In the real world, however, functions are often given by some sort of physical process. For example, a chemist might perform an experiment in which she mixes a solution with a certain concentration, x, of a chemical and then makes some measurement f(x) that depends on the concentration x. The concentration x is the input to this experiment or function. The measurement f(x) is the output or value of the function with input x. Similarly, a market researcher for a chain of stores might run an experiment in which he priced a particular product at p dollars and then measured the number f(p) of units sold at the price p. The price p is the input to this experiment or function. The number, f(p), of units sold is the output or value of the function with input p.

When a function is determined in this way it cannot be computed as precisely as a function defined by some mathematical formula. The instruments used to measure physical quantities like temperature and mass only measure these quantities to within a certain precision. The more precision one needs, the more expensive the instrument. The JAVA applet below simulates a function determined by some physical experiment. You can choose any value that you want for the independent variable or input by typing it in the upper box. Then run the experiment by clicking the button. The result will appear below the button.

The output will have six digits that are correct. For example, the output 1.23456 would mean that the actual value is 1.23456 after rounding to six digits. The output 1.2345 is the same as 1.23450.

Challenge Problem:

Find the best estimate that you can for the derivative of the function given by the JAVA applet above for the input 1.