Your Practice Paper – Analysis and Approaches SL for IBDP Mathematics
� For a hemisphere of radius r : 1 . 2 .
2
3� r : Total surface area
2
2� r : Curved surface area
2 3
3 .
3 � r : Volume
13 Differentiation
� Derivatives of a function y � f ( x)
:
1 .
2 .
3 . dy � f�( x)
: First derivative dx
2 d y d �d �
2 � d d d
y� �� f��( x)
: Second derivative x x � x � n d y n dx
( n)
� : n -th derivative f
( x)
� Rules of differentiation : n n 1
� � � � f ( x) � p( x) � q( x) � f ( x) � p ( x) � q ( x) f ( x) x f ( x) nx �
� � �
|
f ( x)
� sin x
� f
�( x)
� cos x
|
f ( x)
�cp( x)
� f
�( x)
� cp�( x)
|
|
f ( x)
� cos x
� f
�( x)
� � sin x
|
f ( x)
� p( q( x))
� f
�( x)
� p�( q( x))
� q�( x)
|
|
1 f ( x)
� tan x
� f
�( x)
�
2 cos x
|
f ( x)
� p( x) q( x)
� f
�( x)
� p�( x) q( x)
� p( x) q�( x)
|
f ( x) � e x � f �( x)
� e 1 f ( x) � ln x � f �( x)
� x
x px ( ) f( x)
� qx ( ) p�( x) q( x) � p( x) q�( x)
� f�( x) �
2
( qx ( ))
� Relationships between graph properties and the derivatives :
1 . f�( x) � 0 for a �x � b : f( x ) is increasing in the interval 2 . f�( x) � 0 for a �x � b : f( x ) is decreasing in the interval 3 . f�( a) � 0: ( a, f ( a )) is a stationary point of f( x ) 4 . f�( a) � 0 and f� ( x) changes from positive to negative at x� a: ( a, f ( a )) is a maximum point of f( x )
5 . f�( a) � 0 and f� ( x) changes from negative to positive at x� a: ( a, f ( a )) is a minimum point of f( x )
6 . f��( a) � 0 and f�� ( x) changes sign at x� a: ( a, f ( a )) is a point of inflexion of f( x )
12
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