Cambridge Additional Mathematics

A,(a¡f(a))

x=a

y = f(x)

point of

contact

tangent

gradientmT

normal

(1 2),

y

x

f(x) = x + 1 2

1

O

point of inflection

Applications of differential calculus (Chapter 14) 369

TANGENTS

Thetangentto a curve at point A is the best approximating straight line to the curve at A.

In cases we have seen already, the tangenttouchesthe curve.

For example, consider tangents to a circle or a quadratic.

However, we note that for some functions:

² The tangent may intersect the curve again somewhere else.

² It is possible for the tangent to pass through the curve at the point

of tangency. If this happens, we call it apoint of inflection.

Consider a curve y=f(x).

If A is the point withx-coordinatea, then the gradient of the

tangent to the curve at this point is f^0 (a)=mT.

The equation of the tangent is

y¡f(a)=f^0 (a)(x¡a)

or y=f(a)+f^0 (a)(x¡a).

Example 1 Self Tutor

Find the equation of the tangent to f(x)=x^2 +1at the point where x=1.

Since f(1) = 1 + 1 = 2, the point of contact is (1,2).

Now f^0 (x)=2x,somT=f^0 (1) = 2

) the tangent has equation y=2+2(x¡1)

which is y=2x.

A Tangents and normals

Points of inflection

are not required for

the syllabus.

4037 Cambridge

cyan magenta yellow black Additional Mathematics

(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\CAM4037\CamAdd_14\369CamAdd_14.cdr Friday, 4 April 2014 5:58:50 PM BRIAN