A-level Mathematics/OCR/C1/Appendix A: Formulae

By the end of this module you will be expected to have learned the following formulae:

The Laws of Indices

${\sqrt {xy}}={\sqrt {x}}\times {\sqrt {y}}$
${\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}$
${\frac {a}{b+{\sqrt {c}}}}={\frac {a}{b+{\sqrt {c}}}}\times {\frac {b-{\sqrt {c}}}{b-{\sqrt {c}}}}={\frac {a(b-{\sqrt {c}})}{b^{2}-c}}$

If f(x) is in the form $a(x+b)^{2}+c$

Axis of Symmetry = ${\frac {-b}{2a}}$

$ax^{2}+bx+c=0\,$ becomes $a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c$

The solutions of the quadratic $ax^{2}+bx+c=0$ are: $x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$
The discriminant of the quadratic $ax^{2}+bx+c=0$ is $b^{2}-4ac$

$m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$

The equation of a line passing through the point $\left(x_{1},y_{1}\right)$ and having a slope m is $y-y_{1}=m\left(x-x_{1}\right)$ .

Lines are perpendicular if $m_{1}\times m_{2}=-1$

$d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}$

$\left({\frac {{x_{1}}+{x_{2}}}{2}};{\frac {{y_{1}}+{y_{2}}}{2}}\right)$

Area=\pi r^{2}\,

$Circumference=2\pi r\,$

$\left(x-h\right)^{2}+\left(y-k\right)^{2}=r^{2}$ , where (h,k) is the center and r is the radius.

Derivative of a constant function:
${\frac {dy}{dx}}\left(c\right)=0$
The Power Rule:
${\frac {dy}{dx}}\left(x^{n}\right)=nx^{n-1}$
The Constant Multiple Rule:
${\frac {dy}{dx}}cf\left(x\right)=c{\frac {dy}{dx}}f\left(x\right)$
The Sum Rule:
${\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)+g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)+{\frac {dy}{dx}}g\left(x\right)$
The Difference Rule:
${\frac {dy}{dx}}{\begin{bmatrix}f\left(x\right)-g\left(x\right)\end{bmatrix}}={\frac {dy}{dx}}f\left(x\right)-{\frac {dy}{dx}}g\left(x\right)$

If $f'\left(c\right)=0$ and $f''\left(c\right)<0$ , then c is a local maximum point of f(x). The graph of f(x) will be concave down on the interval.
If $f'\left(c\right)=0$ and $f''\left(c\right)>0$ , then c is a local minimum point of f(x). The graph of f(x) will be concave up on the interval.
If $f'\left(c\right)=0$ and $f''\left(c\right)=0$ and $f'''\left(c\right)\neq 0$ , then c is a local inflection point of f(x).

This is part of the C1 (Core Mathematics 1) module of the A-level Mathematics text.

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