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Home/Tag: Extension 2
11 03, 2017

## Inequalities

By |2017-03-13T14:20:04+10:00March 11th, 2017|Tags: , |0 Comments

Inequalities Cauchy's Theorem $\frac{a_1+a_2+a_3+...+a_n}{n}=\sqrt[3]{{{a_1}{a_2}{a_3}...{a_n}}}$   further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 + √x = 1 + x1/2 du/dx = ½x–1/2 du = $\frac{1}{{2\sqrt x }}dx$ x = 49 [...]

11 02, 2017

## Complex Proofs

Complex Proofs $z\in C$ such that $\frac{z}{z-i}$ is real. Show that z is imaginary. Let $\frac{z}{z-i}=a$, where a is real. Then $z=a(z-i)$ $z=az-ai$ $ai=az-z$ $ai=z(a-1)$ so $z=\frac{ai}{a-1}$ $z=\frac{a}{a-1}i$ Since a is real, then $\frac{a}{a-1}$ is real, and hence $\frac{a}{a-1}i$ is imaginary thus z is imaginary     [...]

9 02, 2017

## Parabolas

By |2017-02-09T10:58:01+10:00February 9th, 2017|Tags: , , |0 Comments

Hyperbolas   $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ eccentricity: $e=\sqrt{1+\frac{b^2}{a^2}}$ PS = e.PM Foci: $(\pm ae, 0)$ Directrix: $x=\pm\frac{a}{e}$ Asymptotes: $y=\pm\frac{b}{a}x$   $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ when a < b eccentricity: $e=\sqrt{1-\frac{a^2}{b^2}}$ PS = e.PM Foci: $(0,\pm ae)$ Directrix: $y=\pm\frac{b}{e}$     worked examples further examples = 10x + 15 - [...]

7 02, 2017

## Graphs

By |2017-02-07T20:20:59+10:00February 7th, 2017|Tags: , |0 Comments

Graphs Go to the Graphs forum to see more questions or ask your own Topics Covered in Graphs Motion in One Dimension   Formulas Used in Graphs     Graphs Forum Rules of [...]

7 02, 2017

## Harder 3 Unit Topics

By |2017-03-11T11:41:34+10:00February 7th, 2017|Tags: , |0 Comments

Harder 3 Unit Topics Go to the Harder 3 Unit Topics forum to see more questions or ask your own Topics Covered in Harder 3 Unit Topics Inequalities Induction Geometry of Circle   Formulas Used in Harder 3 [...]

7 02, 2017

## Volumes

By |2017-02-07T18:36:19+10:00February 7th, 2017|Tags: , |0 Comments

Volumes Go to the Volumes forum to see more questions or ask your own   Topics Covered in Volumes Motion in One Dimension Formulas Used in Volumes       Volumes Forum [...]

7 02, 2017

## t Method

t Method   worked examples Find $\int{\ln x\,dx}$. $=\int{1.\ln x\,dx}$ rewrite as $\int{\ln x\,dx}$ let f(x) = ln x    g'(x) = 1 select f(x) = ln x and g'(x) = 1 when selecting which part to make f(x) and which part to make g'(x), we want [...]

7 02, 2017

## Integration Recurrence

Integration Recurrence worked examples Find $\int{\ln x\,dx}$. $=\int{1.\ln x\,dx}$ rewrite as $\int{\ln x\,dx}$ let f(x) = ln x    g'(x) = 1 select f(x) = ln x and g'(x) = 1 when selecting which part to make f(x) and which part to make g'(x), we want the [...]

6 02, 2017

## Argand Diagrams

Argand Diagrams If $z=a+b i$, then its conjugate $\overline{z}=a-b i$ the conjugate of z is reflected in the Real axis worked examples further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 + [...]
Motion in One Dimension Proof of $a=v.\frac{dv}{dx}$ we know $a=\frac{dv}{dt}$ and $v=\frac{dx}{dt}$ $a=v.\frac{dv}{dt}.\frac{1}{v}$ since $v=\frac{dx}{dt}$, then $\frac{1}{v}=\frac{dt}{dx}$ $a=v.\frac{dv}{dt}.\frac{1}{v}$ ⇒ $a=v.\frac{dv}{dt}.\frac{dt}{dx}$ ∴ $a=v.\frac{dv}{dx}$ worked examples further examples = 10x + 15 - 7x + 14 = 3x + 29 harder examples u = 1 [...]