Cambridge Additional Mathematics

(singke) #1
346 Introduction to differential calculus (Chapter 13)

Example 5 Self Tutor


If y=3x^2 ¡ 4 x, find
dy
dx

and interpret its meaning.

As y=3x^2 ¡ 4 x,
dy
dx
=6x¡ 4.

dy
dx
is: ² the gradient function or derivative of y=3x^2 ¡ 4 x from which the gradient of
the tangent at any point on the curve can be found
² the instantaneous rate of change ofywith respect tox.

Example 6 Self Tutor


Find f^0 (x) for f(x) equal to:

a 5 x^3 +6x^2 ¡ 3 x+2 b 7 x¡^4
x

+^3
x^3

a f(x)=5x^3 +6x^2 ¡ 3 x+2
)f^0 (x) = 5(3x^2 ) + 6(2x)¡3(1)
=15x^2 +12x¡ 3

b f(x)=7x¡
4
x

+
3
x^3
=7x¡ 4 x¡^1 +3x¡^3
)f^0 (x) = 7(1)¡4(¡ 1 x¡^2 )+3(¡ 3 x¡^4 )
=7+4x¡^2 ¡ 9 x¡^4

=7+
4
x^2
¡
9
x^4

Example 7 Self Tutor


Find the gradient function of y=x^2 ¡
4
x
and hence find the gradient of the tangent to the function
at the point where x=2.

y=x^2 ¡
4
x
=x^2 ¡ 4 x¡^1

)
dy
dx

=2x¡4(¡ 1 x¡^2 )

=2x+4x¡^2

=2x+
4
x^2

When x=2,
dy
dx
=4+1=5.

So, the tangent has gradient 5.

Remember that
.
_______________
_______________________
1
^^xn=x-n

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_13\346CamAdd_13.cdr Tuesday, 7 January 2014 2:37:24 PM BRIAN

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