Analysis and Approaches HL Practice Paper Book | Page 39

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Limits and Maclaurin Series

� L ’ Hôpital ' s Rule under the conditions of indeterminate forms ⁰

0 ^and ^�

� ^: f ( x) f �( x)

lim � lim x�a g( x) x�a g�( x)

2 3 x x ( 3 )

� f ( x) � f ( 0 ) � xf �( 0 ) � f ��( 0 ) � f ( 0 ) � : Maclaurin series 2 ! 3 !

� Common Maclaurin series :
1 .	2 3 e ^x �1� x� ^x � ^x � 2 ! 3 !
2 .	3 5 7 x x x sin x� x� � � � 3 ! 5 ! 7 !
3 .	2 4 6 x x x cos x �1� � � � ₂ _! ₄ _! ₆ _!
4 .	2 3 4 x x x ln ( 1 � x) � x� � � � 2 3 4	for �1� x � 1
5 .	3 5 7 x x x arctan x� x� � � � 3 5 7	for �1� x � 1
6 .	n ( n)( n �1 ) 2 ( n)( n �1 )( n � 2 ) 3 ( 1 � x) �1� nx � x � x � ( 2 )( 1 ) ( 3 )( 2 )( 1 )		for �1� x � 1

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Differential Equations

dy

� dx ^� f ( x, y)

: First order differential equation

� Solving ^d y � f ( x) g( y) by separating variables : dx dy 1

� f ( x) g( y) � d y f ( x) dx d x

� � g( y)

�

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