Let say we have a function

*f*(

*x*). If we integrate

*f*(

*x*) with respect to

* x*, we get the antiderivative

* F*(

*x*)

* + c* as follow:

where

* c* is a constant.

Now, consider you do not know the function

*f*(

*x*) but you need to sketch the graph of

*F*(

*x*) from the graph of

*f*(

*x*) only, taking

*c *= 0. How are you going to start?

Let's take a look at the graph below first:

Curve f (blue) is the curve for

*f*(

*x*) and curve F (maroon) is the curve of

*F*(

*x*).

Line L is the tangent at point B, with equation

*y = mx + c*, where

*m *is the slope of tangent and

*c *is the

*y*-intercept of the tangent.

Point A is a point at curve

*f*(

*x*) and point B is the corresponding point at curve

*F*(

*x*) having the same

*x*-coordinate of point A.

Now, try to move the point A and observe the following:

(a) As point A falls above

*x*-axis, curve

*F*(

*x*) is increasing.

(b) As point A falls on the

*x*-axis, curve

*F*(

*x*) reaches to its extremum (maximum/minimum).

(c) As point A falls below the

*x*-axis, curve

*F*(

*x*) is decreasing.

(d) As point A falls on the extremum point, point B is a point of inflection, which curve

*F*(

*x*) changes its concavity.

Take note that the

*y*-coordinate of point A is exactly the

**slope of tangnet** **(***m*) of point B!

Perhaps you can try to draw any new curve

*f*(

*x*) and find its integral

*F*(

*x*), taking

*c* = 0?

Betty, Created with GeoGebra