P1 Revision Summary - WordPress.com · 2013-04-01 · FORMULAE Length of arc Area of sector...
Transcript of P1 Revision Summary - WordPress.com · 2013-04-01 · FORMULAE Length of arc Area of sector...
QUADRATICSy = ax2 + bx+ c
Solving Quadratic Equation
Factorisation QuadraticFormula
Completing the Square
Completing the Square
y = x2 − 6x+ 3
= ( )2 +3x− 3 −32
= (x− 3)2 − 9 + 3
= (x− 3)2 − 6
2
y = 2x2 − 3x+ 5
= 2
�x2 − 3
2x+
5
2
�
2
= 2
�( )2 +
5
2
�x− 3
4 −�3
4
�2
= 2
�(x− 3
4)2 − 9
16+
5
2
�
= 2
�(x− 3
4)2 − 31
16
�
= 2
�x− 3
4
�2
− 31
8
Completing the Square
y = −x2 − 5x+ 2
= −(x2 + 5x− 2)
x+5
2−�5
2
�2
= −�( )2 − 2
�2
= −��
x+5
2
�2
− 25
4− 2
�
= −��
x+5
2
�2
− 33
4
�
= −�x+
5
2
�2
+33
4
y = −2x2 + 4x− 1
= −2
�x2 + 2x− 1
2
�
= −2
�( )2 − 1
2
�2
x+ 1 −12
= −2
�(x+ 1)2 − 1− 1
2
�
= −2
�(x+ 1)2 − 3
2
�
= −2(x+ 1)2 + 3
y = x2 − 6x+ 3
= (x− 3)2 − 6
y = 2x2 − 3x+ 5
= 2
�x− 3
4
�2
− 31
8
What you get from completing square
• U shaped because a is positive
• stationary point : min point
• line of symmetry
• domain :
• range :
• U shaped because a is positive
• stationary point : min point
• line of symmetry
• domain :
• range :
(3,−6)�3
4,−31
8
�
x = 3 x =3
4
ROOTS
b2 − 4ac = 0 b2 − 4ac > 0 b2 − 4ac < 0
• equal roots
• line is tangent to the curve
• two distinct roots
• line intersects the curve at two distinct points
• no roots
• does not cut the x-axis
• line does not intersect the curve
SIMULTANEOUS EQUATION
ROOTS
• two real roots
• line intersects the curve at two points
SIMULTANEOUS EQUATION
b2 − 4ac ≤ 0
REDUCIBLE EQUATIONS
FUNCTIONS
COMPOSITE FUNCTIONf(x) = 3x− 2
g(x) = x2 + 1
fg(x) = f(g(x))
= f(x2 + 1)
= 3(x2 + 1)− 2
= 3x2 + 3− 2
= 3x2 + 1
gf(x) = g(f(x))
= g(3x− 2)
= (3x− 2)2 + 1
= 9x2 − 12x+ 4 + 1
= 9x2 − 12x+ 5
ONE TO ONE FUNCTION
Straight LineQuadratic Equation
• Inverse Function exist
1. Complete the square
2. The x-coordinate of the stationary point makes it into a one to one function
3. Inverse Function then exist. You must choose the correct sign. + or - according to the inequality above.
x < A x > Aor
RANGE AND DOMAIN
f(x) f−1(x)
domain range
range domain
x > 3 y > 3
y < 5 x < 5
SKETCH INVERSE FUNCTION
Straight LineQuadratic Equation
• one to one function.
• Calculate x when y = 0
• Calculate y when x =0
1. Complete the square
2. Sketch the graph (one to one function) i.e. either
x < A x > Aor
Show the relationship between and by clearly showing the dotted line
f(x) f−1(x)
y = x
must be a one to one function with the condition given
left side...... or .....right side
COORDINATE GEOMETRY
FORMULAE
Length of line
Midpoint of line
Gradient of line
�(x2 − x1)2 + (y2 − y1)2
�x1 + x2
2,y1 + y2
2
�
m =y2 − y1x2 − x1
PERPENDICULAR
Line AB is perpendicular to the line CD. Find the equation of the perpendicular line.
A
B
C
D
(ii) Find the gradient of ABm
− 1
m(i) the point on the line
A or B− 1
m
y − y1 = − 1
m(x− x1)
PERPENDICULAR BISECTOR
Line AB is perpendicular bisector to the line CD. Find the equation of the perpendicular bisector.
A
B
C
D (ii) Find the gradient of ABm
− 1
m(i) the midpoint of the line CD
− 1
m
y − y1 = − 1
m(x− x1)
M
CIRCULAR MEASURE
FORMULAE
Length of arc
Area of sector
CALCULATOR SET IN RADIANS!
s = rθ
A =1
2r2θ
=1
2rs
o
r
r
s
θ
Chord
OTHER FORMULAE
Cosine Rule
Sine Rule
a b
c
AB
C
c2 = a2 + b2 − 2ab cosC
sinA
a=
sinB
b=
sinC
c
Area of non-right angle triangle
A =1
2ab sinC
SPECIAL ANGLES
π
4
π
4
1
1
√2
π
3
π
6
1
√3
2
when the answer required to be in exact form
TRIGONOMETRY
SIMPLE IDENTITIES
sin2 θ + cos2 θ = 1
tan θ =sin θ
cos θ
use the following identities for proving
GRAPH
• make sure you know how to sketch the original graphs of sin, cos and tan from or
• understand the change in the graph when a, b, c varies
0◦ ≤ θ ≤ 360◦ 0 ≤ θ ≤ 2π
a trig(bx) + c
a amplitude
b cyclec positive shift up
negative shift down
PROPERTIES of TRIG GRAPH
• Sin graph
• Cos graph
−1 ≤ sinx ≤ 1
−1 ≤ cosx ≤ 1
• Tan graph - Make sure you remember where the asymptotes are
SOLVE TRIGONOMETRY EQUATION
1. Put the equation into the form
2. Fix the limit (if necessary)
3. Find basic angle
4. Use ASTC
5. Solve the values of x
sin(ax+ b) = kcos(ax+ b) = ktan(ax+ b) = k
sinx = k
cosx = k
tanx = k
α
SERIESARITHMETIC AND GEOMETRIC PROGRESSION
SERIES
Arithmetic Progression
(AP)
Geometric Progression
(GP)
• nth term
• sum of the first n terms
• nth term
• sum of the first n terms
• sum to infinitySn =
1
2n[2a+ (n− 1)d]
Sn =1
2n[a+ l]
Sn = a(1− rn)
1− r
S∞ =a
1− r
an = a+ (n− 1)d an = arn−1
METHOD
• List down all the information given to you in ‘series’ form
SERIESBINOMIAL EXPANSION
FORMULA
(a+ b)n = an + nC1 an−1b1 + nC2 an−2b2 + nC3 an−3b3 + ....... + nCn−1 abn−1 + bn
OR you can use what you did in P3.
EXPAND UNTIL WHEN?
• Expand..............including and up to the term
• Expand.............. up to the term
• Expand..............where x is sufficiently small for
• Expand.............. the first three non-zero terms
x3
x4
x3
1st term + 2nd term + 3rd term
.....+ ..... x+ ...... x2 + ...... x3
.....+ ..... x+ ...... x2 + ...... x3
.....+ ..... x+ ...... x2
VECTORS
FORMULA
Magnitude of vector
Unit vector
Scalar Product
u = x+ iy
|u| = |x+ iy| =�
x2 + y2
u
|u|
321
−120
= −3 + 4 + 0
= 1
UNDERSTAND THE DIAGRAM
Finding the vectors
O P
QR
S T
UV
−−→V Q =
−−→V R +
−−→RQ
=−→V S +
−→SO +
−−→OR
+−−→RO +
−−→OP +
−−→PQ
DIFFERENTIATION
RULES
EQUATION OF TANGENT
• find the gradient of the tangent
• the point on the line
y − y1 = mT (x− x1)
dy
dx
EQUATION OF NORMAL
• find the gradient of the tangent
• gradient of normal
• the point on the line
dy
dx
y − y1 = − 1
mT(x− x1)
− 1
mT
STATIONARY POINTS
• Calculate the
• Substitute the value of the x-coordinate into and answer > 0
Minimum Point Maximum Point
dy
dx= 0
• Find the x-coordinate and the y-coordinate
d2y
dx2
d2y
dx2
• Calculate the
• Substitute the value of the x-coordinate into and answer < 0
d2y
dx2
d2y
dx2
APPLICATION
• Find the differential equation based on what given in the question
• Solve the differentiation
• Answer the question
INTEGRATION
RULES
AREA BETWEEN THE CURVE AND AXIS
x-axis y-axis
� b
af(x) dy
� b
af(y) dx
AREA BETWEEN TWO CURVES
x-axis y-axis� b
af(x) dy −
� b
ag(x) dy
� b
af(y) dx −
� b
ag(y) dx
top bottom top bottom
VOLUME BETWEEN THE CURVE AND AXIS
x-axis y-axis
V = π
� b
ay2 dx V = π
� b
ax2 dy
VOLUME BETWEEN TWO CURVES
x-axis y-axis
top bottom top bottom
V =
� b
ay2 dx −
� b
ay2 dx V =
� b
ax2 dy −
� b
ax2 dy