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Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 6:49pm On Mar 27, 2015 |
Good afternoon all, in a bid to contribute my own quota to "our" admission, I will be treating some MATHEMATICS topics/questions [I am not promising to be online 24/7, but when tides favour the ocean]. Some of the topics "we" will be considering are, in no particular order: 1. Calculus. 2. Trigonometry. 3. Number base. 4. Co-ordinate geometry. 5. Polynomial. 6. Statistics. 7. Matrices. 8. Algebra. 9. Probability. 10. E.t.c. For any questions/suggestions/observations/e.t.c, please do make it before the first class [23/03/2015]. Note 1. There is no guaranty any of the topics will be among the post utme questions, therefore, I am not liable for any feeling/opinion/e.t.c that may arise from this. 2. I will try my best to make the topics/questions easy to understand, but, typing mathematical symbols/expressions in an online forum is not going to be easy. 3. The fruitfulness of this class depends on my schedule, i.e I can't guaranty everyday lectures. Vote of thanks: Mr. Dopeyomi, thank you for this inspiration. Once again, Happy Sunday to you all. 22nd March, 2015. 13:30:59PM. |
Re: Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 6:53pm On Mar 27, 2015 |
Good evening all, as per my obligation, today will be "our" first class. "We" will be considering differentiation. Note 1: 1. dy/dx can also be represented as y'. 2. d2y/dx2can also be represented as y", till infinity. 3. d/dxmeans "we" are differentiating with respect tox, d/dumeans we are differentiating with respect to . . . . . . . . . . . .very good, u. 4. wrt means with respect to. DIFFERENTIATION. Differentiation means rate of change. General formula: if y = axn, a is the co-efficient of x. n is the power of x. Therefore, dy/dx = power "times" co-effecient of x "times" the power of x "minus" 1. Mathematically, y = axn, dy/dx = naxn - 1. ----------------------------- Example 1. Find y' [it also means dy/dx] if y = 2x. Solution: the co-efficient of x is 2. The power of x is 1. Therefore, y = 2x dy/dx = (1)(2)x1-1 dy/dx = 2x0 dy/dx = 2(1) because x0is 1. dy/dx = 2. -----------------------------. Example 2: find y' if y = 3x4. Solution: y = 3x4 y = 12x3 -----------------------------. Example 3: find y' if y = x . . . . . . . y' = 1. Very good, because, y = x the co-efficient of x is 1. The power of x is 1. Therefore, y = x y' = (1)(1)x1-1 y' = 1x0 y' = 1(1) y' = 1. -----------------------------. Example 4: find y' if y = 2. Solution: y = 2, the question can also be changed into y = 2(1) it can be further changed into y = 2(x0) since, y = 2x0 y' = (0)(2)x0-1 y' = 0x-1 y' = 0. Note 2: The derivative of a constant is 0. i.e d/dx (7) = 0 d/dx (10) = 0, but, d/dx (7x) = 7 d/dx (10x) = 10. -----------------------------. Summary: 1. Differentiation means rate of change. 2. If y = axn, then dy/dx = naxn-1. 3. The differentiation of constant values is 0. E.g, y = 6, dy/dx = 0, y = 1000000000 dy/dx = 0. -----------------------------. Assignment: find dy/dx if y = 1. 20x 2. -50x3 3. -25x-2 4. 30/x2, [Hint: 1/x2is x-2]. Any question? Good night class. 23rd March, 2015. 21:35:45 |
Re: Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 6:58pm On Mar 27, 2015 |
Good morning to you all, how was the night? Yesterday, we were considering differentiation, we were able to conclude that: 1. Differentiation means rate of change. 2. If y = axn, then dy/dx = naxn-1. 3. The differentiation of a constant is zero. Today, we will be considering another aspect of differentiation. DIFFERENTIATION OF POLYNOMIALS. To differentiate a polynomial, we differentiate each term in turn. General form: if y = axb+ cxd+ exf+ . . ., then dy/dx = baxb-1+ dcxd-1+ fexf-1+ . . .. . . . . . . . . . . . . . . . . . . . . . . . . Example 1: if y = x4+ 5x3- 4x2+ 7x - 2. Solution: dy/dx = (4)(1)x4-1+ (5)(3)x3-1- (4)(2)x2-1+ (7)(1)x1-1- 0. dy/dx = 4x3+ 15x2- 8x + 7. . . . . . . . . . . . . . . . . . . . . . . . . Example 2: if y = 2x5+ 4x4- x3+ 3x2- 5x + 7, find an expression for dy/dx and the value of dx/dx at x = 2. Solution: So, first of all, dy/dx = . . . . . . . Very good, because, if y = 2x5+ 4x4- x3+ 3x2- 5x + 7. Then, y' = 10x4+ 16x3- 3x2+ 6x - 5. Substituting x = 2, we have, y' = 10(2)4+ 16(2)3- 3(2)2+ 6(2) - 5. Therefore, y' = 283. . . . . . . . . . . . . . . . . . . . . . . . . Example 3: if y = 2x2- 5x + 3, then dy/dx = 4x - 5. This double statement can be written as a single statement by putting 2x2- 5x + 3 in place of y in dy/dx. i.e d/dx (2x2- 5x + 3) = 4x - 5. . . . . . . . . . . . . . . . . . . . . . . . . Example 4: In the same way, d/dx (4x3- 7x2+ 2x + 3) = Solution: 12x2- 14x + 2. Note: Either of the two methods is acceptable: it is just a case of which is the more convenient in any situation. . . . . . . . . . . . . . . . . . . . . . . . . Summary: 1. To differentiate a polynomial, we differentiate each term in turn. 2. if y = axb+ cxd+ exf+ . . ., then dy/dx = baxb-1+ dcxd-1+ fexf-1+ . . .. . . . . . . . . . . . . . . . . . . . . . . . . Assignment: 1. No 39, 1994. If y = 3t3+ 2t2- 7t + 3, find dy/dt at t = -1. (a) -1 (b) 1 (c) -2 (d) 2. 2. No 19, 2002. If y = x2- 1/x, find dy/dx (a) 2x + 1/x2(b) 2x + x2(c) 2x - 1/x2(d) 2x - x2 3. No 36, 2008. Find the derivative of y = x7- x5 (a) x (x2 - 1) (b) 3x (x[sup]2- 1) (c) 3x2- 1 (d) 7x6- 5x4. . . . . . . . . . . . . . . . . . . . . . . . . Things to note before the next class: 1. There is difference between (a) sinx - cosx (b) sinx + cosx (c) sinx cosx, the difference is, (a) &(b) are polynomials because they are connected by either + or - while (c) is connected by multiplication. Another method is used to solve c. 2. If y = sinx, dy/dx = cosx. 3. If y = cosx, dy/dx = -sinx. Have a splendid day. Any question? 24th March, 2015. 07:07:33 |
Re: Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 7:02pm On Mar 27, 2015 |
Good evening all, how is your day fairing? In our last class, we were considering differentiation of polynomials. We were able to achieve that 1. To differentiate a polynomial, we differentiate each term in turn. 2. If y = axb+ cxd+ . . ., then, dy/dx = baxb-1+ dcxd-1+ . . .. Now, we will be considering a different aspect: HIGHER DERIVATIVES. If y = 2x4- 5x3+ 3x2- 2x + 4, then, by the previous method: dy/dx = 8x3- 15x2+ 6x - 2. This expression for dy/dx is itself a polynomial in powers of x and can be differentiated in the same way as before, i.e. we can find the derivative of dy/dx. d/dx [dy/dx] is written d2y/dx2and is the second derivative of y wrt x (spoken as "dee two y by dee x squared". So, in this example, we have: y = 2x4-5 x3+ 3x2- 2x + 4. dy/dx or y' = 8x3- 15x2+ 6x - 2. d2y/dx2or y" = 24x2- 30x + 6. We could, if necessary, find the third derivative of y in the same way: d3y/dx3or y"' = . . . . . . . . . . . . . Very good, y"' = 48x - 30 . . . . . . . . . . . . . . . . . . . . . . . . Summary: if y = axb+ cxd+ exf+ gxh+ . . . Then, dy/dx = baxb-1+ dcxd-1+ fexf-1+ hgxh-1+ . . . Also, y" = (b-1)baxb-1-1+ (d-1)dcxd-1-1+ (f-1)fexf-1-1+ (h-1)hgxh-1-1+ . . . . . . . . . . . . . . . . . . . . . . . . . . . Things to note before next class: 1. Y = sinx dy/dy = cosx 2. Y = cosx dy/dx = -sinx 3. Y = tanx dy/dx = sec2x. . . . . . . . . . . . . . . . . . . . . . . . . Assignment: Pick up JAMB mathematics past question and solve anything "solvable" in differentiation. Any question?. Have a blissful night rest. 24th March, 2015. 19:22:36 |
Re: Mathematics (basic Differentiation Class And Skill) by Avast(m): 7:49pm On Mar 27, 2015 |
GOOOOOOOOOD WORK |
Re: Mathematics (basic Differentiation Class And Skill) by Nobody: 7:54pm On Mar 27, 2015 |
this is so cool man!!I'm loving Maths. |
Re: Mathematics (basic Differentiation Class And Skill) by Girltee1(f): 9:10pm On Mar 27, 2015 |
Nice job, man. |
Re: Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 11:45pm On Mar 27, 2015 |
shadowgwalker: Avast: Girltee1:check the UI 2015/2016 for current update or better still, you can follow me to be updated. |
Re: Mathematics (basic Differentiation Class And Skill) by thankyouJesus(m): 1:51pm On Mar 03, 2016 |
I need to continue this, not for JAMBITES but to stimulate interest and reduce anxiety. |
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