DERIVATIVES I

Tangent to a curve at a point

Search of the tangent to a curve at a point

Historically the derivative arises to solve the problem of drawing the tangent to a plane curve at one at one of its points.

1.- Draw in your notebook the graph of the applet, draw the tangents at several points (A, B, C and D, for example) and write in your work book how you think the tangent to a curve at one of its points is drawn.

Characteristics of the tangent to a curve at a point

2.- Watch the tangents at different points of the curve, in particular at the points A, B, C, D and answer if the following statements are true or false indicating why.

a) For a straight line to be a tangent to a curve at a point  P it is enough that it passes through this point. 

b) The straight line tangent to a curve at a point P can only have one point of contact with it, which is precisely P.

c) There is always a neighbourhood of the point of tangency in which the tangent straight line and the curve only have this point in common.       

d) The straight line tangent to a curve at a point leaves the curve in one of the half planes in which it divides the plane.    

e) There is always a neighbourhood of the point of tangency in which the tangent straight line leaves the curve entirely in one of the half planes in which it divides the plane. 

Write what you think defines the straight line tangent to a curve at a point. 

Approximation of the tangent   to a curve at a point

You will have seen that it is not easy to give a definition of a tangent to a curve at a point which serves for all cases. 

3.- Watch the straight line secants to the curve which pass through the point P, when Q approaches P ( that is to say when h tends towards zero).

You can press the Clear button or the secondary mouse button when you need to, to erase the trail that the the straight line secants leave. The Init button restores the initial conditions.

a) Put the point P at x = 1 and watch the secants at the right and then at the left.

b) Repeat it for x = 0; x = 1, x = -2, x = 2, ...

c) Reproduce the process with a ruler in your work book, draw with a pencil the secants which pass through the point P and draw the tangent.

Definition of a tangent to a curve at a point

It can be said that the straight line tangent at a point of a curve is the limit of the secants when Q tends towards P.

4.- Watch the succession of secants in the following cases.

a) Modifying the number of secants for the same point P.

b) Modifying the value of h, between -1 and 1, for the same point P.

c) Modifying the point P.

Author: Juan Madrigal Muga

	Ministerio de Educación, Cultura y Deporte. Año 2000