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7/22/2019 Core 3 Revision Sheet.doc
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Core 3 Revision Sheet
Functions
The domainis the inputof the function
The rangeis the outputof the function
Modulus functions
x means the magnitude of x and is always positive.
When sketching modulus functions, the graph will be reflected in the x axis.
When solving modulus equations, remember to find the solutions when it is positive and
negative.
Example
olve 3x ! "
x # $ ! " %& x # $ ! '"
x ! () x ! '*
Composite Functions
When finding composite functions, apply the functions one at a time, starting with the
function closest to x.
Example
f+x ! x- $ g+x ! x /
0ind fg+x fg+x ! f+x /
! +x /- $
! x- ()x 1/ $
! x- ()x 12
To find the range of composite functions, find the range of the first function being applied
then use that as the domain of the next function being applied. This output will give the
range of the overall composite function.
Inverse Functions
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Remember that an inverse function can only be found if the original function is one-to-one.
This can be achieved by restricting the domain.
Example
f+x ! 1
32
+xx
find f 1 +x
let y !1
32
+
x
xy+x # ( ! 1x $
yx # y ! 1x $
yx # 1x ! $ y
x+y # 1 ! $ y
x !2
3
+
yy
Therefore f 1 +x !2
3
+
x
x
Drawing graphs of inverse functions
&eflect the original graph in the line y ! x
3ou may be asked to draw inverse graphs of exponential, natural logarithms and
trigonometric graphs. 4ake sure you are familiar with them.
Differentiation
5hain &ule dx
du
du
dy
dx
dy
=
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6roduct &uledx
duv
dx
dvu
dx
uvd+=
)(
7uotient &ule
2v
dx
dvu
dx
dud
v
u
dy
d =
' this is in your formula book
x as a function of ydx
dy !
Integration
Integration b substitution
determine your value for u
0ind and rearrange it into the form dx ! du
ubstitute du in for dx and simplify where necessary
8f necessary, substitute any x values in terms of u
9ow integrate your new function and substitute x back in at the end if required
8f it is a definite integral, you must either rearrange the limits for u or put the answer
in terms of x before substituting in the limits of x.
Example
dxx
x +23
22
u ! $x- 1dx
du!:x dx !
x6
1du substitute this into the integral
and cancel
u
x2x
x6
1du
u3
1du ! 2
1
3
1
u du ! 32
u 21
c %& u3
2 c
in terms of x 3
223 2 +x c
Integration b parts
dxdx
duvuvdx
dx
dvu = this is in your formula book
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;etermine your value for u and your value fordx
dv
HINT the value for u should get simplerwhen differentiated
0inddxdu and v
ubstitute the values into the formula which should leave you with a simpler integral
at the end to work out. 8f not, integrate by parts again.
Example
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Exponentials and %ogarithms
4ake sure you recognise and could sketch the exponential and natural logarithm graphs,
including transformations of them.
;ifferentiating exponential functions= y ! e kxdx
dy! ke kx
8ntegrating exponential functions= ! c
;ifferentiating natural logarithms= y ! ln xdx
dy!x
1
8ntegrating natural logarithms= dx ! ln x c
pecial 0unctions dx ! ln +f+x c, if f+x > )
&rigonometr
Reciprocal Functions
cosec
sin
1= sec
cos
1= cot
tan
1=
3ou must be able to sketch the graphs of these functions
&rigonometric Identities
tan !
cossin cot !
sincos
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sin +2 cos 2 ! ( +? cos sec 2 ! ( tan 2
sin +2 cos 2 ! ( +? sin cosec 2 ! ( cot 2
3ou must learn all these identities.
Inverse &rigonometric Functions
3ou must be able to recognise and sketch the following inverse trigonometric graphs=
y ! sin 1 x for '(@ x @ ( y ! cos 1
x for '(@ x @ ( y ! tan 1 x for
'A @ x @ A
&emember, the domain and range are the opposite for the inverse functions.
The domain must be restricted in to make it a one'to'one function in order to be able to find
its inverse.
Differentiating &rigonometric Functions
xxdx
dcos)(sin = xx
dx
d 2sec)(tan =
xxdx
dsin)(cos =
Integrating &rigonometric Functions
ckxk
kxdx += sin1
cos cxk
kx += tan1
sec2
ckxk
kxdx += cos1
sin
'ou must learn all of the differentiation and integration trigonometric functions
(umerical methods and solutions
Change of sign method
The equation must be e)ual to *erobefore the change of sign method can be used.
Example
how that the equations y ! x- ( and y !x
1have a solution in the interval ).:2 and ).:B.
0irst make the equations equal to each other.
x- ( !x
1
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4ake it equal to Cero
x- ( 'x
1! )
&earrange if necessary
multiply by x
xD x # ( ! )
f+).:2 ! ).:2D ).:2 # ( ! ').))//:2 f+).:B ! ).:BD ).:B # ( ! ).)(2/)B
change of sign, therefore root in interval ).:2 E x E ).:B
Iterative Methods
This involves programming your calculating so that the process is repeated to find a
sequence of approximations to the required solution +converges to the solution.
3ou need to be able to decide if the convergence produces a cobweb or staircase diagram,
and be able to sketch it.
Mid!ordinate rule
++++=
b
a nnyyyyhydx
2
1
2
3
2
3
2
1 ..... where h !n
ab
Simpson+s rule
odds evens
[ ])...(2)...(43
2421310 +++++++++= nnnb
ayyyyyyyy
hydx
These formulae are both in your formula book.