Week+05+Sept+15-19

September 15 and 16 - Derivative Functions Numerically and Graphically
media type="custom" key="1998772"

September 18th 2008
The class started with a pop quiz. Each student selected a definition of the derivative problem to work; if successful it is not required of the student to do another. If unsuccessful, students will be required to work another, unitl they are able to work one correctly. Problems were of the type shown here: Following the quiz we moved on to prove the power rule ; a shortcut for finding the derivative of a power function.

Power Rule says that to find the derivative of a power function 1) The "new" coefficient is the product of the "old" coefficient and the exponent 2) The "new" exponent is the "old exponent reduced by one. Example: If f(x) = 5x⁴ then f '(x) = 20x³.

19th September 2008
Finishing the material in section 3.4, we talked about all the different notations for the derivative. These are summarized in the graphic below. The symbol dy/dx comes from as shown in the derivation of the derivative of ax^n in yesterday's notes.

After time was allotted for students to do a second definition of the derivative pop quiz, discussion began on the relationships between the concept of the derivative and the topic of particle motion which is the focus of section 3.5. The objective of this lesson is given an equation for the displacement of a moving object, find an equation for its velocity and an equation for its acceleration, and use the equations to analyze the motion. In chapter 1 we had analyzed graphs f position versus time, and found that the instantaneous rate of change, or slope of the tangent line at a point, represented the instantaneous velocity at that point. Combining this information with our understanding of the derivative of a function as the instantaneous rate of change of that function leads us the the conclusion that if x(t) represents the position o a particle at any time t, then the derivative x'(t) must represent the instantaneous velocity of the particle at any time t. Likewise, if v(t) is the velocity at time t, v '(t) must represent the instantaneous rate of change of velocity at any time t, or the acceleration. These results can be summarized in the diagrams below.