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. $x^a{x}^b=x^{a+b}$
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 $a(x+b{)}^2+c$

1. -b is the axis of symmetry
2. c is the maximum or minimum y value

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

Completing the Square

$a{x}^2+bx+c=0$ becomes $a{(x+\frac{b}{2a})}^2-\frac{b^2}{4a}+c$

The Quadratic Formula

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

Errors

1. $Absoluteerror=valueobtained-truevalue$
2. $Relativeerror=\frac{absoluteerror}{truevalue}$
3. $Percentageerror=relativeerror\times 100$

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 $(x_1,y_1)$ and having a slope m is $y-y_1=m(x-x_1)$.

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

$(\frac{x_1+x_2}{2};\frac{y_1+y_2}{2})$

General Circle Formulae

$Area=\pi r^2$

$Circumference=2\pi r$

Equation of a Circle

${(x-h)}^2+{(y-k)}^2=r^2$, where (h,k) is the center and r is the radius.

Differentiation

Differentiation Rules

1. Derivative of a constant function:

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

1. The Power Rule:

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

1. The Constant Multiple Rule:

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

1. The Sum Rule:

$\frac{dy}{dx}\left[\begin{array}{c}f\left(x\right)+g\left(x\right)\end{array}\right]=\frac{dy}{dx}f\left(x\right)+\frac{dy}{dx}g\left(x\right)$

1. The Difference Rule:

$\frac{dy}{dx}\left[\begin{array}{c}f\left(x\right)-g\left(x\right)\end{array}\right]=\frac{dy}{dx}f\left(x\right)-\frac{dy}{dx}g\left(x\right)$

Rules of Stationary Points

• If $f^{\prime }\left(c\right)=0$ 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 $f^{\prime }\left(c\right)=0$ 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 $f^{\prime }\left(c\right)=0$ and $f^{″}\left(c\right)=0$ and $f^{‴}\left(c\right)\ne 0$, then c is a local inflection point of f(x).