The Power to Investigate

The concepts come alive with interactive exploration.

Example 1: Calculus

We know that Which values of alpha make ? Explain your answer.

The graph of sin(x) shows that the areas under the curve alternate between positive and negative sections. Since the negative area cancels the positive area, is zero. The same is also true when sin(x) is integrated from zero to any integral multiple of 2 pi. Because there will always be exactly as many areas above as below the x axis, they will always cancel each other out. A quick check with Mathematica confirms this.

Suppose that you run a diner that serves a popular $20 dinner special and you average one hundred customers a night. You estimate that for each dollar you raise the price of the special, you will lose roughly three customers. What is the best price to charge for dinner if you want to maximize the revenue?

Start by defining a function we will call revenue[x], where the price you charge for dinner is $20 + x. Your nightly revenue will be the price per dinner times the number of dinners you serve.

Now you can run some figures. Let's plot a graph of the revenue.

The maximum looks like it is around 7 or 8; let's check.

6	2132
7	2133
8	2128

So the best price to charge for the dinner, considering revenue alone, is $7.

What if your estimate was wrong, and you actually lost five customers per additional dollar? You can define another revenue function and find out.

Then, as the graph shows, you are already at the ideal price!