Calculus summary

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A function , (f) ,
can have a limit at
'
a' even if flat does not exist

A function (f) , ,
can only have a limit at '
a' if (f) app roches the same value from the left
and the
right
IF a function , (f) ,
is continuous at ✗ =
a
,
the limit at '
a' is flat
Notation is VERY important


Examples ( different cases ) Gradient of a
straight line
04 Yi
, Yz -




m =
D)(
Xz -
Xi
linear
+
Iim (2×-5) ve ;
increasing function
K > 3
decreasing function
-




s substitute given x -

value ve ;

=
213 ) 5 A line has
Str constant gradient
-


. a

=
1


Average gradient of a curve


Fraction determine the
average gradient between two points
✗ 2- 4
1.) Factorise
/ im ✗ -
2
2.) cancel out

Yz -
Y,
✗ →z
M=

#( 2) xz -
x
C x + ,



tim
⇐ -

2)
✗ → 2

Average rate of change
Iim ( ✗ + 2) •
Becomes linear gives the average velocity
✗ → 2
find the
average gradient
05
4
given
=
i
t


sctz ) -
Slt , )
D; - A =
tz -
t
2×-5
/ im K 1- 1
K sa

tangent to a curve
✗ ( 2
-



§z ) '

lim take out x as a common
average gradient of a curve
given by two points ;
✗ →a
x (I t
¥) factor in mum . & denom . m
=
f- ( Kth ) -

f- ( X )
'
h
falls away
2

the will be found making h small
gradient very
=
1


=
2 :
h so
FCK th ) -


f- ( X )
MT = / im
h
h → 0
limit does not exist
✗ +3
Iim the Derivative of a function
✗ →3 X -
3
Notation ; FCK ) f- (K)
'
s


dy / dx
=
8- s denom . =/ O y
s




: does not exist DX ( FCK ))
f- ( Kth )
-
f (X )
f- lim first principle
'

C)C) =
h
h → 0
method .