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. $y\left(t\right)=y_0e^{kt}\,$, where y(t) is the final value, y₀ 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

$\int^b_a y dx \approx \frac{1}{3}h\left\{\left(y_0 + y_n\right) + 4\left(y_1 + y_3 + \ldots + y_{n-1}\right) + 2\left(y_2 + y_4 + \ldots + y_{n-2}\right) \right\}$

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.