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derivative and the slope of a curve

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In this explanation, I am being asked to discuss the relationship between the slope of a secant line, the slope of a tangent line and the derivative AND in addition, I must explain the relationship between the area of a finite number of rectangles under a curve and an infinite number of rectangles under a curve and the definite integral.

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Solution Summary

The relationship between the derivative and the slope of a curve is established.

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Problem:

Explain the relationship between the derivative and the slope of a curve and between the definite integral and the area under a curve

Solution:

a) Derivative and slope of a curve.

Let's consider a curve defined by a continuous function
(1)
and a fixed point (A) on this curve, whose coordinates are (x0, y0):

Another point (B) of coordinates (x, y) can move on this curve, so that the line AB denoted as (d) is varying its position, defined for example by the fixed point (A) and the variable slope ().
The relation between the slope of this line and the coordinates of (A) and (B) is
(2)
Let's consider now that the point (B) moves toward the fixed point (A). As the point (B) gets ...

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