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. $y=f(-x)$ is a reflection of $y=f\left(x\right)$ through the y axis.
3. $y=\left|\begin{array}{c}f\left(x\right)\end{array}\right|$ is a reflection of $y=f\left(x\right)$ when y < 0, through the x-axis.
4. $y=f\left(\left|\begin{array}{c}x\end{array}\right|\right)$ 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. $y=f(x-h)$ is a translation of f(x) by h units to the right.
2. $y=f(x+h)$ 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

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

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

Integration

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

For volumes of revolution:

• $V_x=\pi \underset{a}{\overset{b}{\int }}{y}^{2}dx$
• $V_y=\pi \underset{c}{\overset{d}{\int }}{x}^{2}dy$

Numerical Methods

Simpson's Rule $\underset{a}{\overset{b}{\int }}ydx\approx \frac{1}{3}h\{(y_0+y_n)+4(y_1+y_3+\dots +y_{n-1})+2(y_2+y_4+\dots +y_{n-2})\}$

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