Advanced Higher Maths: Formulae

Advanced Higher Maths Advanced Higher Maths: Formulae 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. Essential Trigonometric Identities: (from Intermediate 2 and Higher) Essential Formulae to know by heart for the exam (G) cos 2 A + sin 2 A = 1 sin A tan A = cos A
Compound sin( A ± B ) = sin A cos B ± cos A sin B
Angle cos( A ± B) = cos A cos B m sin A sin B
Double sin(2 A) = 2 sin A cos A Angle cos(2 A) = cos 2 A - sin 2 A
Squared cos 2 x = 1 (1 + cos 2 x ) Links between ratios Other useful ones that may be useful for homework/classwork etc. 1 + tan 2 A = sec 2 A
cot 2 A + 1 = cosec 2 A (A) tan( A ± B ) =
tan(2 A ) =
tan A ± tan B (R) 1 m tan A tan B
2 tan A (R) 1 - tan 2 A
2 2 1 2 sin x = (1 - cos 2 x) Unit 1.1: Binomial Theorem æ n ö
The coefficient of the r th term in the binomial expansion ( x + y ) n is ç ÷ x n -r y r è r ø æ n ö
n ! n C r = ç ÷ =
è r ø r !( n - r )! Unit 2.3: Complex Numbers For the complex number, ,
z = a + bi
· the modulus is given by z = a 2 + b 2 · and the argument is given by tan q =
b a
-p < q < p
De Moivre’s Theorem says that for any z = r (cosq + i sin q ) , then z n = r n (cos nq + i sin nq ) Newbattle Community High School D Watkins 2011 Advanced Higher Maths Units 1.2 and 2.1: Differentiation f ( x)
tan x
sec x
cosec x
cot x
f '( x ) sec 2 x sec x tan x -cosec x cot x - cosec 2 x ln f ( x ) f '( x ) f ( x ) f ( x)
sin ­1 x cos ­1 x tan ­1 x To differentiate an inverse function: f '( x )
1 1­ x 2 -1 1­ x 2 1 1 + x 2 d -1 1 f ( x ) = (A) dx
f '( f -1 ( x)) Parametric Equations (where x = f (t ), y = g (t ) ):
· dy
Gradient (direction of movement) = =
dx
· Speed =
· d 2 y d æ dy ö dt = ç ÷´
dx 2 dt è dx ø dx
2 dy dt dx dt dx 2 dt
( ) + ( ) dy dt
d 2 y x& &&
y - y& &&
x =
(A) 2
3 dx
x& or Units 1.3 and 2.2: Integration (G) Essential Integrals to Learn f ( x)
ò f ( x ) dx 2 sec x
tan x + C tan x
ln sec x + C f '( x ) f ( x ) ln f ( x ) + C 1 1 + x 2 1 1 - x
tan ­1 x + C 2 sin ­1 x + C (A) Could use substitution if needed:
f ( x)
ò f ( x) dx 1 a + x 2 1 1 a
2
2
a -x
tan -1 ( a x ) + C sin -1 ( a x ) + C 2 (R) To save you time in hard questions for homework/classwork, no need to memorise: f ( x)
ò f ( x) dx cosecx
cot x
- ln cosec x + cot x + C ln sin x + C sec x
ln sec x + tan x + C
b Volume of solid of revolution f(x) about x axis: V = p ò f ( x ) 2 dx
a Newbattle Community High School D Watkins 2011 Advanced Higher Maths Unit 2.4: Sequences and Series Arithmetic Series
un = a + (n - 1) d
Geometric Series un = ar n -1 S n = 1 2 n(2a + (n - 1)d )
S n =
a (1 - r n ) 1 - r a S¥ =
1 - r
r ¹ 1 r < 1 In particular, you are supposed to know that as a consequence of the last formula (A): 1 = 1 + r + r 2 + r 3 + ... 1 - r
1 2 = 1 a (1 - ba + ( b a ) - ...) a + b
and
and also lim(1 + 1 n ) n = e
n ®¥
(also mentioned specifically on syllabus) (R) Important Identities n
n
1 n( n + 1) 2 k =1 n
1 k 2 = n( n + 1)(2n + 1) å 6 k =1 å k =
å 1 = n
k =1 (note: this is named specifically on syllabus) (A) n
å k
3
= 1 4 n 2 (n + 1) 2 (note: this is named specifically on syllabus) (A) k =1 2 æ n ö
(also note: this is the same as ç å k ÷ : realising this may help memorise it) è k =1 ø
Unit 3.3: Maclaurin Series f ( x) = f (0) + f ¢(0) +
f ¢¢(0) 2 f ( n ) (0) n x + ... +
x + ... (G) 2!
n ! and in particular: Very useful to memorise (A): x 2 x 3 x n e x = 1 + x +
+
+ ... +
+ ... 2! 3!
n ! x 3 x 5 x 7 sin x = x +
+ ... 3! 5! 7! x 2 x 4 x 6 cos x = 1 +
+ ... 2! 4! 6! Newbattle Community High School On syllabus but less essential (R): x 3 x5 x 7 +
+ ... 3
5
7 x 2 x 3 x 4 ln(1 + x ) = x +
+ ... 2
3
4 tan -1 x = x -
D Watkins 2011 Advanced Higher Maths Unit 3.1: Vectors, Lines and Planes (G) a · b = a b cos q
Angle between two vectors: (Higher) Equations of a line: Parametric form x = a + tl y = b + tm ( x = a + t d ) z = c + tn
Symmetric/Cartesian form x - a y - b z - c =
=
= t l
m
n
Equations of a plane: æ l ö
Normal n is çç m ÷÷
çn÷
è ø Vector equation x • n = a • n or (x - a) • n = 0
Point on line = P (with position vector a) Symmetric/Cartesian lx + my + nz = k
where k = a • n Parametric
x = a + mb + lc (b and c are any two non­parallel vectors in plane) Angle between two planes = Angle between their normals Angle between line and plane = (Angle between n and d) – 90° Cross product: Scalar triple product: i
j
k a ´ b = a1
a2
a 3 b1
b2
b3 a1
a2
a 3 a g(b ´ c ) = b1
b2
b 3 c1
c2
c3 Newbattle Community High School D Watkins 2011 Advanced Higher Maths Unit 3.2: Matrices (G) 2×2 matrices 3×3 matrices Determinant and Inverse 1 æ d -b ö
det A = ad - bc and A -1 =
ç
÷
ad - bc è -c a ø æ a b ö
A = ç
÷
è c d ø æ a b c ö
e
ç
÷
A = ç d e f ÷ det A = a
h
çg h i ÷
è
ø f
i
-b
d
f
g
i
+ c d
e g
h
( AB ) -1 = B -1 A-1 Transformation Matrices æ 1 0 ö
Reflection in x­axis ç
÷ , è 0 -1ø æ a 0 ö
Enlargement by scale factor a ç
÷ , è 0 a ø æ -1 0 ö
Reflection in y­axis ç
÷
è 0 1 ø æ cosq - sin q ö
Rotation by θ degrees ç
÷
è sin q cosq ø Unit 3.4: Differential Equations (G) For ò P ( x ) dx dy + P ( x) y = Q ( x ) , the Integrating Factor I(x) is e
dx
and the solution is given by I ( x ) y = I ( x)Q ( x ) dx
ò Second Order Differential Equations Nature of roots Two distinct real m and n Form of general solution y = Ae mx + Be nx Real and equal m y = Ae mx + Bxe mx Complex conjugate m = p ± iq
y = e px ( A cos q + B sin qx ) Newbattle Community High School D Watkins 2011