/Tag: Extension
23 04, 2019

Vectors

By |2019-04-29T15:49:55+10:00April 23rd, 2019|Tags: , |0 Comments

Vectors Go to the Forum to see more questions or ask your own Topics in Vectors Introduction Two Dimensions Component Form Scalar Products Projection Proofs Formulas in Vectors Column Vector $$\left( {\begin{array}{ccccccccccccccc}a\\b\end{array}} \right)$$ represents (a, b) Component Form $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{a} = x\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{i} [...]

11 03, 2019

Absolute Functions

By |2019-03-29T08:21:51+10:00March 11th, 2019|Tags: , , |0 Comments

Absolute Functions Go to the Forum to see more questions or ask your own Formulas used in Absolute Functions y= |x| D: (-∞, ∞) R: [0, ∞) y= |ƒ(x)| ƒ(x) = (x - 2)(x + 1) When graphing the absolute value of a function, any negatives will become positive, [...]

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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