A-level Mathematics/OCR/C3/Formulae

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

Transformations of Graphs

Reflection

$y=-f\left(x\right)\,$ is a reflection of $y=f\left(x\right)\,$ through the x axis.
$y=f\left(-x\right)\,$ is a reflection of $y=f\left(x\right)\,$ through the y axis.
$y={\begin{vmatrix}f\left(x\right)\end{vmatrix}}$ is a reflection of $y=f\left(x\right)\,$ when y < 0, through the x-axis.
$y=f\left({\begin{vmatrix}x\end{vmatrix}}\right)$ is a reflection of $y=f\left(x\right)\,$ when x < 0, through the y-axis.
$y=f^{-1}\left(x\right)\,$ is a reflection of $y=f\left(x\right)\,$ through the line y = x.
Note: $f^{-1}\left(x\right)$ exists only if $f\left(x\right)$ is bijective, that is, one-to-one and onto.

Stretching

$y=af\left(x\right)\,$ is stretched toward the x-axis if $0<a<1\,$ and stretched away from the x-axis if $a>1\,$ . In both cases the change is by a units.
$y=f\left(bx\right)\,$ is stretched away from the y-axis if $0<b<1\,$ and stretched toward the y-axis if $b>1\,$ . In both cases the change is by b units.

Translations

$y=f\left(x-h\right)\,$ is a translation of f(x) by h units to the right.
$y=f\left(x+h\right)\,$ is a translation of f(x) by h units to the left.
$y=f\left(x\right)+k\,$ is a translation of f(x) by k units upwards.
$y=f\left(x\right)-k\,$ is a translation of f(x) by k units downwards.

Natural Functions

$e^{\ln x}=\ln e^{x}=x\,$
$y\left(t\right)=y_{0}e^{kt}\,$ , where y(t) is the final value, $y_{0}$ is the initial value, k is the growth constant, t is the elapsed time.
$k=-{\frac {\ln 2}{half-life}}$ , k for calculations involving half-lives.

Trigonometry

Reciprocal Trigonometric Functions and their Inverses

$\sec \theta \equiv {\frac {1}{\cos \theta }}$
$\operatorname {cosec} \ \theta \equiv {\frac {1}{\sin \theta }}$
$\cot \theta \equiv {\frac {1}{\tan \theta }}\equiv {\frac {\cos \theta }{\sin \theta }}$
$\sec ^{2}\theta \equiv 1+\tan ^{2}\theta$
$\operatorname {cosec} ^{2}\ \theta \equiv 1+\cot ^{2}\theta$

Angle Sum and Difference Identities

$\sin(A\pm B)=\sin(A)\cos(B)\pm \cos(A)\sin(B)\,$
$\cos(A\pm B)=\cos(A)\cos(B)\mp \sin(A)\sin(B)\,$
$\tan(A\pm B)={\frac {\tan(A)\pm \tan(B)}{1\mp \tan(A)\tan(B)}}$

Note: The sign $\mp$ means that if you add the angles (A+B) then you subtract in the identity and vice versa. It is present in the cosine identity and the denominator of the tangent identity.

Double Angle Identities

$\sin 2A\equiv 2\sin A\cos A$
$\cos 2A\equiv \cos ^{2}A-\sin ^{2}A\equiv 1-2\sin ^{2}A\equiv 2\cos ^{2}A-1$
$\tan 2A\equiv {\frac {2\tan A}{1-\tan ^{2}A}}$

Combination of Trigonometric Functions

Using radians r = amplitute α = phase. $r={\sqrt {a^{2}+b^{2}}}$

a\sin x+b\cos x=r\cdot \sin(x+\alpha )\,

where

\alpha =\arcsin {\frac {b}{r}}

$a\sin x+b\cos x=r\cdot \cos(x-\alpha )\,$

where

\alpha =\arccos {\frac {b}{r}}

Differentiation

If $y=\operatorname {e} ^{kx}\,$ , then ${\frac {dy}{dx}}=k\operatorname {e} ^{kx}$
If $y=\ln x\,$ , then ${\frac {dy}{dx}}={\frac {1}{x}}$
If $y=f(x).g(x)\,$ , then ${\frac {dy}{dx}}=f^{'}(x)g(x)+g^{'}(x)f(x)$
If $y={\frac {f(x)}{g(x)}}$ , then ${\frac {dy}{dx}}={\frac {f^{'}(x)g(x)-g^{'}(x)f(x)}{\left\{g(x)\right\}^{2}}}$
${\frac {dy}{dx}}={\frac {1}{\frac {dx}{dy}}}$
If $y=f[g(x)]\,$ , then ${\frac {dy}{dx}}=f^{'}[g(x)].g^{'}(x)$
${\frac {dy}{dt}}={\frac {dy}{dx}}.{\frac {dx}{dt}}$

Integration

$\int \operatorname {e} ^{kx}\,dx={\frac {1}{k}}\operatorname {e} ^{kx}+c$
$\int {\frac {1}{x}}\,dx=\ln \left|x\right|+c$

For volumes of revolution:

$V_{x}=\pi \int _{a}^{b}y^{2}\,dx$
$V_{y}=\pi \int _{c}^{d}x^{2}\,dy$

Numerical Methods

Simpson's Rule $\int _{a}^{b}ydx\approx {\frac {1}{3}}h\left\{\left(y_{0}+y_{n}\right)+4\left(y_{1}+y_{3}+\ldots +y_{n-1}\right)+2\left(y_{2}+y_{4}+\ldots +y_{n-2}\right)\right\}$

where $h={\frac {b-a}{n}}$ and n is even

This is part of the C3 (Core Mathematics 3) module of the A-level Mathematics text.

This article is issued from Wikibooks. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.