A-level Mathematics/OCR/C2/Appendix A: Formulae

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

Remainder Theorem
If you have a polynomial f(x) divided by x - c, the remainder is equal to f(c). Note if the equation is x + c then you need to negate c: f(-c).

The Factor Theorem
A polynomial f(x) has a factor x - c if and only if f(c) = 0. Note if the equation is x + c then you need to negate c: f(-c).

The Laws of Exponents

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

Logarithmic Function
The inverse of $$y = b^x\,$$ is $$x = b^y \,$$ which is equivalent to $$y = \log_b x\,$$

Change of Base Rule: $$\log_a x\,$$ can be written as $$\frac { \log_b x}{ \log_b a}$$

Laws of Logarithmic Functions
When X and Y are positive.


 * $$\log_bXY = \log_bX + \log_bY\,$$
 * $$\log_b \frac{X}{Y} = \log_bX - \log_bY\,$$
 * $$\log_b X^k = k \log_bX\,$$

Conversion of Degree Minutes and Seconds to a Decimal
$$X + \frac{Y}{60}+ \frac{Z}{3600}$$ where X is the degree, y is the minutes, and z is the seconds.

Arc Length
$$s= \theta r\,$$ Note: θ need to be in radians

Area of a Sector
$$Area = \frac{1}{2}r^2 \theta$$Note: θ need to be in radians.

Important Trigonometric Values
You need to have these values memorized.

The Law of Cosines
$$a^2=b^2 + c^2 - 2bc \cos \alpha \,$$

$$b^2=a^2 + c^2 - 2ac \cos \beta \,$$

$$c^2=a^2 + b^2 - 2ab \cos \gamma \,$$

The Law of Sines
$$\frac {a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac {c}{\sin \gamma}$$

Area of a Triangle
$$Area = \frac{1}{2}bc \sin \alpha \,$$

$$Area = \frac{1}{2}ac \sin \beta \,$$

$$Area = \frac{1}{2}ab \sin \gamma \,$$

Trigonometric Identities
$$\sin ^2 \theta + \cos ^2 \theta = 1 \,$$

$$tan \theta = \frac{\sin \theta}{\cos \theta}$$

Integration Rules
The reason that we add a + C when we compute the integral is because the derivative of a constant is zero, therefore we have an unknown constant when we compute the integral. $$\int x^n\, dx = \frac{1}{n+1} x^{n+1} + C,\ (n \ne -1)$$

$$\int kx^n\, dx = k \int x^n\, dx$$

$$\int \left\{ f^'(x) + g^'(x)\right\}\, dx = f(x) + g(x) + C$$

$$\int \left\{ f^'(x) - g^'(x)\right\}\, dx = f(x) - g(x) + C$$

Rules of Definite Integrals

 * 1) $$\int_{a}^{b} f \left ( x \right )\ dx = F \left ( b \right ) - F \left ( a \right )$$, F is the anti derivative of f such that F' = f
 * 2) $$\int_{a}^{b} f \left ( x \right )\ dx = - \int_{b}^{a} f \left ( x \right )\ dx$$
 * 3) $$\int_{a}^{a} f \left ( x \right )\ dx = 0$$
 * 4) Area between a curve and the x-axis is $$\int_{a}^{b} y\, dx\ ( \mbox{for}\ y \ge 0)$$


 * 1) Area between a curve and the y-axis is $$\int_{a}^{b} x\, dy\ ( \mbox{for}\ x \ge 0)$$
 * 2) Area between curves is $$\int_{a}^{b}\begin{vmatrix} f\left(x\right) - g\left(x\right) \end{vmatrix} dx$$

Trapezium Rule
$$\int_a^b y \,dx \approx \frac{1}{2} h \left \{\left (y_0 + y_n \right ) + 2\left (y_1 + y_2 + \ldots + y_{n-1} \right ) \right\}$$

Where: $$h = \frac{b-a}{n}$$

Midpoint Rule
$$\int_a^b f \left (x \right ) \,dx \approx = h \left [ f \left (x_1 \right ) + f \left (x_2 \right ) + \ldots + f \left (x_n \right )\right ]$$

Where: $$h = \frac{b-a}{n}$$ n is the number of strips.

and $$x_i = \frac{1}{2} \left [ \left( a +\left \{i - 1 \right \} h \right) + \left (a + ih \right) \right]$$