Questions on differentiation from first principles, and continuity and differentiability.
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\n$a= \\;$[[1]] and $h=\\; $[[2]]
Input your estimation to $5$ decimal places: [[3]]
True value is: [[4]] (input to 5 decimal places).
\n ", "showCorrectAnswer": true, "marks": 0}], "statement": "Use the approximation $f(a+h) \\approx f(a)+hf^{\\prime}(a)$ to estimate \\[\\var{a+h}^{\\frac{\\var{m}}{\\var{n}}} \\]for a suitable function $f(x)$.
", "tags": ["application of differentiation", "approximation", "approximation of the value of a function using the tangent", "approximations", "calculus", "Calculus", "checked2015", "equation of tangent", "first order approximation", "functions", "maclaurin series", "MacLaurin series", "mas1601", "MAS1601", "tangent equation"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n \t\t3/07/2012
Added tags
\n \t\t20/06/2012:
\n \t\t
Added tags.
Got rid of request for 5dps for the function!
\n \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Approximate $f(x)=(a+h)^{m/n}$ by $f(a)+hf^{\\prime}(a)$ to 5 decimal places and compare with true value.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "We have $f(x)=x^\\frac{\\var{m}}{\\var{n}}$ and $a= \\var{a}$, $h=\\var{h}$.
Note that $\\var{a}^\\frac{1}{\\var{n}}=\\var{c}$ and so using the approximation :
$f(a+h)\\approx f(a)+hf^{\\prime}(a)$ and $f^{\\prime}(x) = \\frac{\\var{m}}{\\var{n}}\\simplify[std]{x^({m-n}/{n})}$
we have:
\\[\\simplify[std]{{a+h}^({m}/{n})}\\approx \\simplify[simplifyFractions]{{a}^({m}/{n})+{h}*({m}/{n})*{a}^({p}/{n})}=\\simplify[std,!sqrtSquare]{{c}^{m}+ {h}*({m}/{n})*{c}^{m-n}}=\\var{est}\\]
to 5 decimal places.
The true value to 5 decimal places is {tr}.
The gradient $m =$ [[0]] (input your answer to 3 decimal places).
\n\t\t\tThe equation of the chord is $y=ax+b$ where:
\n\t\t\t$a= \\;$[[1]] and $b=\\; $[[2]]
\n\t\t\tEnter both values $a$ and $b$ correct to 3 decimal places.
\n\t\t\t \n\t\t\t", "steps": [{"type": "information", "prompt": "Given two points $(a,f(a))$ and $(a+h,f(a+h))$ on the graph of the function $y=f(x)$.
Then the chord is the straight line between these two points and has the equation \\[y-f(a)=m(x-a)\\] where $m$ is the gradient of the chord.
The gradient is given by dividing the change in $y$ by the change in $x$.
Hence for this example \\[m = \\frac{f(a+h)-f(a)}{h} = \\frac{f(\\var{a+h})-f(\\var{a})}{\\var{h}}\\]
Let $f(x)=\\simplify[std]{(x+{b})^{n}}$. What are the gradient and equation of the chord between $(\\var{a},f(\\var{a}))$ and $(\\simplify[std]{{a}+{h}},f(\\simplify[std]{{a}+{h}}))$?
\n\tYou can get help by clicking on Show steps. If you do so you will lose 1 mark.
\n\t \n\t", "tags": ["Calculus", "MAS1601", "Newton quotient", "Steps", "checked2015", "chord", "equation of a chord", "equation of a straight line", "function", "functions", "gradient of chord", "straight line"], "rulesets": {"std": ["all", "!collectNumbers"], "dpoly": ["std", "fractionNumbers"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n\t\t20/06/2012:
\n\t\tAdded tags.
\n\t\tDecided to include Steps as a tag. Perhaps the presence of Steps can be searched for in another way?
\n\t\t03/07/2012:
Added tags.
\n\t\t31/07/2012:
\n\t\tSteps issue resolved.
\n\t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "Given $f(x)=(x+b)^n$. Find the gradient and equation of the chord between $(a,f(a))$ and $(a+h,f(a+h))$ for randomised values of $a$, $b$ and $h$.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "Given two points $(a,f(a))$ and $(a+h,f(a+h))$ on the graph of the function $y=f(x)$.
Then the chord is the straight line between these two points and has the equation \\[y-f(a)=m(x-a)\\] where $m$ is the gradient of the chord.
The gradient is given by dividing the change in $y$ by the change in $x$.
Hence for this example \\[m = \\frac{f(a+h)-f(a)}{h} = \\frac{f(\\var{a+h})-f(\\var{a})}{\\var{h}} = \\var{d1} = \\var{val}\\] to 3 decimal places.
Hence the equation of the chord is of the form $y=\\var{d1}x+b$ for some constant $b$.
But we know that when $x=\\var{a}$ then $y=f(\\var{a}) = \\var{a+b}^\\var{n}=\\var{(a+b)^n}$
So \\[b=\\var{(a+b)^n}-\\var{d1}\\times\\var{a} = \\var{d}=\\var{val1}\\] to 3 decimal places
[[0]]
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", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (2 correct out of 4) covering properties of continuity and differentiability. Selection of questions from a pool.
\nCan choose true and false for each option. Also in one test run the second choice was incorrectly entered, rest correct, but the feedback indicates that the third was wrong.
"}, "advice": "You should be able to work out the correct answers from your notes.
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\"", "description": "", "name": "f5"}, "h": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..4)", "description": "", "name": "h"}, "f2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If for some sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$, then the function $f$ is continuous at $c$.
\"", "description": "", "name": "f2"}, "f3": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If $f(x) \\\\to \\\\ell$ as $x \\\\to c$ and if $g(x) \\\\to m$ as $x \\\\to c$, then $\\\\dfrac{f(x)}{g(x)} \\\\to \\\\dfrac{\\\\ell}{m}$ as $x \\\\to c$.
\"", "description": "", "name": "f3"}, "tr5": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"There exists a function $f$ such that the limit of $f(x)$ as $x \\\\to c$ exists and $f(c)$ exists, but $f$ is not continuous at $c$.
\"", "description": "", "name": "tr5"}, "ch1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "if(t=1,tr1,if(t=2,tr2,tr3))", "description": "", "name": "ch1"}, "tr6": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If the limit of $f(x)$ as $x \\\\to c$ exists and the limit is $f(c)$, then $f$ is continuous at $c$.
\"", "description": "", "name": "tr6"}, "tr2": {"templateType": "long string", "group": "Ungrouped variables", "definition": "\"If a function $f$ is continuous at $c$, then for any sequence {$x_n$} converging to $c$, the sequence {$f(x_n)$} converges to $f(c)$.
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", "tags": ["checked2015", "continuous", "convergence", "convergent sequences", "limits", "sequence", "sequences"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Multiple response question (2 correct out of 4) covering properties of continuity and limits of functions. Selection of questions from a pool.
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