Scribe Post 11/16

We began class with the excitement of the follow-up test looming over us. Since Mr. O'B was leaving early, we would save that for the 2nd half of the class. 

We began by looking at this function, and trying to think about what the absolute value signs do to this derivative. To try and understand this equation better, and see how the absolute value sign would affect it's derivative, we split it to make it a piecewise function:

Now, Using our derivative knowledge that (lnx)' = 1/x, we can simplify the equation:

Therefore, 

The absolute value signs don't affect the derivative of ln, however adding absolute value signs to the equation allows us to use negative values for x.

Next, we looked into finding anti-derivatives. We looked an example which read: 

ex/ Find the function f(x) whose derivative is f '(x)= x^2, and which passes through (2,4).

After some thought, we came to realize that the anti-derivative of f'(x) was



 Anti-derivatives must have a constant attached to them because the derivative of any constant is just 0. Unless, that is, we have a condition for our equation. In this case, the condition is that the equation must pass through (2,4). To find our constant, we can plug in our given values:

Since C is 4/3, 

Using this new idea, we looked at a couple examples of some anti derivatives:

Lastly, before taking the Follow-up test we learned of a new Theorem:

Mean Value Theorem

If f(x) is a continuous function on [a,b] and a differentiable function on (a,b), then there is at least 1 value of c in [a,b] such that: 

The example that Mr. O'B used to put the MVT into easier terms is the following:

If you travel 50 miles to Augusta and it takes exactly 1 hour, at some point in the trip the car  must have been going exactly 50 mph. Let's think about this graphically:

What the Mean Value Theorem tells us is that as long as f(x) is continuous on the interval [a,b] and differentiable on the interval (a,b), then a tangent line to the point c will have a secant line with the exact same slope at the point c. 

For another explanation on the MVT, and an applet to help understand it's uses: Click Here!

Don't forget to do IW #3, study for the quiz, and have a nice Turkey Day.