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M May
straight line equationsgradientpoints of intersectionparallel lines and perpendicular linesvectors and directed line segmentsscalar product
Notes on Points
M May
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
M May
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
M May
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......
M May
trigonometric functions
radians
trigonometric graphs
solve trigonometric equations
compound angles
wave function
Notes on Trigonometry
M May
•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
M May
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
M May
€
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
M May
Differerentiation
Integration
polynomials
trigonometric functions
Area / Rate of change / Curve sketching
chain rule
Notes on Calculus
M May
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)[ ]
M May
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)
M May
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
M May
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
M May
Geometry /Symmetry
minimum / maximum
centre, radius
standard equations
points of intersection
tangents
Notes on Parabolae /
Circles
M May
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 )
M May
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
M May
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.
M May
Those bacteria!
Napiers shortcuts! / focus on indices
Notes onRecurrence Relations
Logarithms / Indices
M May
€
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 = ......
M May
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!