A maths - Integration
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Ok, need help with integrating some trigo functions
sin5x cos x dx (between pi/3 and 0)
6 cos 2x cos x dx (pi/2 and -pi/2)
Got stuck for both questions
Next,
A curve is such that d2y/dx2 = 6x -2. The gradient of the curve at the point (2,-9) is 3
Express y in terms of x
Show that the gradient of the curve is never less than -16/3
This is very urgent. Having prelims in 1 days time, so please help!
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Originally posted by anpanman:
Ok, need help with integrating some trigo functions
sin5x cos x dx (between pi/3 and 0)
6 cos 2x cos x dx (pi/2 and -pi/2)
Got stuck for both questions
Next,
A curve is such that d2y/dx2 = 6x -2. The gradient of the curve at the point (2,-9) is 3
Express y in terms of x
Show that the gradient of the curve is never less than -16/3
This is very urgent. Having prelims in 1 days time, so please help!
For integration at O level, you do not have product rule (and for trigo you can only integrate sin cos power of 1 and sec ^2 only). Thus you will have to rewrite the expression in a suitable form before integrating.
1)Using the fact that sin A + sin B = 2 sin[(A+B)/2]cos [(A-B)/2]
Let sin(5x)cos(x) = 0.5(sin A + sin B)
(A+B)/2 = 5x
A+B = 10x
(A-B)/2 = x
A-B = 2x
A = 6x and B = 4x
Thus sin(5x)cos(x) can be rewritten as 0.5sin6x + 0.5sin4x
Therefore int[ sin(5x)cos(x)]
= int [ 0.5sin(6x) + 0.5sin(4x)]
= [(0.5)([-cos 6x]/6 + (0.5)[-cos 4x]/4] your limits
= your answer
2) Same concept as (1). Use the appropriate factor formula.
3)dy/dx = 3x^2- 2x +c
At (2,-9), dy/dx = 3
3 = 3(4)-2(2)+c
c= -5
Therefore dy/dx = Gradient(G) = 3x^2-2x-5
Min/Max of anything occur when d(anything)/d(whatever) = 0
The "anything" we are interested in is the gradient (G).
Thus max/min of G will occur when dG/dx = 0
dG/dx = 6x -2
When 6x-2 = 0
x = 1/3
When x = 1/3, G = 3(1/3)^2-2(1/3)-5 = -16/3
d^2G/dx^2 = 6 (min)
Minimum value of G is -16/3.
Therefore blah blah blah(shown)
oppsss... forgot to to find y for you but you can right? Just integrate dy/dx and sub in (2,-9) to solve the constant.
adjusted some typos as well.
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alright TS. for integration questions,
do it by integration by parts. Then apply reduction formular..
ill upload my solutions here:
Please do let me know if they helped. thanks.
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Hi,
Integration by parts is only in A level syllabus.
Also, reduction formula was in F maths syllabus, but now taken out of H2 syllabus
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Originally posted by eagle:
Hi,
Integration by parts is only in A level syllabus.
Also, reduction formula was in F maths syllabus, but now taken out of H2 syllabus
hi eagle,
i dont mean to hijack or anything. But whats F maths? Ive searched it on google and wikipedia but i dont get any results back
haha. just curious. Thanks anyways.
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Hi,
There used to be two maths subjects at A-level before 2007, namely mathematics and further mathematics (or F maths). Thanks!
Cheers,
Wen Shih -
Originally posted by OHSheet:
hi eagle,
i dont mean to hijack or anything. But whats F maths? Ive searched it on google and wikipedia but i dont get any results back
haha. just curious. Thanks anyways.
No problems :) I shall give a brief history.
F Maths stands for Further Mathematics. It's in the syllabus way before H2 Maths was implemented in 2006, and the first H2 A levels in 2007. Before H2 syllabus was introduced, Maths was divided into C Maths and F Maths. Students (like me) who took both were referred to as those taking "double maths".
The most usual combination at that time for science combi was (1) Maths C, triple sciences and (2) double maths, Physics Chemistry, and (3) double maths, Physics Economics.
C Maths and F maths syllabus contains pure maths, statistics and mechanics (with calculus). A strong understanding was required to do the mechanics question. Yours truly excelled in mechanics questions (near 99% in scores), but was sadly not as good for statistics (about 80+% only
). However, most students would regard statistics as easier because it is much more straightforward. Then again, the mechanics in Maths, coupled with Physics (and S paper) propelled me to a level which I could handle University year 1 physics with ease. It's not as easy nowadays.Nowadays, for H2 Maths, it combines elements of C Maths and F Maths. One of the killer, 3D Trigo and etc, was taken out of syllabus, but another just as equally killer 3D Vectors (planes) was put in. Method of differences was F Maths, but is extremely simple. So was correlation and linear regression, t-distribution in statistics. Recurrence relations wasn't in Maths last time if I remember correctly (or was it?). Basically, there's not much difference in the standards between C maths and H2 maths I would say.
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Hello Hello, Thanks for the help.
I have 4 questions which I want to verify the answer with you guys. You need not do all questions, you can simply do 1 or 2 or as many as you want.
1. Differentiate 2x cos x with respect to x. Hence, or otherwise evaluate x sin x dx (between pi/3 and 0)
My ans 0.342
2. Given that y = x^2 (2x + 3)^0.5, show that dy/dx = (5x^2 + 6x)/(2x+3)^0.5. Hence/otherwise evaluate 5x^2 + 6x / 2 (2x +3)^0.5 dx between 4 and 0
My ans 26.5 (don't remember getting this answer during class practice. Can someone check the accuracy of my ans?
3 given that y = x / (5x - x^2)^0.5, find dy/dx and evaluate 4x / (((5x - x^2)^3)^0.5) dx between 4 and 1
my ans : 2.4
lastly
show that d/dx (x/ 3+2x) = 3/ (3+2x)^2. Hence / otherwise evaluate (3/ (3+2x))^2 dx. between 4 and 1.
My ans 0.491
Sorry for the trouble because I do not have the answer sheet as my teacher did not distribute it out. Again, you can do any question you'd like... hope to know my answers accuracy by 12 am ?
Thanks!
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hi!
Thanks eagle and thanks wee_ws.
btw yes. TS, your answers are right.
I'll upload my solutions for you.
Q1
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Q2
----------------------------------------------------
Q3
----------------------------------------------------
Q4
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ehh.. TS. im sorry. at the bottom of q3, my scanner had it cut off O_o.
But nvm. You can work that out. and your answers are all right btw.
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Originally posted by eagle:
Recurrence relations wasn't in Maths last time if I remember correctly (or was it?). Basically, there's not much difference in the standards between C maths and H2 maths I would say.
Hi,
Recurrences is a new topic.
Thanks!
Cheers,
Wen Shih