{ "document_id": 493, "page": 2, "width_pt": 612, "height_pt": 792, "engine": "paddleocr_vl", "source_run_id": 618, "item_count": 14, "counts_by_type": { "footer": 2, "formula": 5, "heading": 2, "text": 5 }, "build_ms": 1, "items": [ { "id": "p2.1", "order": 1, "type": "text", "bbox": { "l": 43.68, "t": 692.88, "r": 77.28, "b": 680.4 }, "text": "Name", "children": [] }, { "id": "p2.2", "order": 2, "type": "text", "bbox": { "l": 324.72, "t": 744, "r": 375.84, "b": 725.76 }, "text": "Rules:", "attrs": { "block_label": "display_formula" }, "children": [] }, { "id": "p2.3", "order": 3, "type": "heading", "bbox": { "l": 231.12, "t": 722.16, "r": 391.92, "b": 708.72 }, "text": "Practice Test 4 Business Calculus", "attrs": { "block_label": "paragraph_title" }, "level": 3, "children": [] }, { "id": "p2.4", "order": 4, "type": "formula", "bbox": { "l": 408.24, "t": 749.04, "r": 597.12, "b": 654.48 }, "text": "\\begin{aligned} &\\int x^{n}\\, dx = \\frac{x^{n+1}}{n+1} + C \\qquad \\int dx = x + C \\\\ &\\int e^{x}\\, dx = e^{x} + C \\\\ &\\int \\frac{1}{x}\\, dx = \\int x^{-1}\\, dx = \\ln|x| + C \\end{aligned}", "attrs": { "block_label": "display_formula", "latex": "\\begin{aligned} &\\int x^{n}\\, dx = \\frac{x^{n+1}}{n+1} + C \\qquad \\int dx = x + C \\\\ &\\int e^{x}\\, dx = e^{x} + C \\\\ &\\int \\frac{1}{x}\\, dx = \\int x^{-1}\\, dx = \\ln|x| + C \\end{aligned}", "notation": "latex", "alt": "begin aligned & the integral of x to the power of n, dx equals x to the power of n plus 1 over n plus 1 plus C the integral of dx equals x plus C & the integral of e to the power of x, dx equals e to the power of x plus C & the integral of 1 over x, dx equals the integral of x to the power of minus 1, dx equals ln x plus C end aligned", "mathml": "&∫xndx=xn+1n+1+C∫dx=x+C&∫exdx=ex+C&∫1xdx=∫x−1dx=ln|x|+C" }, "children": [] }, { "id": "p2.5", "order": 5, "type": "text", "bbox": { "l": 43.68, "t": 651.12, "r": 508.32, "b": 638.88 }, "text": "MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.", "children": [] }, { "id": "p2.6", "order": 6, "type": "text", "bbox": { "l": 43.68, "t": 629.52, "r": 88.56, "b": 618.24 }, "text": "Evaluate.", "children": [] }, { "id": "p2.7", "order": 7, "type": "formula", "bbox": { "l": 74.88, "t": 615.84, "r": 469.68, "b": 433.68 }, "text": "\\begin{aligned} &1)\\ \\int(7x^{2}+1)\\,\\mathrm{d}x\\\\ &\\quad A)\\,14x+C\\qquad B)\\,x+C\\qquad \\textcircled{C}\\,\\frac{7}{3}x^{3}+x+C\\qquad D)\\,\\frac{7}{3}x^{3}+C\\\\ &=7\\,\\frac{x^{2+1}}{2+1}+\\ x+C\\ =\\ 7\\,\\frac{x^{3}}{3}+x+C\\\\[4pt] &2)\\ \\int 20\\,\\mathrm{d}x\\\\ &\\quad A)\\,10x^{2}\\qquad \\textcircled{B)}\\,20x+C\\qquad C)\\,0\\qquad D)\\,20+C\\\\ &=20x+C \\end{aligned}", "attrs": { "block_label": "display_formula", "latex": "\\begin{aligned} &1)\\ \\int(7x^{2}+1)\\,\\mathrm{d}x\\\\ &\\quad A)\\,14x+C\\qquad B)\\,x+C\\qquad \\textcircled{C}\\,\\frac{7}{3}x^{3}+x+C\\qquad D)\\,\\frac{7}{3}x^{3}+C\\\\ &=7\\,\\frac{x^{2+1}}{2+1}+\\ x+C\\ =\\ 7\\,\\frac{x^{3}}{3}+x+C\\\\[4pt] &2)\\ \\int 20\\,\\mathrm{d}x\\\\ &\\quad A)\\,10x^{2}\\qquad \\textcircled{B)}\\,20x+C\\qquad C)\\,0\\qquad D)\\,20+C\\\\ &=20x+C \\end{aligned}", "notation": "latex", "alt": "begin aligned &1 the integral of of 7 x squared plus 1, d x & A, 14 x plus C B, x plus C textcircled C, 7 over 3 x cubed plus x plus C D, 7 over 3 x cubed plus C & equals 7, frac x squared plus 1 2 plus 1 plus x plus C equals 7, x cubed over 3 plus x plus C 4 pt &2 the integral of 20, d x & A, 10 x squared textcircled B, 20 x plus C C, 0 D, 20 plus C & equals 20 x plus C end aligned", "mathml": "&1) ∫(7x2+1)dx&A)14x+CB)x+C\\textcircledC73x3+x+CD)73x3+C&=7x2+12+1+ x+C = 7x33+x+C[4pt]&2) ∫20dx&A)10x2\\textcircledB)20x+CC)0D)20+C&=20x+C" }, "children": [] }, { "id": "p2.8", "order": 8, "type": "formula", "bbox": { "l": 508.08, "t": 624.24, "r": 565.68, "b": 571.92 }, "text": "_{1)}\\underline{\\mathrm{C}}", "attrs": { "block_label": "display_formula", "latex": "_{1)}\\underline{\\mathrm{C}}", "notation": "latex", "alt": "sub 1 underline C", "mathml": "1)C―" }, "children": [] }, { "id": "p2.9", "order": 9, "type": "text", "bbox": { "l": 508.08, "t": 508.56, "r": 564.96, "b": 462.48 }, "text": "2) $ B $", "children": [] }, { "id": "p2.10", "order": 10, "type": "heading", "bbox": { "l": 508.56, "t": 415.44, "r": 564.72, "b": 391.68 }, "text": "3) D", "level": 3, "children": [] }, { "id": "p2.11", "order": 11, "type": "formula", "bbox": { "l": 77.52, "t": 413.04, "r": 543.36, "b": 317.28 }, "text": "\\begin{aligned} &\\text{3)}~~\\int 7x^{-3}\\,\\mathrm{d}x \\qquad\\qquad \\text{3)}~\\underline{~\\text{D}~}\\\\ &\\text{A)}~x+7C \\qquad \\text{B)}~-21x+C \\qquad \\text{C)}~7x^{3} \\qquad \\textcircled{D}~-\\dfrac{7}{2x^{2}}+C\\\\ &=7\\,\\frac{x^{-3+1}}{-3+1}+C=7\\frac{x^{-2}}{-2}+C=-\\frac{7}{2}\\cdot\\frac{1}{x^{2}}+C \\end{aligned}", "attrs": { "block_label": "display_formula", "latex": "\\begin{aligned} &\\text{3)}~~\\int 7x^{-3}\\,\\mathrm{d}x \\qquad\\qquad \\text{3)}~\\underline{~\\text{D}~}\\\\ &\\text{A)}~x+7C \\qquad \\text{B)}~-21x+C \\qquad \\text{C)}~7x^{3} \\qquad \\textcircled{D}~-\\dfrac{7}{2x^{2}}+C\\\\ &=7\\,\\frac{x^{-3+1}}{-3+1}+C=7\\frac{x^{-2}}{-2}+C=-\\frac{7}{2}\\cdot\\frac{1}{x^{2}}+C \\end{aligned}", "notation": "latex", "alt": "begin aligned & 3 the integral of 7 x to the power of minus 3, d x 3 underline D & A x plus 7 C B minus 21 x plus C C 7 x cubed textcircled D minus dfrac 7 2 x squared plus C & equals 7, x to the power of minus 3 plus 1 over minus 3 plus 1 plus C equals 7 x to the power of minus 2 over minus 2 plus C equals minus 7 over 2 times 1 over x squared plus C end aligned", "mathml": "&3)  ∫7x−3dx3)  D ―&A) x+7CB) −21x+CC) 7x3\\textcircledD −72x2+C&=7x−3+1−3+1+C=7x−2−2+C=−72·1x2+C" }, "children": [] }, { "id": "p2.12", "order": 12, "type": "formula", "bbox": { "l": 74.64, "t": 296.88, "r": 557.28, "b": 60.24 }, "text": "\\begin{align*} &4)\\quad\\int 12x^{3}\\sqrt{x}\\,\\mathrm{d}x &&4)\\ \\underline{\\quad A\\quad}\\\\ &\\textcircled{A}\\ \\frac{8}{3}x^{9/2}+C \\qquad \\text{B)}\\ \\frac{11}{5}x^{9/2}+C \\qquad \\text{C)}\\ \\frac{24}{7}x^{9/2}+C \\qquad \\text{D)}\\ \\frac{2}{9}x^{9/2}+C\\\\ &=12\\int x^{3}\\cdot x^{\\frac{1}{2}}\\,\\mathrm{d}x=12\\int x^{3+\\frac{1}{2}}\\,\\mathrm{d}x=\\\\ &=12\\int x^{\\frac{7}{2}}\\,\\mathrm{d}x=12\\,\\frac{x^{\\frac{7}{2}+1}}{\\frac{7}{2}+1}+C\\\\ &=\\frac{12x^{\\frac{9}{2}}}{\\frac{9}{2}}+C=\\overset{4}{\\cancel{12}}\\cdot\\frac{2}{\\underset{3}{\\cancel{9}}}x^{\\frac{9}{2}}+C=\\frac{8}{3}x^{\\frac{9}{2}}+C \\end{align*}", "attrs": { "block_label": "display_formula", "latex": "\\begin{align*} &4)\\quad\\int 12x^{3}\\sqrt{x}\\,\\mathrm{d}x &&4)\\ \\underline{\\quad A\\quad}\\\\ &\\textcircled{A}\\ \\frac{8}{3}x^{9/2}+C \\qquad \\text{B)}\\ \\frac{11}{5}x^{9/2}+C \\qquad \\text{C)}\\ \\frac{24}{7}x^{9/2}+C \\qquad \\text{D)}\\ \\frac{2}{9}x^{9/2}+C\\\\ &=12\\int x^{3}\\cdot x^{\\frac{1}{2}}\\,\\mathrm{d}x=12\\int x^{3+\\frac{1}{2}}\\,\\mathrm{d}x=\\\\ &=12\\int x^{\\frac{7}{2}}\\,\\mathrm{d}x=12\\,\\frac{x^{\\frac{7}{2}+1}}{\\frac{7}{2}+1}+C\\\\ &=\\frac{12x^{\\frac{9}{2}}}{\\frac{9}{2}}+C=\\overset{4}{\\cancel{12}}\\cdot\\frac{2}{\\underset{3}{\\cancel{9}}}x^{\\frac{9}{2}}+C=\\frac{8}{3}x^{\\frac{9}{2}}+C \\end{align*}", "notation": "latex", "alt": "begin align* &4 the integral of 12 x cubed the square root of x, d x &&4 underline A & textcircled A 8 over 3 x to the power of 9 over 2 plus C B 11 over 5 x to the power of 9 over 2 plus C C 24 over 7 x to the power of 9 over 2 plus C D 2 over 9 x to the power of 9 over 2 plus C & equals 12 the integral of x cubed times x to the power of 1 over 2, d x equals 12 the integral of x cubed plus 1 over 2, d x equals & equals 12 the integral of x to the power of 7 over 2, d x equals 12, x to the power of 7 over 2 plus 1 over 7 over 2 plus 1 plus C & equals 12 x to the power of 9 over 2 over 9 over 2 plus C equals overset 4 cancel 12 times frac 2 underset 3 cancel 9 x to the power of 9 over 2 plus C equals 8 over 3 x to the power of 9 over 2 plus C end align*", "mathml": "4)∫12x3xdx4) A―\\textcircledA 83x9/2+CB) 115x9/2+CC) 247x9/2+CD) 29x9/2+C=12∫x3·x12dx=12∫x3+12dx==12∫x72dx=12x72+172+1+C=12x9292+C=124·293x92+C=83x92+C" }, "children": [] }, { "id": "p2.13", "order": 13, "type": "footer", "bbox": { "l": 58.56, "t": 55.44, "r": 215.76, "b": 43.2 }, "text": "Business Calculus~ Nena Kabranski", "attrs": { "block_label": "footer" }, "children": [] }, { "id": "p2.14", "order": 14, "type": "footer", "bbox": { "l": 540.72, "t": 55.44, "r": 549.12, "b": 43.68 }, "text": "1", "attrs": { "block_label": "number" }, "children": [] } ], "ai_page_review": { "model": "claude-opus-4-8", "applied": true, "changed": true, "reviewed_at": "2026-09-02T17:20:12+00:00" } }