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

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

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

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

If f(x) is in the form ${\displaystyle a(x+b)^{2}+c}$

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

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

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

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

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

${\displaystyle m={\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}}$

The equation of a line passing through the point ${\displaystyle \left(x_{1},y_{1}\right)}$ and having a slope m is ${\displaystyle y-y_{1}=m\left(x-x_{1}\right)}$.

Lines are perpendicular if ${\displaystyle m_{1}\times m_{2}=-1}$

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}}}$

${\displaystyle \left({\frac {{x_{1}}+{x_{2}}}{2}};{\frac {{y_{1}}+{y_{2}}}{2}}\right)}$

${\displaystyle Area=\pi r^{2}\,}$

${\displaystyle Circumference=2\pi r\,}$

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

1. Derivative of a constant function:

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

1. The Power Rule:

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

1. The Constant Multiple Rule:

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

1. The Sum Rule:

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

1. The Difference Rule:

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

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