Cambridge International Mathematics

Atangentto a curve is a straight line whichtouchesthe curve.

Oblique means

at an angle to

horizontal and

vertical.

Other curves can also have tangents. For

example, the quadratic function shown has

a horizontal tangent (1) at the vertex and

an oblique tangent (2) at point A.

Example 8 Self Tutor

From the accurate graph ofy=x^2 ,

estimate the gradient of the tangent

at the point:

a O(0,0) b A(1,1)

a At O, the tangent is horizontal and so the gradient is 0.

b At A(1,1), the tangent has gradient¼^21 ¼ 2 :

EXERCISE 23D

Print off the worksheet to answer this exercise.

1 As accurately as possible, find the gradient of the tangent to:

a y=x^2 at the point A(¡ 1 ,1) b y=x^2 at the point B(2,4)

c y=x^3 at the point C(0,0) d y=

2

x

at the point D(1,2)

e y=

6

x

at the point E(2,3) f y=2x at the point F(0,1).

2 Copy and complete:

a The gradient of the tangent to a curve at a turning point is ................

b The tangent to y=x^3 at the origin tells us that tangents to curves can ................

the curve.

D TANGENTS TO CURVES [3.5]

point of contact

tangent

We are familiar with a

tangent to a circle which

touches the circle at a

singlepoint of contact.

A

vertex

(1)

(2)

GRAPHING

PACKAGE

PRINTABLE

WORKSHEET

O

y

x

2

1

A

This occurs at special points called points of inflection.

480 Further functions (Chapter 23)

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