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Monday 10/19
We started with two problems of the day:





We then defined Differentiability and proceeded to explore different functions that are non-differentiable at X=0.

We then worked problems finding the value of functions to be differentiable at certain x values.



Tuesday 10/20
First, we began with the problem of the day(below) working to solve for constants a and b in this piecewise function. In order to solve a problem like this, it depends on the correlation between continuity and differentiability. In order for this function to be differentiable at x = c, then it must be continuous at x =c as well so the first equation in this system is derived from the the second rule in the definition of continuity (reference definitions and theorems page). Then the second equation in this system is derived from the derivative equations for the original piecewise function but remember not to include the value at this point in the domain in these equations until we know its differentiable. Then since the definition of differentiability says that the derivative is continuous, we can show with limits that the derivative contains this point, and you get the second equation. Solve these equations for a and b. REMEMBER: if the equation is differentiable, it must be continuous. This is the same type of problem as the daily problem except it was in the homework.

In addition, we discussed the derivative equations for a^x and how it relates to e and the natural log of x. We also found the derivative formula for logarithms of any base using the change of base theorem.

Wednesday 10/21
Our homework last night was four AP problems on Continuity and Differentiability. Since they were set prior to all of our births, we were not allowed to use a calculator. Here are the problems and their solutions.



Warning! Keep the graders happy by writing wonderful limit notation whenever the terms CONTINUITY or DIFFERENTIABILITY are asked.



We are unable to answer the b) part of this problem because we have not learned the Mean Value Theorem ye t ! I'm not certain why we didn't do part c)



We had to re-write this function in piecewise form before doing anything else because we have no rules for the derivative of an absolute value function.

We also had to remember a trig identity: sin (-A) = - sin(A)

Following our AP Problem experience we turned out attention to a few tangent line problems.This first one was suggested by Sean and modified by Ms. Gentry.

Some logarithm properties to remember:

To evaluate log ( base 16) of 4 think what power of 16 is 4?Because 4 is the square root of 16, 4 is 16 to the one half power. Consequently log (base 16) of f is one half.

ln(16) = ln (4²) and ln(4²) = 2ln(4) by the exponent to coefficient property of logarithms.

The next problem required some trigonometry review.

First up, arcsin(1/2) means find A if sinA is equal to 1/2 with a restriction on A of -pi/2 to pi/ 2. Remembering one's important triangles gives the answer that arcsin(1/2) = pi/6.

We needed aour problem solving skills for the last two problems. Here they are!



HAPPY STUDYING! Check out the Derivative Rules page for all the derivatives you need to memorize.

Thursday 10/22
duh duh duh duuuuuuuuuhhhh. test #5 day.

Friday 10/23
Today we learned how to find the derivative of non functions. It was interesting.