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MS 2001: Autumn Examination

July 5, 2011 in MS 2001 | Leave a comment

Please find the solutions to the Summer exam here. Note that these also include the marking scheme — numbers in bold brackets indicate marks, i.e. [3] implies three marks.
You will need to do exercises to prepare for your repeat exam. All of the following are of exam grade. If you have any questions please do not hesitate to use the comment function on the bottom:

MS 2001: Summer 2011 Solutions.

June 23, 2011 in MS 2001 | Leave a comment

Find solutions to the summer exam here.

MS 2001: Sample Paper?

February 21, 2011 in MS 2001 | 2 comments

EDIT: The following tables will be attached to your exam paper:

I had considered drafting a sample paper for ye however on mature reflection I’ve realised that this is unnecessary. The main reason for giving a sample is to show the layout of the paper so there are no nasty surprises and because the exam layout will be the same as in years 2007 — 2010, there is no need for a sample paper.

Past Papers

http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/Autumn/MS2001Aut2010.pdf

— except Q. 1(d)

http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

— except Q. 1(d)

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MS 2001: Differentiability Question

December 9, 2010 in MS 2001 | Leave a comment

This is a note written to address an issue we had in Tueday’s tutorial. Specifically the method I have shown you for doing one of the types of differentiability questions is somewhat flawed. There are a number of further assumptions that we need to make in order to make the analysis correct. I apologise for this oversight. However all is not lost – we can still ‘fix’ our (easier) method by taking these additional assumptions into account.

This question will now not be examinable in your summer exam.

The Question

Let f:\mathbb{R}\rightarrow \mathbb{R} be defined by:

f(x):=\left\{\begin{array}{cc}x^2&\text{ if }x\geq0\\ 0&\text{ if }x<0\end{array}\right.

Show that f is differentiable but not twice differentiable.

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MS 2001: Week 12 (Test)

December 9, 2010 in MS 2001 | Leave a comment

Firstly there are two tutorials next week – Monday and Tuesday at the same time and place as the the usual lecture/ tutorial.

Test Results

First of all results are down the bottom. You are identified by the last four digits of your student number. The scores are itemized as you can see. At the bottom there are some average scores. Finally the last column displays your total Continuous Assessment mark (out of 25).

If you would like to see your paper or have it discussed please email me.

Solutions

Test A and Test B.

Stud. Id Q 1(a)/3 Q 1(b)/2 Q 2/4 Q 3/3.5 Test 2 Percent CA Result
9705 2.5 2 4 1 9.5 76 18
1351 0.5 2 2 1 5.5 44 15
9822 3 2 4 1 10 80 19
2081 0.5 0.5 4 2 7 56 9
6454 1.5 2 3 2 8.5 68 13.5
7784 2.5 0.5 3.5 1 7.5 60 9.5
7238 1.5 2 4 1 8.5 68 12
8225 2.5 2 4 2 10.5 84 19.5
5757 2.5 2 4 3.5 12 96 24.5
2471 2 1.5 3 1 7.5 60 13
0869 1.5 2 0 1 4.5 36 10.5
1341 3 2 0 2 7 56 13
9056 2 0.5 4 1 7.5 60 14
7327 0 0 0 0 0 0 4
6188 2 2 4 2 10 80 17
7303 2.5 0.5 4 1 8 64 13.5
3831 2 2 2 1 7 56 16
3024 1 2 3 1 7 56 8
1947 0 0.5 0 0 0.5 4 2.5
2332 0 0.5 0 2 2.5 20 7.5
9423 2.5 2 0 1 5.5 44 10.5
5026 0 0.5 0 0 0.5 4 2.5
2366 2 2 4 0 8 64 18
2185 3 2 4 1 10 80 21
9014 1.5 2 4 1 8.5 68 14.5
3921 0 0 0 1 1 8 4
0166 0 0 0 0 0 0 7.5
8705 2.5 2 4 1 9.5 76 15.5
5321 2.5 0.5 4 2 9 72 10
1701 0 1 4 2 7 56 14
6218 0 2 0 1 3 24 11
4967 1 2 2.5 0 5.5 44 8.5
4761 1 2 3 0 6 48 12
5243 – – – – – – 0*
1863 2.5 2 4 3.5 12 96 22
3995 0 1.5 4 1 6.5 52 8.5
5154 0.5 1.5 0 0 2 16 3
0385 2 2 2 2 8 64 15
9687 2.5 2 4 2 10.5 84 17.5
5642 0 2 2 2 6 48 17.5
7478 2 2 4 2 10 80 21
7029 – – – – – – 3*
8026 0 0.5 0 1 1.5 12 4.5
4575 1 1.5 4 1 7.5 60 18
3845 1.5 2 2 2 7.5 60 14.5
0672 2.5 2 4 1 9.5 76 20.5
8793 2.5 2 4 1 9.5 76 15.5
7144 2.5 1.5 4 2 10 80 17.5
8108 0 0 0 0 0 0 5
3631 3 1 4 0 8 64 18
6302 1.5 1 4 0 6.5 52 12.5
1043 3 2 4 1 10 80 17.5
5904 2.5 1.5 0 2 6 48 15
4257 2 2 4 2 10 80 22.5
9063 2 2 4 1 9 72 21.5
3673 0.5 2 4 1 7.5 60 14
4482 0.5 2 0 1 3.5 28 9.5
4645 0.5 1.5 0 1 3 24 12.5
5527 0 0 0 0 0 0 1
8172 2 2 1 1 6 48 13
6838 0 0 0 0 0 0 6
1817 1.5 2 4 2 9.5 76 21.5
9738 – – – – – – 3*
0511 1.5 2 4 1 8.5 68 18.5
7324 2 2 3 1 8 64 19
6511 0 0 0 0 0 0 3
0492 0 0.5 0 0 0.5 4 2.5
9501 2 0.5 0 1 3.5 28 4.5
0684 0 0 0 0 0 0 0
Ave 1.41 1.41 2.36 1.09 6.27 50.18 12.25
Ave % 46.97 70.45 59.09 31.17

MS 2001: Week 11

December 2, 2010 in MS 2001 | 2 comments

On Monday we completed our introduction to horizontal asymptotes and vertical asymptotes.

On Wednesday we did some examples of curve sketching and applied max/ min problems.

Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Stationary Points are points a\in\mathbb{R} where the derivative of a differentiable function f:\mathbb{R}\rightarrow\mathbb{R}, f'(a)=0.

When asked to find the critical points of a function defined on the entire real line (rather than just on a closed interval [a,b]), the ‘endpoints’, \pm\infty are not considered critical points.

Convex is concave up and concave is concave down.

Q. 11-17  from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise4.pdf

Other Exercise Sheets – Questions on Asymptotes

From section 4 Q. 5 from Problems, find the vertical asymptotes of the the functions 5(b) [(7-10), (13), (15-16), (23)] and Q. 5(c) [except (30-31)]

Past Exam Papers

Q. 1 (d) from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 1(d) from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 2(a),  from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 2(a), 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 2(a)  from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 2(a), 6(b) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams2002/Maths_Stds/ms2001.pdf

Q. 6(b) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf

All of this except Q. 1(d) [this is the Autumn 2010 paper which wasn’t on the library website earlier in the year]

http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/Autumn/MS2001Aut2010.pdf

MS 2001: Curve Sketching Examples

December 2, 2010 in MS 2001 | Leave a comment

Example 1 (Asymptotes)

Let

f(x)=\frac{3x^2-2x+2}{x^2-x-2}

What is the domain of f? What are the roots and y-intercepts of f? Find the horizontal and vertical asymptotes of f.

Solution The domain of f is the set on which f is defined. As a quotient of continuous functions, f is defined when the denominator is not zero:

x^2-x-2\neq0,

\Rightarrow (x-2)(x+1)\neq 0.

That is the domain of f is \mathbb{R}\backslash\{-1,2\} (all the real numbers except -1 and 2. Note now that

f(x)=\frac{3x^2-2x+2}{(x-2)(x+1)}).

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MS 2001: Evaluation Feedback

December 1, 2010 in MS 2001 | Leave a comment

Just a chance for me to give some feedback on some of your most common… feedback. Of course your opinions are correct – they are your opinions. Here are my opinions on some of your opinions. The worst thing (and the failure of teaching evaluation), is that ye will never see the fruits of your criticisms 😦  – i.e. when I have made changes and improvements ye will no longer be my students.

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MS 2001 Announcement: Lecture Swap, Review Week and Test

November 29, 2010 in MS 2001 | Leave a comment

Firstly; there will be no MS 2001 lecture on Monday 6 December at 3 p.m. Instead you will have an MS 2003 lecture at this time in WG G 08. The 12 p.m. MS 2003 lecture on Wednesday December 1 in WG G 08 will now be an MS 2001 lecture.   Indeed it will be the final MS 2001 lecture as Wednesday 8 December is a test day and the week after is review week.The morning lecture at 9 a.m. on Wednesday 1 December will still go ahead.

The Tuesday 7 December tutorial goes ahead as normal.

In terms of revision week, there will be tutorials held on Monday 13 and Tuesday 14 December at the usual times in the usual places of the Monday 3 p.m lecture and Tuesday 1 p.m. tutorial respectively.

For those students who have not been able to attend Tuesday’s tutorial please see your email.

MS 2001: Week 10

November 25, 2010 in MS 2001 | Leave a comment

Firstly; there will be no MS 2001 lecture on Monday 6 December at 3 p.m. Instead you will have an MS 2003 lecture at this time in WG G 08. The 12 p.m. MS 2003 lecture on Wednesday December 1 in WG G 08 will now be an MS 2001 lecture.   Indeed it will be the final MS 2001 lecture as Wednesday 8 December is a test day and the week after is review week. The morning lecture at 9 a.m. on Wednesday 1 December will still go ahead.

On Monday we wrote down the Second Derivative Test and the First Derivative Test. We showed that the First Derivative Test is superior as it can correctly handle all of the functions that the Second Derivative Test can and more (functions with vanishing second derivative and also functions that have points that are not differentiable).
On Tuesday we did Q. 1 & 2 from the sample. It was clear the sample test is too long and I will ensure that the actual test (Wednesday 8 December) isn’t as long.
On Wednesday we defined what it means for the graph of a function to be concave up or concave down. We defined a point of inflection to be a point on the graph of a function where the concavity changes. We then said that we had a lot of tools that we could use to help sketch the graph of a function, and the final one we would examine would be asymptotes. We introduced the horizontal asymptote.
Problems

You need to do exercises – all of the following you should be able to attempt. Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Q. 10 from http://euclid.ucc.ie/pages/staff/wills/teaching/ms2001/exercise4.pdf

Other Exercise Sheets – Questions on the Second Derivative Test and Asymptotes

Section 4 Q. 4-5 from Problems

Past Exam Papers

Stationary Points are points a\in\mathbb{R} where the derivative of a differentiable function f:\mathbb{R}\rightarrow\mathbb{R}, f'(a)=0.

When asked to find the critical points of a function defined on the entire real line (rather than just on a closed interval [a,b]), the ‘endpoints’, \pm\infty are not considered critical points.

Convex is concave up and concave is concave down.

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2010/MathsStds/MS2001Sum2010.pdf

Q . 5 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/MS2001s09.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2009/MathsStds/Autumn/MS2001A09.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2008/Maths_Stds/MS2001Sum08.pdf

Q. 3(b), 5  from http://booleweb.ucc.ie/ExamPapers/Exams2008/MathsStds/MS2001a08.pdf

Q. 5 from http://booleweb.ucc.ie/ExamPapers/exams2007/Maths_Stds/MS2001Sum2007.pdf

Q. 5(b),  from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/MS2001Sum06.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2006/Maths_Stds/Autumn/ms2001Aut.pdf

Q. 4(b),5(b), 6(a)  from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001.pdf

Q. 5(b), 6(a) from http://booleweb.ucc.ie/ExamPapers/Exams2005/Maths_Stds/MS2001Aut05.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/ms2001s2004.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2004/Maths_Stds/MS2001aut.pdf

Q. 4(a), 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/MS2001.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2003/Maths_Studies/ms2001aut.pdf

Q. 5(b) from http://booleweb.ucc.ie/ExamPapers/exams2002/Maths_Stds/ms2001.pdf

Q. 4(a) from http://booleweb.ucc.ie/ExamPapers/exams2001/Maths_studies/MS2001Summer01.pdf

Q. 4(a), 6(a) from http://booleweb.ucc.ie/ExamPapers/exams/Mathematical_Studies/MS2001.pdf


From the Class

1. Prove Theorem 5.2.2 (b)