[<< wikibooks] 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 ==

x

a

x

b

=

x

a
+
b

{\displaystyle x^{a}x^{b}=x^{a+b}\,}

x

a

x

b

=

x

a
−
b

{\displaystyle {\frac {x^{a}}{x^{b}}}=x^{a-b}}

x

−
n

=

1

x

n

{\displaystyle x^{-n}={\frac {1}{x^{n}}}}

(

x

a

)

b

=

x

a
b

{\displaystyle \left(x^{a}\right)^{b}=x^{ab}}

(

x
y

)

n

=

x

n

y

n

{\displaystyle \left(xy\right)^{n}=x^{n}y^{n}}

(

x
y

)

n

=

x

n

y

n

{\displaystyle \left({\frac {x}{y}}\right)^{n}={\frac {x^{n}}{y^{n}}}}

x

a
b

=

x

a

b

{\displaystyle x^{\frac {a}{b}}={\sqrt[{b}]{x^{a}}}}

x

0

=
1

{\displaystyle x^{0}=1\,}

x

1

=
x

{\displaystyle x^{1}=x\,}

== The Laws of Surds ==

x
y

=

x

×

y

{\displaystyle {\sqrt {xy}}={\sqrt {x}}\times {\sqrt {y}}}

x
y

=

x

y

{\displaystyle {\sqrt {\frac {x}{y}}}={\frac {\sqrt {x}}{\sqrt {y}}}}

a

b
+

c

=

a

b
+

c

×

b
−

c

b
−

c

=

a
(
b
−

c

)

b

2

−
c

{\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}}}

== Polynomials ==

=== Parabolas ===
If f(x) is in the form

a
(
x
+
b

)

2

+
c

{\displaystyle a(x+b)^{2}+c}

-b is the axis of symmetry
c is the maximum or minimum y valueAxis of Symmetry =

−
b

2
a

{\displaystyle {\frac {-b}{2a}}}

=== Completing the Square ===

a

x

2

+
b
x
+
c
=
0

{\displaystyle ax^{2}+bx+c=0\,}
becomes

a

(

x
+

b

2
a

)

2

−

b

2

4
a

+
c

{\displaystyle a\left(x+{\frac {b}{2a}}\right)^{2}-{\frac {b^{2}}{4a}}+c}

=== The Quadratic Formula ===
The solutions of the quadratic

a

x

2

+
b
x
+
c
=
0

{\displaystyle ax^{2}+bx+c=0}
are:

x
=

−
b
±

b

2

−
4
a
c

2
a

{\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}}

The discriminant of the quadratic

a

x

2

+
b
x
+
c
=
0

{\displaystyle ax^{2}+bx+c=0}
is

b

2

−
4
a
c

{\displaystyle b^{2}-4ac}

== Errors ==

A
b
s
o
l
u
t
e

e
r
r
o
r
=
v
a
l
u
e

o
b
t
a
i
n
e
d
−
t
r
u
e

v
a
l
u
e

{\displaystyle Absolute\ error=value\ obtained-true\ value}

R
e
l
a
t
i
v
e

e
r
r
o
r
=

a
b
s
o
l
u
t
e

e
r
r
o
r

t
r
u
e

v
a
l
u
e

{\displaystyle Relative\ error={\frac {absolute\ error}{true\ value}}}

P
e
r
c
e
n
t
a
g
e

e
r
r
o
r
=
r
e
l
a
t
i
v
e

e
r
r
o
r
×
100

{\displaystyle Percentage\ error=relative\ error\times 100}

== Coordinate Geometry ==

=== Gradient of a line ===

m
=

y

2

−

y

1

x

2

−

x

1

{\displaystyle 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

)

{\displaystyle \left(x_{1},y_{1}\right)}
and having a slope m is

y
−

y

1

=
m

(

x
−

x

1

)

{\displaystyle y-y_{1}=m\left(x-x_{1}\right)}
.

=== Perpendicular lines ===
Lines are perpendicular if

m

1

×

m

2

=
−
1

{\displaystyle m_{1}\times m_{2}=-1}

=== Distance between two points ===

d
=

(

x

2

−

x

1

)

2

+
(

y

2

−

y

1

)

2

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

=== Mid-point of a line ===

(

x

1

+

x

2

2

;

y

1

+

y

2

2

)

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

=== General Circle Formulae ===

A
r
e
a
=
π

r

2

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

C
i
r
c
u
m
f
e
r
e
n
c
e
=
2
π
r

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

=== Equation of a Circle ===

(

x
−
h

)

2

+

(

y
−
k

)

2

=

r

2

{\displaystyle \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 ===
Derivative of a constant function:

d
y

d
x

(
c
)

=
0

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

The Power Rule:

d
y

d
x

(

x

n

)

=
n

x

n
−
1

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

The Constant Multiple Rule:

d
y

d
x

c
f

(
x
)

=
c

d
y

d
x

f

(
x
)

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

The Sum Rule:

d
y

d
x

[

f

(
x
)

+
g

(
x
)

]

=

d
y

d
x

f

(
x
)

+

d
y

d
x

g

(
x
)

{\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)}

The Difference Rule:

d
y

d
x

[

f

(
x
)

−
g

(
x
)

]

=

d
y

d
x

f

(
x
)

−

d
y

d
x

g

(
x
)

{\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)}

=== Rules of Stationary Points ===
If

f
′

(
c
)

=
0

{\displaystyle f'\left(c\right)=0}
and

f
″

(
c
)

<
0

{\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

f
′

(
c
)

=
0

{\displaystyle f'\left(c\right)=0}
and

f
″

(
c
)

>
0

{\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

f
′

(
c
)

=
0

{\displaystyle f'\left(c\right)=0}
and

f
″

(
c
)

=
0

{\displaystyle f''\left(c\right)=0}
and

f
‴

(
c
)

≠
0

{\displaystyle f'''\left(c\right)\neq 0}
, then c is a local inflection point of f(x).