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Paper 1 – Gradients of Tangents and Normals
Example x
Let f ( x) � e sin x .
( a ) Find f� ( x) .
( b ) ( i ) Find the gradient of the tangent to the curve of f at
� x � .
2
[ 2 ]
( ii ) Hence , find the gradient of the normal to the curve of f at
� x � .
2 [ 4 ]
Solution
( a ) |
x x f �( x) �( e )( sin x) � ( e
)( cos x)
|
( M1 ) for product rule |
|
f �( x) �e x ( sin x � cos x) |
A1 |
( b ) |
( i ) |
The gradient of the tangent |
|
|
��
� � f ��
� � 2
�
|
|
�
2 � � �
�
�e �sin
�cos
� � 2
2�
|
( M1 ) for substitution |
|
�
4.810477381
|
|
�
4.81
|
A1 |
[ 2 ]
10
( ii ) The gradient of the normal �1
� � f �
� � � �
� 2 � �1
� ( M1 ) for valid approach 4.810477381 �� 0.2078795764 �� 0.208
A1
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