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Logarithmic Laws
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Worked Examples on Functions
1. Determine the Domain and/or Range of the given functions, as required:
\((a)\:\:After\:sketching\:the\:function\:f(t)\:=\:−t^2\:+\:2,\:find\:the\:Range\:of\:the\:function\:when\:the\:Domain\:of\:f(t)\:is\:t\:∈\:(−∞,\:∞). \)
\((b)\:After\:sketching\:the\:function\:f(t)\:=\:−2t\:−\:1,\:find\:the\:Range\:when\:the\:Domain\:of\:f(t)\:is\:t\:∈\:[−1,\:3].\)
\((c)\:Find\:the\:Range\:of\:the\:function\:f(t)\:=\:3t\:−\:3\:when\:the\:Domain\:of\:f(t)\:is\:t\:∈\:[0,\:2.5).\)
\((d)\:Find\:the\:Range\:of\:the\:function\:f(t)\:=\:t^2\:+\:2t\:+\:3\:when\:the\:Domain\:of\:f(t)\:is\:t\:∈\:(−∞,\:∞). \)
\((e)\:After\:sketching\:the\:function\:f(t)\:=\:t/2\:−\:3/2,\:find\:the\:Domain\:of\:the\:function\:when\:the\:Range\:of\:f(t)\:is\:f(t)\:∈\:(−1,\:2].\:\)
\(\:2.\:Draw\:the\:graph\:of\:f(x)\:=\:x^2\:+\:2x\:−\:3,\:first\:by\:completing\:the\:square,\:and\:then\:by\:finding\:intercepts\:and\:turning\:points\)
\( Question 3: Evaluate \)
\((a)\:sin(\:5π/6),\:cos(\:5π/6),\:tan(\:5π/6).\:\)
\((b)\:sin(\:5π/4),\:cos(\:5π/4),\:tan(\:5π/4).\)
In-Class Practice on Functions
\( 1.\:Sketch\:the\:following\:functions.\:State\:the\:domain\:and\:range\:for\:each\:function.\:How \\ \:does\:the\:form\:of\:the\:function\:change\:if\:the\:function\:is\:shifted\:one\:unit\:to\:the\:right\:and \\ \:down\:by\:two\:units?\:Sketch\:the\:new\:function\:on\:the\:same\:axes. \)
\((a)\:y\:=\:3x\:−\:2\:\)
\((b)\:f(x)\:=\:−x^2\:+\:5x\:−\:4\:\)
\((c)\:f(x)\:=\:(x\:+\:2)^3\:−\:5\:\)
\((d)\:y\:=\:x^3\:−\:x\:+\:1\:\)
\(\:\:2.\:Using\:trig\:identities\:and\:standard\:triangles,\:find\:values\:for\:the\:following:\:\)
\((a)\:sin(π/6)\)
\((b)\:cos\:(2π/3)\:\)
\((c)\:cos(5π/12)\)
Problem-Solving Practice Question 1 Elementary Functions
\((a)\:For\:the\:function\:f(x)\:=\:3x^2\:−\:2x\:+\:1,\:evaluate\:(i)\:f(0)\:\:(ii)\:f(1)\:\:(iii)\:f(2)\:\:\)
\((b)\:Suppose\:g(x)\:=\:x\:+\:1;\:now\:evaluate\:(i)\:f(g(0))\:(ii)\:f(g(1))\:(iii)\:f(g(2))\:\)
\((c)\:Sketch\:the\:graphs\:of\:f(x)\:and\:f(g(x)).\:What\:is\:the\:relationship\:between\:f(x)\:and\:f(g(x))?\:\)
\((d)\:Suppose\:h(x)\:=\:f(x)\:−\:5.\:Evaluate\:h\:at\:x\:=\:0,\:1,\:2.\:Sketch\:h(x).\:What\:is\:the\:relationship\:between\:f(x)\:and\:h(x)?\:\)
\(\:(e)\:For\:the\:table\:of\:data\:given\:here,\:sketch\:the\:graph\:of\:y.\:\begin{bmatrix}x&0&1&2&3&4&5&6&7&8&9&10\\y&0&1.2&2&2&1.2&0&-1.2&-2&-2&-1.2&0\end{bmatrix}\:\)
\((f)\:Sketch\:the\:graph\:m(x)\:=\:sin\:x,\:for\:x\:∈\:[0,\:2π].\:\)
\((g)\:Could\:we\:model\:the\:data\:in\:(e)\:using\:a\:shifted\:and/or\:stretched\:trigonometric\:function?\\ \:What\:would\:the\:function\:be?\:What\:is\:the\:amplitude?\:If\:you\:have\:one,\:use\:your\:graphing\:calculator\\ \:to\:plot\:this\:function\:as\:well\:as\:the\:data\:in\:(e)\:-\:so\:you\:can\:see\:how\:closely\:(or\:not)\:your\\ \:function\:matches\:the\:data)\:\)
Problem-Solving Practice Question 2 Motion and Trigonometry
\( (a)\:Suppose\:an\:object\:is\:moving\:in\:a\:straight\:line,\:and\:its\:position\:from\:the\:origin\:at\:any\\ \:time\:t\:seconds\:is\:given\:by\:f(t)\:=\:t^2\:−\:8t\:+\:7,\:in\:centimetres.\)
\((i)\:What\:is\:the\:initial\:position\:of\:the\:object?\:\)
\((ii)\:What\:is\:the\:position\:after\:each\:of\:1,\:2,\:3,\:4\:and\:5\:seconds?\:\)
\((iii)\:What\:is\:the\:average\:velocity\:during\:the\:3rd\:and\:4th\:seconds?\:\)
Problem-Solving Practice Question 3 Equation of Straight Line
\(\:\:(a)\:Suppose\:a\:road\:with\:a\:uniform\:grade\:is\:to\:be\:constructed\:between\:points\:A\:and\:B\:where\\ \:point\:A\:is\:at\:distance\:0\:and\:elevation\:450\:metres,\:and\:point\:B\:is\:at\:distance\:500\:metres\:and\\ \:elevation\:500\:metres.\:\)
\((i)\:Find\:the\:equation\:of\:the\:line\:joining\:points\:A\:and\:B.\:\)
\((ii)\:What\:is\:the\:elevation\:of\:the\:point\:C\:that\:is\:on\:this\:line\:and\:is\:a\:horizontal\:distance\:100\:metres\:from\:A?\:\)
\(\:(iii)\:How\:far\:from\:A\:(horizontally)\:is\:point\:D\:if\:D\:has\:elevation\:480\:metres?\:\)
\(\:(iv)\:What\:is\:the\:angle\:of\:elevation\:of\:the\:road?\:\)
Home
Tips for new Engineers
Free Useful Programs and Information Sources
University Website Tips and Tricks
Surviving University
Learning Resources
First Year Engineering
>
Engineering Sustainability and Professional Practice
>
What is this course about?
Concepts
>
Cultural Awareness
Wicked Problems
EWB Challenge
Foundation of Engineering Design
>
What is this course about?
Energy in Engineering Systems
>
What is this course about?
Engineering Computation
>
What is this course about?
Statics
>
Sine and Cosine Law
Mathematics: Single variable Calculus and differential equations
>
What is this course about?
Logarithmic Laws
Week One:
Workshop One:
Identity Sheet
Introductory Engineering Mathematics
>
Week1
Engineering Computation
>
week 3
Other
>
The Business of IT
>
Helpful links
Notes
Practice Final Exam
Information Security
Databases
Networks