UBC Wiki - User contributions [en]

Template:MER Category parent page

2018-05-27T14:42:24Z

CarmenBruni: Attempt 7 maybe?...

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* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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==== Tags Dictionary ====
<categorytree mode=categories hideroot=on>MER Parent Tag</categorytree>


[[Category:MER Parent Tag]]
__HIDDENCAT__ __NOTOC__ __NOEDITSECTION__
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:39:43Z

CarmenBruni:

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* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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[[Category:MER Parent Tag]]
__HIDDENCAT__ __NOTOC__ __NOEDITSECTION__
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:36:57Z

CarmenBruni: Attempt 5...

<includeonly>{{DISPLAYTITLE:<span style="position: absolute; clip: rect(1px 1px 1px 1px); clip: rect(1px, 1px, 1px, 1px);">{{FULLPAGENAME}}</span>}}{{#ifgroup:science|
* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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[[Category:MER Parent Tag]]
__HIDDENCAT__ __NOTOC__ __NOEDITSECTION__
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:31:10Z

CarmenBruni: Attempt 4...

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* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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[[Category:MER Parent Tag]]
__HIDDENCAT__ __NOTOC__ __NOEDITSECTION__
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:28:12Z

CarmenBruni: Attempt 3...

<includeonly>{{DISPLAYTITLE:<span style="position: absolute; clip: rect(1px 1px 1px 1px); clip: rect(1px, 1px, 1px, 1px);">{{FULLPAGENAME}}</span>}}{{#ifgroup:science|
* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:25:57Z

CarmenBruni: Attempt 2...

<includeonly>{{DISPLAYTITLE:<span style="position: absolute; clip: rect(1px 1px 1px 1px); clip: rect(1px, 1px, 1px, 1px);">{{FULLPAGENAME}}</span>}}{{#ifgroup:science|
* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
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titlematch=Math Exam Resources/Courses/%
nottitlematch=Math Exam Resources/Courses/%/%
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<noinclude>[[Category:MER Template]]{{documentation}}</noinclude>

Template:MER Category parent page

2018-05-27T14:25:28Z

CarmenBruni: Attempting to fix the replaceintitle error in Tags Dictionary.

<includeonly>{{DISPLAYTITLE:<span style="position: absolute; clip: rect(1px 1px 1px 1px); clip: rect(1px, 1px, 1px, 1px);">{{FULLPAGENAME}}</span>}}{{#ifgroup:science|
* [[Science:MER|Back to the MER portal]]
* [[Science:MER/Tags|Back to the Tags overview]]
* [[Science:Math Exam Resources|Back to the main page]]
}}
= {{PAGENAME}} =

{{{content}}}

{{#dpl:
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=== Quick Navigation ===

[[Science:Math Exam Resources|Back to the main page]]
<dpl>
titlematch=Math Exam Resources/Courses/%
nottitlematch=Math Exam Resources/Courses/%/%
replaceintitle=$Math Exam Resources/Courses/$,
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__HIDDENCAT__ __NOTOC__ __NOEDITSECTION__
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Category:MER Tag Taylor series

2018-04-20T13:25:03Z

CarmenBruni:

{{MER Category page|parent_category=Sequences and series|content={{MER External credits|towhom=CarmenBruni}}

Also, check out [[:Category:MER Tag Linear approximation|Linear approximation]].

A '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page26DerivativesUsingTaylorSeries.mp4 derivatives using Taylor series]

{{MER Video
|title=How to integrate and differentiate power (or Taylor) series
|id=qPl9nr8my2Q
|towhom=PatrickJMT
}}

{{MER Video
|title=How to multiply and divide power (or Taylor) series
|id=_L2VcGjp670
|towhom=PatrickJMT
}}

{{MER Video
|title=The Taylor series for <math> \sin(x) </math>
|id=dp2ovDuWhro
|towhom=PatrickJMT
}}

{{MER Video
|title=Examples of Taylor series (Part 1)
|id=cjPoEZ0I5wQ
|towhom=PatrickJMT
}}

{{MER Video
|title=Examples of Taylor series (Part 2)
|id=Os8OtXFBLkY
|towhom=PatrickJMT
}}

{{MER Video
|title=An explanation on Taylor's Remainder Theorem
|id=yJWFAG1g4Jw
|towhom=PatrickJMT
}}

{{MER Video
|title=An important variation
|text=A slight variation on Taylor's remainder theorem (and probably the one that would be used on exams). This is sometimes called Taylor's inequality or the Lagrange form of the remainder.
|id=9LnCcAoEHG4
|towhom=PatrickJMT
}}
}}

Category:MER Tag Geometric series

2018-04-20T13:24:49Z

CarmenBruni:

{{MER Category page|parent_category=Sequences and series
|content={{MER External credits|towhom=CarmenBruni}}

A '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page1BasicSeries.mp4 summing geometric series]

{{MER Video
|title=A quick review
|text=A review on summation notation.
|id=hEPk36Yncxg
|towhom=PatrickJMT
}}

A two part video series on geometric series (these also include the divergence test which if you haven't seen yet you will see later).

{{MER Video
|title=Geometric series (Part 1)
|id=xjmy5hkZccY
|towhom=PatrickJMT
}}

{{MER Video
|title=Geometric series (Part 2)
|id=C8piSCOdo1Y
|towhom=PatrickJMT
}}

{{MER Video
|title=Geometric series with rational numbers
|text=Another example of geometric series expressing decimals as rational numbers.
|id=7tDK_UjdWOs
|towhom=PatrickJMT
}}
}}

Category:MER Tag Fundamental theorem of calculus

2018-04-20T13:23:17Z

CarmenBruni:

{{MER Category page|parent_category=Differentiate|content=
{{About|the single variable fundamental theorem of calculus||MER_Tag_Fundamental_theorem_of_calculus_(disambiguation)|Here}}
{{MER External credits|towhom=CarmenBruni}}

Links updated April 20th, 2018.

Here is a '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page5FundamentalTheoremofCalculus.mp4 differentiating integrals.]

{{MER Video
|title=Explanation (Part 1)
|text=A video explaining the Fundamental Theorem of Calculus (Part 1).
|id=PGmVvIglZx8
|towhom=PatrickJMT
}}

{{MER Video
|title=Explanation (Part 2)
|text=A video explaining the Fundamental Theorem of Calculus (Part 2).
|id=nHnZVFeQvNQ
|towhom=PatrickJMT
}}

{{MER Video
|title=Introductory examples
|text=Here are some introductory examples.
|id=aw_VM_ZDeIo
|towhom=PatrickJMT
}}
}}

Category:MER Tag Sequences

2018-04-20T13:22:15Z

CarmenBruni:

{{MER Category page|parent_category=Sequences and series
|content=

{{MER Video
|title=An introduction to sequences
|id=9K1xx6wfN-U
|towhom=PatrickJMT
}}

=== Cobwebbing ===

{{MER External credits|towhom=CarmenBruni}}

Last updated links on April 20, 2018.

The following '''pencasts''' are

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page13ConvergentSequencePt1.mp4 A cobwebbing example in disguise solved using the monotone convergence theorem]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page15ConvergentSequencePt2.mp4 A formal proof of the above video]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page17ConvergentSequencePt3.mp4 Bad things happen if you don't check for convergence first!]

[https://www.livescribe.com/int/player/lsnotesdesktop.htm?cak=6a89cc5a3e4a31c6;docId=2722771;pageId=120617001 Using cobwebbing to solve an example similar to the above]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page19Stability.mp4 Stability, fixed points, equilibria, steady states, monotonic, oscillatory and other buzz words.]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page20Cobwebbing.mp4 Cobwebbing by hand]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page25CobwebbingPt2.mp4 One last cobwebbing example]
}}

Category:MER Tag Sequences

2018-04-20T13:21:50Z

CarmenBruni:

{{MER Category page|parent_category=Sequences and series
|content=

{{MER Video
|title=An introduction to sequences
|id=9K1xx6wfN-U
|towhom=PatrickJMT
}}

=== Cobwebbing ===

{{MER External credits|towhom=CarmenBruni}}

The following '''pencasts''' are

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page13ConvergentSequencePt1.mp4 A cobwebbing example in disguise solved using the monotone convergence theorem]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page15ConvergentSequencePt2.mp4 A formal proof of the above video]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page17ConvergentSequencePt3.mp4 Bad things happen if you don't check for convergence first!]

[https://www.livescribe.com/int/player/lsnotesdesktop.htm?cak=6a89cc5a3e4a31c6;docId=2722771;pageId=120617001 Using cobwebbing to solve an example similar to the above]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page19Stability.mp4 Stability, fixed points, equilibria, steady states, monotonic, oscillatory and other buzz words.]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page20Cobwebbing.mp4 Cobwebbing by hand]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page25CobwebbingPt2.mp4 One last cobwebbing example]
}}

Category:MER Tag Riemann sum

2018-04-20T13:20:10Z

CarmenBruni:

{{MER Category page|parent_category=Integrate|content=
Typically, the following formulas will be useful

:<math>
\sum_{i=1}^{n}i = \frac{n(n+1)}{2} \qquad \sum_{i=1}^{n}i^{2} = \frac{n(n+1)(2n+1)}{6} \qquad \sum_{i=1}^{n}i^{3} = \left(\frac{n(n+1)}{2}\right)^{2}
</math>

{{MER External credits|towhom=CarmenBruni}}

Some '''pencasts''' [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page3RiemannSums.mp4 on Riemann sum notation] and [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page4RiemannSumsPt2.mp4 on converting Riemann sums to definite integrals].

{{MER Video
|title=The basics of the definite integral and Riemann sums
|id=LkdodHMcBuc
|towhom=PatrickJMT
}}

{{MER Video
|title=The rectangle method
|text=Approximating area under a curve (definite integral) using rectangles.
|id=Gi8_a7NlKAQ
|towhom=PatrickJMT
}}

{{MER Video
|title=An example computation (Part 1)
|text=A two part series on computing definite integrals using Riemann sums.
|id=gFpHHTxsDkI
|towhom=PatrickJMT
}}

{{MER Video
|title=An example computation (Part 2)
|text=A two part series on computing definite integrals using Riemann sums.
|id=GE4OLfmJ8P8
|towhom=PatrickJMT
}}
}}

Category:MER Tag Integration by parts

2018-04-20T13:19:43Z

CarmenBruni:

{{MER Category page|parent_category=Integrate|content=
{{MER External credits|towhom=CarmenBruni}}

A '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page11IntegrationByParts.mp4 Integration by Parts] (there is a homemade term that the author calls 'the table method' just to help with remembering and with speed).

{{MER Video
|title=The main ideas of integration by parts
|text=This video explains by example the idea behind integration by parts.
|id=dqaDSlYdRcs
|towhom=PatrickJMT
}}

{{MER Video
|title=Definite integral
|text=Here's an example of a definite integral.
|id=zGGI4PkHzhI
|towhom=PatrickJMT
}}

{{MER Video
|title=Integrate by parts twice
|text=If at first you don't succeed, try try again! Here is an example of doing integration by parts twice.
|id=_MuKOm8IHBA
|towhom=PatrickJMT
}}

{{MER Video
|title=An Example
|text=Another example of using integration by parts twice.
|id=L8HoALsls-E
|towhom=PatrickJMT
}}
}}

Category:MER Tag Solid of revolution

2018-04-20T13:18:02Z

CarmenBruni:

{{MER Category page|parent_category=Applications of integration|content=
{{MER External credits|towhom=CarmenBruni}}

A '''pencast''' series on the solid of revolution.

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page7.png the area between two curves (No volume)]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page8AreaBetweenTwoCurvesPt2.mp4 Area between two curves with respect to the y-axis.] Includes a review of completing the square.

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page9VolumesOfRevolutionPt1.mp4 Volume of revolution around axis parallel to x-axis]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page10VolumesOfRevolutionPt2.mp4 Volume of revolution around axis parallel to y-axis]
}}

Category:MER Tag Area between two curves

2018-04-20T13:16:19Z

CarmenBruni:

{{MER Category page|parent_category=Applications of integration|content=
{{MER External credits|towhom=CarmenBruni}}

Links updated as of April 20th, 2018.

A '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page7.png the area between two curves (No volume for some reason - working on it!)]

A '''pencast''' on [https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page8AreaBetweenTwoCurvesPt2.mp4 Area between two curves with respect to the y-axis.] Includes a review of completing the square.

{{MER Video
|title=Basics
|text=A video explaining how to compute the area between two curves
|id=DRFyNHdVgUA
|towhom=PatrickJMT
}}

{{MER Video
|title=Using the ''y'' coordinate (part 1)
|text=Here is a two part video on computing the area between two curves using the ''y'' coordinate.
|id=70NQ3ISYihw
|towhom=PatrickJMT
}}

{{MER Video
|title=Using the ''y'' coordinate (part 2)
|text=Here is a two part video on computing the area between two curves using the ''y'' coordinate.
|id=Xne6Hv9useE
|towhom=PatrickJMT
}}
}}

Category:MER Parent Tag Sequences and series

2018-04-20T13:06:42Z

CarmenBruni:

{{MER Category parent page|content=
{{MER External credits|towhom=CarmenBruni}}

A series where the author blindly goes through series problems and attempts to solve them while discussing his thought process. These used to be pencasts but have since been converted to videos and have had the sites changed to be updated as of April 20th, 2018.

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page21HowToDetermineConvergentSeriesPt1.mp4 How to Determine Convergent Series Part 1]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page22HowToDetermineConvergentSeriesPt2.mp4 How to Determine Convergent Series Part 2]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page23HowToDetermineConvergentSeriesPt3.mp4 How to Determine Convergent Series Part 3]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page24HowToDetermineConvergentSeriesPt4.mp4 How to Determine Convergent Series Part 4]

{{MER Video
|title=Starter video on summation notation basics
|id=hEPk36Yncxg
|towhom=PatrickJMT
}}

{{MER Video
|title=An introduction to series
|id=haK3oC0L_a8
|towhom=PatrickJMT
}}

{{MER Video
|title=The geometric series and the divergence test (Part 1)
|id=xjmy5hkZccY
|towhom=PatrickJMT
}}

{{MER Video
|title=The geometric series and the divergence test (Part 2)
|id=C8piSCOdo1Y
|towhom=PatrickJMT
}}

{{MER Video
|title=An example showing divergence using the definition (ie by partial sums)
|id=MDYb5DnRH2c
|towhom=PatrickJMT
}}

{{MER Video
|title=An example of a telescoping series.
|id=UpauFx8fJ24
|towhom=PatrickJMT
}}

{{MER Video
|title=The integral test for convergence
|id=ojztrQMqLgE
|towhom=PatrickJMT
}}

{{MER Video
|title=The limit comparison test and the direct comparison test (Part 1)
|id=xesQnFWw8f8
|towhom=PatrickJMT
}}

{{MER Video
|title=The limit comparison test and the direct comparison test (Part 2)
|id=8eCFY82HkRA
|towhom=PatrickJMT
}}

{{MER Video
|title=A first example using the ratio test
|id=iy8mhbZTY7g
|towhom=PatrickJMT
}}

{{MER Video
|title=A second example using the ratio test
|id=gay5CEXnkhA
|towhom=PatrickJMT
}}

{{MER Video
|title=Alternating series (Part 1)
|id=-K9Qt6YUIrI
|towhom=PatrickJMT
}}

{{MER Video
|title=Alternating series (Part 2)
|id=zgTRNpKcenE
|towhom=PatrickJMT
}}

{{MER Video
|title=Practice your intuition
|id=DvadVYHf3pM
|towhom=PatrickJMT
}}

{{MER Video
|title=A video culminating all of the above.
|id=6hOeqjoQvNw
|towhom=PatrickJMT
}}
}}

Science talk:Math Exam Resources/Courses/MATH101/April 2012/Question 04 (b)

2018-04-20T13:04:26Z

CarmenBruni: Talk page autocreated when first thread was posted

Thread:Science talk:Math Exam Resources/Courses/MATH101/April 2012/Question 04 (b)/Last Step

2018-04-20T13:04:25Z

CarmenBruni: New thread: Last Step

Probably should factor out the ''n'' in the final step before evaluating the limit. Should make it easier for students to realize the final answer is valid.

Science:Math Exam Resources/Courses/MATH101/April 2012/Question 04 (b)/Solution 1

2018-04-20T13:02:33Z

CarmenBruni:

We want to use the Ratio test, so we look at the limit:

<math> \begin{align}
\lim_{n\to \infty} \left|\frac{a_{n+1}}{a_{n}}\right|&=\lim_{n\to \infty} \frac{\frac{(2n+2)!}{((n+1)^2+1)((n+1)!)^2}}{\frac{(2n)!}{(n^2+1)(n!)^2}}\\
&=\lim_{n\to \infty}\frac{(2n+2)!(n^2+1)(n!)^2}{(2n)!((n+1)^2+1)((n+1)!)^2}\\
&=\lim_{n\to \infty} \frac{(2n+2)!}{(2n)!}\cdot \frac{(n^2+1)}{(n+1)^2+1} \cdot \left(\frac{n!}{(n+1)!}\right )^2\\
&=\lim_{n\to \infty}(2n+2)(2n+1)\cdot \frac{n^2+1}{n^2+2n+2}\cdot \frac{1}{(n+1)^2}\\
&=\lim_{n\to \infty} \frac{2n+2}{n+1}\cdot \frac{2n+1}{n+1}\cdot \frac{n^2+1}{n^2+2n+2}=2\cdot 2 \cdot 1=4
\end{align}</math>

Since this limit is greater than <math> 1 </math>, by the Ratio test, the series <math>\sum_{n=1}^\infty a_n</math> '''diverges'''.

Category:MER Parent Tag Integrate

2018-04-20T12:59:15Z

CarmenBruni:

{{MER Category parent page|content=
{{MER External credits|towhom=CarmenBruni}}

The following series of '''pencasts''' are done as follows. The author takes questions he has never seen before and the author tries to articulate his thought process while solving the questions. The links have been updated as of April 20th, 2018 to the author's new website.

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page12HowToIntegratePt1.mp4 How to Integrate Part 1]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page14HowToIntegratePt2.mp4 How to Integrate Part 2]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page16HowToIntegratePt3.mp4 How to Integrate Part 3]

[https://cs.uwaterloo.ca/~cbruni/videos/Math103JanApr2014/Page18HowToIntegratePt4.mp4 How to Integrate Part 4]

=== All Integration Questions ===

{{#dpl:
|category=MER Tag Substitution{{!}}MER Tag Integration by parts{{!}}MER Tag Trigonometric integral{{!}}MER Tag Trigonometric substitution{{!}}MER Tag Improper integral{{!}}MER Tag Partial Fractions{{!}}MER Tag Properties of the definite integral{{!}}MER Tag Riemann sum
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|columns=4
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}}

Science:Math Exam Resources/Courses/MATH152/April 2010/Question A 27/Solution 1

2016-04-10T21:54:35Z

CarmenBruni:

If we have a linear system

:<math>
A\mathbf{x}=\mathbf{b}
</math>

then Matlab can solve this via the \ operator. Therefore if we type

x=A\b;

then the vector ''x'' produced by Matlab is the solution to the linear system. (Note that this is not a typo. The command is A\b and not b\A)

'''Note.''' Since ''A'' is invertible, it is possible to get the solution by typing

x=inv(A)*b;

but you are generally discouraged from using the inv() operation in Matlab to solve linear systems. (That's because calculating the inverse of a matrix ''A'' is an overkill here, there is faster ways of finding ''x'' that don't require the matrix inverse.)

Science:Math Exam Resources/Courses/MATH100/December 2014/Question 03 (f)/Solution 1

2015-09-04T18:29:56Z

CarmenBruni:

The function <math>\arcsin</math> has domain <math>[-1,1]</math> and range <math>[-\pi/2,\pi/2]</math>, so our answer must lie in this range.

* We know that <math>\sin(x) = \sin(x-2\pi)</math>; so <math>\sin(9\pi/11)=\sin(31\pi/11)</math>.
* Also we know that <math>\sin(x)=\sin(\pi-x)</math> and so, <math>\sin(9\pi/11)=\sin(2\pi/11)</math>.

Since <math>-\pi/2\le 2\pi/11 \le \pi/2</math>, we conclude that

<math>\begin{align}
\arcsin\left(\sin(\frac{31\pi}{11})\right) &= \arcsin\left(\sin(\frac{2\pi}{11})\right)\\&= \frac {2\pi}{11}\end{align}</math>

Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (b)

2015-05-26T04:54:21Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Limit of a square root function]]

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Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (j)

2015-05-26T04:51:01Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER QGT flag]]





[[Category:MER Tag Continuous compound interest]]

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Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (k)

2015-05-26T04:11:05Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER QGT flag]]





[[Category:MER Tag Chain rule]]

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Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (m)

2015-05-26T04:04:15Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER QGT flag]]





[[Category:MER Tag Linear approximation]]

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Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (m)/Solution 1

2015-05-26T04:03:20Z

CarmenBruni:

We are looking at the linear approximation of <math>f(x) = \sin x</math> about the point <math>0</math>. Hence <math>a = 0</math> and the linear approximation is given by

:<math>
\begin{align}
L(x) &= f(a) + f'(a)(x-a) \\
&= \sin(0) + \cos(0)(x-0) \\
&= 0 + (1)(x) \\
&= x
\end{align}
</math>

We now want to estimate the error of this approximation at the point <math>x = 0.12</math>. From the formula for the error of a linear approximation we find that

:<math>\begin{align}
|\sin(0.12) - 0.12| &= |\sin(0.12) - L(0.12)| \\
&= |f(x) - L(x)| \\
&\leq \frac{M(x-a)^2}{2} \\
&= \frac{M(0.12-0)^2}{2} \\
&= 0.0072M
\end{align}</math>

Finally it remains to calculate <math>M</math>, which is the largest absolute value of the second derivative of <math>f(x)</math> in the interval <math>[a, x] = [0, 0.12]</math>. Doing the math we obtain <math>|f''(x)| = |-\sin(x)| \leq \sin(0.12)</math> (the number 1 could also be used as an upper bound though it is not as tight of an upper bound), therefore the final answer is

:<math>
|\sin(0.12) - 0.12| \leq 0.0072M = 0.0072\sin(0.12)
</math>

Science:Math Exam Resources/Courses/MATH104/December 2014/Question 02

2015-05-26T03:57:41Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER QGT flag]]





[[Category:MER Tag Limit definition of the derivative]]

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Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (d)

2015-05-11T16:21:43Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Steady state and equilibrium]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (c)

2015-05-11T16:20:55Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Steady state and equilibrium]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (a)

2015-05-11T16:19:18Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Steady state and equilibrium]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (b)

2015-05-11T16:19:13Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Steady state and equilibrium]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (a)

2015-05-11T16:16:17Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Steady states]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (a)/Solution 1

2015-05-11T16:15:49Z

CarmenBruni:

By the definition of steady state, we look for the value of <math>c</math> such that <math>\frac{dc}{dt}=0, \quad \text{or } P(c)-D(c)=0.</math> Thus we need to solve

:<math>\begin{align}
P_m \frac{c^2}{k^2+c^2}&=rc\\
P_m \cdot c^2 &=rc(k^2+c^2)\\
P_m \cdot c^2 &=k^{2}rc+rc^3\\
P_m \cdot c^2 -k^{2}rc - rc^{3} &=0\\
c(P_{m}c-k^2r-rc^{2})&=0.
\end{align}</math>

Substitute values of <math>P_m=5</math>, <math>k=10</math> and <math>r=1/5</math> and simplify the expression, we obtain

:<math>
\begin{align}
c(P_{m}c-k^2r-rc^{2})&=0\\
c(5c-20-c^{2}/5)&=0\\
\frac{-c}{5}(-25c+100+c^{2})&=0\\
\frac{1}{5}c(c-5)(c-20)=0
\end{align}
</math>

which gives <math>c=0, \ 5, \ 20</math> are the steady states.

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 02 (a)/Solution 1

2015-05-11T16:10:53Z

CarmenBruni:

By the definition of steady state, we look for the value of <math>c</math> such that <math>\frac{dc}{dt}=0, \quad \text{or } P(c)-D(c)=0.</math> Thus we need to solve

<math>\begin{align}
P_m \frac{c^2}{k^2+c^2}&=rc\\
P_m \cdot c^2 &=rc(k^2+c^2)\\
P_m \cdot c^2 &=k^{2}rc+rc^3\\
P_m \cdot c^2 -k^{2}rc - rc^{3} &=0\\
c(P_{m}c-k^2r-rc^{2})&=0.
\end{align}</math>

Substitute values of <math>P_m</math>, <math>k</math> and <math>r</math> and simplify the expression, we obtain

:<math>
\begin{align}
\frac{1}{5}c(c-5)(c-20)=0
\end{align}
</math>

which gives <math>c=0, \ 5, \ 20</math> are the steady states.

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 03

2015-05-11T16:06:36Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QBH flag]][[Category:MER QBS flag]][[Category:MER RT flag]]





[[Category:MER Tag Optimization]]

{{MER Question page}}

Science talk:Math Exam Resources/Courses/MATH102/December 2014/Question C 03

2015-05-11T16:06:09Z

CarmenBruni: Talk page autocreated when first thread was posted

Thread:Science talk:Math Exam Resources/Courses/MATH102/December 2014/Question C 03/More Hints and More details

2015-05-11T16:06:09Z

CarmenBruni: New thread: More Hints and More details

In a solution this long we need more hints. Perhaps some of the special triangles or something to help lead on the solution.

Also the work for <math> \varphi </math> should be shown and not just 'similarly'.

Lastly some note should be made that critical points also occur where the denominator is 0 (though in this case this is impossible).

Science:Math Exam Resources/Courses/MATH102/December 2014/Question C 01

2015-05-11T15:58:13Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Quotient rule]]
[[Category:MER Tag Implicit differentiation]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 07

2015-05-11T15:54:34Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Related rates]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 07/Solution 1

2015-05-11T15:53:54Z

CarmenBruni:

We draw a diagram as shown:

[[File:December 2014 Math 102 Final Exam Question B7.png|thumb|Graphs| 400px]]

Suppose the finish line is located on the positive side of the <math>x</math>-axis. Assume the angle between the line pass through the runner and the origin and the positive x-axis is <math>\theta</math>, then we have <math>\tan(\theta)=\frac{y}{x}</math>, where <math>(x,\ y)</math> is the runner’s position. When the runner crosses the finish line, <math>\theta=0, \quad \theta'=\frac{1}{10},</math> We also get <math>x'=0</math> as the elliptical track is perpendicular to the <math>x</math>-axis at the finish point so there is no change in <math>x</math>. Thus, at the finish point, the velocity of the runner can be described solely as the change in the y-axis. Also it follows <math>y'=v</math>. Further, we obtain <math>x=30</math> from the equation of the ellipse <math>x^2 + \dfrac{y^2}{4} =900</math> by substituting <math>y=0</math> and noting that thus <math>x=\pm 30</math>. Then we differentiate <math>\tan(\theta)</math> with respect to time to get

<math>\begin{align}
\sec^2(\theta)\theta'&=\frac{y'x-yx'}{x^2}\\
1\cdot\theta'&=\frac{y'x-yx'}{x^2}\\
\theta'&=\frac{v\cdot30-0}{30^2}\\
&=\frac{v}{30}
\end{align}</math>

Then <math>v=30\theta'=3m/s</math>

Note: The third line of the equation above is where the observation that <math>x'=0</math> is applied into the question as indicated by the substitution of <math>y'</math> with v, the variable of <math>x</math> with 30, and the rate of <math>x'</math> with 0.

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 06

2015-05-11T15:49:38Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Linear approximation]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 05

2015-05-11T15:48:39Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER RT flag]]





[[Category:MER Tag Euler's method]]

{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04

2015-05-11T15:47:04Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QGS flag]][[Category:MER CT flag]]







{{MER Question page}}

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04/Solution 1

2015-05-11T15:45:46Z

CarmenBruni:

The general expression of a cosine function is <math>T(t)=A+B\cos(\omega t+\varphi).</math> Suppose <math>A</math>, <math>B</math> are both positive. Here the lowest temperature is <math>2^o</math> C and highest is <math>10^o</math> C, then

:<math>A-B=2, A+B=10,</math>

so <math>A=6</math> and <math>B=4</math>. The value <math>2\pi/\omega</math> is the period of this function. As it describes the temperature changes throughout a day which lasts for 24 hours, we have <math>\frac{2\pi}{\omega}=24, \quad \omega=\frac{2\pi}{24}.</math> From the problem statement we know the lowest temperature is obtained at 7:00, by symmetry the highest is around 19:00. Then to shift the maxima from <math>0</math> to <math>19</math> we subtract <math>t</math> by 19. So <math>\varphi=-19\frac{2\pi}{24}</math> and the final function is <math>T(t)=6+4\cos\left(\frac{\pi}{12}t-\frac{\pi}{12}19\right).</math>

Note the expression of the solution is not unique, it can be written in infinitely many ways, such as <math>T(t)=6+4\cos\left(\frac{\pi}{12}t+\frac{\pi}{12}5\right),</math>

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04/Solution 1

2015-05-11T15:41:58Z

CarmenBruni:

The general expression of a cosine function is <math>T(t)=A+B\cos(\omega t+\varphi).</math> Suppose <math>A</math>, <math>B</math> are both positive. Here the lowest temperature is <math>2^o</math> C and highest is <math>10^o</math> C, then

:<math>A-B=2, A+B=10,</math>

so <math>A=6</math> and <math>B=4</math>. The value <math>2\pi/\omega</math> is the period of this function. As it describes the temperature changes throughout a day which lasts for 24 hours, we have <math>\frac{2\pi}{\omega}=24, \quad \omega=\frac{2\pi}{24}.</math> From the problem statement we know the lowest temperature is obtained at 7:00, by symmetry the highest is around 19:00. Then to shift the maxima from <math>0</math> to <math>19</math> we subtract <math>t</math> by 19. So <math>\varphi=-19\frac{2\pi}{24}</math> and the final function is <math>T(t)=6+4\cos(\frac{\pi}{12}t-\frac{\pi}{12}19).</math>

Note the expression of the solution is not unique, it can be written in infinitely many ways, such as <math>T(t)=6-4\cos(\frac{2\pi}{24}t-\frac{2\pi}{24}7),</math> or <math>T(t)=6+4\cos(\frac{2\pi}{24}t+\frac{2\pi}{24}5),</math>

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04/Hint 1

2015-05-11T15:40:29Z

CarmenBruni:

What is the general form of a cosine function? How do we stretch or shift a cosine (or sine) function?

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 04/Statement

2015-05-11T15:38:31Z

CarmenBruni:

The daily temperature in Vancouver during the month of March varies from an average low of <math>2^{\circ}</math> C to an average high of <math>10^{\circ}</math> C. It is coolest just before sunrise which is roughly 7 AM. Construct a function ('''using cosine''') that provides a good description of the temperature <math>T(t)</math> throughout a day in March where <math>t</math> is measured in hours from 12 AM.

Science:Math Exam Resources/Courses/MATH102/December 2014/Question B 03

2015-05-11T15:36:46Z

CarmenBruni:

[[Category:MER QGQ flag]][[Category:MER QGH flag]][[Category:MER QBS flag]][[Category:MER RT flag]]





[[Category:MER Tag Tangent line]]

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