Advanced Higher Maths: Formulae - Newbattle Community High ...

Advanced Higher Maths: Formulae
Advanced Higher Mathematics
Green (G): Formulae you absolutely must memorise in order to pass Advanced
Higher maths. Remember you get no formula sheet at all in the exam!
Amber (A): You don’t have to memorise these formulae, as it is possible to derive
them from scratch in the exam. But… it will save you a lot of time if you do choose
to memorise them, and I advise that you do.
Red (R): Don’t worry about memorising these. Just use this sheet to help jog your
memory in classwork and homework. One or two of these formulae are on the
syllabus, but are sufficiently obscure that I don’t think it essential to memorise them.
Trigonometric Identities: (from Intermediate 2 and Higher)
Links
between
ratios
Essential Formulae to know
off by heart for the exam (G)
2 2
cos Asin A 1
Other useful ones that may
be useful for
homework/classwork etc.
2 2
1 tan A sec A (A)
2 2
cot A 1 cosec A (R)
sin A
tan A
cos A
Squared
2 1
cos x
2
(1 cos 2 x)
2 1
sin x
2
(1 cos 2 x)
Compound sin( A B) sin Acos B
cos Asin
B
tan A tan B
tan( A B)
(R)
Angle cos( A B) cos Acos B
sin Asin
B
1 tan A tanB
Double sin(2 A) 2sin Acos
A
2tanA
Angle (A) tan(2 A)
(R)
2 2
2
cos(2 A) cos Asin
A
1 tan A
Exact Values(you should know all these, though there is no non-calculator paper, unlike Higher)
0
3
6 4 3 2 2
2
sin 0
1 1 3
2 2 2
1 0 1 0
cos 1
3 1 1
2 2 2
0 1 0 1
tan 0
1
3
1 3 undef. 0 undef. 0
Unit 2.3: Complex Numbers
For the complex number, z a bi,
the modulus is given by
2 2
z a b
b
and the argument is given by tan
a
The conjugate is z a
bi
Negative facts:
sin(
) sin( )
cos(
) cos( )
tan(
) tan(
)
De Moivre’s Theorem says that
for any z r(cos
isin ) , then n n
z r (cos n
isin n
) ( n )
Newbattle Community High School © D Watkins 2014

Advanced Higher Mathematics
Units 1.2 and 2.1: Differentiation
f( x) f '( x)
tan x
sec
sec x sec xtan
x
cosec x cosec xcot
x
cot x
2
cosec
x
ln f( x)
f '( x)
f( x)
2
x
f ( x) f '( x)
sin
cos
tan
-1
-1
-1
x
x
x
1
1
x
1
2
1
x
1
2
1
x
2
To differentiate an inverse function:
dx 1
dy
dy (A)
dx
Parametric Equations (where x f(), t y g()
t ):
Gradient (direction of movement) =
dy dx
Speed =
2 2
dt dt
2
d y x y
y x
2 3
dx
x
Units 1.3 and 2.2: Integration
dy
dx
dy
dt
dx
dt
(G) Essential Integrals to Learn
f ( x) f ( x)
dx
sec
2
x tan x
C
tan x ln sec x C
f '( x)
f ( x)
1
2
1 x
1
1 x
2
ln f ( x)
C
tan
sin
-1
-1
x C
x C
(A) Could use substitution if needed:
f ( x) f ( xdx )
1 1 x tan C
1
1
x
sin C
2 2
a x
a
1
2 2
a x a a
(R) To save you time in hard questions for
homework/classwork, no need to memorise:
f ( x) f ( x)
dx
cosecx ln cosec xcot
x C
cot x ln sin x C
sec x ln sec x
tan x C
Volume of solid of revolution f(x) between a and b:
b
2
2
About x axis: V f( x)
dx About y axis: V
f( y)
dy
Unit 1.4: Properties of functions
a
Odd function: f ( x) f( x)
Even function: f ( x) f( x)
(180° rotational symmetry) (line symmetry about the y-axis)
Newbattle Community High School © D Watkins 2014
b
a

Advanced Higher Mathematics
Unit 2.4: Sequences and Series
Arithmetic Series Geometric Series
u a( n1)
d u ar
n
n
1
a(1 r )
Sn
2
n(2 a( n1) d) Sn
r 1
1
r
a
S
r 1
1
r
In particular, you are supposed to know that as a consequence of the last formula:
1
2 3
1 r r r ...
1 r
1
and also lim(1 ) n e (A)
n
n
n
n1
Important Identities
n
n
1
k n ( n
1)
1 n
k 1
2
k 1
n
2 1
k n ( n 1)(2 n
1) (note: this is named specifically on syllabus) (A)
6
k 1
n
n
2
3
1 2 2
k
k
4
n ( n1)
k1 k1
(also named specifically on syllabus) (A)
Unit 3.3: Maclaurin Series
( n)
f(0) 2 f (0) n
f( x) f(0) f(0) x ... x ... (G)
2! n!
and in particular:
Very useful to memorise (A):
2 3
n
x x x x
e 1 x ... ...
2! 3! n!
3 5 7
x x x
sin x x ...
3! 5! 7!
2 4 6
x x x
cos x 1 ...
2! 4! 6!
On syllabus but less essential (R):
3 5 7
1
x x x
tan x x ...
3 5 7
2 3 4
x x x
ln(1 x) x ...
2 3 4
Newbattle Community High School © D Watkins 2014

Advanced Higher Mathematics
Unit 1.1: Binomial Theorem
The coefficient of the r th term in the binomial expansion ( x y) n is
n
C
r
n
n!
r r!( n
r)!
n
x
r
nr
y
r
Unit 3.1: Vectors, Lines and Planes
Angle between two vectors: (Higher)
ab a b cos
Equations of a 3d line: through ( x1, y1, z
1)
and with direction vector d aibjck
Parametric form
Symmetric form
x x1
at
x x1 y y1 z
z
y y1
bt ( x atd )
1
( t)
a b c
z z ct
1
Equations of a plane:
l
Normal n is
m
n
Point on line = P (with position vector a)
Vector equation Symmetric/Cartesian Parametric (A)
x•n
a•n lx my nz k
x abc
where k a•n
(b and c are any two nonparallel
vectors in plane)
Angle between two lines = Acute angle between their direction vectors
Angle between two planes = Acute angle between their normals
Angle between line and plane = 90° – (Acute angle between n and d)
i j k
Cross (vector) product: ab
a1 a2 a3
b b b
1 2 3
Scalar triple product:
a a a
1 2 3
a ( bc)
b b b (A)
1 2 3
c c c
1 2 3
(this is important but is only rated amber, as you wouldn’t have to remember it, as any
exam question would have to tell you a
( b
c ))
Newbattle Community High School © D Watkins 2014

Advanced Higher Mathematics
Unit 3.2: Matrices
2×2 matrices
3×3 matrices
a
b
A
c
d
a b c
A d e f
g h i
Determinant and Inverse
1 1 d b
det A ad bc and A
ad bc c
a
e f d f d e
det A a b c
h i g i g h
( AB)
B A
( AB) T B T A
T
det AB det Adet
B
1 1 1
(A)
Transformation Matrices
cos
sin
Anti-CW Rotation by θ degrees
sin
cos
, Reflection in y-axis 1 0
0 1
a
0
Enlargement by scale factor a
0
a , Reflection in x-axis 1 0
0 1
Unit 3.4: Differential Equations (G)
dy
( )
For Pxy ( ) Qx ( )
dx , the Integrating Factor I(x) is
and the solution is given by I( xy ) IxQxdx ( ) ( )
e P x dx
Second Order Differential Equations
COMPLEMENTARY FUNCTION (Homogeneous Equations)
Nature of roots
Form of general solution
Two distinct real m and n
mx
y Ae
nx
Be
Real and equal m y ( A
Bx) e
mx
Complex conjugate m p iq y e px ( Acosqx
Bsin qx)
PARTICULAR INTEGRAL (Inhomogeneous Equations)
Right-hand side contains… For Particular Integral, try…
sin ax or cos ax y Pcos ax
Qsin
ax
e ax
ax
y Pe
Linear expression y ax b
y Px
Q
Quadratic expression
2
y ax bx
c
2
y Px Qx
R
Newbattle Community High School © D Watkins 2014