Cambridge Additional Mathematics

1

y

x

y= xln¡

O

&__Q.,-1*

372 Applications of differential calculus (Chapter 14)

Example 4 Self Tutor

Show that the equation of the tangent to y=lnx at the point where y=¡ 1 is y=ex¡ 2.

When y=¡ 1 , lnx=¡ 1

) x=e¡^1 =^1 e

) the point of contact is

¡ 1

e,¡^1

¢

.

Now f(x)=lnx has derivative f^0 (x)=^1

x

) the tangent at

¡ 1

e,¡^1

¢

has gradient

1

1

e

=e

) the tangent has equation y=¡1+e(x¡^1 e)

which is y=ex¡ 2

13 Find the equation of:

a the tangent to f:x 7 !e¡x at the point where x=1

b the tangent to y=ln(2¡x) at the point where x=¡ 1

c the normal to y=ln

p

x at the point where y=¡ 1.

Example 5 Self Tutor

Find the equation of the tangent to y= tanx at the point where x=¼ 4.

When x=¼ 4 , y= tan¼ 4 =1

) the point of contact is (¼ 4 ,1).

Now f(x) = tanx has derivative f^0 (x) = sec^2 x

) the tangent at (¼ 4 ,1) has gradient sec^2 ¼ 4 =(

p

2)^2 =2

) the tangent has equation y=1+2(x¡¼ 4 )

which is y=2x+(1¡¼ 2 )

14 Show that the curve with equation y=

cosx

1 + sinx

does not have any horizontal tangents.

15 Find the equation of:

a the tangent to y= sinx at the origin

b the tangent to y= tanx at the origin

c the normal to y= cosx at the point where x=¼ 6

d the normal to y=

1

sin(2x)

at the point where x=¼ 4.

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\372CamAdd_14.cdr Friday, 10 January 2014 10:32:06 AM BRIAN