By failing to prepare, you are preparing to fail.
/Tag: Extension
29 07, 2018

## Oblique Asymptotes

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

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

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

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

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 = [...]

27 04, 2017

## Pascal’s Triangle

Pascal's Triangle To construct a Pascal's Triangle, start with 1 at the top, a single number All rows must start and end with 1, so the second row will have two numbers : 1 and 1 The third row - first number is 1, to get the next number, you [...]

25 04, 2017

## Complementary Inverse Trig

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 [...]

22 03, 2017

## Proving Inequalities

Proving Inequalities worked examples Prove $\left( {a + b} \right)\left( {\frac{1}{a} + \frac{1}{b}} \right) \ge 4$ for a > 0 and b > 0. Consider (√a – √b)2 ≥ 0 since all square numbers are positive, it is a true statement to say that any perfect square will be larger than [...]