Summary - Mathematics (Paper 1 & 2)

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Determine the derivative




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by first principles /
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f- ( x)
'
= derivative To find the gradient at a point
ie : instantaneous gradient
function

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¥ derivative
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* find x AND sub x into f- (x) to → y
OF TANGENT PO l
GIVEN x - A f " (x) = 0 find x


gRAD1fNT→DERlVATJ
o
,


x find derivative
°
sub x into floc) to find y
f (x)
'
→




X SU b MT x y Into a sub x into f- (x)
xtc •
find y Moc t C
-

im
y
-




[ find C ]




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NB @ the
turning TO FIND EQUATION GIVEN 3C -
INT



in
f (x)
'
= 0




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. -




OWEN Tp ( m X n)

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POLYNOMIALS
NB DIFFERENCE OF CUB ft




x = -
l X =3 x = -
1



LIMITS




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REMAINDER THEOREM
factor IIe




÷
I remainder




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x # =
x
DETERMINE GRADIENT / RATE OF CHANGE

× fc # f =
remainder
X remainder =
f- ( x ) with subbed in # -



h

find




m=imo""i'-
to A
# THE RATE of CHANGE AT x / DERIVATIVE




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OF f- (x) / GRADIENT OF TANGENT
FACTOR THEOREM


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.




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OPTIMISATION
a C ) b ( ) C C ) of factor
RATE OF CHANGE
Y
AVE xz -
x ,




x = 8 X =


If at if a -




MODE TABLE ENTER f- ( x )


I W


i'"%"
AVERAGE GRADIENT U



mi :
"

"




aq"""P
IF THERE IS ONLY ONE TURNING POINT
( xiy )

y
JC z
-
X I

, 3
f- (x) =
ax tbx
2
tcxtd → I
f- C )
'
=3 ax
2
t 2b x t C → A
f
'
( x) 70
bax t 2b / f- ' ( x ) co

t.ch
"
f- ( x) = →
M
NEE , ,




i¥""/
f- Cx)
'
-
I




/
Mt
, m -




f- (a) 9 I feel d l
l " Cx)
l f-
I y
I m t

I r f Cx) M




"




\
O
f- " (x)
-
-




int
x
I
-




I

'
f-
"
Cock O
face, - F
,
l
TP derivative f
''
(x) > O
l T
i ,
TP f (x)
NB f ( x)
'
→ PARABOLA
f
'
( x) → GRADIENT OF f- (x)



GRAPH OF DERIVATIVE SECOND DERIVATIVE
f- (x) → Str line
"



•
f ( x)
'
> O (above x -
AXIS )
f- Cx)
"
.
f.(x) increases → 70
original graph increases f- Cx) Mt
above x -
axis

f ( x) so ( below x axis)
'
of (x) (x) co
' "
decreases → f
-
•



original graph decreases f-(x) m below x axis
- -




of ( x) (x) int
''
'
TP → f O → x
f (x) = O [ x Ints ]
' - -
• -




TPS of f- (x) are m - O
f '(x) → f (SC) = O
"
• TP of

POINT OF INFLECTION OF f-(x)

, FUNCTIONS



lfcxt-acx.p/
l#J.:i . : n:n....÷.
i i ai :oi
PARABOLA
'

or




:

! !!
TPC Fa if C- Fa ) ) x
-
-


,
y -




a -




"


Domain ' er

. .

RANGE
ma,nt V. YE Caio )
, p




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"" "
.




' I (x p ) tq
hors Ita Als y = -



p →




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:
p)
2
y
= a (x -
ta

2x Ints and Op




✓
-




y
= a ( x -
sci ) (x -
xz )


ii. iinei.mn : ai
'




÷÷:÷÷÷÷
x -
P
a. b tq
y
=



at
TOP HEAVY ASYMPTOTE y q
-
-




b 71

STRAIGHT LINE




.
.




#
a -




o c b C l
M = 92-42
Xz - X l




( → VERTICAL SHIFT ( E)
( Oic) y -
int