**I am emailing a link of this to everyone on the class list every Friday afternoon. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.ie and I will add you to the mailing list.**

## Homework

Please find the Homework. Before you open it don’t be too alarmed: you only have to do ONE of the NINE options. All of the options are about differential calculus:

- Nowhere-Continuous Functions
- Intermediate Value Theorem
- Fixing Nasty Functions and Making them Nice
- Summarise Limits & Continuity
- “Leaving Cert Questions”
- L’Hopital’s Rule
- Linear Algebra
- Summarise Differentiation
- Extrema of Functions of Several Variables with an Application to Statistics
*— contains a typo. Q.9 (a) should read “***Show that (1, 1) and (−1,- 1) are local minima…**” not “Show that (1, 1) and (−1, 1) are local minima…”. Also you can find the partial derivatives and*without multiplying out …*

I am not going to pretend that this is an easy assignment, but I will say that clear and logical thinking will reveal that the solutions and answers aren’t ridiculously difficult: a keen understanding of the principles of differential calculus should see you through.

Roughly, I have gone with less thinking & more writing or more thinking & less writing but half the battle here is picking an option that you think you can do well.

The final date for submission is 01 February 2013 and you can hand up early if you want. You will be submitting to the big box at the School of Mathematical Science. If I were you I would aim to get it done and dusted early.

Note that you are will be free to collaborate with each other and use references but this must be indicated on your hand-up in a declaration. Evidence of copying or plagiarism will result in divided marks or no marks respectively. You will not receive diminished marks for declared collaboration or referencing although I demand originality of presentation. If you have a problem interpreting any question feel free to approach me, comment on the webpage or email.

Ensure to put your name, student number, module code (MS 2001), and your declaration on your homework.

## Lectures

We have finished up to but not including Section 3.2

## Tutorials

Remember you can ask whatever you want in tutorials. If you have questions about the test or past exam papers work away.

### Tutorial 7 Question Bank

Questions 3, 5, 7, 8, 13, 16, 17 from Exercise Sheet 3.

Question 3 from MS2001: Problems (after page 102 in the notes)

Questions 14, 20, 29, 35 from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 6 Question Bank

Questions 1, 2, 3 (i), (iii), 4, 7 (i), 8 (a) (i), (ii), 9, 13 (i) from Exercise Sheet 3.

Questions 35 from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 5 Question Bank

Questions 6 (iv), (v), (vii), 8, 9 from Exercise Sheet 2 — but don’t worry about removable and essential discontinuities (after page 62 in the notes).

Question 1 from MS2001: Problems (after page 108 in the notes)

Questions 1, 2, 3, 10, 11, 12, 13, 27 from the Additional but Harder Exercises for Definitions II (two after page 108 in the notes).

### Tutorial 4 Question Bank

Question 10 from Exercise Sheet 1 (after page 59 in the notes)

Questions 5, 6 (i), (ii), (iii), 7 from Exercise Sheet 2 — but don’t worry about removable and essential discontinuities (we are using the terms *skip-discontinuity *and *blow-up*) (after page 62 in the notes)

Question 1 from MS2001: Exercises (before page 63 in the notes)

Questions 23, 31- 37 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 3 Question Bank

Question 9 from Exercise Sheet 1 (after page 59 in the notes)

Questions 1 – 4 from Exercise Sheet 2 (after page 62 in the notes)

Questions 4 from MS2001: Exercises (before page 63 in the notes)

Questions 27 – 30 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 2 Question Bank

Questions 4, and 6 – 8 from Exercise Sheet 1 (after page 59 in the notes).

Questions 1 – 3 from MS2001: Exercises (before page 63 in the notes)

Questions 17 – 22 and 24 – 26 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

### Tutorial 1 Question Bank

Questions 1, 2 and 5 from Exercise Sheet 1 (after page 59 in the notes).

Questions 1 to 16 from the Additional but Harder Exercises for Definitions I (just before page 60 in the notes).

## 4 comments

Comments feed for this article

November 19, 2012 at 8:38 am

Student 24Hi J.P.,

Just a quick question about the Homework. In Question 9, you have down:

Show that and are local minima of the function of two variables:

.

I tried to work out the 2nd point, , but could not get to make it a minimum.

I got it equal to zero when the point was , so I put the equation into Wolfram Alpha to see what it plotted and this is what came out

http://www.wolframalpha.com/input/?i=plot+f%28x%2Cy%29%3Dx%5E4%2By%5E4-4xy%2B1

Is the question meant to read or is correct?

November 19, 2012 at 8:41 am

J.P. McCarthyStudent 24,

You are correct this is a typo.

Regards,

J.P.

December 8, 2012 at 11:24 am

Student 27I have a query regarding question 5 on the assignment. If i was to make up a set of worked solution i presume i would have to break down the answer step by step regarding what rules i used. however when you mentioned to make remarks about algebra, is that in reference to what algebraic techniques were used to solve the equations?

Also do you want every single step in each solution broken down and explained? i would appreciate if you could clarify this.

Thank You

December 8, 2012 at 11:41 am

J.P. McCarthyI presume you are doing part (b):

“make up a set of worked solutions of these questions for a Leaving Cert class. This means that you can’t just give the solutions: you must also explain the concepts involved and perhaps make remarks along the way about algebra, etc.”

Try to think in terms of ideas rather than “rules”. Illustrate the ideas using pictures where possible but be precise when you need to be. It is up to you to decide which steps are going to be routine to an average Leaving Cert higher level class and which need a bit more explanation. Once you have explained, say, the product rule, you won’t have to explain it again but maybe write down the product rule formula each time and just say that we are using the product rule?

If you assume (that the class knows) too much you will almost certainly lose marks but if you assume too little your project will be longer. Obviously everyone has different teaching styles; for example in the tutorials when I answer a question not only do I answer the question but I often write down the definitions we are using, the theorems/facts we are using and sometimes I try and draw a picture to explain what we are doing/have done. It is all about striking a balance and obviously it is subjective and open to interpretation how well you have done — with myself having the final word obviously.

Regarding algebra: I don’t want to see any mistakes or “rule-type” algebra; e.g. “bring over…”, “cancel…”, etc are all rubbish. Numbers aren’t “brought over”, numbers don’t “cancel”, etc. Functions are applied to both sides of an equation and numbers can equal one and zero, etc. Another example: if you need to solve a cubic equation explain that you are using the factor theorem. If you are just solving … well you can probably assume that they know what they know what they are doing.

However if you are doing Part (a) you don’t need to explain all the ideas, but you must justify every step you make:

“give the solutions quoting the relevant deﬁnitions/theorems/results/facts/etc.

from MS2001 that makes your solutions correct.”

Regards,

J.P.