MS2001: Week 11 & 12

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

These Weeks

In lectures, we finished off the notes.

Review Week

I will hold a tutorial on Tuesday at the same time and place. I should have the results of Test 2 by then.

Appraisal

Thank you very much for your work on the appraisal. Sorry about all the hairstyles.

Problems

You need to do exercises – all of the following you should be able to attempt.

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.

Finding the limit as $x\rightarrow\pm\infty$ means find an asymptotic.

‘Horizontal’ asymptotes are asymptotics.

Convex is concave up and concave is concave down.

Do as many as you can/ want in the following order of most beneficial:

Wills’ Exercise Sheets

Q. 11 – 17 from Exercise Sheet 4.

More Exercise Sheets

Section 4; Q. 6,7 from Problems.

Past Exam Papers

I’ve only just noticed that these aren’t necessarily available off campus — I will fix this in future. Although we have the tools to answer the italicised questions I won’t be asking questions of this particular form.

Q. 1 (d) from Summer 2010.

Q. 1 (d) from Autumn 2010.

Q. 1 (d) from Summer 2009.

Nothing from Autumn 2009.

Q. 1 (d) from Summer 2008.

Q. 1(d) from Autumn 2008.

Q. 1(e) from Summer 2007.

Q. 1 (e) from Autumn 2007.

Q. 2(a) from Summer 2006.

Q. 2(a) Autumn 2006.

Q. 2(a) from Summer 2005.

Q. 6(b) from Autumn 2005.

Q. 6(b) from Summer 2004.

Q. 6(b) from Autumn 2004.

Q. 6(b) from Summer 2003.

Q. 6(b) from Autumn 2003.

Q. 6(b) from Summer 2002.

Nothing from Summer 2001.

Q. 6(b) from Summer 2000.

From the Class

1. Show that

$\lim_{x\rightarrow 2}\left(\frac{3x^2-2x+2}{x^2-x-2}\right)=\pm\infty$ .

2. First example in section 4.7.

6 comments

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May 17, 2012 at 4:27 pm

Student 16

I was just wondering in the exam do all questions carry equal mark?
i.e. does question 1 carry the same marks as the other four questions?

May 17, 2012 at 4:28 pm

J.P. McCarthy

Question 1 is compulsory.

All questions are worth 25 marks.

Regards,
J.P.

May 21, 2012 at 1:23 pm

Student 20

Sorry to be annoying you just wondering if you could answer my query on p.83-Solution To The Question.
Does testing the limit of f ‘ (x) prove differentiability? Then what does testing the limit of f(x) mean?please.

May 21, 2012 at 1:51 pm

Suppose you are given a function defined piecewise like

$\displaystyle f(x)=\left\{\begin{array}{cc}u(x) & \text{ for }x<a \\ v(x) & \text{ for }x\geq a\end{array}\right.$

Usually (i.e. if this question is on your paper), restricted to their own domains, both $u(x)$ and $v(x)$ will be both continuous and differentiable. Two questions now pop up:

1. Do the graphs 'line up' at $x=a$ — is $f$ continuous?

2. If they do, do they line up smoothly with a well defined tangent at $x=a$ — is $f$ differentiable at $x=a$ ?

To answer the first question we must show/ask that

$\displaystyle \lim_{x\rightarrow a}f(x)=f(a)$ .

We proved that a limit exists if and only if the left- and right-hand limits exist and are equal. Hence we calculate

$\displaystyle \lim_{x\rightarrow a^-}f(x)=\lim_{x\rightarrow a}u(x)$ and

$\displaystyle \lim_{x\rightarrow a^+}f(x)=\lim_{x\rightarrow a}v(x)$ .

If they are equal then the function is continuous… assuming (as will be the case), that they are in turn equal to $f(a)$ : in this case $v(a)$ . Otherwise $f$ is not continuous.

Now the second question. We have two main ideas here:

(A): If the function is not continuous then it is not differentiable.

On the other hand, if $f$ is continuous, we write down $f'(x)$ for $x\neq a$ :

$\displaystyle f'(x)=\left\{\begin{array}{cc}u'(x) & \text{ for }x<a \\ v'(x) & \text{ for }x<a\end{array}\right.$

We proved in class (the proposition on p.82) that if $f$ is continuous and in addition

$\displaystyle L=\lim_{x\rightarrow a}f'(x)$

exists then $f$ is differentiable. Otherwise $f$ is not differentiable. We calculate $L$ in the same way by looking at left- and right-hand limits.

(B) Alternatively we calculate

$\displaystyle \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$ via

$\displaystyle \lim_{h\rightarrow 0^-}\frac{f(a+h)-f(a)}{h}=\lim_{h\rightarrow 0^-}\frac{u(a+h)-v(a)}{h}$ and

$\displaystyle \lim_{h\rightarrow 0^+}\frac{f(a+h)-f(a)}{h}=\lim_{h\rightarrow 0^+}\frac{v(a+h)-u(a)}{h}$ .

Again, if these left- and right-hand derivatives exist and are equal, then the function is differentiable. Otherwise it is not differentiable.

Regards,

May 22, 2012 at 2:16 pm

Student 22

I just want to ask if all of the proofs of the propositions in ms2001 are examinable.I am aware that the main theorems like Rolles theorem are examinable, and I know that it is important for us to know and understand the results from the proofs of the propositions but could we be asked to prove all of them in the exam? For example, the proof of propostion 3.1.8. or 3.2.3.?

May 22, 2012 at 2:17 pm

You will not be asked to prove any of the propositions from the notes.

However show, demonstrate, etc. are all asking for proof. Essentially everything in maths is proof but in the sense of can you prepare certain propositions/theorems? The answer is no.

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MS2001: Week 11 & 12

These Weeks

Review Week

Appraisal