File size: 46,435 Bytes
8ea0712 |
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": 6,
"source_file": "/hkfs/work/workspace/scratch/tum_fmp0582-dndworkspace/自己训练lora/prepare/data/math/Mu-Math.json",
"selected_indices": [
2,
8,
15,
46,
82,
92,
101,
102,
105,
112,
135,
157,
158,
162,
176,
207,
211,
214,
222,
224,
225,
229,
241,
252,
254,
257,
263,
272,
354,
360,
367,
370,
372,
423,
426,
434,
446,
448,
459,
485,
489,
509,
520,
532,
533,
538,
541,
544,
546,
560,
562,
564,
570,
585,
602,
612,
625,
628,
636,
648,
652,
667,
671,
677,
716,
720,
728,
730,
738,
746,
753,
758,
761,
775,
778,
779,
804,
830,
856,
862,
877,
890,
893,
899,
908,
909,
914,
918,
932,
958,
984,
993,
1000,
1006,
1016,
1023,
1026,
1032,
1034,
1048
],
"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": "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": "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": "Consider the differential equation $\\frac{ d y }{d x}=\\frac{ 4+y }{ x }$. Find the particular solution $y=f(x)$ to the given differential equation with the initial condition $f(3)=-3$.",
"response": "$y$ = $\\frac{|x|}{3}-4$",
"system": ""
},
{
"prompt": "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": "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": "Solve the integral: $\\int{\\frac{ -9 \\cdot \\sqrt[3]{x} }{ 9 \\cdot \\sqrt[3]{x^2}+3 \\cdot \\sqrt{x} } d x}$.",
"response": "$\\int{\\frac{ -9 \\cdot \\sqrt[3]{x} }{ 9 \\cdot \\sqrt[3]{x^2}+3 \\cdot \\sqrt{x} } d x}$ =$-\\left(C+\\frac{1}{3}\\cdot\\sqrt[6]{x}^2+\\frac{2}{27}\\cdot\\ln\\left(\\frac{1}{3}\\cdot\\left|1+3\\cdot\\sqrt[6]{x}\\right|\\right)+\\frac{3}{2}\\cdot\\sqrt[6]{x}^4-\\frac{2}{3}\\cdot\\sqrt[6]{x}^3-\\frac{2}{9}\\cdot\\sqrt[6]{x}\\right)$",
"system": ""
},
{
"prompt": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 2 \\cdot \\pi \\cdot x, & 0 \\le x \\le 1 \\\\ 0, & x>1 \\end{cases}$.",
"response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{2\\cdot\\left(\\alpha\\cdot\\sin\\left(\\alpha\\right)+\\cos\\left(\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+2\\cdot\\left(\\sin\\left(\\alpha\\right)-\\alpha\\cdot\\cos\\left(\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ 10 \\cdot x-8 }{ \\sqrt{2 \\cdot x^2+4 \\cdot x+10} } d x}$.",
"response": "$\\int{\\frac{ 10 \\cdot x-8 }{ \\sqrt{2 \\cdot x^2+4 \\cdot x+10} } d x}$ =$\\frac{1}{\\sqrt{2}}\\cdot\\left(10\\cdot\\sqrt{x^2+2\\cdot x+5}-18\\cdot\\ln\\left(\\left|2+2\\cdot x+2\\cdot\\sqrt{x^2+2\\cdot x+5}\\right|\\right)\\right)+C$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ 1 }{ \\sin(8 \\cdot x)^5 } d x}$.",
"response": "This is the final answer to the problem: $C+\\frac{1}{128}\\cdot\\left(2\\cdot\\left(\\tan(4\\cdot x)\\right)^2+6\\cdot\\ln\\left(\\left|\\tan(4\\cdot x)\\right|\\right)+\\frac{1}{4}\\cdot\\left(\\tan(4\\cdot x)\\right)^4-\\frac{2}{\\left(\\tan(4\\cdot x)\\right)^2}-\\frac{1}{4\\cdot\\left(\\tan(4\\cdot x)\\right)^4}\\right)$",
"system": ""
},
{
"prompt": "Solve the integral: $\\int{\\frac{ 1 }{ \\sin(8 \\cdot x)^5 } d x}$.",
"response": "This is the final answer to the problem: $C+\\frac{1}{128}\\cdot\\left(2\\cdot\\left(\\tan(4\\cdot x)\\right)^2+6\\cdot\\ln\\left(\\left|\\tan(4\\cdot x)\\right|\\right)+\\frac{1}{4}\\cdot\\left(\\tan(4\\cdot x)\\right)^4-\\frac{2}{\\left(\\tan(4\\cdot x)\\right)^2}-\\frac{1}{4\\cdot\\left(\\tan(4\\cdot x)\\right)^4}\\right)$",
"system": ""
},
{
"prompt": "What is the general solution to the differential equation $\\frac{ d y }{d x}=-3 \\cdot y+12$ for $y>4$?",
"response": "$y$ = $C\\cdot e^{-3\\cdot x}+4$",
"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": "Use the method of Lagrange multipliers to maximize $U(x,y)=8 \\cdot x^{\\frac{ 4 }{ 5 }} \\cdot y^{\\frac{ 1 }{ 5 }}$; $4 \\cdot x+2 \\cdot y=12$.",
"response": "Answer: maximum $16.715$ at $P(2.4,1.2)$",
"system": ""
},
{
"prompt": "Find $\\frac{ d y }{d x}$ for $y=x \\cdot \\arccsc(x)$.",
"response": "$\\frac{ d y }{d x}$= $-\\frac{x}{|x|\\cdot\\sqrt{x^2-1}}+\\arccsc(x)$",
"system": ""
},
{
"prompt": "Find the 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": "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": "Find the derivative of $f(x)=\\frac{ 1 }{ 15 } \\cdot \\left(\\cos(x)\\right)^3 \\cdot \\left(\\left(\\cos(x)\\right)^2-5\\right)$.",
"response": "This is the final answer to the problem: $f'(x)=-\\frac{\\left(\\cos(x)\\right)^2}{15}\\cdot\\left(3\\cdot\\sin(x)\\cdot\\left(\\left(\\cos(x)\\right)^2-5\\right)+\\cos(x)\\cdot\\sin(2\\cdot x)\\right)$",
"system": ""
},
{
"prompt": "Find the 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 $\\sqrt[4]{90}$ with accuracy $0.0001$.",
"response": "This is the final answer to the problem: $3.0801$",
"system": ""
},
{
"prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ x^4 }{ 4 }-\\frac{ 11 }{ 3 } \\cdot x^3+15 \\cdot x^2+17$.",
"response": "The point(s) where the function has a local minimum:$P(6,89), P(0,17)$ \nThe point(s) where the function has a local maximum:$P\\left(5,\\frac{1079}{12}\\right)$",
"system": ""
},
{
"prompt": "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": "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": "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": "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 gradient: $f(x,y)=\\frac{ \\sqrt{x}+y^2 }{ x \\cdot y }$.",
"response": "$\\nabla f(x,y)$ =$\\left\\langle\\frac{1}{2\\cdot x\\cdot y\\cdot\\sqrt{x}}-\\frac{\\sqrt{x}+y^2}{y\\cdot x^2},\\frac{2}{x}-\\frac{\\sqrt{x}+y^2}{x\\cdot y^2}\\right\\rangle$",
"system": ""
},
{
"prompt": "Find the Fourier series of the periodic function $f(x)=\\frac{ x^2 }{ 2 }$ in the interval $-2 \\cdot \\pi \\le x<2 \\cdot \\pi$ if $f(x)=f(x+4 \\cdot \\pi)$.",
"response": "The Fourier series is: $\\frac{2\\cdot\\pi^2}{3}+\\sum_{n=1}^\\infty\\left(\\frac{8\\cdot(-1)^n}{n^2}\\cdot\\cos\\left(\\frac{n\\cdot x}{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Compute $\\int_{0}^{\\frac{ 1 }{ 3 }}{e^{-\\frac{ x^2 }{ 3 }} d x}$ with accuracy $0.00001$.",
"response": "This is the final answer to the problem: $0.32926$",
"system": ""
},
{
"prompt": "Washington, D.C. is located at $39$ deg N and $77$ deg W. Assume the radius of Earth is $4000$ mi. Express the location of Washington, D.C. in spherical coordinates (use radians).",
"response": "$P\\left(r,\\theta,\\varphi\\right)$ = $P(4000,-1.34,0.89)$",
"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": "Evaluate $I=\\int{\\frac{ 1 }{ x^3+8 } d x}$.",
"response": "This is the final answer to the problem: $I=\\frac{\\sqrt{3}}{12}\\cdot\\arctan\\left(\\frac{x-1}{\\sqrt{3}}\\right)+\\frac{1}{24}\\cdot\\ln\\left(\\frac{(x+2)^2}{x^2-2\\cdot x+4}\\right)+C$",
"system": ""
},
{
"prompt": "Find the derivative of the function $y=\\frac{ 2 \\cdot \\csc(x)-7 \\cdot \\sin(x) }{ 4 \\cdot \\left(\\cos(x)\\right)^5 }-\\frac{ 3 }{ 5 } \\cdot \\cot(2 \\cdot x)$.",
"response": "$y'$=$\\frac{6}{5\\cdot\\left(\\sin(2\\cdot x)\\right)^2}+\\frac{28\\cdot\\left(\\cos(x)\\right)^6-25\\cdot\\left(\\cos(x)\\right)^4-2\\cdot\\left(\\cos(x)\\right)^6\\cdot\\left(\\csc(x)\\right)^2}{4\\cdot\\left(\\cos(x)\\right)^{10}}$",
"system": ""
},
{
"prompt": "Calculate integral: $\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$. $\\left(M=4\\right)$, $\\left(N=5\\right)$, $\\left(p=2\\right)$, $\\left(q=9\\right)$, $\\left(m=2\\right)$.",
"response": "$\\int{\\frac{ M \\cdot x+N }{ \\left(x^2+p \\cdot x+q\\right)^m } d x}$ =$C+\\frac{x+1}{128+16\\cdot(x+1)^2}+\\frac{\\sqrt{2}}{64}\\cdot\\arctan\\left(\\frac{1}{2\\cdot\\sqrt{2}}\\cdot(x+1)\\right)-\\frac{2}{8+(x+1)^2}$",
"system": ""
},
{
"prompt": "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": "Find the Fourier series expansion of the function $f(x)=\\frac{ x }{ 2 }$ with the period $4$ at interval $[-2,2]$.",
"response": "The Fourier series is: $f(x)=\\sum_{n=1}^\\infty\\left(\\frac{(-1)^{n+1}\\cdot2}{\\pi\\cdot n}\\cdot\\sin\\left(\\frac{\\pi\\cdot n\\cdot x}{2}\\right)\\right)$",
"system": ""
},
{
"prompt": "Given two functions $f(x)=\\sqrt{x^2-1}$ and $g(x)=\\sqrt{3-x}$\n\n1. compute $f\\left(g(x)\\right)$\n2. compute $\\frac{ f(x) }{ g(x) }$ and find the domain of the new function.",
"response": "1. the new function $f\\left(g(x)\\right)$ is$f\\left(g(x)\\right)=\\sqrt{2-x}$\n2. the function $\\frac{ f(x) }{ g(x) }$ is $\\frac{\\sqrt{x^2-1}}{\\sqrt{3-x}}$, and the domain for the function is $1\\le x<3 \\lor x\\le-1$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$.",
"response": "$\\int{\\frac{ x+2 }{ \\sqrt{6+10 \\cdot x+25 \\cdot x^2} } d x}$ =$\\frac{1}{25}\\cdot\\sqrt{6+10\\cdot x+25\\cdot x^2}+\\frac{9}{25}\\cdot\\ln\\left(1+5\\cdot x+\\sqrt{1+(5\\cdot x+1)^2}\\right)+C$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$.",
"response": "$\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$ =$\\sqrt{x^2+1}+\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x^2+1}-1}{\\sqrt{x^2+1}+1}\\right|\\right)+C$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$.",
"response": "$\\int{\\frac{ \\sqrt{1+x^2} }{ x } d x}$ =$\\sqrt{x^2+1}+\\frac{1}{2}\\cdot\\ln\\left(\\left|\\frac{\\sqrt{x^2+1}-1}{\\sqrt{x^2+1}+1}\\right|\\right)+C$",
"system": ""
},
{
"prompt": "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": "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": "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": "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": "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": "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": "Given that $\\frac{ 1 }{ 1-x }=\\sum_{n=0}^\\infty x^n$ , use term-by-term differentiation or integration to find power series for function $f(x)=\\ln(x)$ centered at $x=1$ .",
"response": "$\\ln(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^n\\cdot\\frac{(x-1)^{n+1}}{n+1}\\right)$",
"system": ""
},
{
"prompt": "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": "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": "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": "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": "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": "Solve $\\sin(x)+\\cos(x)-2 \\cdot \\sqrt{2} \\cdot \\sin(x) \\cdot \\cos(x)=0$.",
"response": "This is the final answer to the problem: $x=(-1)^{n+1}\\cdot\\frac{\\pi}{6}-\\frac{\\pi}{4}+\\pi\\cdot n, x=\\frac{\\pi}{4}+2\\cdot\\pi\\cdot k$",
"system": ""
},
{
"prompt": "Evaluate the function at the indicated values $f(-3)$, $f(2)$, $f(-a)$,$-f(a)$, and $f(a+h)$. \n\n$f(x)=|x-1|-|x+1|$",
"response": "This is the final answer to the problem: $f(-3)$=$2$\n\n$f(2)$=$-2$ \n\n$f(-a)$=$|a+1|-|-a+1|$ \n\n$-f(a)$= $-|a-1|+|a+1|$ \n\n$f(a+h)$=$|a+h-1|-|a+h+1|$",
"system": ""
},
{
"prompt": "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": "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 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": "Compute the integral $\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$.",
"response": "$\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$ =$C-\\frac{1}{4}\\cdot\\left(\\frac{3}{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}+3\\cdot\\ln\\left(\\frac{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}-1}{1+e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}\\right)\\right)$",
"system": ""
},
{
"prompt": "Compute the integral $\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$.",
"response": "$\\int{\\frac{ -3 }{ e^{4 \\cdot x}+\\sqrt{1+e^{8 \\cdot x}} } d x}$ =$C-\\frac{1}{4}\\cdot\\left(\\frac{3}{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}+3\\cdot\\ln\\left(\\frac{e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}-1}{1+e^{4\\cdot x}+\\sqrt{1+e^{8\\cdot x}}}\\right)\\right)$",
"system": ""
},
{
"prompt": "Find the Taylor series of the given function $f(x)=\\cos(x)$ centered at the indicated point: $a=\\frac{ \\pi }{ 2 }$ .",
"response": "$\\cos(x)$ =$\\sum_{n=0}^\\infty\\left((-1)^{n+1}\\cdot\\frac{\\left(x-\\frac{\\pi}{2}\\right)^{2\\cdot n+1}}{(2\\cdot n+1)!}\\right)$",
"system": ""
},
{
"prompt": "Find the derivative of the function: $y=-3 \\cdot x^{\\sqrt[3]{2 \\cdot x}}$.",
"response": "$\\frac{ d y }{d x}$ =$-\\left(\\frac{3\\cdot\\sqrt[3]{2}}{x^{\\frac{2}{3}}}+\\frac{\\sqrt[3]{2}\\cdot\\ln(x)}{x^{\\frac{2}{3}}}\\right)\\cdot x^{\\sqrt[3]{2}\\cdot\\sqrt[3]{x}}$",
"system": ""
},
{
"prompt": "Let $R$ be the region bounded by the graphs of $y=\\frac{ 1 }{ x+2 }$ and $y=-\\frac{ 1 }{ 2 } \\cdot x+3$.\n\nFind the volume of the solid generated when $R$ is rotated about the vertical line $x=-3$.",
"response": "The volume of the solid is $292.097$ units³.",
"system": ""
},
{
"prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y)=\\frac{ A \\cdot x+B \\cdot y }{ C \\cdot x+D \\cdot y }$.",
"response": "$f_{xx}(x,y)$=$\\frac{2\\cdot C\\cdot(B\\cdot C-A\\cdot D)\\cdot y}{(C\\cdot x+D\\cdot y)^3}$$f_{xy}(x,y)$=$f_{yx}(x,y)$=$\\frac{-(B\\cdot C-A\\cdot D)\\cdot(C\\cdot x-D\\cdot y)}{(C\\cdot x+D\\cdot y)^3}$ \n\n$f_{yy}(x,y)$=$-2\\cdot D\\cdot\\frac{B\\cdot C\\cdot x-A\\cdot D\\cdot x}{(C\\cdot x+D\\cdot y)^3}$",
"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": "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": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{7}$ with accuracy 0.0001.",
"response": "This is the final answer to the problem: $1.9129$",
"system": ""
},
{
"prompt": "Using the series expansion for the function $(1+x)^m$ calculate approximately $\\sqrt[3]{7}$ with accuracy 0.0001.",
"response": "This is the final answer to the problem: $1.9129$",
"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": "Find the Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ 5 \\cdot \\pi \\cdot x, & 0 \\le x \\le 4 \\\\ 0, & x>4 \\end{cases}$.",
"response": "$q(t)$ = $\\int_0^\\infty\\frac{\\left(5\\cdot\\left(4\\cdot\\sin\\left(4\\cdot\\alpha\\right)\\cdot\\alpha+\\cos\\left(4\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot t\\right)+5\\cdot\\left(\\sin\\left(4\\cdot\\alpha\\right)-4\\cdot\\alpha\\cdot\\cos\\left(4\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot t\\right)\\right)}{\\alpha^2}d \\alpha$",
"system": ""
},
{
"prompt": "Find the equations of the common tangent lines to the following ellipses:\n\n$\\frac{ x^2 }{ 6 }+y^2=1$ \n\n$\\frac{ x^2 }{ 4 }+\\frac{ y^2 }{ 9 }=1$",
"response": "This is the final answer to the problem: $2\\cdot x+y-5=0 \\lor 2\\cdot x+y+5=0 \\lor 2\\cdot x-y-5=0 \\lor 2\\cdot x-y+5=0$",
"system": ""
},
{
"prompt": "Find the local minimum and local maximum values of the function $f(x)=\\frac{ 3 }{ 4 } \\cdot x^4-10 \\cdot x^3+24 \\cdot x^2-4$.",
"response": "The point(s) where the function has a local minimum:$P(8,-516), P(0,-4)$ \nThe point(s) where the function has a local maximum:$P(2,24)$",
"system": ""
},
{
"prompt": "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 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": "Compute the integral: $\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$.",
"response": "$\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$ =$C-\\frac{1}{2}\\cdot\\left(\\cot(4\\cdot x)\\right)^5-\\frac{5}{2}\\cdot\\cot(4\\cdot x)-\\frac{5}{3}\\cdot\\left(\\cot(4\\cdot x)\\right)^3$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$.",
"response": "$\\int{\\frac{ 10 }{ \\sin(4 \\cdot x)^6 } d x}$ =$C-\\frac{1}{2}\\cdot\\left(\\cot(4\\cdot x)\\right)^5-\\frac{5}{2}\\cdot\\cot(4\\cdot x)-\\frac{5}{3}\\cdot\\left(\\cot(4\\cdot x)\\right)^3$",
"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": "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": "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": "Sketch the curve: \n\n$y=25 \\cdot x^2 \\cdot e^{\\frac{ 1 }{ 5 \\cdot x }}$ \n\nSubmit as your final answer:\n\n1. The domain (in interval notation)\n2. Vertical asymptotes\n3. Horizontal asymptotes\n4. Slant asymptotes\n5. Intervals where the function is increasing\n6. Intervals where the function is decreasing\n7. Intervals where the function is concave up\n8. Intervals where the function is concave down\n9. Points of inflection",
"response": "1. The domain (in interval notation): $(-\\infty,0)\\cup(0,\\infty)$\n2. Vertical asymptotes: $x=0$\n3. Horizontal asymptotes: None\n4. Slant asymptotes: None\n5. Intervals where the function is increasing: $\\left(\\frac{1}{10},\\infty\\right)$\n6. Intervals where the function is decreasing $\\left(0,\\frac{1}{10}\\right), (-\\infty,0)$\n7. Intervals where the function is concave up: $(0,\\infty), (-\\infty,0)$\n8. Intervals where the function is concave down: None\n9. Points of inflection: None",
"system": ""
},
{
"prompt": "Calculate the second-order partial derivatives. (Treat $A$,$B$,$C$,$D$ as constants.)\n\n$f(x,y,z)=\\arctan(x \\cdot y \\cdot z)$.",
"response": "$f_{xx}(x,y,z)$=$\\frac{-2\\cdot x\\cdot y^3\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$$f_{xy}(x,y,z)$=$f_{yx}(x,y,z)$=$\\frac{z-x^2\\cdot y^2\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yy}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y\\cdot z^3}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{yz}(x,y,z)$=$f_{zy}(x,y,z)$=$\\frac{x-x^3\\cdot y^2\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{zz}(x,y,z)$=$\\frac{-2\\cdot x^3\\cdot y^3\\cdot z}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$ \n\n$f_{xz}(x,y,z)$=$f_{zx}(x,y,z)$=$\\frac{y-x^2\\cdot y^3\\cdot z^2}{\\left(1+x^2\\cdot y^2\\cdot z^2\\right)^2}$",
"system": ""
},
{
"prompt": "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 Fourier integral of the function $q(x)=\\begin{cases} 0, & x<0 \\\\ \\pi \\cdot x, & 0 \\le x \\le 2 \\\\ 0, & x>2 \\end{cases}$.",
"response": "$q(x)$ = $\\int_0^\\infty\\left(\\frac{\\left(2\\cdot\\alpha\\cdot\\sin\\left(2\\cdot\\alpha\\right)+\\cos\\left(2\\cdot\\alpha\\right)-1\\right)\\cdot\\cos\\left(\\alpha\\cdot x\\right)+\\left(\\sin\\left(2\\cdot\\alpha\\right)-2\\cdot\\alpha\\cdot\\cos\\left(2\\cdot\\alpha\\right)\\right)\\cdot\\sin\\left(\\alpha\\cdot x\\right)}{\\alpha^2}\\right)d\\alpha$",
"system": ""
},
{
"prompt": "Compute the integral: $\\int{\\frac{ 1 }{ (x+4) \\cdot \\sqrt{x^2+2 \\cdot x+5} } d x}$.",
"response": "$\\int{\\frac{ 1 }{ (x+4) \\cdot \\sqrt{x^2+2 \\cdot x+5} } d x}$ =$C+\\frac{1}{\\sqrt{13}}\\cdot\\ln\\left(\\sqrt{13}-4-x-\\sqrt{x^2+2\\cdot x+5}\\right)-\\frac{1}{\\sqrt{13}}\\cdot\\ln\\left(4+\\sqrt{13}+x+\\sqrt{x^2+2\\cdot x+5}\\right)$",
"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": "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 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": ""
}
]
} |