.54166667$",
"system": ""
},
{
"prompt": "The velocity of a bullet from a rifle can be approximated by $v(t)=6400 \\cdot t^2-6505 \\cdot t+2686$ where $t$ is seconds after the shot and $v$ is the velocity measured in feet per second. This equation only models the velocity for the first half- second after the shot: $0 \\le t \\le 0.5$ What is the total distance the bullet travels in $0.5$ sec?",
"response": "The total distance is:$796.54166667$",
"system": ""
},
{
"prompt": "Find the Maclaurin series of $F(x)=\\arcsin(x)$ ( $F(x)=\\int_{0}^x{f(t) d t}$) by integrating the Maclaurin series of $f(t)=\\frac{ 1 }{ \\sqrt{1-t^2} }$ ( $f(t)=\\sum_{k=0}^\\infty\\left(\\frac{ 1 }{ 2^{2 \\cdot k-1} \\cdot k } \\cdot \\frac{ (2 \\cdot k-2)! }{ \\left((k-1)!\\right) \\cdot (k-1)! } \\cdot t^{2 \\cdot k}\\right)$) term by term. If is not strictly defined at zero, you may substitute the value of the Maclaurin series at zero.",
"response": "$\\arcsin(x)$ =$\\sum_{k=0}^\\infty\\left(\\frac{1}{2^{2\\cdot k-1}\\cdot k}\\cdot\\frac{(2\\cdot k-2)!}{\\left((k-1)!\\right)\\cdot(k-1)!}\\cdot\\frac{x^{2\\cdot k+1}}{2\\cdot k+1}\\right)$",
"system": ""
},
{
"prompt": "Given functions $p(x)=\\frac{ 1 }{ \\sqrt{x} }$ and $m(x)=x^2-4$,\nstate the domain of each of the following functions\nusing interval notation:\n\n1. $\\frac{ p(x) }{ m(x) }$\n2. $p\\left(m(x)\\right)$\n3. $m\\left(p(x)\\right)$",
"response": "1. Domain of $\\frac{ p(x) }{ m(x) }$: $(0,2)\\cup(2,\\infty)$\n2. Domain of $p\\left(m(x)\\right)$: $(-\\infty,-2)\\cup(2,\\infty)$\n3. Domain of $m\\left(p(x)\\right)$: $(0,\\infty)$",
"system": ""
},
{
"prompt": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{29}$ with accuracy 0.0001.",
"response": "This is the final answer to the problem: $3.0723$",
"system": ""
},
{
"prompt": "Find the Taylor series for $f(x)=\\frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $x\\cdot\\left(1-3\\cdot(x+1)+6\\cdot(x+1)^2-10\\cdot(x+1)^3+\\cdots\\right)$=\r\n= $-1 + 4 (x + 1) - 9 (x + 1)^2 + 16 (x + 1)^3$\r\n",
"system": ""
},
{
"prompt": "Find the Taylor series for $f(x)=\\frac{ x }{ (2+x)^3 }$, centered at $x=-1$. Write out the sum of the first four non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $x\\cdot\\left(1-3\\cdot(x+1)+6\\cdot(x+1)^2-10\\cdot(x+1)^3+\\cdots\\right)$=\r\n= $-1 + 4 (x + 1) - 9 (x + 1)^2 + 16 (x + 1)^3$\r\n",
"system": ""
},
{
"prompt": "Consider the function $f(x)=\\frac{ 1 }{ 2 } \\cdot x^5+2 \\cdot x$. Let $g$ denote the inverse of $f$. Find the derivative $g'(2.5)$ using the theorem $g'(c)=\\frac{ 1 }{ f'\\left(g(c)\\right) }$.",
"response": "$g'(2.5)$ =$\\frac{2}{9}$",
"system": ""
},
{
"prompt": "Make full curve sketching of $y=\\ln\\left(\\left|\\frac{ 3 \\cdot x-2 }{ 3 \\cdot x+2 }\\right|\\right)$. Submit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem:\n\n1. The domain (in interval notation) $\\left(-\\infty,-\\frac{2}{3}\\right)\\cup\\left(-\\frac{2}{3},\\frac{2}{3}\\right)\\cup\\left(\\frac{2}{3},\\infty\\right)$\n2. Vertical asymptotes $x=\\frac{2}{3}, x=-\\frac{2}{3}$\n3. Horizontal asymptotes $y=0$\n4. Slant asymptotes None\n5. Intervals where the function is increasing $\\left(\\frac{2}{3},\\infty\\right), \\left(-\\infty,-\\frac{2}{3}\\right)$\n6. Intervals where the function is decreasing $\\left(-\\frac{2}{3},\\frac{2}{3}\\right)$\n7. Intervals where the function is concave up $\\left(-\\frac{2}{3},0\\right), \\left(-\\infty,-\\frac{2}{3}\\right)$\n8. Intervals where the function is concave down $\\left(0,\\frac{2}{3}\\right) \\cup \\left(\\frac{2}{3}, \\infty\\right)$\n9. Points of inflection $P(0,0)$",
"system": ""
},
{
"prompt": "Find the sum of the $\\sum_{n=0}^\\infty\\left(\\frac{ (-1)^n }{ (2 \\cdot n+1)! }\\right)$ with estimate error $0.01$.",
"response": "This is the final answer to the problem: $\\frac{101}{120}$",
"system": ""
},
{
"prompt": "Consider points $P$$P(3,7,-2)$ and $Q$$P(1,1,-3)$. Determine the angle between vectors $\\vec{OP}$ and $\\vec{OQ}$. Express the answer in radians rounded to two decimal places.",
"response": "$\\theta$ =$0.91$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=\\frac{ x^3 }{ 6 \\cdot (x+3)^2 }$. \n\nProvide the following:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation): $(-1\\cdot\\infty,-3)\\cup(-3,\\infty)$\n2. Vertical asymptotes: $x=-3$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: $y=\\frac{x}{6}-1$\n5. Intervals where the function is increasing: $(-3,0), (0,\\infty), (-\\infty,-9)$\n6. Intervals where the function is decreasing: $(-9,-3)$\n7. Intervals where the function is concave up: $(0,\\infty)$\n8. Intervals where the function is concave down: $(-3,0), (-\\infty,-3)$\n9. Points of inflection: $P(0,0)$",
"system": ""
},
{
"prompt": "Write the Taylor series for the function $f(x)=x \\cdot \\sin(2 \\cdot x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).",
"response": "This is the final answer to the problem: $0+2\\cdot\\pi\\cdot(x-\\pi)+\\frac{4}{2}\\cdot(x-\\pi)^2$",
"system": ""
},
{
"prompt": "Calculate $\\sqrt[3]{30}$ with estimate error $0.001$, using series expansion.",
"response": "This is the final answer to the problem: $\\frac{755}{243}$",
"system": ""
},
{
"prompt": "The surface of a large cup is formed by revolving the graph of the function $y=0.25 \\cdot x^{1.6}$ from $x=0$ to $x=5$ about the $y$-axis (measured in centimeters). Find the curvature $\\kappa$ of the generating curve as a function of $x$.",
"response": "$\\kappa$=$\\frac{30}{x^{\\frac{2}{5}}\\cdot\\left(25+4\\cdot x^{\\frac{6}{5}}\\right)^{\\frac{3}{2}}}$",
"system": ""
},
{
"prompt": "Make full curve sketching of $y=2 \\cdot \\arcsin\\left(\\frac{ 1-7 \\cdot x^2 }{ 1+7 \\cdot x^2 }\\right)$. Submit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "This is the final answer to the problem: \n1. The domain (in interval notation) $(-1\\cdot\\infty,\\infty)$\n2. Vertical asymptotes $\\text{None}$\n3. Horizontal asymptotes $y=-\\pi$\n4. Slant asymptotes $\\text{None}$\n5. Intervals where the function is increasing $(-\\infty,0)$\n6. Intervals where the function is decreasing $(0,\\infty)$\n7. Intervals where the function is concave up $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down $\\text{None}$\n9. Points of inflection $\\text{None}$",
"system": ""
},
{
"prompt": "Use the table of integrals to evaluate the integral $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$. \n\nUse this link to access the table of integrals: [Table of Integrals](https://openstax.org/books/calculus-volume-2/pages/a-table-of-integrals)",
"response": "1. Submit the formula used: $\\int{\\left(\\cos(u)\\right)^3 d u}=\\frac{ 1 }{ 3 } \\cdot \\left(2+\\left(\\cos(u)\\right)^2\\right) \\cdot \\sin(u)+c, \\int{\\left(\\sin(u)\\right)^n \\cdot \\left(\\cos(u)\\right)^m d u}=-\\frac{ \\left(\\sin(u)\\right)^{n-1} \\cdot \\left(\\cos(u)\\right)^{m+1} }{ n+m }+\\frac{ n-1 }{ n+m } \\cdot \\int{\\left(\\sin(u)\\right)^{n-2} \\cdot \\left(\\cos(u)\\right)^m d u}$ (For example: to evaluate $\\int{(x+3)^2 d x}$ you would use and submit the formula $\\int{u^n d u}=\\frac{ u^{n+1} }{ n+1 }+C$).\n2. $\\int{\\left(\\sin(y)\\right)^2 \\cdot \\left(\\cos(y)\\right)^3 d y}$=$-\\frac{\\sin(y)\\cdot\\left(\\cos(y)\\right)^4}{5}+\\frac{1}{5}\\cdot\\frac{1}{3}\\cdot\\left(2+\\left(\\cos(y)\\right)^2\\right)\\cdot\\sin(y)+c$",
"system": ""
},
{
"prompt": "Find the first derivative of the function: $y=\\left(3 \\cdot a^2-2 \\cdot a \\cdot b \\cdot x+\\frac{ 5 }{ 3 } \\cdot b^2 \\cdot x^2\\right) \\cdot \\sqrt[3]{\\left(\\frac{ a }{ 3 }+\\frac{ b }{ 3 } \\cdot x\\right)^2}$.",
"response": "The first derivative is:$\\frac{40\\cdot b^3\\cdot x^2}{9\\cdot3^{\\frac{2}{3}}\\cdot\\sqrt[3]{a+b\\cdot x}}$",
"system": ""
},
{
"prompt": "Find the first derivative of the function: $y=\\left(3 \\cdot a^2-2 \\cdot a \\cdot b \\cdot x+\\frac{ 5 }{ 3 } \\cdot b^2 \\cdot x^2\\right) \\cdot \\sqrt[3]{\\left(\\frac{ a }{ 3 }+\\frac{ b }{ 3 } \\cdot x\\right)^2}$.",
"response": "The first derivative is:$\\frac{40\\cdot b^3\\cdot x^2}{9\\cdot3^{\\frac{2}{3}}\\cdot\\sqrt[3]{a+b\\cdot x}}$",
"system": ""
},
{
"prompt": "Solve the following system of equations:\n\n$x+y=\\frac{ 2 \\cdot \\pi }{ 3 }$ \n\n$\\frac{ \\sin(x) }{ \\sin(y) }=2$",
"response": "This is the final answer to the problem: $x=\\frac{\\pi}{2}+\\pi\\cdot k, y=\\frac{\\pi}{6}-\\pi\\cdot k$",
"system": ""
},
{
"prompt": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ with the horizontal. What is the maximum range? Round your answer to one decimal digit.",
"response": "Answer: $22092.5$ m",
"system": ""
},
{
"prompt": "A particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.",
"response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$",
"system": ""
},
{
"prompt": "A particle moves along the plane curve C described by $\\vec{r}(t)=t \\cdot \\vec{i}+t^2 \\cdot \\vec{j}$. Find the length of the curve over the interval $[0,2]$.",
"response": "$s$ = $\\frac{4\\cdot\\sqrt{17}+\\ln\\left(4+\\sqrt{17}\\right)}{4}$",
"system": ""
},
{
"prompt": "A certain bacterium grows in culture in a circular region. The radius of the circle, measured in centimeters, is given by $r(t)=6-\\frac{ 5 }{ t^2+1 }$, where $t$ is time measured in hours since a circle of a $1$-cm radius of the bacterium was put into the culture.\n\n1. Express the area of the bacteria, $A(t)$, as a function of time.\n2. Find the exact and approximate area of the bacterial culture in $3$ hours.\n3. Express the circumference of the bacteria, $C(t)$, as a function of time.\n4. Find the exact and approximate circumference of the bacteria in $3$ hours.",
"response": "This is the final answer to the problem:\n\n1. $A(t)$ = $\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)^2$square centimeters\n2. The exact area of the bacterial culture in $3$ hours = $\\frac{121}{4}\\cdot\\pi$ square centimeters. The approximate area of the bacterial culture in $3$ hours = $95.03317777$ square centimeters.\n3. $C(t)$ = $2\\cdot\\pi\\cdot\\left(6-\\frac{5}{t^2+1}\\right)$centimeters\n4. The exact circumference of the bacteria in $3$ hours = $11\\cdot\\pi$ centimeters. The approximate circumference of the bacteria in $3$ hours = $34.55751919$ centimeters.",
"system": ""
},
{
"prompt": "Find the derivative of the function: $f\\left(\\alpha\\right)=\\ln\\left(\\sqrt{\\frac{ 3-\\sin\\left(\\alpha\\right) }{ 4+2 \\cdot \\sin\\left(\\alpha\\right) }}\\right)$ at $\\alpha=\\frac{ \\pi }{ 4 }$.",
"response": "$f'\\left(\\frac{ \\pi }{ 4 }\\right)$ =$-0.2848$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$.",
"response": "$\\int{\\frac{ \\sin(x)^4 }{ \\cos(x) } d x}$ =$C-\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{1-\\sin(x)}{1+\\sin(x)}\\right|\\right)-\\frac{1}{3}\\cdot\\left(\\sin(x)\\right)^3-\\sin(x)$",
"system": ""
},
{
"prompt": "Write the Taylor series for the function $f(x)=-2 \\cdot x \\cdot \\sin(x)$ at the point $x=\\pi$ up to the third term (zero or non-zero).",
"response": "This is the final answer to the problem: $2\\cdot\\pi\\cdot(x-\\pi)+2\\cdot(x-\\pi)^2$",
"system": ""
},
{
"prompt": "Find the Fourier series of the function $f(x)=\\frac{ -1 }{ 2 } \\cdot x$ in the interval $[-2,2]$.",
"response": "The Fourier series is: $\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Compute the derivative $y=\\arctan\\left(\\frac{ 4-\\cos\\left(\\frac{ x }{ 2 }\\right) }{ 1+4 \\cdot \\cos\\left(\\frac{ x }{ 2 }\\right) }\\right)$.",
"response": "$y'$= $\\frac{\\sin\\left(\\frac{x}{2}\\right)}{2+2\\cdot\\left(\\cos\\left(\\frac{x}{2}\\right)\\right)^2}$",
"system": ""
},
{
"prompt": "Find $\\frac{ d y }{d x}$, given $y=\\tan(2 \\cdot v)$ and $v=\\arctan(2 \\cdot x-1)$.",
"response": "This is the final answer to the problem: $\\frac{dy}{dx}=\\frac{2\\cdot x^2-2\\cdot x+1}{2\\cdot\\left(x-x^2\\right)^2}$",
"system": ""
},
{
"prompt": "Let $z=e^{1-x \\cdot y}$, $x=t^{\\frac{ 1 }{ 3 }}$, $y=t^3$. Find $\\frac{ d z }{d t}$.",
"response": "$\\frac{ d z }{d t}$ =$\\frac{-(10\\cdot e)}{3}\\cdot e^{-t^3\\cdot\\sqrt[3]{t}}\\cdot t^2\\cdot\\sqrt[3]{t}$",
"system": ""
},
{
"prompt": "Find the Taylor series of the given function $f(x)=\\cos(x)$ centered at the indicated point: $a=\\frac{ \\pi }{ 2 }$ .",
"response": "$\\cos(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^{n+1}\\cdot\\frac{\\left(x-\\frac{\\pi}{2}\\right)^{2\\cdot n+1}}{(2\\cdot n+1)!}\\right)$",
"system": ""
},
{
"prompt": "Find zeros of $f(x)=\\sin(x)+\\sin(2 \\cdot x)+2 \\cdot \\sin(x) \\cdot \\sin(2 \\cdot x)-2 \\cdot \\cos(x)-\\cos(2 \\cdot x)$.",
"response": "This is the final answer to the problem: $x_1=-\\frac{\\pi}{2}+2\\cdot\\pi\\cdot n, x_2=-\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_3=\\frac{2\\cdot\\pi}{3}+2\\cdot\\pi\\cdot n, x_4=(-1)^n\\cdot\\frac{\\pi}{6}+\\pi\\cdot n$",
"system": ""
},
{
"prompt": "Find the Fourier series of the function $\\psi(x)=e^{-x}$ in the interval $(-2 \\cdot \\pi,\\pi \\cdot 2)$.",
"response": "The Fourier series is: $e^{-x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{4}+\\sum_{n=1}^\\infty\\left(\\frac{(-1)^n}{4+n^2}\\cdot\\left(2\\cdot\\cos\\left(\\frac{n}{2}\\cdot x\\right)+n\\cdot\\sin\\left(\\frac{n}{2}\\cdot x\\right)\\right)\\right)\\right)$",
"system": ""
},
{
"prompt": "Find the Fourier series of the function $\\psi(x)=2 \\cdot e^{-2 \\cdot x}$ in the interval $(-\\pi,\\pi)$.",
"response": "The Fourier series is: $2\\cdot e^{-2\\cdot x}=\\frac{\\left(e^{2\\cdot\\pi}-e^{-2\\cdot\\pi}\\right)}{\\pi}\\cdot\\left(\\frac{1}{2}+\\sum_{n=1}^\\infty\\left(\\frac{2\\cdot(-1)^n\\cdot\\left(2\\cdot\\cos(n\\cdot x)+n\\cdot\\sin(n\\cdot x)\\right)}{4+n^2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Evaluate $\\int\\int\\int_{E}{\\left(x^3+y^3+z^3\\right) d V}$, where $E$=$\\left\\{(x,y,z)|0 \\le x \\le 2,0 \\le y \\le 2 \\cdot x,0 \\le z \\le 4-x-y\\right\\}$.",
"response": "$I$ = $\\frac{112}{5}$",
"system": ""
},
{
"prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x^2+2}$ with the center $a=0$.",
"response": "$\\sqrt{x^2+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^{2\\cdot n}\\right)$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$.",
"response": "$\\int{\\frac{ -12 }{ \\sin(6 \\cdot x)^6 } d x}$ =$C+2\\cdot\\cot(6\\cdot x)+\\frac{2}{5}\\cdot\\left(\\cot(6\\cdot x)\\right)^5+\\frac{4}{3}\\cdot\\left(\\cot(6\\cdot x)\\right)^3$",
"system": ""
},
{
"prompt": "Find the Taylor series for $f(x)=\\ln\\left(\\sqrt{3} \\cdot x+\\sqrt{1+3 \\cdot x^2}\\right)$, centered at $x=0$. Write out the sum of the first four non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $\\sqrt{3} x - (\\sqrt{3} x^3)/2 + (27 \\sqrt{3} x^5)/40 - (135 \\sqrt{3} x^7)/112 + O(x^9)$",
"system": ""
},
{
"prompt": "Find the Taylor series for $f(x)=\\ln\\left(\\sqrt{3} \\cdot x+\\sqrt{1+3 \\cdot x^2}\\right)$, centered at $x=0$. Write out the sum of the first four non-zero terms, followed by dots.",
"response": "This is the final answer to the problem: $\\sqrt{3} x - (\\sqrt{3} x^3)/2 + (27 \\sqrt{3} x^5)/40 - (135 \\sqrt{3} x^7)/112 + O(x^9)$",
"system": ""
},
{
"prompt": "Given $g(x)=\\frac{ 1 }{ 3 } \\cdot (a+b) \\cdot x^3+\\frac{ 1 }{ 2 } \\cdot (a+b+c) \\cdot x^2-(a+b+c+d) \\cdot x+a \\cdot b \\cdot c \\cdot d$, simplify the derivative of $g(x)$ if $x^2+x=a+b$.",
"response": "This is the final answer to the problem: $g'(x)=(a+b)^2+c\\cdot x-(a+b+c+d)$",
"system": ""
},
{
"prompt": "A projectile is shot in the air from ground level with an initial velocity of $500$ m/sec at an angle of $60$ deg with the horizontal. At what time is the maximum range of the projectile attained? ",
"response": "$t$ = $88.37$",
"system": ""
},
{
"prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of the function $x^{\\frac{ 1 }{ 3 }}$ with the center $a=27$.",
"response": "$x^{\\frac{ 1 }{ 3 }}$ =$\\sum_{n=0}^\\infty\\left(3^{1-3\\cdot n}\\cdot C_n^{\\frac{1}{3}}\\cdot(x-27)^n\\right)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$.",
"response": "$\\int{\\frac{ 20 \\cdot \\cos(-10 \\cdot x)^3 }{ 21 \\cdot \\sin(-10 \\cdot x)^7 } d x}$ =$C+\\frac{1}{21}\\cdot\\left(\\frac{1}{2}\\cdot\\left(\\cot(10\\cdot x)\\right)^4+\\frac{1}{3}\\cdot\\left(\\cot(10\\cdot x)\\right)^6\\right)$",
"system": ""
},
{
"prompt": "Find the centers of symmetry of the curve of $f(x)=\\left(\\sin(x)\\right)^2$.",
"response": "This is the final answer to the problem: $\\left(k\\cdot\\pi+\\frac{\\pi}{4},\\frac{1}{2}\\right)$",
"system": ""
},
{
"prompt": "Find the equations of the common tangent lines to the following ellipses:\n\n$\\frac{ x^2 }{ 6 }+y^2=1$ \n\n$\\frac{ x^2 }{ 4 }+\\frac{ y^2 }{ 9 }=1$",
"response": "This is the final answer to the problem: $2\\cdot x+y-5=0 \\lor 2\\cdot x+y+5=0 \\lor 2\\cdot x-y-5=0 \\lor 2\\cdot x-y+5=0$",
"system": ""
},
{
"prompt": "A box is to be made with the following properties:\n\nThe length of the base, $l$, is twice the length of a width $w$.\n\nThe cost of material to be used for the lateral faces and the top of the box is three times as the cost of the material to be used for the lower base.\n\nFind the dimensions of the box in terms of its fixed volume $V$ such that the cost of the used material is the minimum.",
"response": "This is the final answer to the problem: $y=\\frac{2}{3}\\cdot\\sqrt[3]{\\frac{4\\cdot V}{3}}, w=\\frac{1}{2}\\cdot\\sqrt[3]{\\frac{9\\cdot V}{2}}, l=\\sqrt[3]{\\frac{9\\cdot V}{2}}$",
"system": ""
},
{
"prompt": "Use the substitution $(b+x)^r=(b+a)^r \\cdot \\left(1+\\frac{ x-a }{ b+a }\\right)^r$ in the binomial expansion to find the Taylor series of function $\\sqrt{x+2}$ with the center $a=0$.",
"response": "$\\sqrt{x+2}$ =$\\sum_{n=0}^\\infty\\left(2^{\\frac{1}{2}-n}\\cdot C_{\\frac{1}{2}}^n\\cdot x^n\\right)$",
"system": ""
},
{
"prompt": "Find the radius of convergence and sum of the series: $\\frac{ 3 }{ 2 }+\\frac{ 3 \\cdot x }{ 1 \\cdot 3 }+\\frac{ 3 \\cdot x^2 }{ 1 \\cdot 2 \\cdot 4 }+\\cdots+\\frac{ 3 \\cdot x^n }{ \\left(n!\\right) \\cdot (n+2) }+\\cdots$ .",
"response": "This is the final answer to the problem: 1. Radius of convergence:$R=\\infty$\n2. Sum: $f(x)=\\begin{cases}\\frac{3}{x^2}+\\frac{3\\cdot x\\cdot e^x-3\\cdot e^x}{x^2},&x\\ne0\\\\\\frac{3}{2},&x=0\\end{cases}$",
"system": ""
},
{
"prompt": "Find the Fourier series of the periodic function $f(x)=x^2$ in the interval $-\\pi \\le x<\\pi$ if $f(x)=f(x+2 \\cdot \\pi)$.",
"response": "The Fourier series is: $x^2=\\frac{\\pi^2}{3}-4\\cdot\\left(\\frac{\\cos(x)}{1^2}-\\frac{\\cos(2\\cdot x)}{2^2}+\\frac{\\cos(3\\cdot x)}{3^2}-\\cdots\\right)$",
"system": ""
},
{
"prompt": "Find the tangential and normal components of acceleration for $\\vec{r}(t)=\\left\\langle a \\cdot \\cos\\left(\\omega \\cdot t\\right),b \\cdot \\sin\\left(\\omega \\cdot t\\right) \\right\\rangle$ at $t=0$ with positive coefficients.",
"response": "This is the final answer to the problem:\n\n$a_{t}$: $0$ \n\n$a_{N}$: $\\omega^2\\cdot|a|$",
"system": ""
},
{
"prompt": "Find the area of the surface formed by rotating the arc of the circle $x^2+y^2=1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x+y=1$.",
"response": "This is the final answer to the problem: $\\frac{4\\cdot\\pi-\\pi^2}{\\sqrt{2}}$",
"system": ""
},
{
"prompt": "Find the area of the surface formed by rotating the arc of the circle $x^2+y^2=1$ between the points $(1,0)$ and $(0,1)$ in the first quadrant, around the line $x+y=1$.",
"response": "This is the final answer to the problem: $\\frac{4\\cdot\\pi-\\pi^2}{\\sqrt{2}}$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ -2 }{ \\left(4+\\sqrt{x}\\right) \\cdot \\sqrt{x-x^2} } d x}$.",
"response": "$\\int{\\frac{ -2 }{ \\left(4+\\sqrt{x}\\right) \\cdot \\sqrt{x-x^2} } d x}$ = $C-\\frac{8}{\\sqrt{15}}\\cdot\\arctan\\left(\\frac{1}{\\sqrt{15}}\\cdot\\left(1+4\\cdot\\tan\\left(\\frac{1}{2}\\cdot\\arcsin\\left(\\sqrt{x}\\right)\\right)\\right)\\right)$",
"system": ""
},
{
"prompt": "Find the second derivative $\\frac{d ^2y}{ d x^2}$ of the function $x=\\left(4 \\cdot \\sin(t)\\right)^3$, $y=2 \\cdot \\sin(2 \\cdot t)$.",
"response": "$\\frac{d ^2y}{ d x^2}$=$\\frac{\\left(2304\\cdot\\left(\\sin(t)\\right)^3-1536\\cdot\\sin(t)\\right)\\cdot\\cos(2\\cdot t)-1536\\cdot\\left(\\sin(t)\\right)^2\\cdot\\cos(t)\\cdot\\sin(2\\cdot t)}{7077888\\cdot\\left(\\cos(t)\\right)^3\\cdot\\left(\\sin(t)\\right)^6}$",
"system": ""
},
{
"prompt": "Sketch the curve: \n\n$y=25 \\cdot x^2 \\cdot e^{\\frac{ 1 }{ 5 \\cdot x }}$ \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "1. The domain (in interval notation): $(-\\infty,0)\\cup(0,\\infty)$\n2. Vertical asymptotes: $x=0$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(\\frac{1}{10},\\infty\\right)$\n6. Intervals where the function is decreasing $\\left(0,\\frac{1}{10}\\right), (-\\infty,0)$\n7. Intervals where the function is concave up: $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down: None\n9. Points of inflection: None",
"system": ""
},
{
"prompt": "Find the derivative of $y=\\sin(2 \\cdot x) \\cdot \\cos(3 \\cdot x)-\\frac{ \\ln(x-1) }{ \\ln(x+1) }+c$",
"response": "This is the final answer to the problem: $y'=2\\cdot\\cos(5\\cdot x)-\\sin(3\\cdot x)\\cdot\\sin(2\\cdot x)-\\frac{(x+1)\\cdot\\ln(x+1)-(x-1)\\cdot\\ln(x-1)}{(x-1)\\cdot(x+1)\\cdot\\left(\\ln(x+1)\\right)^2}$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{22 \\cdot \\cot(-11 \\cdot x)^5 d x}$.",
"response": "$\\int{22 \\cdot \\cot(-11 \\cdot x)^5 d x}$ =$C+\\frac{1}{2}\\cdot\\left(\\cot(11\\cdot x)\\right)^4+\\ln\\left(1+\\left(\\cot(11\\cdot x)\\right)^2\\right)-\\left(\\cot(11\\cdot x)\\right)^2$",
"system": ""
},
{
"prompt": "Determine the interval(s) on which $f(x)=x^3 \\cdot e^{-x}$ is decreasing.",
"response": "The function is decreasing on the interval(s) $(3,\\infty)$.",
"system": ""
},
{
"prompt": "Consider points $A$$P(3,-1,2)$, $B$$P(2,1,5)$, and $C$$P(1,-2,-2)$.\n\n1. Find the area of parallelogram ABCD with adjacent sides $\\vec{AB}$ and $\\vec{AC}$.\n2. Find the area of triangle ABC.\n3. Find the distance from point $A$ to line BC.",
"response": "1. $A$=$5\\cdot\\sqrt{6}$\n2. $A$=$\\frac{5\\cdot\\sqrt{6}}{2}$\n3. $d$=$\\frac{5\\cdot\\sqrt{6}}{\\sqrt{59}}$",
"system": ""
},
{
"prompt": "Find the dimensions of the box whose length is twice as long as the width, whose height is $2$ inches greater than the width, and whose volume is $192$ cubic inches.",
"response": "The dimensions of the box are $6, 4, 8$",
"system": ""
},
{
"prompt": "Find the dimensions of the box whose length is twice as long as the width, whose height is $2$ inches greater than the width, and whose volume is $192$ cubic inches.",
"response": "The dimensions of the box are $6, 4, 8$",
"system": ""
},
{
"prompt": "Use the second derivative test to identify any critical points of the function $f(x,y)=x^2 \\cdot y^2$, and determine whether each critical point is a maximum, minimum, saddle point, or none of these.",
"response": "Maximum:NoneMinimum: $P(0,0)$ \n\nSaddle point: None\n\nThe second derivative test is inconclusive at: None",
"system": ""
},
{
"prompt": "Find and classify all critical points of the function $f(x,y)=x \\cdot y \\cdot (1-7 \\cdot x-9 \\cdot y)$.",
"response": "Points of local minima: None.\n\nPoints of local maxima: $P\\left(\\frac{1}{21},\\frac{1}{27}\\right)$.\n\nSaddle points: $P\\left(0,\\frac{1}{9}\\right), P\\left(\\frac{1}{7},0\\right), P(0,0)$.",
"system": ""
},
{
"prompt": "Find and classify all critical points of the function $f(x,y)=x \\cdot y \\cdot (1-7 \\cdot x-9 \\cdot y)$.",
"response": "Points of local minima: None.\n\nPoints of local maxima: $P\\left(\\frac{1}{21},\\frac{1}{27}\\right)$.\n\nSaddle points: $P\\left(0,\\frac{1}{9}\\right), P\\left(\\frac{1}{7},0\\right), P(0,0)$.",
"system": ""
},
{
"prompt": "When hired at a new job selling electronics, you are given two pay options:\n\nOption A: Base salary of $20\\ 000$ USD a year with a commission of $12$ percent of your sales.\n\nOption B: Base salary of $26\\ 000$ USD a year with a commission of $3$ percent of your sales.\n\nHow much electronics would you need to sell for Option A to produce a larger income? Give your answer either exactly or rounded to two decimal places.",
"response": "This is the final answer to the problem: $66666.67$",
"system": ""
},
{
"prompt": "Find the extrema of a function $y=\\frac{ x^4 }{ 4 }-\\frac{ 2 \\cdot x^3 }{ 3 }-\\frac{ x^2 }{ 2 }+2$. Then determine the largest and smallest value of the function when $-2 \\le x \\le 4$.",
"response": "This is the final answer to the problem: \n\n1. Extrema points: $P\\left(\\frac{2-2\\cdot\\sqrt{2}}{2},1.969\\right), P(0,2), P\\left(\\frac{2+2\\cdot\\sqrt{2}}{2},-1.8023\\right)$\n2. The largest value: $\\frac{46}{3}$\n3. The smallest value: $-1.8023$",
"system": ""
}
]
} |