C2 Question on Differentiation

Hi All,
I'm studying C2 differentiation and think I've just about cracked it, but I just can't work out the 'maximum' bit at the end.
Basis of question is that Perimeter of shape of rectangle with semicircular top is 40cm, and that area = 40r-2r^2-1/2Pir^2. I've worked it all out and have found that r =40/4+Pi.
When I now put this back into the original formula substituting r for 40/4+Pi, I get A = 40 x 40/4+Pi - 2 x (40/4+Pi)^2 - 1/2 Pi (40/4+Pi)^2. And then I can't get any further. The next bit SHOULD be A = 1600/4+Pi - (2 + 1/2 Pi) (40/4+Pi)^2, but I have no idea how you get there. I'm teaching myself A Level Maths and I think I've missed out on some basic adding and multiplication of fractions and powers. Please can someone show me the steps one by one, and advise me on what I can study further to get to grips with these formulas?
Many thanks
J

Reply 1

Theafricanlegend

15

Original post by jrowe

Hi All,
I'm studying C2 differentiation and think I've just about cracked it, but I just can't work out the 'maximum' bit at the end.
Basis of question is that Perimeter of shape of rectangle with semicircular top is 40cm, and that area = 40r-2r^2-1/2Pir^2. I've worked it all out and have found that r =40/4+Pi.
When I now put this back into the original formula substituting r for 40/4+Pi, I get A = 40 x 40/4+Pi - 2 x (40/4+Pi)^2 - 1/2 Pi (40/4+Pi)^2. And then I can't get any further. The next bit SHOULD be A = 1600/4+Pi - (2 + 1/2 Pi) (40/4+Pi)^2, but I have no idea how you get there. I'm teaching myself A Level Maths and I think I've missed out on some basic adding and multiplication of fractions and powers. Please can someone show me the steps one by one, and advise me on what I can study further to get to grips with these formulas?
Many thanks
J

ill give you the general gist, when working a maximum. differentiate the equation and then make it equal to zero. If you have 2 solutioins differentiate the equation again to get the second derivative and plug in the solutions one at a time, if its less than zero it is a maximum. If it is more than zero it is a minimum. Hope that helps.. are you yr 11?

(edited 11 years ago)

Reply 2

Lord of the Flies

19

Original post by jrowe

Hi All,
I'm studying C2 differentiation and think I've just about cracked it, but I just can't work out the 'maximum' bit at the end.
Basis of question is that Perimeter of shape of rectangle with semicircular top is 40cm, and that area = 40r-2r^2-1/2Pir^2. I've worked it all out and have found that r =40/4+Pi.
When I now put this back into the original formula substituting r for 40/4+Pi, I get A = 40 x 40/4+Pi - 2 x (40/4+Pi)^2 - 1/2 Pi (40/4+Pi)^2. And then I can't get any further. The next bit SHOULD be A = 1600/4+Pi - (2 + 1/2 Pi) (40/4+Pi)^2, but I have no idea how you get there. I'm teaching myself A Level Maths and I think I've missed out on some basic adding and multiplication of fractions and powers. Please can someone show me the steps one by one, and advise me on what I can study further to get to grips with these formulas?
Many thanks
J

Start by factoring

(\dfrac{40}{4}+\pi)^2

. Also, you should simplify all the fractions.

Reply 3

jrowe

OP

My knowledge of some of the basics is so bad, especially when it comes to division. So (40/4+Pi) can be changed to (40/4) + Pi and still be the same equation? If so, then ((40/4) + Pi)^2 works out as Pi^2 + 20Pi + 100. Is that right?? What should the next step be?

Reply 4

jrowe

OP

Just looked at my last reply and have worked out that 40/(4+pi) is not the same as (40/4) + pi so please ignore last post! I think I've got some way towards an answer. I realised that the common factor of the equation is (40/(4+pi)) so the equation then becomes 40/(4+pi) [40 - 80/(4+pi) -20pi/(4+pi) ] but how do I then simplify [40 - 80/(4+pi) - 20pi/(4+pi)] ???

Reply 5

jack.hadamard

4

Original post by jrowe

Just looked at my last reply and have worked out that 40/(4+pi) is not the same as (40/4) + pi so please ignore last post! I think I've got some way towards an answer. I realised that the common factor of the equation is (40/(4+pi)) so the equation then becomes 40/(4+pi) [40 - 80/(4+pi) -20pi/(4+pi) ] but how do I then simplify [40 - 80/(4+pi) - 20pi/(4+pi)] ???

\displaystyle \ \ A\ =\ 40 \times \frac{40}{4 + \pi} - 2 \times \left( \frac{40}{4+\pi} \right)^2 - \frac{\pi}{2} \times \left( \frac{40}{4+\pi} \right)^2

\displaystyle \qquad =\ 40 \times \frac{40}{4 + \pi} - \left(2 + \frac{\pi}{2} \right) \times \left( \frac{40}{4 + \pi}\right)^2

Reply 6

jrowe

OP

Ok, this is the bit I don't understand. If you have -2 x (40/(4+pi)^2 - 1/2pi x (40/(4+pi)^2 because there are two brackets the same you can then do ( -2 x - 1/2 pi ) and multiply it by one of the (40/(4+pi)^2. Can someone give me an example of this with much simpler formulas? Why can you get rid of one of the squared brackets? I can see what is happening but I can't understand why. Thank you everyone for your help, I'm starting to get there!

Reply 7

Lord of the Flies

19

Original post by jrowe

If you have -2 x (40/(4+pi)^2 - 1/2pi x (40/(4+pi)^2 because there are two brackets the same you can then do ( -2 x - 1/2 pi ) and multiply it by one of the (40/(4+pi)^2.

Yes except for one thing: it would be a + sign where the x sign is. So (- 2 - 1/2 pi) multiplied by (40/(4 + pi))^2.

Original post by jrowe

Can someone give me an example of this with much simpler formulas?

It is a simple technique called factorisation. In simple terms:

ak+bk=k(a+b)

Where

a,\; b,\; k

can be anything.

You are basically transforming a sum into a product.

Original post by jrowe

Why can you get rid of one of the squared brackets? I can see what is happening but I can't understand why.

You are not exactly getting rid of them, you are simplifying the expression. Here's an example: say I have three apples and three oranges (mathematically: 3 apples + 3 oranges). Factoring would be to say, I have three "apples and oranges" (mathematically: 3 x (apples + oranges)). In the end, it is the same thing. But most often the second expression (the factored one) will be simpler. Clear?

Not to raise an eyebrow but I don't see how you're getting through differentiation without knowing how factorisation works...?

Reply 8

jrowe

OP

Not to raise an eyebrow but I don't see how you're getting through differentiation without knowing how factorisation works...?

Hi, thank you for your reply - raising an eyebrow would be the least expression I'd expect!! I do know basically how to factorise but it doesn't come naturally to me and I sometimes find it difficult to spot the factors if that makes sense?! I think I haven't really understood how important it is to factorise and simplify to obtain an answer.
So... A = 1600/(4+Pi) - (2 + 1/2 Pi) (40/(4+Pi)^2.
A = 1600/(4+Pi) - ((4+Pi)/2) (1600/(4+Pi)^2)
A = 1600/(4+Pi) - 800/(4+Pi)
A = 800/(4+Pi) which is the right answer according to my book!!
My problem is that I don't automatically realise that I need to simplify examples such as (2 + 1/2 Pi) into ((4+Pi)/2) but I suppose it's because I need to always try and reduce down to (4+Pi) to match with the other (4+Pi) so that I can break down the equation even further. It's starting to all make sense now - I think I need to just keep on practising and practising!!! Many thanks for your patience.

C2 Question on Differentiation

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