/Tag: Extension
11 03, 2019

Absolute Functions

By |2019-03-13T17:25:09+10:00March 11th, 2019|Tags: , , |0 Comments

Absolute Functions Go to the Forum to see more questions or ask your own Worked examples Consider the function $$f(x) = x + |x|$$ for –2 ≤ x ≤  2. Sketch the graph of $$y=f(x)$$ showing the endpoints. The best way to approach any graph where x is added to an absolute [...]

29 07, 2018

Oblique Asymptotes

By |2018-07-29T14:56:26+10:00July 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 |2019-03-04T13:55:11+10:00July 20th, 2017|Tags: , , |0 Comments

Induction: Inequalities Go to the Forum to see more questions or ask your own Worked examples Show by induction that 3n ≥ 2n + 5 for n > 1. Show true for n = 2 LHS = 32             RHS = 2(2) + 5 = 9      [...]

9 07, 2017

Finding Coefficients

By |2017-07-09T17:03:48+10:00July 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+10:00July 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 |2019-03-04T13:32:19+10:00July 7th, 2017|Tags: , , |0 Comments

Induction: Divisibility Go to the Forum to see more questions or ask your own Worked examples Use the method of Mathematical Induction to prove that 5n + 3 is divisible by 4 for n ≥ 1.  Show true for n = 151 + 3 = 8since 8 is a multiple of [...]

5 07, 2017

Sums

By |2019-03-04T13:34:29+10:00July 5th, 2017|Tags: , , |0 Comments

Induction: Sums Go to the Forum to see more questions or ask your own Worked examples Use the method of Mathematical Induction to prove that 12 + 22 + 32 + … + n2 = 1/6n(n + 1)(2n + 1) for n ≥ 1. Show true for n = 1 LHS [...]

25 04, 2017

Complementary Inverse Trig

By |2017-06-29T18:00:12+10:00April 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 [...]

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