/Tag: Extension
29 07, 2018

Oblique Asymptotes

By |2018-07-29T14:56:26+00:00July 29th, 2018|Tags: , , , |0 Comments

Oblique Asymptotes Worked Examples Sketch the graph of $$y=\frac{{{x^2}}}{{x - 1}}$$ , showing all the important features. Domain: all real x, x ≠ 1 we have a vertical asymptote at x = 1 Find the intercepts: at x = 0, y = 0 hence passes through the [...]

20 07, 2017

Inequalities

By |2017-07-20T14:48:29+00:00July 20th, 2017|Tags: , , |0 Comments

Induction: Inequalities   Show by induction that 3n ≥ 2n + 5 for n > 1. Step 1: Show true for n = 2 LHS = 32         RHS = 2(2) + 5 = 9                        = 9 9 ≥ 9 hence true for n = 2   Step 2: Assume [...]

9 07, 2017

Finding Coefficients

By |2017-07-09T17:03:48+00:00July 9th, 2017|Tags: , , |0 Comments

Finding Coefficients $${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^c}$$   worked examples Find the coefficient of x4 in the expansion of  $${\left( {x - \frac{2}{x}} \right)^{12}}$$. 12Ck x12 – k (2/x)k = Ax4 using the formula $${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^c}$$ where n = 12, a =  x, b = [...]

9 07, 2017

Independent Term

By |2017-07-09T09:47:57+00:00July 9th, 2017|Tags: , , |0 Comments

Independent Term $${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^0}$$   worked examples Find the term that is independent of x in the expansion of $${\left( {\frac{1}{{{x^3}}} + 2{x^5}} \right)^{16}}$$.   $${}^{16}{C_k}{\left( {\frac{1}{{{x^3}}}} \right)^{16 - k}}{(2{x^5})^k} = A{x^0}$$   using the formula $${}^n{C_r}{(a)^{n - r}}{(b)^r} = A{x^0}$$ where n = 16, a [...]

7 07, 2017

Divisibility

By |2017-10-22T21:23:54+00:00July 7th, 2017|Tags: , , |0 Comments

Induction: Divisibility   worked examples Use the method of Mathematical Induction to prove that 5n + 3 is divisible by 4 for n ≥ 1.  Step 1: Show true for n = 1 51 + 3 = 8 since 8 is a multiple of 4 ∴ true for n [...]

5 07, 2017

Sums

By |2017-11-01T20:56:48+00:00July 5th, 2017|Tags: , , |0 Comments

Induction: Sums worked examples Use the method of Mathematical Induction to prove that 12 + 22 + 32 + … + n2 = 1/6n(n + 1)(2n + 1) for n ≥ 1. Step 1: Show true for n = 1 LHS = 12 = [...]

25 04, 2017

Complementary Inverse Trig

By |2017-06-29T18:00:12+00:00April 25th, 2017|Tags: , , , , |0 Comments

Complementary Inverse Trig $$\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}$$ Proof let y = sin-1 x hence sin y = x we know that sin y = cos (90 - y) (from complementary ratios) hence $$\sin y= \cos\,(\frac{\pi}{2}-y)$$ thus $$x= \cos\,(\frac{\pi}{2}-y)$$ so $$\cos^{-1}x= \frac{\pi}{2}-y$$ and y = sin-1 x $$\cos^{-1}x= \frac{\pi}{2}-\sin^{-1}x$$ ∴ $$\sin^{-1}x+\cos^{-1}x= \frac{\pi}{2}$$   worked [...]

8 02, 2017

Zeroes

By |2017-02-09T07:27:28+00:00February 8th, 2017|Tags: , , |0 Comments

Zeroes  zeroes, are the roots, or solutions to the equation P(x) Number of Roots The maximum number of roots a Polynomial can have is equal to the degree of that polynomial If the degree is odd then there must be at least 1 root - Odd degree polynomials can [...]

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