THE TANGENT TO A CURVE AT A POINT
Analysis
 
1. FINDING THE TANGENT OF A CURVE AT A POINT

Historically, the derivative appeared in order to solve the problem of drawing the tangent to a two-dimensional curve at one of its points. This window shows the graph of a function and suggests where tangents at certain points are to be drawn.

Point P can be moved by using the coloured arrows, by writing in the x-coordinate value a and clicking on Enter or by dragging the yellow point on the X-axis with the mouse.

1.- Copy this graph into your exercise book and draw the tangents at certain points (e.g. A, B, C and D). Write down how you think the tangent to a curve at one of its points is drawn.
2. CHARACTERISTICS OF THE TANGENT TO A CURVE AT A POINT

In this window the tangent to the curve at any point on the graph is shown. We can see that the relative position of the tangent at each point, with respect to the curve, is different depending on the characteristics of that point.

Point P can be moved as in the previous activity.

2.- Look carefully at the tangents at different points, especially at points A, B, C and D. In your exercise book write down whether the following statements are true or false justifying your answers and giving examples.
a) In order for a straight line to be a tangent to a curve at point P it just has to go through that point.
b) The tangent to a curve at point P must only come into contact with the curve at this point.
c) There is always a neighbourhood of point P where the tangent and the curve only have this point in common.
d) The tangent at P divides the plane in half and leaves the curve in one of the half-planes.
e) There is always a neighbourhood of P on the curve which is found in one of the half-planes created by the tangent.

3. Once you have finished answering these questions in your exercise book write down your definition of the tangent to a curve at a point.
3. INTRODUCTION TO THE TANGENT OF A CURVE AT A POINT

You will have noticed that it is not easy to give a definition for a tangent to a curve at point P which is always true. However, it is easy to define the secant which goes through points P and Q, as the straight line which goes through these two points. In this window we are going to use the secants to help us to work out the definition of the tangent.

In order to make Q closer to P use the h arrows or the animate button. The pause button will stop the animation.

4.- Look carefully at the secants at the curve which go through point P  when Q approaches P (i.e. when h tends to zero).

Click on the Clear button to clear the trace left by the secants. The Init button will restore the initial situation.

5.- Place point P at a = 1 and look carefully at the secants on the right (h>0) and then on the left (h<0). Which straight line are they approaching?

6.- Repeat the exercise for a = 0, a = -1, a = -2 etc. Note that in each case Q approaches P on both the left and the right, the line itself is the limit.

7.- Now look carefully at what happens when a = 2. Explain what happens in your exercise book.

8.- Repeat the process in your exercise book by drawing the secants which go through point P with a ruler and draw the tangent.
4.  THE DEFINITION OF A TANGENT TO A CURVE AT A POINT

The tangent to a curve at a point can be defined as the limiting position of the secants when Q approaches P.

 

You can choose the number of secants drawn between P and Q.

Use the zoom to move closer to or further away from the tangent point.

9.- Look carefully at the sequence of secants in the following examples.

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

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

c) Moving point P.

The animate button shows how point Q approaches P as well as zooming in on the point.
10.- Show that for any point (except a=2) the secants always approach the tangent when Q approaches P.
       
           
  Juan Madrigal Muga
 
Spanish Ministry of Education. Year 2001
 
 

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