A terrific example of piecewise functions is our graduated income tax system. In that system, the more you make…the higher percentage you pay. However, you DO NOT pay the higher percentage on all of your income. In the two FAQ’s below,we take a look how all of this works.
In Section 5.3, you are asked to find the rate in a sinking fund. Using the ordinary annuity formula results in an equation that is very difficult to solve. Instead, try graphing each side to the equation and locating the point of intersection.
In many investment problems, you are given an amount of money and asked what will it accumulate to in a certain amount of time at some interest rate. Essentially, these problems are asking you to find the future value of the amount of money. Depending on how that money accumulates, you might use one of several different formulas.
The MathFAQ below looks at how you calculate present value in the context of different type of interest/
Points of Inflection are locations on a graph where the concavity changes. In the case of the graph above, we can see that the graph is concave down to the left of the inflection point and concave down to the right of the infection point. We can use the second derivative to find such points as in the MathFAQ below.
What is the significance of this point? On both sides of the inflection point, the graph is increasing. This means that as the number of connections increased, so did the revenue from those connections. However, on the left side of the inflection point, the increases in revenue due to increasing connections is getting smaller and smaller. On the right side of the point of inflection, increasing the connections results in larger and larger increases in revenue.
It is easy to confuse the processes for solving for the rate versus the number of years in the compound interest formula. The two MathFAQs compare the process of solving for the rate (using roots or powers) with solving for years (using logarithms)