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

As we have already seen the tangent at a point on a curve is obtained as a limit of the secants at that point.

Use the arrows to select point P and to bring point Q nearer or move it further away. Click on the animate button to make point Q approach point P.

1.- Check that as h tends to zero, i.e. as point Q approaches P, the secant QP moves closer to the tangent.
2. THE GRADIENT OF A STRAIGHT LINE

This window illustrates the method you can use to find the gradient of any straight line.

y = m x + c

 

Move the red point by dragging it with the mouse.

2.- Move the red point and check that the quotient between the segments indicated (in blue and green) is constant for any point which is not on the straight line and is equal to the gradient.

3.- Check that the condition is true for any straight line you choose.

4.- In your exercise book write out the process for finding the gradient of a straight line.
5.- Which value would you give the blue line to make it easier to calculate the gradient?

 

3. THE GRADIENTS OF THE SECANTS

All the secants go through point P (a, f(a))
and through point Q (a+h, f(a+h)).
Therefore, the gradient of the secants is:
6.- Observe how the gradients of the secants change when point Q approaches point P.

7.- Find the gradient of the tangent at the point where the x-coordinate is 1.

8.- Find the gradient of other points: x=2; x=0; x=-1 etc.

9.- Write the equation of the tangent to the curve of the graph at the point x=1.

10.- Write the equation of the tangents at the points where you have found the gradients.

11.- In your exercise book write down how to find the gradient of a tangent and how to find its equation.
 
       
           
  Juan Madrigal Muga
 
Spanish Ministry of Education. Year 2001
 
 

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