September 27, 2010

Alright, this is my first ever scribe post and I'm a bit tired from soccer, but I hope this post will be fine. So we spent the day going over derivatives in four different ways:
1. Verbal
2. Numeric
3. Graphic
4. Algebraic

1. We verbally stated that the derivative of a function at a point is the "instantaneous" rate of change of the function at that point. Knowing that the derivative is a rate of change, we then said that a derivative was the slope of the tangent line to a function at a point (use local linearity) and from physics the velocity at a moment and the acceleration at a moment (which is a derivative of a derivative).

2. Next came the numeric. By looking at a table of the function , we discovered symmetric difference quotients and one-sided difference quotients. The one-sided quotients and the symmetric quotient were not equal but all three of them can be used to estimate the derivative. We also learned that the handy nDeriv on our calculators was actually the symmetric difference quotient which is expressed by the formula:

3. To look at the derivative graphically we looked at question #5 from the test. So we graphed the function and plotted a point A at (1, f(1)). Next we made a second point B on the function in a random spot and created a line that connected the two lines, or, the secant line. We also created the tangent line of the function at the point A and found the slope of the tangent line which was . This slope is the derivative of the function and can be estimated by using the secant line if the second point on on the function is really close, but not equal to, the first point, A in this case. Here is what it should look like:


4. When it came to algebraically, we went over another problem from the test, #16c, and a few example problems. From our example problems we discovered two forms to find the derivative algebraically:

(In the first equation, the h can be substituted with )

If you still have questions about the derivative (which you probably will), these links might help:
http://www.sosmath.com/calculus/diff/der00/der00.html
http://people.hofstra.edu/stefan_waner/realworld/tutorials/frames2_4.html

There is no new assignment for homework but remember that supercorrections are due Wednesday.