and

In this part and the next, we explore the same concepts in the context of two additional examples. Our first example was one for which you might or might not have recognized the symbolic formulas for the various functions involved. In this part you will certainly recognize all the formulas that appear. In the next part, you will almost certainly not recognize the antiderivative formula produced by your computer algebra system. The point of these examples is that it doesn't matter whether the formulas are familiar or even whether they exist -- our first general principle is a true statement about continuous functions in general, and the second is true about differentiable functions in general.

1. In your worksheet, define
   

   Graph this function.

2. Define the corresponding function F(x), and graph it. Explain why it has the form it has -- from the graph of f, not from whatever simplified symbolic form F(x) might have.

3. Differentiate F(x), and confirm that the result is f(x).

4. Define g(x) as the appropriate integral of f '. Graph g, and compare the graph with that of f. Confirm that g(x) = f(x) - f(0).

5. (optional) Explain why F is the solution of the initial value problem
   

6. (optional) Explain why g is the solution of the initial value problem