Cambridge Additional Mathematics

(a a ), 2

y

x

y=x 2

(2 3),

O

(1)

=1

x

x

must be a

factor since we have

the tangent at.

¡ 2

Applications of differential calculus (Chapter 14) 373

Example 6 Self Tutor

Find where the tangent to y=x^3 +x+2 at (1,4) meets the curve again.

Let f(x)=x^3 +x+2

) f^0 (x)=3x^2 +1 and ) f^0 (1)=3+1=4

) the equation of the tangent at (1,4) is 4 x¡y= 4(1)¡ 4

or y=4x:

The curve meets the tangent again when x^3 +x+2=4x

) x^3 ¡ 3 x+2=0

) (x¡1)^2 (x+2)=0

When x=¡ 2 , y=(¡2)^3 +(¡2) + 2 =¡ 8

) the tangent meets the curve again at (¡ 2 ,¡8).

16 a Find where the tangent to the curve y=x^3 at the point where x=2, meets the curve again.

b Find where the tangent to the curve y=¡x^3 +2x^2 +1at the point where x=¡ 1 , meets the

curve again.

17 Consider the function f(x)=x^2 +

4

x^2

.

a Find f^0 (x). b Find the values ofxat which the tangent to the curve is horizontal.

c Show that the tangents at these points are the same line.

18 The tangent toy=x^2 ex atx=1cuts thexandy-axes at A and B respectively. Find the coordinates

of A and B.

Example 7 Self Tutor

Find the equations of the tangents to y=x^2 from the external point (2,3).

Let (a,a^2 ) be a general point on f(x)=x^2.

Now f^0 (x)=2x,sof^0 (a)=2a

) the equation of the tangent at (a,a^2 ) is

y=a^2 +2a(x¡a)

which is y=2ax¡a^2

Thus the tangents which pass through (2,3) satisfy

3=2a(2)¡a^2

) a^2 ¡ 4 a+3=0

) (a¡1)(a¡3) = 0

) a=1or 3

) exactly two tangents pass through the external point (2,3).

If a=1, the tangent has equation y=2x¡ 1 with point of contact (1,1).

If a=3, the tangent has equation y=6x¡ 9 with point of contact (3,9).

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\373CamAdd_14.cdr Wednesday, 8 January 2014 12:09:28 PM BRIAN