Saturday, June 4, 2011

Indefinite Integrals

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:

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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

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