**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.**

## Test

The test will take place at 3 pm in WGB G 05 next **Friday 2 November** in Week 6.

I would’ve intended that everything up to and including section 2.2 is examinable for the test.

You can find the tests I set for the last two years after Exercise Sheet 1 — which is itself after page 49. Also there is a sample from last year here.

As we will not have covered continuity, we will not be able to do the questions two as they are phrased. However we have covered limits.

I will make some brief remarks on these tests now.

### “MS2001: Test 1 A Solutions”

- Note that we have only done Q. 1(a) using the
**Alternate Solution**given. This question is worth 5/12.5 = 40%; the thinking being that to be of passing standard you must be able to do this ‘exercise’ question. - In light of us not covering Continuity on time, the first part of question could read “Find or else explain why it does not exist.” To answer this you would have to look at the left- and right-hand limits and invoke Proposition 2.1.3. The second part might ask you “for what values of does exist”, and you would once more have to invoke Proposition 2.1.3. This question is worth 4/12.5 = 32%; the thinking being to get a first you should be able to do the ‘exercise’ question, and this, an ‘exam’ question.
- For Q. 3 of the test, you need to know and understand the following definitions:
*even, odd, increasing, decreasing, quadratic, roots, polynomial, rational function, absolute value, limit, one-sided limit.*Q.3 is a harder question and the thinking behind this is that you can get 72% a bare first if you get all of Q.1 and Q.2 — but you will have to be even better than this to get a higher mark. You get one mark for one right, two marks for two right and three plus a bonus half-mark for all three right.

### “MS2001: Test 1 A and 1 B Q 3”

I will not ask you for a definition in question two. Again the question could be “Find or else explain why it does not exist” and “for what value(s) of does exist?”

## Lectures

We are in the middle of section 2.3.

## Tutorials

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

### 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

October 31, 2012 at 8:33 pm

Student 23Hi J.P.

In one of the test solutions provided in the booklet it says:

Q: Suppose that is a function, increasing on the interval . Which of the following are true?

Ans: For any , .

Can you explain why this is the case ?

Thanks

October 31, 2012 at 8:39 pm

J.P. McCarthyAlgebraically we have the definition:

is increasing on if for all we have implies that .

Thinking like this we can say that if then at the very least as is the largest element of . Therefore if we have and increasing we have, by definition of increasing,

.

That is the technical answer.

Thinking geometrically helps though. If a function is increasing then the bigger the value of , the bigger the value of . In other words as grows from to the function is increasing so that it's maximum occurs at .

So the biggest can get on is and thus if you pick any we have

.

Regards,

J.P.

November 2, 2012 at 8:51 am

Student 23Hi J.P,

Could I just ask you for the answer of the Question one from last years test the inequality? The question is

.

Do I start this by multiplying both sides by the bottom thing squared?

November 2, 2012 at 8:54 am

J.P. McCarthyI am a little short of time so will point you towards the solution of a similar problem: Q.1 (a) from last year:

Click to access solutions.pdf

The important thing to understand is how we use the properties of the absolute value function to simplify this and that we can only multiply both sides of an inequality by a positive thing… in this case after we write the absolute value of the fraction as a fraction of the absolute values, we multiply across by which is positive.

Regards,

J.P.