f(t)=6sin(t)

{ "query": { "display": "$$f\\left(t\\right)=6\\sin\\left(t\\right)$$", "symbolab_question": "FUNCTION#f(t)=6\\sin(t)" }, "solution": { "level": "PERFORMED", "subject": "Functions & Graphing", "topic": "Functions", "subTopic": "Combination", "default": "\\mathrm{Period}: 2π<br/>\\mathrm{Domain}: -\\infty <t<\\infty <br/>\\mathrm{Range}: -6\\le f(t)\\le 6<br/>\\mathrm{X\\:Intercepts}: (2πn,0),(π+2πn,0),\\mathrm{Y\\:Intercepts}: (0,0)<br/>\\mathrm{Asymptotes}: \\mathrm{None}<br/>\\mathrm{Extreme\\:Points}: \\mathrm{Maximum}(\\frac{π}{2}+2πn,6),\\mathrm{Minimum}(\\frac{3π}{2}+2πn,-6)", "interval": "\\mathrm{Domain}: (-\\infty ,\\infty )<br/>\\mathrm{Range}: [-6,6]" }, "steps": { "type": "interim", "steps": [ { "type": "interim", "title": "Periodicity of $$6\\sin\\left(t\\right):{\\quad}2π$$", "steps": [ { "type": "step", "primary": "Periodicity of $$a\\cdot\\sin\\left(bx\\:+\\:c\\right)\\:+\\:d=\\frac{\\mathrm{periodicity\\:of}\\:\\sin\\left(x\\right)}{|b|}$$", "secondary": [ "Periodicity of $$\\sin\\left(x\\right)\\:$$is $$2π$$" ], "result": "=\\frac{2π}{\\left|1\\right|}" }, { "type": "step", "primary": "Simplify", "result": "=2π" } ], "meta": { "solvingClass": "Function Periodicity", "interimType": "Period Top 1Eq" } }, { "type": "interim", "title": "Domain of $$6\\sin\\left(t\\right)\\::{\\quad}-\\infty\\:<t<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<t<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Range of $$6\\sin\\left(t\\right):{\\quad}-6\\le\\:f\\left(t\\right)\\le\\:6$$", "steps": [ { "type": "definition", "title": "Function range definition", "text": "The set of values of the dependent variable for which a function is defined" }, { "type": "step", "primary": "The range of the basic $$\\sin\\:$$function is $$-1\\le\\:\\sin\\left(t\\right)\\le\\:1$$", "result": "-1\\le\\:\\sin\\left(t\\right)\\le\\:1" }, { "type": "step", "primary": "Multiply the edges of the range by: $$6$$", "result": "-6\\le\\:6\\sin\\left(t\\right)\\le\\:6" }, { "type": "step", "primary": "Therefore the range is", "result": "-6\\le\\:f\\left(t\\right)\\le\\:6" } ], "meta": { "solvingClass": "Function Range", "interimType": "Function Range Simple Trig Top 1Eq" } }, { "type": "interim", "title": "Axis interception points of $$6\\sin\\left(t\\right):\\quad\\:$$X Intercepts$$:\\:\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right),\\:$$Y Intercepts$$:\\:\\left(0,\\:0\\right)$$", "steps": [ { "type": "interim", "title": "$$x-$$axis interception points of $$6\\sin\\left(t\\right):{\\quad}\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right)$$", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "definition", "title": "x-axis interception points definition", "text": "x-intercept is a point on the graph where $$y=0$$" }, { "type": "interim", "title": "$$6\\sin\\left(t\\right)=0:{\\quad}t=2πn,\\:t=π+2πn$$", "input": "6\\sin\\left(t\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6\\sin\\left(t\\right)=0", "result": "\\sin\\left(t\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6\\sin\\left(t\\right)}{6}=\\frac{0}{6}" }, { "type": "step", "primary": "Simplify", "result": "\\sin\\left(t\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\sin\\left(t\\right)=0$$", "result": "t=0+2πn,\\:t=π+2πn", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "t=0+2πn,\\:t=π+2πn" } ], "meta": { "interimType": "Trig General Solutions sin 1Eq" } }, { "type": "interim", "title": "Solve $$t=0+2πn:{\\quad}t=2πn$$", "input": "t=0+2πn", "steps": [ { "type": "step", "primary": "$$0+2πn=2πn$$", "result": "t=2πn" } ], "meta": { "solvingClass": "Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "t=2πn,\\:t=π+2πn" } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Trig Equations" } }, { "type": "step", "result": "\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception X Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMXsqdoP/+t8mkG9iMyF4x8bt5g+KB4e7b6i2FR+k9P7OluKb7H8udBsdDESrRSOTA3g2F2yQn3vCs6qoxcWnKGqGgBg7+Qqjt3KiwKdNp2BLWFmDzgQyN3TmtLk4lh8AZfPH+lyHehZ1t6zfP2gNcYA==" } }, { "type": "interim", "title": "$$y-$$axis interception point of $$6\\sin\\left(t\\right):{\\quad}\\left(0,\\:0\\right)$$", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "definition", "title": "y-axis interception points definition", "text": "$$y$$-intercept is the point on the graph where $$x=0$$" }, { "type": "interim", "title": "Solve $$y=6\\sin\\left(0\\right):{\\quad}0$$", "input": "6\\sin\\left(0\\right)", "steps": [ { "type": "interim", "title": "Use the following trivial identity:$${\\quad}\\sin\\left(0\\right)=0$$", "input": "\\sin\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\sin(x)&x&\\sin(x)\\\\\\hline 0&0&π&0\\\\\\hline \\frac{π}{6}&\\frac{1}{2}&\\frac{7π}{6}&-\\frac{1}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{4π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{2}&1&\\frac{3π}{2}&-1\\\\\\hline \\frac{2π}{3}&\\frac{\\sqrt{3}}{2}&\\frac{5π}{3}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{3π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&\\frac{1}{2}&\\frac{11π}{6}&-\\frac{1}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "Trig Trivial Angle Value Title 0Eq" } }, { "type": "step", "result": "=6\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0" } ], "meta": { "solvingClass": "Trig Evaluate", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Interception Y Points Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMoY5LPa3x5862ED2Fb21Kz+w/rOyd5aPv89U4pRBmcfl6LyUoUL+I7q65gKckzflQfXtjcKfn9o/LtdD1OV+zPmDlwPZY/s+3QwhHk6uKuROF4lNRpq12ZW6pdO2+w5FaHGQeTWtVAb0koXkhtOR8ZA==" } }, { "type": "step", "result": "\\mathrm{X\\:Intercepts}:\\:\\left(2πn,\\:0\\right),\\:\\left(π+2πn,\\:0\\right),\\:\\mathrm{Y\\:Intercepts}:\\:\\left(0,\\:0\\right)" } ], "meta": { "solvingClass": "Function Intersect", "interimType": "Function Intercepts Top 2Eq" } }, { "type": "interim", "title": "Asymptotes of $$6\\sin\\left(t\\right):\\quad\\:$$None", "steps": [ { "type": "interim", "title": "Vertical asymptotes of $$6\\sin\\left(t\\right):{\\quad}$$None", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "step", "primary": "Go over every undefined point $$x=a$$ and check if at least one of the following statements is true:<br/>$${\\quad}\\lim_{x\\to{a^{-}}}f\\left(x\\right)=\\pm\\infty$$<br/>$${\\quad}\\lim_{x\\to{a^{+}}}f\\left(x\\right)=\\pm\\infty$$" }, { "type": "step", "primary": "The function $$6\\sin\\left(t\\right)\\:$$has no undefined points" }, { "type": "step", "result": "\\mathrm{No\\:vertical\\:asymptotes}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Vertical Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OM+9IkDmC/hwmX5Axiwqw5dYmTfaH9H2aPbIKr5Cp0rWPo/JoDPoSVqyvca6Gfq7+bMepVOsDAGapr7bqRnErCjVyCE69kIqeddfE61jK6imGRom3voBZGzBETUfoiuX94" } }, { "type": "interim", "title": "Horizontal Asymptotes of $$6\\sin\\left(t\\right):{\\quad}$$None", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "step", "primary": "Check if at $$t\\to\\pm\\infty$$ the function $$y=6\\sin\\left(t\\right)$$ behaves as a line, $$y=b$$" }, { "type": "interim", "title": "Find an asymptote for $$t\\to-\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{t\\to-\\infty\\:}{f\\left(t\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{t\\to-\\infty\\:}{f\\left(t\\right)}=\\lim_{t\\to-\\infty\\:}{6\\sin\\left(t\\right)}=$$diverges", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(6\\sin\\left(t\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=6\\cdot\\:\\lim_{t\\to\\:-\\infty\\:}\\left(\\sin\\left(t\\right)\\right)" }, { "type": "interim", "title": "Apply Limit Divergence Criterion:$${\\quad}$$diverges", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(\\sin\\left(t\\right)\\right)", "steps": [ { "type": "definition", "title": "Limit Divergence Criterion Test:", "text": "If two sequences exist, $$\\{x_n\\}_{n=1}^{\\infty}$$ and $$\\{y_n\\}_{n=1}^{\\infty}$$ with <br/>$$\\quad\\:x_n\\ne{c}$$ and $$y_n\\ne{c}$$<br/>$$\\quad\\:\\lim_{n\\to\\infty}{x_n}=\\lim_{n\\to\\infty}{y_n}=c$$<br/>$$\\quad\\:\\lim_{n\\to\\infty}{f\\left(x_n\\right)}\\ne\\lim_{n\\to\\infty}{f\\left(y_n\\right)}$$<br/>Then $$\\lim_{x\\to\\:c}f\\left(x\\right)$$ does not exist", "secondary": [ "$$c=-\\infty\\:,\\:x_n=-2nπ,\\:y_n=-\\left(2n+\\frac{1}{2}\\right)π$$" ] }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(-2nπ\\right)=-\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(-2nπ\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=-2π\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "result": "=-2π\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$-c\\cdot\\infty=-\\infty$$", "result": "=-\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(-\\left(2n+\\frac{1}{2}\\right)π\\right)=-\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(-\\left(2n+\\frac{1}{2}\\right)π\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=-π\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(2n+\\frac{1}{2}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=-π\\left(\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is odd<br/>$$a=2,\\:n=1$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sfBo3vAig8R1lz2YYpgaM3DF+eeEKocWhVLM0jshYE/GgWZR3zCAilVrsafPmsT5abA9Nb3zLvO1gWJufFEd2tIwFfVA2P3O5Qj7EDGNsqJdP3yb3suSQQSaGuJSfY5SMA3YT9wXbTpi7kl+cxTnHm7eyE0iYvjxncDM4tBgVfPAJXX4ZFqZyWsvrYx5GanN7Q==" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)=\\frac{1}{2}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=-π\\left(\\infty\\:+\\frac{1}{2}\\right)" }, { "type": "interim", "title": "Simplify $$-π\\left(\\infty\\:+\\frac{1}{2}\\right):{\\quad}-\\infty\\:$$", "input": "-π\\left(\\infty\\:+\\frac{1}{2}\\right)", "result": "=-\\infty\\:", "steps": [ { "type": "interim", "title": "$$\\infty\\:+\\frac{1}{2}=\\infty\\:$$", "input": "\\infty\\:+\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\infty\\:+c=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=-π\\left(\\infty\\:\\right)" }, { "type": "interim", "title": "$$-π\\left(\\infty\\:\\right)=-\\infty\\:$$", "input": "-π\\left(\\infty\\:\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$-c\\cdot\\infty=-\\infty$$", "result": "=-\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=-\\infty\\:" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7wjOBNuWn7+HLVz9dqXFabOAIl1fTgtaFP7RNKlTUGu0gJ/ZZA32ZInFBpDtxBfiKCxkwChozDewfzsMw3fvcsYcC7vzc2ez+8NEkHAK+Yu4eKYFFhGodJZ4mS5Q+F9MLEo2LztLLyi1IGISCO+0Yy+6ON13w6GpTJnZZ8ozeeAK/Mg94S0N9we//Py6WzxN6" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "$$\\lim_{n\\to\\infty}x_n=\\lim_{n\\to\\infty}y_n=c=-\\infty\\:$$" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\sin\\left(\\left(-2nπ\\right)\\right)\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\sin\\left(\\left(-2nπ\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(\\left(-2nπ\\right)\\right)=0,\\:\\forall\\:n\\in\\mathbb{Z}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(0\\right)" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(0\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(0\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\sin\\left(\\left(-\\left(2n+\\frac{1}{2}\\right)π\\right)\\right)\\right)=-1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\sin\\left(\\left(-\\left(2n+\\frac{1}{2}\\right)π\\right)\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(\\left(-\\left(2n+\\frac{1}{2}\\right)π\\right)\\right)=\\left(-1\\right),\\:\\forall\\:n\\in\\mathbb{Z}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(-1\\right)" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(-1\\right)=-1$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(-1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=-1" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=-1" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "$$\\lim_{n\\to\\infty}f\\left(x_n\\right)\\ne\\lim_{n\\to\\infty}f\\left(y_n\\right)$$" }, { "type": "step", "primary": "Therefore $$\\lim_{t\\to\\:-\\infty\\:}\\left(\\sin\\left(t\\right)\\right)$$ is divergent at $$t\\to\\:-\\infty\\:$$" }, { "type": "step", "result": "=6\\mathrm{diverges}" } ], "meta": { "interimType": "Limits Apply Divergence Criterion 0Eq" } }, { "type": "step", "result": "=\\mathrm{diverges}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The result is not a finite constant, therefore" }, { "type": "step", "result": "\\mathrm{No\\:Horizontal\\:Asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$t\\to\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{t\\to\\infty\\:}{f\\left(t\\right)}\\:$$to find b:" }, { "type": "interim", "title": "$$\\lim_{t\\to\\infty\\:}{f\\left(t\\right)}=\\lim_{t\\to\\infty\\:}{6\\sin\\left(t\\right)}=$$diverges", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(6\\sin\\left(t\\right)\\right)", "steps": [ { "type": "interim", "title": "Apply Limit Divergence Criterion:$${\\quad}$$diverges", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(6\\sin\\left(t\\right)\\right)", "steps": [ { "type": "definition", "title": "Limit Divergence Criterion Test:", "text": "If two sequences exist, $$\\{x_n\\}_{n=1}^{\\infty}$$ and $$\\{y_n\\}_{n=1}^{\\infty}$$ with <br/>$$\\quad\\:x_n\\ne{c}$$ and $$y_n\\ne{c}$$<br/>$$\\quad\\:\\lim_{n\\to\\infty}{x_n}=\\lim_{n\\to\\infty}{y_n}=c$$<br/>$$\\quad\\:\\lim_{n\\to\\infty}{f\\left(x_n\\right)}\\ne\\lim_{n\\to\\infty}{f\\left(y_n\\right)}$$<br/>Then $$\\lim_{x\\to\\:c}f\\left(x\\right)$$ does not exist", "secondary": [ "$$c=\\infty\\:,\\:x_n=2nπ,\\:y_n=\\left(2n+\\frac{1}{2}\\right)π$$" ] }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(2nπ\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(2nπ\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=2π\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)" }, { "type": "step", "primary": "Apply the common limit: $$\\lim_{n\\to\\:\\infty\\:}\\left(n\\right)=\\infty\\:$$", "result": "=2π\\cdot\\:\\infty\\:" }, { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\left(2n+\\frac{1}{2}\\right)π\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\left(2n+\\frac{1}{2}\\right)π\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=π\\cdot\\:\\lim_{n\\to\\:\\infty\\:}\\left(2n+\\frac{1}{2}\\right)" }, { "type": "step", "primary": "$$\\lim_{x\\to{a}}[f\\left(x\\right)\\pm{g\\left(x\\right)}]=\\lim_{x\\to{a}}{f\\left(x\\right)}\\pm\\lim_{x\\to{a}}{g\\left(x\\right)}$$<br/>With the exception of indeterminate form", "result": "=π\\left(\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)+\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)\\right)", "meta": { "title": { "extension": "Indeterminate Forms:<br/>$$\\frac{\\pm\\infty}{\\pm\\infty}$$<br/>$$\\frac{0}{0}$$<br/>$$\\pm\\infty\\cdot0$$<br/>$$0^0$$<br/>$$1^{\\pm\\infty}$$<br/>$$\\infty^{0}$$<br/>$$\\infty-\\infty$$" } } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)=\\infty\\:$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(2n\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(ax^{n}+\\cdots+bx+c\\right)=\\infty,\\:a>0,\\:$$n is odd<br/>$$a=2,\\:n=1$$", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sfBo3vAig8R1lz2YYpgaM3DF+eeEKocWhVLM0jshYE/GgWZR3zCAilVrsafPmsT5abA9Nb3zLvO1gWJufFEd2tIwFfVA2P3O5Qj7EDGNsqJdP3yb3suSQQSaGuJSfY5SMA3YT9wXbTpi7kl+cxTnHm7eyE0iYvjxncDM4tBgVfPAJXX4ZFqZyWsvrYx5GanN7Q==" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)=\\frac{1}{2}$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(\\frac{1}{2}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=\\frac{1}{2}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=π\\left(\\infty\\:+\\frac{1}{2}\\right)" }, { "type": "interim", "title": "Simplify $$π\\left(\\infty\\:+\\frac{1}{2}\\right):{\\quad}\\infty\\:$$", "input": "π\\left(\\infty\\:+\\frac{1}{2}\\right)", "result": "=\\infty\\:", "steps": [ { "type": "interim", "title": "$$\\infty\\:+\\frac{1}{2}=\\infty\\:$$", "input": "\\infty\\:+\\frac{1}{2}", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\infty\\:+c=\\infty\\:$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=π\\left(\\infty\\:\\right)" }, { "type": "interim", "title": "$$π\\left(\\infty\\:\\right)=\\infty\\:$$", "input": "π\\left(\\infty\\:\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$c\\cdot\\infty=\\infty$$", "result": "=\\infty\\:" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "result": "=\\infty\\:" } ], "meta": { "solvingClass": "Solver", "interimType": "Generic Simplify Specific 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7DmLgzF/KVij1bsdqAC1OxONDpki66REhuqBXLs4YMK/dd47a0hQ8flDbGsI5To1dnnElq05yDzre8mRMUYzF5z/L0MoYg+CUn6oyL3EO7YppEjsYKnQdDP7MPDbdrF10a2EmYU2451fnE9MJ6JgmPCF1qZmUUVcwDwkdubATOYU=" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "$$\\lim_{n\\to\\infty}x_n=\\lim_{n\\to\\infty}y_n=c=\\infty\\:$$" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(6\\sin\\left(2nπ\\right)\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(6\\sin\\left(2nπ\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(2nπ\\right)=0,\\:\\forall\\:n\\in\\mathbb{Z}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:0\\right)" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:0\\right)=0$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:0\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=6\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=0" } ], "meta": { "interimType": "N/A" } }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(6\\sin\\left(\\left(2n+\\frac{1}{2}\\right)π\\right)\\right)=6$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(6\\sin\\left(\\left(2n+\\frac{1}{2}\\right)π\\right)\\right)", "steps": [ { "type": "step", "primary": "$$\\sin\\left(\\left(2n+\\frac{1}{2}\\right)π\\right)=1,\\:\\forall\\:n\\in\\mathbb{Z}$$", "result": "=\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:1\\right)" }, { "type": "interim", "title": "$$\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:1\\right)=6$$", "input": "\\lim_{n\\to\\:\\infty\\:}\\left(6\\cdot\\:1\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}{c}=c$$", "result": "=6\\cdot\\:1" }, { "type": "step", "primary": "Simplify", "result": "=6", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "result": "=6" } ], "meta": { "interimType": "N/A" } }, { "type": "step", "primary": "$$\\lim_{n\\to\\infty}f\\left(x_n\\right)\\ne\\lim_{n\\to\\infty}f\\left(y_n\\right)$$" }, { "type": "step", "primary": "Therefore $$\\lim_{t\\to\\:\\infty\\:}\\left(6\\sin\\left(t\\right)\\right)$$ is divergent at $$t\\to\\:\\infty\\:$$" }, { "type": "step", "result": "=\\mathrm{diverges}" } ], "meta": { "interimType": "Limits Apply Divergence Criterion 0Eq" } }, { "type": "step", "result": "=\\mathrm{diverges}" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The result is not a finite constant, therefore" }, { "type": "step", "result": "\\mathrm{No\\:Horizontal\\:Asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "result": "\\mathrm{No\\:horizontal\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMW0QiF7WQsnIxF55f0Pihousni/ZoADN866T60JAMQj/T4CYauXUHC1TsdNlLYhWLoikLUK4FEa5HcFqXfsJe/e/zor9OJwenrcQbpYK5/59VjRTOt2PEnLPCyn0y4wdOsIjaxJ4DvjTb2fbKjbvtlQ==" } }, { "type": "interim", "title": "Slant Asymptotes of $$6\\sin\\left(t\\right):{\\quad}$$None", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "step", "primary": "Check if at $$t\\to\\pm\\infty$$ the function $$y=6\\sin\\left(t\\right)$$ behaves as a line, $$y=mx+b\\:$$ where $$m\\neq0$$" }, { "type": "interim", "title": "Find an asymptote for $$t\\to-\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{t\\to-\\infty\\:}{\\frac{f\\left(t\\right)}{t}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{t\\to-\\infty\\:}{\\frac{f\\left(t\\right)}{t}}=\\lim_{t\\to-\\infty\\:}{\\frac{6\\sin\\left(t\\right)}{t}}=0$$", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{6\\sin\\left(t\\right)}{t}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=6\\cdot\\:\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)" }, { "type": "interim", "title": "Apply the Squeeze Theorem:$${\\quad}0$$", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)", "steps": [ { "type": "definition", "title": "Squeeze Theorem:", "text": "Let f, g and h be functions such that for all $$x\\in[a,\\:b]\\:$$(except possibly at the limit point c), <br/>$$f\\left(x\\right)\\le{h\\left(x\\right)}\\le{g\\left(x\\right)}$$<br/>Also suppose that, $$\\lim_{x\\to{c}}{f\\left(x\\right)}=\\lim_{x\\to{c}}{g\\left(x\\right)}=L$$<br/>Then for any $$a\\le{c}\\le{b},\\:\\lim_{x\\to{c}}{h\\left(x\\right)}=L$$", "secondary": [ "$$-1\\le\\:\\sin\\left(t\\right)\\le\\:1$$", "$$\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{-1}{t}\\right)\\le\\:\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)\\le\\:\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{1}{t}\\right)$$" ] }, { "type": "interim", "title": "$$\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{-1}{t}\\right)=0$$", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{-1}{t}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to-\\infty}\\left(\\frac{c}{x^a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sUhR/vrajzWiR3uhW123/P7n2togh//f8yrYiU+dc+7K/M3hESAhJ79Q7OQNfwXSMJMDkcv4dePDnEJ+wTfpdJbtO0Z3AymWP8h7exzfmMiuqo32aoZKzqAM/LEW+8F2ViZSX883BjX2RutQZXLVRJQOk+3P4Ld4FLkFA+bA3lwXdYM3Gmg3zOUWjqFEMvjkOe1xIwoq/BL8h2MCbq1CVjk=" } }, { "type": "interim", "title": "$$\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{1}{t}\\right)=0$$", "input": "\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{1}{t}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to-\\infty}\\left(\\frac{c}{x^a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sUhR/vrajzWiR3uhW123/P6om8/FgAwfFTluLNl5ocmvHI5S0StY1FdtOqqOPr0Te4JMK9mJAiMjl/1mWqGgvKjyKNertMk7CgVOMvzgcgt5ZuJKdCFsPJy1+5gBMEc9djmqm40qzyGVY+WsRGEnpKKyAhmSvtPJX9QXrRLfFSLpANy3COzVotYmBVfa7jOLWWezmh25OJYWjI6t5+TWV14=" } }, { "type": "step", "primary": "By the squeeze theorem: $$\\lim_{t\\to\\:-\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)=0$$", "result": "=0" } ], "meta": { "interimType": "Squeeze Theorem 0Eq", "practiceLink": "/practice/limits-practice?subTopic=Squeeze%20Theorem", "practiceTopic": "Limit Squeeze Theorem", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sUhR/vrajzWiR3uhW123/P5wbxTzFyDqseIF99oyy0gl9BcViVDE1xeyWoJuR43IR+9lknJpqDOgfs8//g58Qqcfz2mYeCmSi9QcTaSCiIFEHLV0KNaUEtRrxxgByI2yJ6Yp+18uqAXrMTpLgOZPq6RK28PwYoM0kk3yqK7RdFxV6m1Z5HoWiHLmO1K5CguAUauxFjpiEkoDxJ5LNXofr1k=" } }, { "type": "step", "result": "=6\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is zero, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "interim", "title": "Find an asymptote for $$t\\to\\infty\\::{\\quad}$$None", "steps": [ { "type": "step", "primary": "Compute $$\\lim_{t\\to\\infty\\:}{\\frac{f\\left(t\\right)}{t}}\\:$$to find m:" }, { "type": "interim", "title": "$$\\lim_{t\\to\\infty\\:}{\\frac{f\\left(t\\right)}{t}}=\\lim_{t\\to\\infty\\:}{\\frac{6\\sin\\left(t\\right)}{t}}=0$$", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{6\\sin\\left(t\\right)}{t}\\right)", "steps": [ { "type": "step", "primary": "$$\\lim_{x\\to{a}}[c\\cdot{f\\left(x\\right)}]=c\\cdot\\lim_{x\\to{a}}{f\\left(x\\right)}$$", "result": "=6\\cdot\\:\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)" }, { "type": "interim", "title": "Apply the Squeeze Theorem:$${\\quad}0$$", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)", "steps": [ { "type": "definition", "title": "Squeeze Theorem:", "text": "Let f, g and h be functions such that for all $$x\\in[a,\\:b]\\:$$(except possibly at the limit point c), <br/>$$f\\left(x\\right)\\le{h\\left(x\\right)}\\le{g\\left(x\\right)}$$<br/>Also suppose that, $$\\lim_{x\\to{c}}{f\\left(x\\right)}=\\lim_{x\\to{c}}{g\\left(x\\right)}=L$$<br/>Then for any $$a\\le{c}\\le{b},\\:\\lim_{x\\to{c}}{h\\left(x\\right)}=L$$", "secondary": [ "$$-1\\le\\:\\sin\\left(t\\right)\\le\\:1$$", "$$\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{-1}{t}\\right)\\le\\:\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)\\le\\:\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{1}{t}\\right)$$" ] }, { "type": "interim", "title": "$$\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{-1}{t}\\right)=0$$", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{-1}{t}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(\\frac{c}{x^a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sezXmz2KkdvoPna6LT17SBl1qwRHakTLQDHfiZZkMhSsHI5S0StY1FdtOqqOPr0Te4JMK9mJAiMjl/1mWqGgvKjyKNertMk7CgVOMvzgcgt5ZuJKdCFsPJy1+5gBMEc9djmqm40qzyGVY+WsRGEnpKLWPy4FLCGIJSrSPa16o6Kn59raIIf/3/Mq2IlPnXPuymezmh25OJYWjI6t5+TWV14=" } }, { "type": "interim", "title": "$$\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{1}{t}\\right)=0$$", "input": "\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{1}{t}\\right)", "steps": [ { "type": "step", "primary": "Apply Infinity Property: $$\\lim_{x\\to\\infty}\\left(\\frac{c}{x^a}\\right)=0$$", "result": "=0" } ], "meta": { "solvingClass": "Limits", "interimType": "Limits", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sezXmz2KkdvoPna6LT17SBkCuJLxsYbeiK2Ak+vp/gy5k3WldPTzMRCmfRYnoIUxcFAsqG8mG4+AKrheu2ZGyPikLL/bRq43OIhscddZMC8wf7LqB9CcyvYCWDsGseX09hi2Sg2N1jZXcumfy0+UpgDs15s9ipHb6D52ui09e0gZpZIi+SX58RwVU3RuStwM/82K6cO8RrGfyfucGJ9J7KQ=" } }, { "type": "step", "primary": "By the squeeze theorem: $$\\lim_{t\\to\\:\\infty\\:}\\left(\\frac{\\sin\\left(t\\right)}{t}\\right)=0$$", "result": "=0" } ], "meta": { "interimType": "Squeeze Theorem 0Eq", "practiceLink": "/practice/limits-practice?subTopic=Squeeze%20Theorem", "practiceTopic": "Limit Squeeze Theorem", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7+jbUbsVfpMbXIJ0UHfO5sezXmz2KkdvoPna6LT17SBlBskwu2sxfLUpi/X6NIfhK/nvRnTkWcyEmnKlaREVZgKban4/SD6XbJPFRGxIaHN4AZUyUYTdLExKbse8TELqREDTDGcaaQdZ2u1WyU+l/7XNS9SX5M3gDB/Er/MAH1V8EuDOVaQvKofqHoY5jNaps/6vuMhuvnWlzao0NsfWFiJSzgI2fCj+uJ7BHl84M3Aw=" } }, { "type": "step", "result": "=6\\cdot\\:0" }, { "type": "step", "primary": "Simplify", "result": "=0", "meta": { "solvingClass": "Solver" } } ], "meta": { "solvingClass": "Limits", "interimType": "Limits" } }, { "type": "step", "primary": "The slope is zero, therefore" }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Horizontal Asymptotes Side 1Eq" } }, { "type": "step", "result": "\\mathrm{No\\:slant\\:asymptote}" } ], "meta": { "interimType": "Slant Asymptotes Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMLXiD7VTAsp80tg/tmv96KDOd8ve6NcfDLFCjdOthzV1kS3dlcCKpQTQcheuut7MkSKdo5EZklerpkvddxX/+4ym7K7hyw99PRPQyF27Tvgdc3K906IO8V93Ej6B9LCCc" } }, { "type": "step", "result": "\\mathrm{None}" } ], "meta": { "solvingClass": "Function Asymptotes", "interimType": "Function Asymptotes Top 2Eq" } }, { "type": "interim", "title": "Extreme Points of $$6\\sin\\left(t\\right):{\\quad}$$Maximum$$\\left(\\frac{π}{2}+2πn,\\:6\\right),\\:$$Minimum$$\\left(\\frac{3π}{2}+2πn,\\:-6\\right)$$", "steps": [ { "type": "definition", "title": "First Derivative Test definition", "text": "Suppose that $$x=c$$ is a critical point of $$f\\left(x\\right)$$ then, <br/>If $$f\\:{^{\\prime}}\\left(x\\right)>0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the right of $$x=c$$ then $$x=c$$ is a local maximum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)<0$$ to the left of $$x=c$$ and $$f\\:{^{\\prime}}\\left(x\\right)>\\:0$$ to the right of $$x=c$$ then $$x=c$$ is a local minimum.<br/>If $$f\\:{^{\\prime}}\\left(x\\right)$$ is the same sign on both sides of $$x=c$$ then $$x=c$$ is neither a local maximum nor a local minimum." }, { "type": "interim", "title": "Find the critical points:$${\\quad}t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn$$", "input": "6\\sin\\left(t\\right)", "steps": [ { "type": "definition", "title": "Critical point definition", "text": "Critical points are points where the function is defined and its derivative is zero or undefined" }, { "type": "interim", "title": "Find where $$f^{\\prime}\\left(t\\right)$$ is equal to zero or undefined", "input": "6\\sin\\left(t\\right)", "result": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn", "steps": [ { "type": "interim", "title": "$$f^{\\prime}\\left(t\\right)=6\\cos\\left(t\\right)$$", "input": "\\frac{d}{dt}\\left(6\\sin\\left(t\\right)\\right)", "steps": [ { "type": "step", "primary": "Take the constant out: $$\\left(a{\\cdot}f\\right)'=a{\\cdot}f'$$", "result": "=6\\frac{d}{dt}\\left(\\sin\\left(t\\right)\\right)" }, { "type": "step", "primary": "Apply the common derivative: $$\\frac{d}{dt}\\left(\\sin\\left(t\\right)\\right)=\\cos\\left(t\\right)$$", "result": "=6\\cos\\left(t\\right)" } ], "meta": { "solvingClass": "Derivatives", "interimType": "Derivatives", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s79Kg+idP5vLVrjUll6eMdYrTNb7/sChChiL46APBO7ZyQp7tdIFyr1eVqMMLZHDTG7JiujyaxykLR7Tv0Nr+wOLDPUIMxWeVHJ2k/ylp55ZNfn8iLQx9FJ8rTInMoxXpsgEaetDNhvhSgg+zfGqgzuQ==" } }, { "type": "interim", "title": "Solve $$6\\cos\\left(t\\right)=0:{\\quad}t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn$$", "input": "6\\cos\\left(t\\right)=0", "steps": [ { "type": "interim", "title": "Divide both sides by $$6$$", "input": "6\\cos\\left(t\\right)=0", "result": "\\cos\\left(t\\right)=0", "steps": [ { "type": "step", "primary": "Divide both sides by $$6$$", "result": "\\frac{6\\cos\\left(t\\right)}{6}=\\frac{0}{6}" }, { "type": "step", "primary": "Simplify", "result": "\\cos\\left(t\\right)=0" } ], "meta": { "interimType": "Divide Both Sides Specific 1Eq", "gptData": "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" } }, { "type": "interim", "title": "General solutions for $$\\cos\\left(t\\right)=0$$", "result": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn", "steps": [ { "type": "step", "primary": "$$\\cos\\left(x\\right)$$ periodicity table with $$2πn$$ cycle:<br/>$$\\begin{array}{|c|c|c|c|}\\hline x&\\cos(x)&x&\\cos(x)\\\\\\hline 0&1&π&-1\\\\\\hline \\frac{π}{6}&\\frac{\\sqrt{3}}{2}&\\frac{7π}{6}&-\\frac{\\sqrt{3}}{2}\\\\\\hline \\frac{π}{4}&\\frac{\\sqrt{2}}{2}&\\frac{5π}{4}&-\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{π}{3}&\\frac{1}{2}&\\frac{4π}{3}&-\\frac{1}{2}\\\\\\hline \\frac{π}{2}&0&\\frac{3π}{2}&0\\\\\\hline \\frac{2π}{3}&-\\frac{1}{2}&\\frac{5π}{3}&\\frac{1}{2}\\\\\\hline \\frac{3π}{4}&-\\frac{\\sqrt{2}}{2}&\\frac{7π}{4}&\\frac{\\sqrt{2}}{2}\\\\\\hline \\frac{5π}{6}&-\\frac{\\sqrt{3}}{2}&\\frac{11π}{6}&\\frac{\\sqrt{3}}{2}\\\\\\hline \\end{array}$$" }, { "type": "step", "result": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Trig General Solutions cos 1Eq" } } ], "meta": { "solvingClass": "Trig Equations", "interimType": "Generic Solve Title 1Eq" } }, { "type": "step", "result": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn" } ], "meta": { "interimType": "Explore Function Slope Zero 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xymbOTBTIQkBEGSNjyYYsjjDErT97kX84sZPuiUzCW6s7owSbZQBVuOtvRJx3LSI/x7ifgj01leBvCgjt/oPHZe5sx9ofChL1R09C2FxYxGw7b/eAUcCiWhKnKgCxH9DYoeKx1MjWVm0Kdyxzxb8CEW68btwYXu7RnF2bfMUQyo0D6XsImsfQ9kwwlZykdzF3r25KB+jBxoNJrTrINu1WPPbyEawxV0/dXBbwtg74kPgUN0sGffhjqw8+xWFcAD71R0QaEv91sNcOmUNqFeVOBahxhTpHDZ1k+ucVsahTHV5a" } }, { "type": "step", "primary": "Identify critical points not in $$f\\left(t\\right)$$ domain" }, { "type": "interim", "title": "Domain of $$6\\sin\\left(t\\right)\\::{\\quad}-\\infty\\:<t<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<t<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "step", "primary": "All critical points are in domain", "result": "t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn" } ], "meta": { "solvingClass": "Function Critical", "interimType": "Critical Points Table Top 1Eq", "gptData": "pG0PljGlka7rWtIVHz2xyoU2cWyPLDgE1QLLHeauuc3PHQdChPJ2JhfqHT+ZU0OMs2QEmolszJqqcFPw9dQfeG06oE/IhPOL1FHTTt/qX1Wjeh7+jKEzLb7VNCEMF3Z/YwBrLvM2HGOB9Om3+sTF+phTIy0qqxfGDy1f5QnA+ICwiNrEngO+NNvZ9sqNu+2V" } }, { "type": "interim", "title": "Domain of $$6\\sin\\left(t\\right)\\::{\\quad}-\\infty\\:<t<\\infty\\:$$", "steps": [ { "type": "definition", "title": "Domain definition", "text": "The domain of a function is the set of input or argument values for which the function is real and defined" }, { "type": "step", "primary": "The function has no undefined points nor domain constraints. Therefore, the domain is", "result": "-\\infty\\:<t<\\infty\\:" } ], "meta": { "solvingClass": "Function Domain", "interimType": "Function Domain Top 1Eq" } }, { "type": "interim", "title": "Find intervals:$${\\quad}$$Increasing$$:2πn\\le\\:t<\\frac{π}{2}+2πn,\\:$$Decreasing$$:\\frac{π}{2}+2πn<t<\\frac{3π}{2}+2πn,\\:$$Increasing$$:\\frac{3π}{2}+2πn<t<2π+2πn$$", "steps": [ { "type": "interim", "title": "Periodicity of $$6\\sin\\left(t\\right):{\\quad}2π$$", "steps": [ { "type": "step", "primary": "Periodicity of $$a\\cdot\\sin\\left(bx\\:+\\:c\\right)\\:+\\:d=\\frac{\\mathrm{periodicity\\:of}\\:\\sin\\left(x\\right)}{|b|}$$", "secondary": [ "Periodicity of $$\\sin\\left(x\\right)\\:$$is $$2π$$" ], "result": "=\\frac{2π}{\\left|1\\right|}" }, { "type": "step", "primary": "Simplify", "result": "=2π" } ], "meta": { "solvingClass": "Function Periodicity", "interimType": "Period Top 1Eq" } }, { "type": "step", "primary": "Combine the critical point(s): $$t=\\frac{π}{2}+2πn,\\:t=\\frac{3π}{2}+2πn\\:$$with the period: $$2πn\\le\\:t<2π+2πn$$" }, { "type": "step", "primary": "The function monotone intervals are:", "result": "2πn\\le\\:t<\\frac{π}{2}+2πn,\\:\\frac{π}{2}+2πn<t<\\frac{3π}{2}+2πn,\\:\\frac{3π}{2}+2πn<t<2π+2πn" }, { "type": "interim", "title": "Check the sign of $$f\\:{^{\\prime}}\\left(t\\right)=6\\cos\\left(t\\right)$$ at each monotone interval", "steps": [ { "type": "interim", "title": "Check the sign of $$6\\cos\\left(t\\right)$$ at $$2πn\\le\\:x<\\frac{π}{2}+2πn:{\\quad}$$Positive", "steps": [ { "type": "step", "primary": "Evaluate the derivative at a point on the interval. Take the point $$t=1+2πn$$ and plug it into $$6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(1+2πn\\right)" }, { "type": "step", "primary": "Since $$f\\left(x\\right)\\:$$is periodic, then $$f\\:{^{\\prime}}\\left(t\\right)\\:$$is also periodic: $$6\\cos\\left(t+2πn\\right)=6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(1\\right)" }, { "type": "step", "primary": "Refine to a decimal form", "result": "3.24181…" }, { "type": "step", "result": "\\mathrm{Positive}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "interim", "title": "Check the sign of $$6\\cos\\left(t\\right)$$ at $$\\frac{π}{2}+2πn<x<\\frac{3π}{2}+2πn:{\\quad}$$Negative", "steps": [ { "type": "step", "primary": "Evaluate the derivative at a point on the interval. Take the point $$t=3+2πn$$ and plug it into $$6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(3+2πn\\right)" }, { "type": "step", "primary": "Since $$f\\left(x\\right)\\:$$is periodic, then $$f\\:{^{\\prime}}\\left(t\\right)\\:$$is also periodic: $$6\\cos\\left(t+2πn\\right)=6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(3\\right)" }, { "type": "step", "primary": "Refine to a decimal form", "result": "-5.93995…" }, { "type": "step", "result": "\\mathrm{Negative}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } }, { "type": "interim", "title": "Check the sign of $$6\\cos\\left(t\\right)$$ at $$\\frac{3π}{2}+2πn<x<2π+2πn:{\\quad}$$Positive", "steps": [ { "type": "step", "primary": "Evaluate the derivative at a point on the interval. Take the point $$t=5+2πn$$ and plug it into $$6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(5+2πn\\right)" }, { "type": "step", "primary": "Since $$f\\left(x\\right)\\:$$is periodic, then $$f\\:{^{\\prime}}\\left(t\\right)\\:$$is also periodic: $$6\\cos\\left(t+2πn\\right)=6\\cos\\left(t\\right)$$", "result": "6\\cos\\left(5\\right)" }, { "type": "step", "primary": "Refine to a decimal form", "result": "1.70197…" }, { "type": "step", "result": "\\mathrm{Positive}" } ], "meta": { "interimType": "Check Positive Negative One Region 2Eq" } } ], "meta": { "interimType": "Checking Positive Negative Derivative 1Eq" } }, { "type": "step", "primary": "Summary of the monotone intervals behavior", "secondary": [ "$$\\begin{array}{|c|c|c|c|c|c|}\\hline &2πn\\le x<\\frac{π}{2}+2πn&x=\\frac{π}{2}+2πn&\\frac{π}{2}+2πn<x<\\frac{3π}{2}+2πn&x=\\frac{3π}{2}+2πn&\\frac{3π}{2}+2πn<x<2π+2πn\\\\\\hline \\mathrm{Sign}&+&0&-&0&+\\\\\\hline \\mathrm{Behavior}&\\mathrm{Increasing}&\\mathrm{Maximum}&\\mathrm{Decreasing}&\\mathrm{Minimum}&\\mathrm{Increasing}\\\\\\hline \\end{array}$$" ] }, { "type": "step", "result": "\\mathrm{Increasing}:2πn\\le\\:t<\\frac{π}{2}+2πn,\\:\\mathrm{Decreasing}:\\frac{π}{2}+2πn<t<\\frac{3π}{2}+2πn,\\:\\mathrm{Increasing}:\\frac{3π}{2}+2πn<t<2π+2πn" } ], "meta": { "interimType": "Function Find Intervals 0Eq" } }, { "type": "step", "primary": "Plug the extreme point $$x=\\frac{π}{2}+2πn\\:$$into $$6\\sin\\left(t\\right)\\quad\\Rightarrow\\quad\\:y=6$$", "result": "\\mathrm{Maximum}\\left(\\frac{π}{2}+2πn,\\:6\\right)" }, { "type": "step", "primary": "Plug the extreme point $$x=\\frac{3π}{2}+2πn\\:$$into $$6\\sin\\left(t\\right)\\quad\\Rightarrow\\quad\\:y=-6$$", "result": "\\mathrm{Minimum}\\left(\\frac{3π}{2}+2πn,\\:-6\\right)" }, { "type": "step", "result": "\\mathrm{Maximum}\\left(\\frac{π}{2}+2πn,\\:6\\right),\\:\\mathrm{Minimum}\\left(\\frac{3π}{2}+2πn,\\:-6\\right)" } ], "meta": { "solvingClass": "Function Extreme", "interimType": "Extreme Points Table Top 1Eq" } } ] }, "plot_output": { "meta": { "plotInfo": { "variable": "t", "plotRequest": "6\\sin(t)" }, "showViewLarger": true } } }