Summary Calculus (derivatives) summarized

CALCULUS first Principles



average gradient :
CUBIC FUNCTION GRAPHS
f- ( x) = -2×2
f- ( x -1h) =
-2 ( x -1h)2




!
•
m = Yz Yi
Xz -
Xi
floc -1h) floc)
f. ( x) / im
-

=



h→o h
subinx values to get values
y
• - -




y =
acx -
Xi )( x xz)(x xD
-
-




f- Loc)
'
=
Iim -
2( oath)2 _ (-2×2)
Me
h -0
derivative : h
d DY a




q•c2
NOTATION : f-
'

Ix) or
dx doc or DX -2 ( xthlx.tn) +2×2
or
f- ( x) / im
'
=

≥

y=a( x -
xD ( x xD - h→o h


No No 's -
: variable in denominator; brackets ; x
x
f
'

( x) = Iim -2 ( x2 + och + och + h2 ) +2×2
h→0
h



( x2 + zxhth ) -12×2
≥
f-
'
( x) =
/ im -2




q)^
h -70 h
points of inflection concave acceleration
y=alx+pP+q
derive twice f- ( x)
'
=
/ im -2×2 Hoch 2h2 -12×2

f"@@
• - -




h -70 h
'

solve for 0C

*



F'(x) =/ im
h→o H

f ( oc) -4×-210)
'
'
=


tangents
* NB gradient
: = derivative
i. floc) = -43C



curve sketching : equations of graphs :
1. find 4- intercept

sub co cords into equations
•
-



2. find oc
-

intercept

Use the derivative to make an
equation
'




3. find derivative . make it -0
-
{ solve foroc

solve simultaneously
'



4. sub x values into OG equation and solve for y



( these are the turning points) y
: ACK -

xD ( x -
✗ 2) ( x -
✗ 3)


?⃝