MATH6040: Winter 2019, Week 10

Test 2

On Chapter 3, on at 5 pm (not 4 pm) Monday 25 November, Week 11 in Melbourne Hall. You will have one full hour.

The BioEng2B tutorial will take place from 4 pm sharp to 4.45 pm in B263 on that day.

Chapter 3: Differentiation is going to be examined. A Summary of Differentiation (p.147-8): you will want to know this stuff very well. You will be given a copy of these tables.

There is a sample test on p. 149 of the notes.

I strongly advise you that attending tutorials alone will not be sufficient preparation for this test and you will have to devote extra time outside classes to study aka do exercises.

Between tutorials and private study you really should aim to have completed as least the following:

P. 113, Q. 1-6 (not 5c or 6iii)
P. 119, Q. 1-4
P. 127, Q. 1-4
P. 138, Q. 1-3
P. 146, Q. 1-4
The Sample Test

There are more questions in most of these exercises.

If you are having any problem, take a photo of your work and email me your question.

Week 10

We started Chapter 4 on (Further) Integration with a revision of antidifferentiation, and had a look at Integration by Parts. We used implicit differentiation to differentiate inverse sine.

Week 11

We have our test on Monday, then we will look at completing the square, and work on Tuesday and Thursday.

Week 12

We will look at centroids of laminas and centres of gravity of solids of revolution. Any spare lecture time will be given over to tutorial time.

Week 13

There is an exam paper at the back of your notes — I will go through this on the board in the lecture times (in the usual venues):

Monday 16:00
Tuesday 09:00
Thursday 09:00

We will also have tutorial time in the tutorial slots. You can come to as many tutorials as you like.

Monday at 09:00 in B180
Monday at 17:00 in B189
Wednesday at 10:00 in F1. 3

The exam is on Friday 13 December.

Academic Learning Centre

If you are a little worried about your maths this semester, perhaps after the Quick Test or in general, I would just like to remind you about the Academic Learning Centre. Most students received slips detailing areas of maths that they should brush up on. The timetable is here.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

CIT Mathematics Exam Papers

These are not always found in your programme selection — most of the time you will have to look here.

Student Resources

Please see the Student Resources tab on the top of this page for information on the Academic Learning Centre, etc.

4 comments

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November 22, 2019 at 2:46 pm

Student

Hi J.P.,

I am stuck with this partial differentiation question. Question is
$\displaystyle y= u + ve^w$ ,
and they want $\displaystyle \frac{\partial y}{\partial w}$ .

Can you help me with this?
Thanks ! Regards.

November 22, 2019 at 2:54 pm

J.P. McCarthy

So we must understand that $y$ is a function of three variables:

$y=y(u,v,w)=u+v\cdot e^w$ .

$e\approx 2.718$ , and $x\mapsto e^x$ is the exponential function.

If we are finding $\displaystyle \frac{\partial y}{\partial w}$ , this is differentiating with respect to $w$ , while keeping all other variables constant.

So with respect to $w$ we see:

$\displaystyle y=\underbrace{u}_{\text{const.}}+\underbrace{v}_{\text{const.}}\cdot \underbrace{e^w}_{\text{function}}$

So kind of like, say, in one variable calculus:

$y(x)=3+4\cdot e^x$ $\displaystyle\Rightarrow \frac{dy}{dx}=0+4\cdot e^x=4e^x$ ,

we have

$\displaystyle \frac{\partial y}{\partial w}=0+v\cdot e^w=ve^w$ ,

as the derivative of $e^x$ is itself.

Regards,
J.P.

November 24, 2019 at 8:00 pm

How do I finish off Q. 5 from the Sample Test?

November 24, 2019 at 8:13 pm

To make more sense of what you are doing you need to calculate the volume:

$V_0=\pi(0.008)^2(0.018)\approx 3.619\times 10^{-6}$ .

This is the calculation of $V$ .

So you have, correctly, what I might denote:

$\Delta V_r\approx 4.52\times 10^{-9}$ and $\Delta V_h\approx 4.02\times 10^{10}$ .

These are, respectively, the error in calculation of $V$ due to the error in the measurement of $r$ , and the error in the calculation of $V$ due to the error in the measurement of $h$ .

Therefore we further approximate the error in the calculation of the volume, due to the errors in measurement as:

$\Delta V\approx \Delta V_r+\Delta V_h\approx 4.926\times 10^{-9}$ ,

and full marks would be given for:

$V=(3.619\times 10^{-6}\pm 4.926\times 10^{-9})\text{ m}^3$ ,

in other words:

$V=V_0\pm \Delta V$ . Full marks.

In an ideal world you can do things slightly better.

The approximation of the error is rough, however, and it is good practise to round the error to one significant figure:

$V=(3.619\times 10^{-6}\pm 5\times 10^{-9})\text{ m}^3$ .

Round $V_0$ to the same precision… it already is… and consider presenting as

$V=(3.619\times 10^{-6}\pm 0.005 \times 10^{-6})\text{ m}^3$ .

Then consider:

$V=(3.619\pm 0.005)\times 10^{-6}\text{ m}^3$ .

Then consider that maybe $\text{m}^3$ might not be the optimal unit. Note that
$\text{m}^3=(100\text{ cm})^3=10^6\text{ cm}\Rightarrow 10^{-6}\text{ m}^3=\text{cm}^3$ , so an even better presentation might be:

$V=(3,619\pm 5)\text{ cm}^3$ .

But remember, I will give full marks for

$V=(3.619\times 10^{-6}\pm 4.926\times 10^{-9})\text{ m}^3$ .

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MATH6040: Winter 2019, Week 10

Test 2

Week 10

Week 11

Week 12

Week 13

Academic Learning Centre

Study

CIT Mathematics Exam Papers

Student Resources

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J.P. McCarthy Maths

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MATH6040: Winter 2019, Week 10

Test 2

Week 10

Week 11

Week 12

Week 13

Academic Learning Centre

Study

CIT Mathematics Exam Papers

Student Resources

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J.P. McCarthy Maths

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