A-level Mathematics/OCR/C3/Formulae

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

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, $$y_0$$ 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.

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

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
For volumes of revolution:
 * $$ \int \operatorname{e}^{kx}\, dx = \frac{1}{k}\operatorname{e}^{kx} + c $$
 * $$ \int \frac{1}{x}\, dx = \ln \left|x\right| + c$$
 * $$ V_x = \pi \int_{a}^{b} y^2\, dx$$
 * $$ V_y = \pi \int_{c}^{d} x^2\, dy$$

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