Precal Blog

Today in Pre-Calc, we spent most of the class working on Activity 2.2 in our packets, problems 1-3. These problems dealt with another type of exponential function:

f(x)= (1+1/x)^x

The first problem dealt with a real life application problem dealing with compound interest. We were told to complete a table for the function:

f(x)=100(1+.08/x)^x

We needed to figure out the values for your balance annually, semiannually, quarterly, monthly, daily, and hourly. Here are the values for each of the time periods:

Annual: $108.00; Semiannual: $108.16; Quarterly: $108.24; Monthly: $108.30; Daily: $108.33; Hourly: $108.33

The basic pattern in this function would be it increases at an exponential rate but after a certain time period, it will begin to level off. The number that this function begins to level off at would be $108.33. These types of functions are used quite frequently by almost every banker.

To figure out how to get there, we built up the function. We started off by creating a table and a graph for the function:

f(x)=1/x

That graph made an L-shape in the first quadrant. There was a horizontal asymptote at the y=.01 line. This is because the f(x) value will never get to .01, it will just approach it. Then, we derived another function, which was:

f(x)=1+1\x

This graph also produced an L-shaped curve in the first quadrant. However, the horizontal asymptote instead of being at .1, was at 1.01. This was because it will approach 1.01, but never quite reach that value. This leads us to the function I mentioned earlier:

f(x)=(1+1/x)^x

Unlike the first two graphs, this one produced a half-upside down bowl shaped curve. Sorry for the explanation of the shape. Anyways, this graph is shaped like this because your number will start low and gradually climb higher due to the exponent. The “limit” of this function is approximately 2.718….. This is actually a special number many bankers use, and its symbol is e.

On a side note, we also spent a good part of the class period talking about the new iPad and doing some random background checks. Fun times. Anyways, tomorrow’s scribe is ……………….. Emily M.