Derivatives

Today we started unit two: Derivatives.

Verbal: The derivative of a function at a point is the "instantaneous" rate of change of the function at that point. It is the slope of the tangent line at the point on the function's graph. In physics, the derivative finds the velocity at a moment in time.
Ex.) rise/run, displacement/time, bytes/sec, words/min...

Numeric: We used#4 from the test:

We used difference quotients to find approximate values. The difference quotient is obtained by subtracting the twoY1

values and dividing by the two X values. Using (X,Y1) values on each side of one gives the symmetric difference quotient.

To use nDeriv, the syntax is nDeriv(e^(X),X,1)
We found out that nDeriv is actually finding the symmetric difference quotient to give us the value. It is not, as Mr. O'Brien said, a "magical elephant" performing calculations.

Graphic:
We used #5 from the test:

We graphed with GeoGebra, adding in values (1, f(1)) and (4, f(4)). Connecting this lines gave us a secant line.

The slope shown above is the average rate of change: 2.28

The graph above shows the secant line in purple, and the tangent line in green. As we move the second point on the secant line closer to the first point, the slope of the secant line approaches the slope of the tangent line.

Algebraic:

This is the limit of the slope of the secant line as the second point approaches the point of tangency.

So, as an example:

We then did an example with 16c on our test. WolframAlpha confirms that the derivative of g(x)=1/x is 1/4. WolframAlpha can also find it using nDeriv.

A second form of the derivative definition:

Explained by the scanned graph (click for a larger image):

The "trick":

Links: A PDF showing an example of a derivative found algebraically can be found here. A good walkthrough of the whole idea is here.

Homework for next class is to finish Supercorrections. We will have Tuesday in class to work on them.

Marnie is the scribe for next class.