must have + c lor. The first part on differentiation only serves as a guide for the integration in the second part. In A-level and higher, we always leave the answer with + c unless the integrands are given.
A good analogy has already been explained by eagle: A square is a rectangle but a rectangle may not be a square.
Similary, given that y=2x + 3, you will know that gradient is 2 by differentiating but if you are given only 2 as the gradient, there can be infinitely many equation that satisfy the solution. (i.e. y=2x + 4, y=2x-1000000)
(i) Differentiate y = 2x + 3
(ii) Hence, integrate 2 dx.
Talking about integration,
my FM teacher did post me one question before:
no reversal or whatsoever:
Try this: integrate from pie/2 to 0 (tanx)/(1+tanx) dx. the answer is pie/2.
Originally posted by Uncertain:Talking about integration,
my FM teacher did post me one question before:
no reversal or whatsoever:
Try this: integrate from pie/2 to 0 (tanx)/(1+tanx) dx. the answer is pie/2.
What is the most elegant method?
The shortest one I can think of at the moment is multiply by cos x top and bottom, then multiply (cos x - sin x) top and bottom, then can simply to 0.5 tan 2x + 0.5 - 0.5 sec 2x
Which when we integrate, gives us -0.25 ln |cos 2x| + 0.5x - 0.25 ln |sec 2x + tan 2x|
When x = 0, we get 0
When x = pi/2, we get -0.25 ln 1 + 0.25 pi - 0.25 ln 1 = pi/4
My answer is pi/4.... Is it wrong? :(
Originally posted by weewee:must have + c lor. The first part on differentiation only serves as a guide for the integration in the second part. In A-level and higher, we always leave the answer with + c unless the integrands are given.
A good analogy has already been explained by eagle: A square is a rectangle but a rectangle may not be a square.
Similary, given that y=2x + 3, you will know that gradient is 2 by differentiating but if you are given only 2 as the gradient, there can be infinitely many equation that satisfy the solution. (i.e. y=2x + 4, y=2x-1000000)
Hi Weewee,
May I refer you to Shing Lee Additional Mathematics Textbook used in many government schools, page 422, the sentence on the lower part of the page "When more information is given, the value of c can be found".
Thank you for your kind attention.
Regards,
ahm97sic
Originally posted by Ahm97sic:Hi Weewee,
May I refer you to Shing Lee Additional Mathematics Textbook used in many government schools, page 422, the sentence on the lower part of the page "When more information is given, the value of c can be found".
Thank you for your kind attention.
Regards,
ahm97sic
Weewee is talking about the actual way of doing it. It's because this way of questioning is rather weird.... And not much explanation is being given... It's only a subset of what is integration... Typical at O levels to teach things only halfway... hence making the student relearn everything at A levels or uni.... -.-"
Weewee, what ahm97sic is trying to say is that the extra information is manifested in the original equation. Normally, when we get +c, and if we are given a point, we can find c. For this case, instead of being given a point, we are given the relation between y and x => this is equivalent to being given a point to solve for c. The only thing that is confusing about ahm97sic's explanation is the 2nd part, where we are actually not really finding y, but rather, things 7y/6.
Originally posted by Ahm97sic:Hi Weewee,
May I refer you to Shing Lee Additional Mathematics Textbook used in many government schools, page 422, the sentence on the lower part of the page "When more information is given, the value of c can be found".
Thank you for your kind attention.
Regards,
ahm97sic
Yeah since more information is NOT given in this case(questions you posed in post #1), the value of c cannot be found.
The information given should be a set of points (x,y) in order to find the value of c.
(i) Differentiate y = 2x + 3 and Differentiate y = 2x + 100 with respect to x. Hence, integrate 2 dx.
Originally posted by weewee:
Yeah since more information is NOT given in this case(questions you posed in post #1), the value of c cannot be found.
Hi Weewee,
For the reversal method that is used in the "O" level add maths questions, the original y is alawys given as my examples shown to illustrate the reversal method.
The questions in my earlier post are actually taken from an IP school. Indeed, you are correct. The orginal y is not given and hence we need to add the constant c in the answers.
It is my oversight. Please accept my apologies.
Thank you for your kind attention.
Regards,
ahm97sic