Demo - supplement to some lecture slides
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We have learned to take derivative of a function at a point. Now we need to distinguish the following:
Derivative at a point
Derivative as a function (arbitrary points)
The following may help you to memorize the first principle and the concept of derivative:
- Slope of secant line approaches to slope of the tangent line (see the picture)
- Limit of the ratio of change:
Notations for derivative (for a function):
For the derivative function, we can use any one of the following notations
For derivative at a particular point, we can use any one of the following notations
A further comment:
Around 1950, a MIT Professor, Struik Dirk, taught his student in this way about tangent line:
"The tangent passes through two consecutive points of the curve".
As for computer graphics, they consists of pixels, which can be think of as some tiny dots. A secant line passing two adjacent pixels on a curve would be a very good approximation of a tangent line. The following picture will illustrate the idea, if you can imagine that Q approaches to P all the way to the closest dot on the curve (dotted approximation of the continuous curve) .
As an exercise, use the function, take a sequence of values in the following as the coordinates of the move point Q:
1, 0.8, 0.6, 0.4, 0.3, 0.25, 0.21, 0.201,
then calculate the correspondingvalues and equation of the corresponding secant lines and their slopes. After that, construct a table and observe that the slope of the secant lines indeed approaches to the slope of the tangent line and the derivative .
If you have any doubts, please contact me or post messages in the topic forum.
Example (Implicit differentiation):
. Find the slope of tangent at .
Plug inabove to get .
In general,Solve for . , get
The picture of the curve looks like this: