Analysis and Approaches for IBDP Maths Ebook 1 | Página 153

50
Example
Paper 1 Section A – Gradients of
tangents and normals
Front Page
13
x
Let f ( x) � e sin x . Find the gradient of the normal to the curve of f at
Solution
�
x � .
2
[6]
f�
( x)
( x
x
� e )(sin x) � ( e )(cos x)
(M1) for product rule
�e x (sin x � cos x)
A2
�
f �
� �
� �
� 2 �
�
2
� � � �
�e
�sin �cos
�
(M1) for substitution
� 2 2�
�
e 2
�
The gradient of the normal
�1
�
(M1) for negative reciprocal
��
�
f � � �
� 2 �
1
��
�
A1 N3
2
e
[6]
Exercise 50
3�
1. Let f ( x) � 3xcos
x . Find the gradient of the normal to the curve of f at x � .
2
[6]
�x
2. Let f ( x) � e cos x . Find the gradient of the normal to the curve of f at x � 2�
.
[6]
2
3. Let f ( x) � cos( x ) . Find the value of a if the gradient of the tangent to the curve of f at
x� a is � 2a , where 1�a
� 2.
[6]
2
4. Let g( x) � x ln x . Find the value of a if the gradient of the tangent to the curve of g at
x� a is 3a .
[6]
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