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

1. $y=-f\left(x\right)\,$ is a reflection of $y=f\left(x\right)\,$ through the x axis.
2. $y=f\left(-x\right)\,$ is a reflection of $y=f\left(x\right)\,$ through the y axis.
3. $y={\begin{vmatrix}f\left(x\right)\end{vmatrix}}$ is a reflection of $y=f\left(x\right)\,$ when y < 0, through the x-axis.
4. $y=f\left({\begin{vmatrix}x\end{vmatrix}}\right)$ is a reflection of $y=f\left(x\right)\,$ when x < 0, through the y-axis.
5. $y=f^{-1}\left(x\right)\,$ 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

1. $y=af\left(x\right)\,$ 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.
2. $y=f\left(bx\right)\,$ 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

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

Natural Functions

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