As he has told us before, he will be taking the raw score of our marks from within the class, but he has not told us before about how the marks will be manipulated to better represent our progress as best as possible. He asked other AP teachers how they have achieved such appropriate representation, but he found them to be rather arbitrary or completely asinine. Examples of such include the utilization of multipliers, or in one case using the square root of the class average as a multiplier. But Mr. K's method is rather different, and if my explanation does not suffice, I do believe that Mr. K will outline his method once again on Monday to the whole class.
Class marks are often illustrated as a normal distribution, where quantities are represented by a distribution that indicates many quantities to be centralized around the mean (average) value, but become more infrequent as they approach the extrema. In other words, there will only be a few really low or really high marks, but there will be more values in a certain area as marks become more stabilized towards the average. This can be shown by the graph below (also known as a 'bell curve')
Mr. K has compiled our pre-calculus 40 marks, and graphed them as a normal distribution curve (like the one above). He calculated the mean value of the distribution, otherwise known as our class average, and also found the first and second standard deviations (in a normal distribution, standard deviations show which area of values a quantity resides or is spread out relative to the mean). Naturally, most of our marks would go down from our pre-calculus marks, so the distribution of our marks in AP calculus is expected to appear shifted to have a lower mean and a lower centralized curve. While our pre-calculus class average might be around 70-80, our AP calculus class average might even be around 60-70, which does not fairly represent our potential as students in mathematics. An example of such curves can be seen below, where the graph on the right (the solid blue graph) can be an example pre-calculus distribution, whereas the slightly transparent red curve on the left depicts an example AP calculus distribution of our raw marks. 
Mr. K plans to apply a translational transformation to the AP calculus curve to nearly match that of the pre-calculus class. In this way, the class' overall achievement in the course can more fairly be represented relative to how we performed in calculus relative to the mean after it has been shifted. That's mostly everything, but that is not nearly the end of this scribe post.
Exponential function differentiation:
It's finally time for another one of those great "ta-da" class' that makes the teacher want to present the board like their advertising a bottle of shampoo and proclaim "isn't that cool? ehh? ehh?" First off, we established the fact that we began to determine rules for differentiating (finding the derivative of a function) basic functions, but we have not encountered differentiation of exponential equations. What a beautiful segue into the inevitable lesson for the day; learning the process and foundation for differentiating exponential functions.
Mr. K announced that we would need our calculators for this lesson, and that we want to be able to determine the derivative of any exponential function. To do this, we input the following equations into our calculators:
y1 = Ax <-- The parent exponential function
y2 = [A(x + h) - Ax]/h <-- The derivative of the parent function using the limit definition.
We then defined the function by setting A = 2 (using the [sto>] and [alpha] keys), and since we want h to approach 0 but not equal zero (as that would cause an error in the calculator) we need to use a value extremely close to 0, which in this case was 0.001. So far we had A = 2 and h = 0.001 set in our calculators, and before we graphed them we turned off y2 momentarily, so that we could fully focus on analyzing what's happening in y1 and to the values of y1 and y2 in a table. Here is what we viewed as we graphed the seemingly simple function Ax:
We then deduced that we have to try and look at it algebraically, but first Mr. K digressed and we began to analyze the table values for y1 and y2 in [2nd][graph]. From this, the class noticed that there was a doubling effect happening for each subsequent value of f'(x), starting from 0.69317. Now, the relationship between the parent function f(x) and it's derivative f'(x) is shown in this table for when A=2, in which case there is a multiplier of 0.69317 (as can be seen when x=0) Mr. K then told us to look at the two graphs (turn on y2) and try to identify how y1 is related to y2 graphically. Graeme suggested that the derivative appears to be a translation of the parent function, but by referring back to the table of values it became apparent that this was not true. Instead, Mr. K asked if there was any way to determine this multiplier, which was determined in a similar fashion as calculating the common ratio of a geometric sequence (since geometric sequences are exponential in essence). We then added another function into our list of functions in our calculator; y3 = y2 / y1. This allowed us to determine the fact that each respective y value is multiplied by a common factor of 0.69317 (as we determined before) and that this factor is equivalent to f(0). This meaning that the value of the derivative at x=0 is in fact the multiplier. Followed this was a table we set up ourselves to show the multiplier values (c) for specific bases.
Base ------ c
2 ---------- 0.693
3 ---------- 1.099
pi --------- 1.145
5 ---------- 1.609
We can easily determine that by looking at our data tables, each output of the parent function is simply multiplied by the multiplier that is given by when f'(x)=0. This can be shown symbolically by:
f'(x) = c * bx
But now we encountered a question that probably riddled many of the students in the class. How can we find a perfect base number such that c will be equal to 1? Well, we can see that 3 is slightly too large a base to give a multiplier of 1, but 2 is quite a bit smaller to do so as well. We played around with numbers around 2.7 and determined that the number that gives a multiplier is 2.718281828459, otherwise known as e. e is the perfect number that we were searching for, and once we adopted it's value as A, we received a multiplier of 1.0001. It was not exactly equal to one since when we provided a value for h close to 0, we only approximated 0 thereby giving a selected margin of accruacy or a chosen error value accompanying our calculations.
We also determined that e is the perfect number through the use of algebra:
We solved for this equation to arrive at e by inserting the graph as y4 in our calculators, and solving for the root of the equation or by viewing the table of values when x=0 to see if it in fact arrives at a multipler of 1. We then learned that we could then say two things about e:
- the perfect base for e x retains a tangent line of slope 1 at x = 0
- e = lim(x->0) (1+x) ^ (1/x) Now comes an extremely important constituent of the rule for differentiation exponential equations. That is to find a rule, or a true relationship between the different bases and their respective values of c. Our main goal at this point was to determine a rule that would determine what c is for absolutely any base. To further develop the aforementioned base / multiplier table, we added values for bases 6 and 7.
Base ------ c
2 ---------- 0.693
3 ---------- 1.099
pi --------- 1.145
5 ---------- 1.609
6 ---------- 1.791
7 ---------- 1.946
Under deep analysis of this given data, it became apparent that there was an underlying logarithmic connection. We attempted to model a natural logarithm regression, and it turned out that it was an exact fit. We even entered the tested base values into the natural logarithm, and the respective multipliers determined previously were a match. Now since the natural log of the base, (ln(a)), gave us the multiplier, c, then:
This ultimately gave us the final key to determining how to differentiate an exponential function. The following equations represent the rules for determining the derivative of an exponential function:
