To end this unit we are going to focus on the relationship between the definite integral and finding antiderivatives.

1.- Choose any values for a and b (where a<b) and place x between a and b. Change the value of x and observe the values of the expression I(x). What does this expression represent?

2.- Place a and b so that the function f(x) is always positive over the interval [a,b]. Give x and a the same value and start to move x towards the right. Whilst x remains on the interval [a,b] what can be said about the function I(x)?

3.- Now place a and b such that the function f(x) is always negative on the interval [a,b] and repeat the above. What can be said about the function I(x) now?

4.- Your conclusions to the two questions above should remind you of a well known theorem. Can you remember which one it is?

The answer to the last question makes us think that there must be an important relationship between f(x) and the integral function I(x). Let's see this in the following activity:

1.- Move x between a and b. The yellow line is the graph of f(x). the turquoise line is the graph of I(x) and the blue line is the tangent at each point to I(x). What is the relationship between the red shaded area that appears and the value of each point of the function I(x)?

2.- Look carefully at the values of f(x) and the gradient of the tangent to I(x) at each point. What conclusion can we draw from this?