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Math (Calculus) P3

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What did you guys think of it? I can't believe during the exam I couldn't figure out how to do the question where you had to prove using the mean value theorem that (b-a)/b<lnb/a<(b-a)/a. I spent a ton of time trying to do that finding a c such that c is between those two constants, but just could not get c to be lnb/a.

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Guest SNJERIN

That question about the MVT for me was the easiest to prove, but the last question!!!. It took me a lot of time and still couldn't manage it.  

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Which part of the last question was the problem? The only thing was a step I used to show that the sequence is decreasing that I don't think was valid, but that was a mistake rather than difficulty.

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Just now, keegansimyh said:

Which part of the last question was the problem? The only thing was a step I used to show that the sequence is decreasing that I don't think was valid, but that was a mistake rather than difficulty.

First of all, they ask you explain why the series is an alternating series. The second part was to prove that Un+1 is less than Un using the substitution given. I used the substitution but still ended up with another integral form of the sequence, which did not actually help me in verifying that  Un+1 is less than Un. The second thing was to prove that the integral or the sequence converges to 0. In this case I tried to compare the integral with the integral of 1/x from nπ to (n+1)π and took the limit to infinity and I proved that it is actually equals to zero. I am not sure if this is actually correct. 

Anyways, I struggle with part c, which is to prove that the series is less than 1.65. How do you do that. 

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7 minutes ago, keegansimyh said:

I showed that some partial sum (I chose the 15th) is about 1.59 using the GDC, then used the alternating series estimation theorem to show that it can't be more than 1.65.

Oh, I see. Thats a nice way of showing it. BTW, what is "alternating series estimation theorem" ? Do you mean the truncating error or is it a theorem that is beyond the scope of the syllabus ?

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Yes, the truncating error, where the error is less than the absolute value of the next term. And no, it is very much in the syllabus, mentioned explicitly in the guide :D

Edited by keegansimyh

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10 minutes ago, keegansimyh said:

Yes, the truncating error, where the error is less than the absolute value of the next term. And no, it is very much in the syllabus, mentioned explicitly in the guide :D

Yep, thanks! 

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2 hours ago, keegansimyh said:

What did you guys think of it? I can't believe during the exam I couldn't figure out how to do the question where you had to prove using the mean value theorem that (b-a)/b<lnb/a<(b-a)/a. I spent a ton of time trying to do that finding a c such that c is between those two constants, but just could not get c to be lnb/a.

 

You should look at the domains for a and b for hints to do this question. It took time. Watched a video on MVT made by a guy called Andrew Chambers, therefore, same method was used to come up with the answer. 

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I think overall this was a fine paper. Except for the last Question, too much of integration and tried to use the integral test but it was to no avail. I will average out a 6 for Math, Paper 1 wasn't the best but paper 2 was the best and P3 was kind of like in the middle, I wouldn't be surprised if I dropped to a mid 5. BTW Could you do the differential equations Part ii to find x and y. I just ended up with my integrating factor multiplied throughout and then failed to integrate. Many in my class found that particularly difficult. 

Edited by Awesomeness

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49 minutes ago, Awesomeness said:

I think overall this was a fine paper. Except for the last Question, too much of integration and tried to use the integral test but it was to no avail. I will average out a 6 for Math, Paper 1 wasn't the best but paper 2 was the best and P3 was kind of like in the middle, I wouldn't be surprised if I dropped to a mid 5. BTW Could you do the differential equations Part ii to find x and y. I just ended up with my integrating factor multiplied throughout and then failed to integrate. Many in my class found that particularly difficult. 

Oh, for that DF,  the integrating factor is equal to e^(x^2) (which I hope is the one you got). You then multiply both sides of the equation with this integrating factor. Afterwards, everything becomes easy. Actually I was surprised to see that it is worth 9 points. I have seen much harder DF which are worth less marks. 

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1 hour ago, Awesomeness said:

I think overall this was a fine paper. Except for the last Question, too much of integration and tried to use the integral test but it was to no avail. I will average out a 6 for Math, Paper 1 wasn't the best but paper 2 was the best and P3 was kind of like in the middle, I wouldn't be surprised if I dropped to a mid 5. BTW Could you do the differential equations Part ii to find x and y. I just ended up with my integrating factor multiplied throughout and then failed to integrate. Many in my class found that particularly difficult. 

i think the differential equation was dy/dx=x/y-xy, with a point (0,2)

dz/dx+2xz=2x

integrating factor = e^ ∫ 2x=e^(x^2)

e^(x^2)z= e^(x^2)*2x

e^(x^2)(y^2)=e^(x^2)+C

y^2=1+C/[e^(x^2)]

y^2=1+3/[e^(x^2)]

y=+/- [sqrt (1+3/[e^(x^2)])]

 

what did you do for the last question (5)? i really wasn't sure what to do with that one. i hope i got some credit... 

Edited by hpnottv

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4 minutes ago, hpnottv said:

i think the differential equation was dy/dx=x/y-xy, with a point (0,2)

dz/dx+2xz=2x

integrating factor = e^ ∫ 2x=e^(x^2)

e^(x^2)z= e^(x^2)*2x

e^(x^2)(y^2)=e^(x^2)+C

y^2=1+C/[e^(x^2)]

y^2=1+3/[e^(x^2)]

y=+/- [sqrt (1+3/[e^(x^2)])]

 

Just to point out that y>0 for all values of x. Thus, I hope you have removed the +/- sign in your final answer. 

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I did Q4 by separation of variables and got more or less the same answer, but not sure about the fraction

Also Q1 is the approximation larger or smaller than the actual value? Over at reddit they seem split about the issue

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18 minutes ago, wazzupworld26 said:

I did Q4 by separation of variables and got more or less the same answer, but not sure about the fraction

Also Q1 is the approximation larger or smaller than the actual value? Over at reddit they seem split about the issue

if i remember correctly, the approximation is larger. i confirmed it by calculating the approximation and actual value. (i don't remember the actual values, but it was something like .762 and .760). what's the link to the reddit?

 

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10 hours ago, Haitham Wahid said:

Oh, for that DF,  the integrating factor is equal to e^(x^2) (which I hope is the one you got). You then multiply both sides of the equation with this integrating factor. Afterwards, everything becomes easy. Actually I was surprised to see that it is worth 9 points. I have seen much harder DF which are worth less marks. 

 
 

Yep I got the Integral factor as that but after that I was Completely blank I tried to use integration by parts but yeah. I should have substituted my y^2 with z earlier rather than later in my differential equation.

Edited by Awesomeness

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