� Relationship 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 )
� Tangents and normals : 1 . f� ( a)
: Slope of tangent at x� a
2 .
�1 f�( a)
: Slope of normal at x� a
3 . y � f ( a) � f �( a)( x � a) : Equation of tangent at x� a
4 . y f 1 ( a � � �
� ) � � ( ) f ( a) x � �
� a � �
: Equation of normal at x� a
16
Integration and Trapezoidal Rule
� Integrals of a function y � f ( x) :
1 . � f ( x) dx : Indefinite integral of f( x ) b
2 . � f ( x ) d x : Definite integral of f( x ) from a to b a
� Rules of integration : 1 .
� � �
1 n �1 n n�1 x dx � x �C
2 . ( p�( x) � q�( x)) d x � p( x) � q( x) �C
3 . cp�( x) d x �cp( x) �C b
� � f ( x ) d x : Area under the graph of f( x ) and above the x -axis, between x� a a and x � b www . seprodstore . com
11