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Re: Nairaland Mathematics Clinic by Nobody: 7:40pm On Jan 09, 2013 |
ositadima1: Same method, I'll do it later. |
Re: Nairaland Mathematics Clinic by ositadima1(m): 7:44pm On Jan 09, 2013 |
doubleDx: Ok, man |
Re: Nairaland Mathematics Clinic by Nobody: 7:47pm On Jan 09, 2013 |
2nioshine: d truth is that i actually tried same method as u did bt i got stock somewere after those identity stuff(factorizatn)...must confes av been seen as d bst n ma sku...bt i off my cap 4 u tnks.@ositadime i apreciate u also You are welcome bro. Maybe you made a mistake when you were collecting like terms. It happens all the time! 1 Like |
Re: Nairaland Mathematics Clinic by Hardfact: 8:14pm On Jan 09, 2013 |
doubleDx:Kudos! @All how about solving this: If f(x) = a(x^3) + 2(x^2) + bx + c, and f(0) = f(1) = 4, and f(-1) = 8, Find: a, b, and c. |
Re: Nairaland Mathematics Clinic by Nobody: 9:17pm On Jan 09, 2013 |
ositadima1: a+b+c = 8 ...equation 1 a^2+b^2+c^2 =16 ...equation 2 a^3+ b^3+c^3 = 64 ...equation 3 Before we proceed. We have to create an expression for a^3 + b^3 in terms of (a + b) Using addition and substraction method to factorize the polynomial a^3 + a^3 yields: a^3 + b^3 can be expressed as : (a + b)^3 - 3ab (a + b) Also a^2 + y^2 can be expressed as : (a + b)^2 - 2ab Now, having created an expressing for the above. From equation 1, (a + b) = (8 - c) ....equation 4 Substituting the expression for a^2 + b^2 in equation 2 yields : (a + b)^2 - 2ab + c^2 = 16 .... equation 5 Substituting the expression for a^3 + b^3 in equation 3 yields: (a + b)^3 - 3ab(a + b) + c^3 = 64 .... equation 6 Since we already know the expression for a + b = (8 - c) in terms of c. We can now find an expression for ab from equation 5 so we can substitute back in equation 6 to give us the values of c. Computing ab in terms of c, subtitute a + b = (8 - c) in equation 5: (8 - c)^2 - 2ab + c^2 = 16 2ab = (8 - c)( 8 - c) - 16 + c^2 2ab = 64 -16c + c^2 - 16 +c^2 2ab = 2c^2 - 16c + 48 :. ab = c^2 - 8c + 24 Having known the expression for ab interms of c as written above. We can now substitute both expression (i.e the for (a + b) and ab) in terms of c in equation 6. a + b = (8 - c) and ab = c^2 - 8c + 24 (a + b)^3 - 3ab (a + b) + c^3 = 64 substituting yields : (8 - c)^3 - 3 ( c^2 - 8c + 24) (8 - c) + c^3 = 64 (8 - c)(64 -16c + c^2) - 3 ( c^2 - 8c + 24) (8 - c) + c^3 = 64 Expanding and collecting like terms yields: 3c^3 - 24c^2 - 72c - 64 = 64 Simplifying further => c^3 - 8c^2 - 24c = 128/3 ^The above polynomial has no perfect factor and cannot be factorize, so I guess the answer should be a little complex. I'll compute it for the values of c later. |
Re: Nairaland Mathematics Clinic by biolabee(m): 10:07pm On Jan 09, 2013 |
Hardfact: f(0) = 0 + 0+ 0+ c = 4 thus c = 4 f(1) = a + 2 + b + 4 = 4 thus a+b = -2 A f(-1) = -a + 2 - b + 4 = 8 thus -a-b = 2 B Hmmm The equation has a fault somewhere as the simultaneous equations A & B are similar 1 Like |
Re: Nairaland Mathematics Clinic by lanre074(m): 7:38am On Jan 10, 2013 |
ositadima1:tnx bros i gat it |
Re: Nairaland Mathematics Clinic by Richiez(m): 12:22pm On Jan 10, 2013 |
biolabee: f(0) = 0 + 0+ 0+ c = 4 thus c = 4 f(1) = a + 2 + b + 4 = 4 thus a+b = -2 A f(-1) = -a + 2 - b + 4 = 8 thus -a-b = 2 B Hmmm The equation has a fault somewhere as the simultaneous equations A & B are similar yep i also got c=4 but it seems the values of f(1) and f(-1) generates inconsistent simultaneous eqns |
Re: Nairaland Mathematics Clinic by Hardfact: 1:17pm On Jan 10, 2013 |
biolabee:Thanks for the effort. Lifted it from somewhere. Anyone else? |
Re: Nairaland Mathematics Clinic by kasbeats(m): 4:16pm On Jan 10, 2013 |
guys please help me with this,i v been stuck on it for a while naw....... Prove that sin A-sin B=2cos(A+B)/2.sin (A-B)/2.......hope u'll understand it like dis...... |
Re: Nairaland Mathematics Clinic by digitalwale: 4:52pm On Jan 10, 2013 |
1000 naira GLO recharge card for anyone who can prove the below mathematical expression. Given that Q = xyz Prove that the CURL of GRAD Q = 0 |
Re: Nairaland Mathematics Clinic by Nobody: 5:04pm On Jan 10, 2013 |
digitalwale: 1000 naira GLO recharge card for anyone who can prove the below mathematical expression.Use the concept of DOROGO and arrive at the result. |
Re: Nairaland Mathematics Clinic by ositadima1(m): 6:02pm On Jan 10, 2013 |
digitalwale: 1000 naira GLO recharge card for anyone who can prove the below mathematical expression. Game on, let me chop first... |
Re: Nairaland Mathematics Clinic by ositadima1(m): 6:42pm On Jan 10, 2013 |
digitalwale: 1000 naira GLO recharge card for anyone who can prove the below mathematical expression. The scalar Q=xyz grad=(d/dx, d/dy, d/dz) Where d/dx, d/dy, d/dz are partial derivatives grad(xyz)=[d(xyz)/dx, d(xyz)/dy, d(xyz)/dz] grad(xyz)=[yz, xz, xy] We now have a vector...[yz, xz, xy] curl[yz, xz, xy]= grad x [yz, xz, xy] where x is a vector cross curl[yz, xz, xy]= [d/dx, d/dy, d/dz] x [yz, xz, xy] Matrix form for d curl curl[yz, xz, xy]= | i j k | |d/dx d/dy d/dz| | yz xz xy | Breaking down d matrix... curl[yz, xz, xy]= [d(xy)/dy-d(xz)/dz]i-[d(xy)/dx-d(yz)/dz]j+[d(xz)/dx-d(yz)/dy]k curl[yz, xz, xy]=[(x-x)i-(y-y)j+(z-z)k] curl[yz, xz, xy]=[(0)i-(0)j+(0)k] curl[yz, xz, xy]=[0]=0 3 Likes |
Re: Nairaland Mathematics Clinic by ositadima1(m): 7:10pm On Jan 10, 2013 |
^^^Na phone I use type oo, no light for my area. |
Re: Nairaland Mathematics Clinic by Richiez(m): 7:44pm On Jan 10, 2013 |
kasbeats: guys please help me with this,i v been stuck on it for a while naw....... before attempting this question, one has to know the basic rules of sine addition, and the derivation of the factor formulae NOW; Sin(P+Q) = SinPCosQ + CosPSinQ......................(1) Sin(P-Q) = SinPCosQ - CosPSinQ......................(2) Subtracting eqn(2) from eqn(1), Sin(P+Q) - Sin(P-Q) = CosPSinQ + CospSinQ Sin(P+Q) - Sin(P-Q) = 2CosPSinQ ....................(3) Now we have to do a little trick by letting; P+Q = A ............................................(4) P-Q = B ............................................(5) adding eqns (4) and (5) we get; 2P = A + B P = (A+B)/2 ........................................(6) subtracting eqn(5) from eqn(4) we get; 2Q = A - B Q = (A-B)/2 ........................................(7) By appropriate substitution for A,B,P and Q in equation(3) we get ; SinA - SinB = 2[Cos(A+B)/2*Sin(A-B)/2] Q. E. D 2 Likes |
Re: Nairaland Mathematics Clinic by biolabee(m): 7:47pm On Jan 10, 2013 |
nice one guys... |
Re: Nairaland Mathematics Clinic by Nobody: 9:56pm On Jan 10, 2013 |
Nice work people! kasbeats: guys please help me with this,i v been stuck on it for a while naw....... Richiez has given a solution to your question. Lemme add another method => sin A - sin B = 2cos(A+B )/2 . sin (A-B )/2 Lets take this from the right hand side: 2cos(A+B )/2 . sin (A-B )/2 , simplifying yields => 2cos(A/2+B/2 ) . sin(A/2 - B/2) Lets put P = A/2 and Q = B/2 2cos(A/2+B/2 ) . sin(A/2 - B/2) becomes : 2cos(P +Q ) . sin( P - Q) Since sin (P - Q) = sinP cosQ - cosP sinQ & cos (P + Q) = cosP cos Q - sinP sin Q :. RHS => 2cos( P + Q ) . sin (P - Q) => =>2{cosP cosQ - sinP sinQ } . {sinP cosQ - cos P sinQ } Expanding: => 2[sinP cosP cos^2Q - sinQ cosQ cos^2P - sin Q cosQ sin^2P + sinP cosP sin^2Q] Collecting like terms yields: => 2[sinP cosP cos^2Q + sinP cosP sin^2Q - sinQ cosQ cos^2P - sin Q cosQ sin^2P] Factorising: => 2[ sinP cosP (sin^2Q + cos^2Q) - sinQ cosQ (sin^2P + cos^2P) ] Since Sin^2 x + Cos^2 x = 1 substituting yields => = 2[sinP cosP (1) - sinQ cosQ (1)] = 2sinP cosP - 2sinQ cosQ Replacing P = A/2 and Q = B/2 = 2sinA/2 cosA/2 - 2sinB/2 cosB/2 ^The above is an expression for the difference of two half angle for sin A and Sin B respectively. Because : Sin A = Sin(A/2 + A/2) = 2sinA/2 cos A/2 and also sin B = sin (B/2 + B/2 ) = 2sinB/2 cos B/2 :.RHS = Sin A - Sin B Hence => sin A - sin B = 2cos(A+B )/2 . sin (A-B )/2 2 Likes |
Re: Nairaland Mathematics Clinic by Nobody: 1:16am On Jan 11, 2013 |
1. Solve the following simultaneous equations and evaluate (2x^2 - y)/(x - 7). x + y = 9 3x - ³√y = 1 2. Find the second derivative of sin^3(x) cos(3x) Thanks. |
Re: Nairaland Mathematics Clinic by Nobody: 2:41am On Jan 11, 2013 |
Ghettoguru: 1. Solve the following simultaneous equations and evaluate (2x^2 - y)/(x - 7). Question 1 x + y = 9 .....(1) 3x - ³√y = 1 ......(2) from equation (1) y = (9 - x) ..... (3) from equation (2) ³√y = (3x - 1) cubing both sides to get rid of the cube root yields: (³√y)^3 = (3x - 1)^3 y = (3x - 1)(3x - 1)(3x - 1) . Expanding gives : y = 27x^3 - 27x^2 + 9x - 1 .....(4) Substituting (3) in (4) for the values of x gives => 9 - x = 27x^3 - 27x^2 + 9x -1 :. 27x^3 - 27x^2 + 10x - 10 = 0 Factorising the polynomial yields : (x - 1)(27x^2 + 10) = 0 :. x = 1 or 27x^2 = -10 The real value of x = 1, the other two are complex numbers => (1/3)i√(10/3) and -(1/3)i√(10/3) Substitute x = 1 in equation (3) y = 9 - 1 y = 8 Evaluating : (2x^2 - y)/ (x - 7) gives => = (2(1)^2 - 8 )/ (1 - 7). = -6/-6 = 1 :.(2x^2 - y)/(x - 7) => 1 I'll check question 2 later in the day. Sleepy now! 1 Like |
Re: Nairaland Mathematics Clinic by biolabee(m): 6:20am On Jan 11, 2013 |
Ghettoguru: Using producr derivate d(vu) = vdu + udv d\First level d = -3 sin^3x sin 3x + 3 cos x sin ^2 x cos 3x Second level (wey una dey see all dese kain question sef ![]() ![]() ![]() d2x = -9cos3x sin3x-9 sin3x cosx sin^2x -9 cosx sin^2 x sin3x+3 cos 3x(2 cos^2 x - sin^2 x sin x) I don try |
Re: Nairaland Mathematics Clinic by indoorlove(m): 8:48am On Jan 11, 2013 |
Pls this integration has been given me prob. Question 1: integrate 1/cos(x-a)cos(x-b) with respect to x. Question2: Pls, I need explanation on complex analysis with some examples and their solutions. Thanks ! |
Re: Nairaland Mathematics Clinic by Odunnu: 9:01am On Jan 11, 2013 |
I cant even moderate the thread as its way above me ![]() ![]() Ride on guys! 3 Likes |
Re: Nairaland Mathematics Clinic by marcangelo(m): 9:23am On Jan 11, 2013 |
God really bless Some people here wit super brain ohh.. Did u guys study Mathematics as ur field in d university? 1 Like |
Re: Nairaland Mathematics Clinic by Nobody: 9:33am On Jan 11, 2013 |
Ghettoguru: 1. Solve the following simultaneous equations and evaluate (2x^2 - y)/(x - 7). Question 2 f(x) = sin^3 x cos 3x First off, let us simplify cos 3x => cos(3x) = cos 2xcos x - sin 2x sin x = cosx ( 1- 2sin^2 x) - 2sin^2 x cosx = cos x (1- 4sin^2 x) :. Inserting cos 3x into the question yields : f(x) = sin^3x cos x (1- 4sin^2 x) f(x) = cos x sin^3 x - 4sin^5 x cos x Now, we can split the function into two (sin^3 x cos x) and (- 4 sin^5 x cos x) Applying product and chain rule to -4sin^5 x cos x yields : Put u = - 4sin^5 x, du/dx = - 20sin^4 x cos x v = cos x, dv/dx = sin x f'(x) of - 4sin^5 x cos x => = udv/dx + vdu/dx = -4sin^5 x .(-sin x) + cos x (- 20sin^4 x cos x) = 4sin^6 x - 20sin^4 x cos^2 x since cos^2 x = (1 - sin^2 x) substituting yields = 4sin^6 x - 20sin^4 x (1 - sin^2x) = 4sin^6 x - 20sin^4 x + 20sin^6 x = 24sin^6 x - 20sin^4 x .....(1) For the derivative of the second part of f(x) which is cos x sin^3 x Put u = cos x, du/dx = -sin x v = sin^3 x, dv/dx = 3sin^2 x cos x f'(x) for cos x sin^3 x => = udv/dx + vdu/dx f'(x) = cos x . (3sin^2 x cos x) + sin^3 x (-sin x) = 3sin^2 x cos^2 x - sin^4 x since cos^2 x = (1 - sin^2 x) substituting yields => = 3sin^2 x (1 - sin^2 x) - sin^4 x = 3sin^2 x - 3sin^4 x - sin^4 x = 3sin^2 x - 4sin^4 x .... (2) :.f'(x) for the expression cos x sin^3 x - 4sin^5 x cos x => = 3sin^2 x - 4sin^4 x + 24sin^6 x - 20sin^4 x collecting like times => :.The first derivative of the function f(x) => f'(x) = 3sin^2 x - 24sin^4 x + 24sin^6 x Second derivative of f(x) is the derivative of f'(x). Since f'(x) = 3sin^2 x - 24sin^4 x + 24sin^6 x f''(x) = 6sin x cos x - 96sin^3 x cos x + 144sin^5 x cos x. Simplifying => f''(x) = 6sin x cos x (1 - 16sin^2 x + 24sin^4 x) @Ghettoguru, I made some mistakes before. Re-check now, I have corrected them. 1 Like |
Re: Nairaland Mathematics Clinic by Freiburger(m): 9:43am On Jan 11, 2013 |
Nice job guyZ, keep it up. |
Re: Nairaland Mathematics Clinic by KingOfNairaland: 11:34am On Jan 11, 2013 |
[size=15pt]Proof that 12/7 is = 2. Without any reminder or decimal point. [/size] Answers please |
Re: Nairaland Mathematics Clinic by indoorlove(m): 12:06pm On Jan 11, 2013 |
KingOfNairaland: [size=15pt]Proof that 12/7 is = 2. Without any reminder or decimal point. [/size]the roman numeral for 12 is XII and dat of 7 is VII. Therefore: 12/7= XII/VII= X/V( X=10, V=5). X/V=10/5=2ANS. 2 Likes |
Re: Nairaland Mathematics Clinic by Nobody: 12:33pm On Jan 11, 2013 |
@DoubleDx thanks, much appreciated. @All, you are doing a great job. Keep them coming! |
Re: Nairaland Mathematics Clinic by digitalwale: 1:23pm On Jan 11, 2013 |
ositadima1: This is outstanding! Lemme have your phone number |
Re: Nairaland Mathematics Clinic by ositadima1(m): 1:41pm On Jan 11, 2013 |
digitalwale: Can u PM my spoils, Mtn would be nice 1 Like |
Re: Nairaland Mathematics Clinic by digitalwale: 1:48pm On Jan 11, 2013 |
ositadima1: How can I PM you? I tried going to your profile and clicked to send email but the page that came up only showed a prepared and uneditable message. |
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