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Higher Revision Notes

Mathematics

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straight line equationsgradientpoints of intersectionparallel lines and perpendicular linesvectors and directed line segmentsscalar product

Notes on Points

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gradient is vertical /horizontal

€

y = mx + c

€

y − b = m(x − a)€

m =y2 − y1

x2 − x1

Straight line equations

€

(0,c)

€

(a,b)

perpendicular lines

€

m1 × m2 = −1 parallel lines

€

m1 = m2

point of intersection -solve equations simultaneously

Distance Formula

€

d = (x2 − x1)2 + (y2 − y1)

2

midpoint

€

x1 + x2

2,y1 + y2

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

tanθ = m

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Three Dimensions:

Distance Formula

€

x, y,z( )P

€

d = (x2 − x1)2 + (y2 − y1)

2 + (z2 − z1)2

€

x

y

z

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

p

€

=

€

=xi + y j + zk

position vector of P

component form

Scalar product (dot product)

€

u .v = x1x2 + y1y2 + z1z2

€

u .v = u v cosθ

€

u = x 2 + y 2 + z2

magnitude of u

€

u .v = 0 ⇔ θ = 90˚perpendicular

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mediansaltitudes

bisectors

join

ver

tex

to m

idpo

int o

f opp

osite

side

drop perpendicular from vertex

cut in halfanglessides

€

m + n

•m n:

A

CB

€

(b − a) =m

m + n(c − a)

B divides AC in the ratio......

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trigonometric functions

radians

trigonometric graphs

solve trigonometric equations

compound angles

wave function

Notes on Trigonometry

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•360˚€

y = sin x

r

r

r

sin

11 radian

cos

tan

€

=y

r

€

=x

r

€

=y

x

sin2 + cos2 = 1

€

y = cos x

•360˚

•360˚180˚€

y = tan x

tan = sincos

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sin = nCAS

T = sin-1(n)

two values in 1 complete turn

sin(A+B) = sinAcosB + cosA sinB

sin(A-B) = sinA cosB - cosA sinB

cos(A+B) = cosA cosB - sinA sinB

cos(A-B) = cosA cosB + sinA sinB

sin(2A) = 2sinA cosA

cos(2A) = cos2A - sin2Acos(2A) = 2cos2A - 1cos(2A) = 1 - 2sin2A

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€

acos x + bsin x in form

€

k cos(x −α )

€

k cos(x −α )

€

k cosα =

k sinα =

€

k sinα

k cosα= tanα =

CAS

T

€

a

b

€

k = a2 + b2€

=k cos x cosα + k sin x sinα

also in form

€

k sin(x + α )

€

k sin(x −α )

€

k cos(x + α )Reminder:Maximum and Minimum values

ofsinx or cosx are 1 and -121

€

3

11

€

2SohCahToa for exact values

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Differerentiation

Integration

polynomials

trigonometric functions

Area / Rate of change / Curve sketching

chain rule

Notes on Calculus

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rate of change

gradient

gradient of tangent

stationary points: maximum, minimum, inflexion

sketch the curve

displacement / velocity / acceleration

Area under / between curves

‘Undoing’ differentiation

€

dy

dx= f (x)

€

y = f (x)dx∫

€

y = F(x) + C

€

dy

dx

€

f '(x)

€

A = f (x)dxa

b

∫A = F(x)[ ]a

b

A = F(b)[ ] − F(a)[ ]

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Basic functions

€

y = x n

dy

dx= nx n−1

€

y = sin x

dy

dx= cos x

x in radians

€

y = cos x

dy

dx= −sin x

€

y = (ax + b)n

dy

dx= a × n(ax + b)n−1

€

π =180˚

€

y = sin(ax 2 + b)

dy

dx= 2ax cos(ax 2 + b)

€

y = f (g(x))

dy

dx=

dy

du×

du

dx

€

y = (ax 2 + bx)n

dy

dx= n(ax 2 + bx)n−1(2ax + b)

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Always check your integration by differentiating!x in radians

€

sin x∫ dx

= −cos x + C

€

cos x∫ dx

= sin x + C

Reminder:21

€

3

11

€

2

€

tan(π 3) =√ 3

1

€

sin(π 3) =√ 3

2

€

cos(π 3) =1

2€

sin(π 6) =1

2

€

cos(π 6) =√ 3

2

€

tan(π 6) =1

√ 3

€

sin(π 4) =1

√ 2

€

cos(π 4) =1

√ 2

€

tan(π 4) =1

€

sin(ax + b)∫ dx

=−cos(ax + b)

a+ C

€

cos(ax + b)∫ dx

=sin(ax + b)

a+ C

€

(ax + b)∫ndx

=(ax + b)n +1

a(n +1)+ c

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at turning points

€

y = f (x)

dy

dx= f '(x)

€

dy

dx= 0

€

⇒ x = ......, .......solve equation to give

x < ? < ? <

€

dy

dx ±? 0 ±? 0 ±?

/?\ _ /?\ _ /?\

X

Y€

x = ..., ....⇒ y = ..., ....

€

(.., ..) , (.., ..)giving turning points

maximum?/minimum?€

dy

dx

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Geometry /Symmetry

minimum / maximum

centre, radius

standard equations

points of intersection

tangents

Notes on Parabolae /

Circles

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Parabola

polynomial of degree 2Circles

€

y = x 2

€

y = ax 2 + bx + c€

x 2 + y 2 + 2gx + 2 fy + c = 0

€

x 2 + y 2 = r2Centre O(0,0) radius r

€

y = (x − a)2 + bminimum at (a, b)

€

y = k(x − a)(x − b)

cuts the X-axis at (a,0) and (b,0)

€

x 2 + y 2 + 2gx + 2 fy + c = 0

Centre

€

r = g2 + f 2 − cradius

€

y = −(x − a)2 + b

maximum at (a, b)€

(−g, − f )

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Sketching graphs

Given f(x).....

- f(x)

k f(x)

f(x) + b

f(x - a)

f(-x)

f(x + a)

€

k sin(bx − a)

k stretches

b periods in 360˚ or 2π

-a horizontal shift

+a horizontal shift <- ↑ move up

← move left

→ move right

↕ stretch

reflection in X-axis

reflection in Y-axis

amplitude k

period

€

360

bor

2π

b

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Points of intersection:

Solve simultaneous equations (by substitution).

It is a Tangent if two solutions are equal.

Reminder: find discriminant for a quadratic equation.

if zero, then equal roots => tangent

if less than 0, then no roots => no points of intersection

A tangent to a circle meets the radius at 90˚ (perpendicular).

and remember right angles in semicircle.

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Those bacteria!

Napiers shortcuts! / focus on indices

Notes onRecurrence Relations

Logarithms / Indices

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€

un +1 = aun + b

Find how ‘long’ til .....

After 1after 2after 3.....

Limit exists if

€

−1≤ a ≤1

€

L = aL + bLimit

€

L €

u1 = au0 + b

u2 = au1 + b

u3 =State that:

€

⇒ L =b

1− a

Make sure you make most efficient use of your

calculator.

€

un = aun−1

€

un = anu0

€

un = un−1 + b

€

un = bn + c

€

u0 = ...... or u1 = ......

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Logarithms = Indices

€

y = ax

€

x = loga y

€

loge y = ln y

€

log10 y

can use calculator for base e and base 10

non-calculator for other bases

€

log10100 = 2

€

log2 32 = 5

25 = 32

€

loga a =1

€

loga 1 = 0

€

n loga y = loga y n

€

loga xy = loga x + loga y

€

p loga x + q loga y − r loga z

= loga x p + loga y q − loga z r

= loga

x p y q

z r €

loga ( xy) = loga x − loga y

M May

Examination Techniques

Do read each question carefully.

Re-read each question once you have finished to make sure you have answered all parts

appropriately.Make sure you leave enough time to attempt all

questions.Show all working steps.

(particularly the substitution of numbers into formulae)

Having prepared thoroughly, get a good night’s sleep before your exam!