Cambridge Additional Mathematics

If two curves then

they share a common

tangent at that point.

touch

y

x

y=lnx

y=ax 2

O^1 b

374 Applications of differential calculus (Chapter 14)

19 a Find the equation of the tangent to y=x^2 ¡x+9at the point where x=a.

b Hence, find the equations of the two tangents from (0,0) to the curve. State the coordinates of

the points of contact.

20 Find the equations of the tangents to y=x^3 from the external point (¡ 2 ,0).

21 Find the equation of the normal to y=

p

x from the external point (4,0).

Hint: There is no normal at the point where x=0, as this is the endpoint of the function.

22 Find the equation of the tangent to y=ex at the point where x=a.

Hence, find the equation of the tangent to y=ex which passes through the origin.

23 A quadratic of the form y=ax^2 , a> 0 ,

touches the logarithmic function y=lnx as

shown.

a If the x-coordinate of the point of

contact isb, explain why ab^2 =lnb and

2 ab=^1

b

.

b Deduce that the point of contact is

(

p

e,^12 ).

c Find the value ofa.

d Find the equation of the common tangent.

24 Find, correct to 2 decimal places, the angle between the tangents to y=3e¡x and y=2+ex at

their point of intersection.

25 Consider the cubic function f(x)=2x^3 +5x^2 ¡ 4 x¡ 3.

a Show that the equation of the tangent to the curve at the point where x=¡ 1 can be written in

the form y=4¡8(x+1).

b Show that f(x) can be written in the form f(x)=4¡8(x+1)¡(x+1)^2 +2(x+1)^3.

c Hence explain why the tangent is the best approximating straight line to the curve at the point

where x=¡ 1.

26 A cubic has three real roots. Prove that the tangent line at the average of any two roots of the cubic,

passes through the third root.

Hint: Let f(x)=a(x¡®)(x¡ ̄)(x¡°).

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Additional Mathematics
Y:\HAESE\CAM4037\CamAdd_14\374CamAdd_14.cdr Friday, 10 January 2014 10:40:39 AM BRIAN