C1 Revision Guide AWK

8/8/2019 C1 Revision Guide AWK

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C1 REVISION GUIDE

1. SURDSAND INDICES

Natural/Counting numbers, N, make up the set of numbers {1, 2, 3, 4, 5, } Integernumbers, Z, make up the set of numbers { , -3, -2, -1, 0, 1, 2, 3, ) Real numbers, R, make up all numbers that correspond to a point on a number line.

Rational numbers, Q, are numbers that may be written asq

pwherep and q have no common factors.

Irrational numbers are those realnumbers that are not rational.

Surds are roots that cannot be written as a rationalnumber.Manipulating Surds:

Rationalising theDenominator:

Difference of two squares:

In the expression nx ,x is the base and n is the exponent/power/index.Definitions:

Rules:

Manipulation:

PAGE 1 OF 10

ba = ab (Simplify by taking out square factors:

baba =2 )

b

a=

b

a

aa

=2

a = a

b

c

b

a =

b

ac

cb

cb

cb

a

=

cb

cba

2)(

22))(( yxyxyx =+

bababa =+ ))((

10 =a )0( a

aa =1 nmnm aaa +=

nm

n

m

aa

a =

n

n

a

a1

= , or more generally,

nn

a

b

b

a

=

mnnm aa =)( nn aa =

1

( )m

n

m

nn

m

aaa =

=

1

or ( ) n mnm aa ==1


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C1 REVISION GUIDE

Different Bases, Same Powers:

2. ALGEBRAIC EXPRESSIONS

A term consists of the product of numbers and variables.

The number multiplying the variables is the co-efficient.

A polynomial is the collection of terms in the form 011

1 ... axaxaxan

n

n

n ++++

The degree of a polynomial is the highest power ofx. Degree = 1 Linear, 2 Quadratic, 3 Cubic, 4 Quartic.

Products and Factorisations:

A fraction is written as numeratorover a denominator.Adding and Subtracting:

Multiplication/Division :

Nevercancel part of a bracket, only the whole bracket.

Multiplying both the numerator and the denominator by the same number, does not change the value of thefraction.

3. EQUATIONS & QUADRATIC FUNCTIONS

An identity(shown by use of the identity sign ) is an equation that is true ofall values of the variable. An inequality may be true for a setof values of the unknown. When solving equations, always do the same thing to both sides in order to collect all the unknown terms on one

side.

A quadratic equation may always be transformed into the following form: 02 =++ cbxax )0( a . A quadratic equation always has two roots. These roots may both be real and distinct, they may be the same

(repeated root) or they may both be complex and distinct.

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mmm abba )(=

m

m

m

b

a

b

a

=

acabcba +=+ )(

bdbcadacdcba +++=++ ))((

22))(( bababa =+ (The difference of twosquares)

222 2)( bababa += 32233 33)( babbaaba +=

))(()( 223 babababa +=

Note: 22 ba + cannot be factorised. abxbaxbxax +++=++ )())(( 2

bcxbacaxcxbax +++=++ )())(( 2

bdxbcadacxdcxbax +++=++ )())(( 2

bd

bcad

d

c

b

a =

bd

ac

d

c

b

a=

b

a

c

d

b

a

d

c

b

a==


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C1 REVISION GUIDE

Solve:

Complete the Square:

Quadratic Formula:

Sketching Quadratics:

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Factorisation: ==+ xxx 0))(( or =x

...0)1)(1(0)1(0 23 =+== xxxxxxx

Substitution: 0352 31

3

2

=+ xx Let yx =31

0)3)(12(0352 2 =+=+ yyyy

2

1= y or 3=y

2

13

1

= x or 331

=x 81=x or

27=x Turn 02 =++ cbxax into 0)( 2 =+ qpxaThis produces solutions of p

a

qx =

If 1=a then

022

0

22

2 =

+=++ c

bbxcbxx

Find pa, and q by use of identities andcomparison of

coefficients: qpxaxx ++ 22 )(5124

qapapxax ++ 22 2

;4=a ;2

3212 == pap

145 2 == qqap

Equation must be in the form 02 =++ cbxaxThen

a

acbbx

2

42

=

Rearrange

Compare to find values of ba, and c

Substitute into the equation to find two solutions forx.

Be careful with the signs! E.g. ( ) 1644 22 === bb

For graphs that are of cbxaxy ++= 2

If 0>a the graph is -shapedIf 0 0,

042 > acb 042 = acb

042


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C1 REVISION GUIDE

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C1 REVISION GUIDE

4. INEQUALITIES

A linearinequality can be solved by rearranging so that all the terms of the unknown are gathered on one side andthe remainder of the terms lie on the other side.

The inequality sign should only be reversed if both sides have been multiplied or divided by a negative number.

Solving a quadraticinequality :

Interpreting solutions:

5. SIMULTANEOUS EQUATIONS

Simultaneous equations are equations that may be solved simultaneously by one set of solutions.

It is only possibly to solve sets of simultaneous equations if there are as many sets of independent equations asthere are unknown variables. To solve forx andy, we need two equations.

If there are fractions in the equations, multiply throughout by the denominators to end up with equations with onlyinteger coefficients. Similarly, remove any common factors.

Linear: Elimination Method:

Linear: Substitution Method:

If you must solve simultaneous equations where one is linear and one is quadratic, rearrange the linear so that thefirst variable is the subject, substitute this into the quadratic, and solve the resulting quadratic equation. You canthen substitute your2 solutions back into the linear equation to find the associated values.

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Rearrange so that there is a zero on one side of the

inequality and cbxax ++2 on the other.

Solve the 02 =++ cbxax . Sketch a graph to see which regions are required.

Assume a > 0

If solving 02 >++ cbxax then

Solution:>x or


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C1 REVISION GUIDE

6. CO-ORDINATE GEOMETRYAND STRAIGHT LINES

Thex-coordinate of a point is called the abscissa, whilst they-coordinate is called the ordinate.

A line segment is a stretch of line that runs fromA toB.

The mid-point of the lineAB, is the pointequidistant from bothA andB.

The gradient of a line is a measure of how steep the line is.

It can be calculated as the change in y divided by the change inx. Gradient =x

y

If it slopes from bottom left to top right (/), it is a +ve gradient (> 0).

If it slopes from bottom right to top left (\), it is ave gradient (< 0).

Values of gradients:

Forms of Equations:

Equations of a Straight Line:

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If the line is fromA ),( 11 yx toB ),( 22 yx

then

the distanceAB, the distance fromA toB, isgiven by

( ) ( ) 2122

12 yyxxAB +=(Pythagoras)

If the line If the line is fromA ),( 11 yx toB ),( 22 yx then

themid-point ofAB, is given by

mid-point

++=

2,

2

2121 yyxx

A line is such that it includes the two points A ),( 11 yx andB ),( 22 yx .

The gradient,12

12

xx

yym

=

A horizontal line has gradient 0.

A vertical line has gradient (infinity)

The gradient of a line is the tangent of the angle the line makes with the +vedirection of thex-axis.

If two lines have gradients 1m and 2m then

121 =mm the two lines are perpendicular 0( 1 m ,)02 m

21 mm = the two lines are parallel From the equation cmxy += , we may read off the gradient (m) andthe

y-intercept (c)

The general form of a straight line is

0=++ cbyax

where it crosses thex-axis at

0,a

c

and they-axis at

b

c,0

If you know the gradient (m) and one point ),( 11 yx then

theequation of the straight line is

)()( 11 xxmyy =

If you know two points ),( 11 yx and ),( 22 yx then the

equationof the straight line is

12

1

12

1

xxxx

yyyy =


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C1 REVISION GUIDE

7. GRAPHSOF FUNCTIONS

Thex-axis is y = 0 and they-axis is x= 0.

Lines parallel to thex-axis have the equations y = k. Lines parallel to they-axis have the equations x= k.

Standard Graphs:

Other graphs:

Transformation of Graphs:

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2

xy = 3

xy = xy1

=

( )0, >= xxy 3 xy = 2

1

xy =

axfxf + )()( )()( axfxf )()( xafxf

Translation of

a

0Translation of

0

aStretch iny, SF a.

)()( axfxf )()( xfxf

)()( xfxf

Stretch inx, SFa

1Reflection inx = 0 Reflection iny = 0


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C1 REVISION GUIDE

The shape of the2

xy = graph is a parabola. The shape of ax

y1

= an hyperbola.

If one has two equations and one solves them simultaneously, then one is finding the point(s) of intersection of

the two graphs. An asymptote is a vertical/horizontal line that a graph approaches but nevertouches.

8. SEQUENCESAND SERIES: ARITHMETIC SERIES

A sequence is a set of terms, that is in a distinctive order and where each term is obtained by a rule.

Recurrence Relations:

nth Rule:

Notation:

Summation Results:

PAGE 8 OF 10

Define the first term of the sequence. )( 1 au =

Describe the relation between successive terms. ))(( 1 nn ufu =+ Successive answers may various things: Converge towards a specific value, known as the limiting value.

Diverge away from any value, rapidly reaching . Oscillate either side of the limiting value, gradually getting closer.

Oscillate either side of the limiting value, without changing status. Defines every term as a function of its position.

)(nfun = If first level differences are the same, the function )(nf can be foundby:

andnf += )1()(

where =n position of term in sequence =a the first term in the sequence =d the common difference

If the first level differences are different, but the second level differences are the same,then the function )(nf can be found by: f(n) = a + (n 1)d+ (n 1)(n 2)c

where =n position of term in sequence =a the first term of the sequence =d the difference between the first two terms of the sequence =c the second level common difference

=

=+ ++++=

br

ar

bbaar uuuuu 11 ...

where =a first position =b last position

=r

position counter

It is often simply written as b

a

ru

=k

h

k

h

rr

k

h

rr baba )(

=++

k

h

r

k

j

r

j

h

r aaa1

nccn

=1

where c is a constant

=n

m

n

m

rr akka where kis a

constant


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C1 REVISION GUIDE

Successive terms in an arithmetic series (orarithmetic progression) always have the same difference.

Arithmetic Series:

9. DIFFERENTIATION

Differentiation is about finding the rate that one value changes with respect to another. It is a natural extensionof gradients in Co-Ordinate Geometry, which found how they value changed with respect tox on a straight line.

The gradient on a curve changes as x changes, hence the gradient of a curve is equation.

Notation:

Standard Results:

PAGE 9 OF 10

Ifn =the number of terms in the series a = the first term of the series d =the common difference

then nth term = a + (n 1)d

If nS is defined as the sum of the first n terms of a series, then let us

also define the last term (i.e. the nth term) as l= a + (n 1)d

ldldldadaaSn ++++++++= )()2(...)2()(

or adadadldllSn ++++++++= )()2(...)2()(If the above two equations are added

)()()(...)()()(2 lalalalalalaSn ++++++++++++=

)(2 lanSn +=

)))1(((2

dnaan

Sn ++=

))1(2(2

dnan

Sn +=

If this is the first n natural numbers, then a = d = 1 and

2

)1( +=

nnSn or

2

)1(

1

+= nnr

n

If )(xfy = then the first derivative (the gradient) may be written:

dx

dyor )(' xf or

dx

xfd ))((or 'y

The second derivative (found by differentiating the first derivative) may be written: 2

2

dx

ydor )('' xf or dx

xfd ))('(or ''y or 2

2))((

dx

xfdor

dx

dx

dyd

If )(xfy = then

h

xfhxf

dx

dy

h

)()(lim

0

+=

If naxy = then

1= nanxdx

dy

Hence

cy = 0=dx

dy

kxy = kdx

dy=


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C1 REVISION GUIDE

Always rewrite all expressions as the sum of terms in powers ofx before differentiating.

Rememberthat 21

xx = and your otherRules of Indices.

Recognition:

10. INTEGRATION

This it the reverse function to differentiation.

Standard Result:

Notation and further results:

11. PROOF

A proofshould be a set of mathematical statements, each one leading logically from the previous statements,that lead as a whole, from the starting position to the destination.

When asked to show that or prove thatA is related to B in a certain way, the initial statement(s) should refer towhat is already known and the final statement should be that A is related to B in a certain way.

Signs:

PAGE 10 OF 10

Differentiation is required whenever you come across a curve and therequest is to:

find the gradient at

differentiate

find the rate of change

find the tangent to the curve

find the normal to the curve

find the first derivative

nax

dx

dy= c

n

axy

n

++

=+

1

1

)1( n

where c is the arbitrary constant of integration

++=+

cn

axdxax

nn

1

1

)1( n

= gdxxfdxxgxf ()()}()({

= dxxfkdxxkf )()(

implies is implied by

both implies and is impliedby

is a member of the set

therefore because for all values of

C1 Revision Guide AWK

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