4
Sep 17

Geometry related to derivatives

In a sequence of videos I explain the main ideas pursued by the fathers of Calculus - Isaac Newton and Gottfried Wilhelm Leibniz.

Derivative equals speed

Here we assume that a point moves along a straight line and try to find its speed as usual. We divide the distance traveled by the time it takes to travel it. The ratio is an average speed over a time interval. As we reduce the length of the time interval, we get a better and better approximation to the exact (instantaneous) speed at a point in time.

Video 1. Derivative is speed

Position of point as a function of time

Working with the visualization of the point movement on a straight line is inconvenient because it is difficult to correlate the point position to time. It is much better to visualize the movement on the space-time plane where the horizontal axis is for time and the vertical axis is for the point position.

Video 2. Position of point as function of time

Measuring the slope of a straight line

A little digression: how do you measure the slope of a straight line, if you know the values of the function at different points?

Video 3. Measuring the slope of a straight line

Derivative as the slope of a tangent line

This is like putting two and two together: we apply the previous definition to the slope of a secant drawn through two points on a graph. Then it remains to notice that the secant approaches the tangent line, as the second point approaches the first.

Video 4. Derivative as tangent slope

From function to its derivative

This is a very useful exercise that allows later to come up with the optimization conditions, called first order and second order conditions.

Video 5. From function to its derivative

Conclusion

Let $P(t)$ be some function and fix an initial point $t_1$. The derivative $P^\prime(t_1)$ is defined as the limit

$P^\prime(t_1)=\lim_{t_2\rightarrow t_1}\frac{P(t_2)-P(t_1)}{t_2-t_1}.$

When $P(t)$ describes the movement of a point along a straight line, the derivative gives the speed of that point. When $P(t)$ is drawn on a plane, the derivative gives the slope of the tangent line to the graph.

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