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. $\frac{a}{b+\sqrt{c}} = \frac{a}{b+\sqrt{c}} \times \frac{b-\sqrt{c}}{b-\sqrt{c}} = \frac{a(b-\sqrt{c})}{b^2-c}$

Polynomials

Parabolas

If f(x) is in the form a(x + d)² + e

1. -d is the axis of symmetry
2. e is the maximum or minimum value

Axis of Symmetry = $\frac{-b}{2a}$

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 ax² + bx + c = 0 are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
• The discriminant of the quadratic ax² + bx + c = 0 is b² - 4ac

Errors

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

Coordinate Geometry

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

Differentiation Rules

1. Derivative of a constant function:

   $\frac{dy}{dx} \left (c \right) = 0$

2. The Power Rule:

   $\frac{dy}{dx} \left (x^n \right) = nx^{n - 1}$

3. The Constant Multiple Rule:

   $\frac{dy}{dx} c f \left ( x \right ) = c \frac{dy}{dx} f \left ( x \right )$

4. The Sum Rule:

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

5. 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).

This is part of the C1 (Core Mathematics 1) module of the A-level Mathematics text.