A-level Mathematics/OCR/C1/Appendix A: Formulae

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

The Laws of Indices

 * 1) $$x^ax^b = x^{a+b}\,$$
 * 2) $$\frac{x^a}{x^b} = x^{a-b}$$
 * 3) $$x^{-n}=\frac{1}{x^n}$$
 * 4) $$\left(x^a\right)^b = x^{ab}$$
 * 5) $$\left(xy \right)^n = x^n y^n$$
 * 6) $$\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$$
 * 7) $$x^\frac{a}{b} = \sqrt[b]{x^a}$$
 * 8) $$x^0 = 1\,$$
 * 9) $$x^1 = x\,$$

The Laws of Surds

 * 1) $$\sqrt{xy} = \sqrt{x} \times \sqrt{y}$$
 * 2) $$\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$$
 * 3) $$\frac{a}{b+\sqrt{c}} = \frac{a}{b+\sqrt{c}} \times \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{a(b-\sqrt{c})}{b^2-c}$$

Parabolas
If f(x) is in the form $$a(x + b)^2 + c$$ Axis of Symmetry = $$\frac{-b}{2a}$$
 * 1) -b is the axis of symmetry
 * 2) c is the maximum or minimum y value

Completing the Square
$$ ax^2+bx+c=0\,$$ becomes $$a\left(x + \frac{b}{2a}\right)^2 -\frac{b^2}{4a} + c$$

The Quadratic Formula

 * The solutions of the quadratic $$ax^2+bx+c=0$$ are: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
 * The discriminant of the quadratic $$ax^2+bx+c=0$$ is $$b^2 - 4ac$$

Errors

 * 1) $$Absolute\ error = value\ obtained - true\ value$$
 * 2) $$Relative\ error = \frac{absolute\ error}{true\ value}$$
 * 3) $$Percentage\ error = relative\ error \times 100$$

Gradient of a line
$$m=\frac {y_2-y_1}{x_2-x_1}$$

Point-Gradient Form
The equation of a line passing through the point $$\left (x_1, y_1 \right )$$ and having a slope m is $$y - y_1 = m \left ( x - x_1 \right)$$.

Perpendicular lines
Lines are perpendicular if $$m_1 \times m_2=-1$$

Distance between two points
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$

Mid-point of a line
$$\left(\frac {{x_1} + {x_2}}{2} ; \frac {{y_1} + {y_2}}{2}\right)$$

General Circle Formulae
$$Area = \pi r^2\,$$

$$Circumference = 2 \pi r\,$$

Equation of a Circle
$$\left (x - h \right )^2 + \left (y - k \right )^2 = r^2$$, where (h,k) is the center and r is the radius.

Differentiation Rules

 * 1) Derivative of a constant function:

$$\frac{dy}{dx} \left (c \right) = 0$$
 * 1) The Power Rule:

$$\frac{dy}{dx} \left (x^n \right) = nx^{n - 1}$$
 * 1) The Constant Multiple Rule:

$$\frac{dy}{dx} c f \left ( x \right ) = c \frac{dy}{dx} f \left ( x \right )$$
 * 1) The Sum Rule:

$$\frac{dy}{dx} \begin{bmatrix} f \left ( x \right ) + g \left ( x \right ) \end{bmatrix} = \frac{dy}{dx} f \left ( x \right ) +  \frac{dy}{dx} g \left ( x \right )$$
 * 1) The Difference Rule:

$$\frac{dy}{dx} \begin{bmatrix} f \left ( x \right ) - g \left ( x \right ) \end{bmatrix} = \frac{dy}{dx} f \left ( x \right ) -  \frac{dy}{dx} g \left ( x \right )$$

Rules of Stationary Points

 * If $$f' \left ( c \right ) = 0$$ and $$f'' \left ( c \right ) <0$$, then c is a local maximum point of f(x). The graph of f(x) will be concave down on the interval.
 * If $$f' \left ( c \right ) = 0$$ and $$f'' \left ( c \right ) >0$$, then c is a local minimum point of f(x). The graph of f(x) will be  concave up on the interval.
 * If $$f' \left ( c \right ) = 0$$ and $$f \left ( c \right ) = 0$$ and $$f' \left ( c \right ) \ne 0$$, then c is a local inflection point of f(x).