Sept. 29th

We started off class with going back to our 'closing door' example from earlier in the year, only this time we analyzed it using Derivatives:


We then went through and worked on the problems on our own, before OB stepped in and we worked on them together. We also graphed the function d(t), and d'(t) and analyzed the resulting graph.


We then created a table of values for Derivatives we knew and ones we didn't know:



However, we didn't know the values for x^2, x^3, and x^4, so OB showed us an example on the board of how to calculate the derivative of x^2.

Another way we can write Derivatives is as such:

So, all we have to do is substitute x^2 into the values for f(x). When we do that, we have the limit

However, we can't just simply plug in a value for h and solve for the limit, since we'll get 0/0, and we don't like 0/0. It's a rather anti-social value. So, if we just simplify, we get

OMG-ness!!! Could this really be the derivative of x^2?? Why yes indeed! So, OB left us with the equation, and gave us free reign over x^3, to find it ourselves. So, we plugged in x^3 just like we did with x^2, and got 3x^2. We also graphed x^4, and made the graph of its derivative by tracing the slopes of the tangent lines slopes (ie the derivative!!) and compared it to the graph of 4x^3. But this whole 'OB is psychic' thing was a little too unbelievable so…… with our table complete, OB let us in on a little secret to finding the Derivative of any equation. Instead of using the big long limit equation, we can use a smaller equation: If then .

So, with our newfound knowledge, we can calculate the derivative of any function, because when you think about it, 1 can also be written as x^0, which when we use our little magic equation, produced , or 0! And can also be written as , which would make the derivative , or !!!

After this, we got a chance to test our newfound wisdom on this applet.

UPDATE(FIXED): We then checked out two other functions that actually have no derivative at the requested point: If find f'(0), and find g'(2). These two functions have no derivatives at the requested values because the limit of f'(0) Does Not Exist, and the limit of g'(2) is infinite, so there isn't a Derivative for those given points. So beware of other functions like this, especially since our calculators will lie to us and say the Derivative is Zero, when there actually isn't one.

If you'd like to look at the Constant Multiple Rule, where , then check out a proof on page 128 of the textbook.

Unit 2 Assignment #1 is posted below for the previous date.

If you're still having issues with the whole Derivative thing, then here's a link that can help.

Next Scribe is Kayla.