Cambridge Additional Mathematics

Applications of differential calculus (Chapter 14) 371

Example 3 Self Tutor

Find the equations of any horizontal tangents to y=x^3 ¡ 12 x+2.

Since y=x^3 ¡ 12 x+2,

dy

dx

=3x^2 ¡ 12

Horizontal tangents have gradient 0 ,so 3 x^2 ¡12 = 0

) 3(x^2 ¡4) = 0

) 3(x+ 2)(x¡2) = 0

) x=¡ 2 or 2

When x=2, y=8¡24 + 2 =¡ 14

When x=¡ 2 , y=¡8 + 24 + 2 = 18

) the points of contact are (2,¡14) and (¡ 2 ,18)

) the tangents are y=¡ 14 and y=18.

3 Find the equations of any horizontal tangents to y=2x^3 +3x^2 ¡ 12 x+1.

4 Find the points of contact where horizontal tangents meet the curve y=2

p

x+

1

p

x

.

5 Findkif the tangent to y=2x^3 +kx^2 ¡ 3 at the point where x=2has gradient 4.

6 Find the equation of another tangent to y=1¡ 3 x+12x^2 ¡ 8 x^3 which is parallel to the tangent

at (1,2).

7 Consider the curve y=x^2 +ax+b whereaandbare constants. The tangent to this curve at the

point where x=1is 2 x+y=6. Find the values ofaandb.

8 Consider the curve y=a

p

x+pb

x

whereaandbare constants. The normal to this curve at the point

where x=4is 4 x+y=22. Find the values ofaandb.

9 Show that the equation of the tangent toy=2x^2 ¡ 1 at the point wherex=a,is 4 ax¡y=2a^2 +1.

10 Find the equation of the tangent to:

a y=

p

2 x+1 at x=4 b y=

1

2 ¡x

at x=¡ 1

c f(x)=

x

1 ¡ 3 x

at (¡ 1 ,¡^14 ) d f(x)=

x^2

1 ¡x

at (2,¡4).

11 Find the equation of the normal to:

a y=

1

(x^2 +1)^2

at (1,^14 ) b y=

1

p

3 ¡ 2 x

at x=¡ 3

c f(x)=

p

x(1¡x)^2 at x=4 d f(x)=

x^2 ¡ 1

2 x+3

at x=¡ 1.

12 Consider the curvey=a

p

1 ¡bx whereaandbare constants. The tangent to this curve at the point

where x=¡ 1 is 3 x+y=5. Find the values ofaandb.

4037 Cambridge

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Y:\HAESE\CAM4037\CamAdd_14\371CamAdd_14.cdr Wednesday, 8 January 2014 12:04:42 PM BRIAN