We saw further that F'(x) = f(x) -- that is, F is an antiderivative of f.

Another way to say the same thing is this: If we integrate a function f to a variable upper limit x, and then differentiate the resulting function with respect to x, we get back where we started. In short, differentiation undoes (definite) integration.

In this part, we study the effect of doing the operations in the other order: differentiation first, then integration. The result will be similar to that in Part 1, but subtly different.

As in Part 1, we will continue with the example function

Our reason for selecting this function -- apart from familiarity of its graph -- is that you probably do not know a simple formula for an antiderivative. Nevertheless, we were able to construct an antiderivative function without knowing (or finding) such a formula. Of course, you do know how to differentiate this function -- but we will not immediately make use of that knowledge.

1. Define a function g(x) as the integral (to a variable upper limit) of the derivative of f:
   

   Graph both f ' and g, and explain why the graph of g has the form it has. In particular, why are all the values of g negative?

2. Compare the graph of g with the graph of f. How are these graphs similar? How are they different? How do you think the functions g and f are related?

3. We can answer precisely the question of relationship between g and f. From Part 1, we know that g is an antiderivative of f ' -- but, of course, so is f. How are different functions with the same derivative related? Express this relationship in a formula with an as-yet undetermined constant.

4. To determine the constant in the relationship between g and f, we need to know values of g(x) and f(x) at a single value of x. That's no problem with f -- we know a formula for all of its values. For g, there is one value of x at which we can easily calculate g(x): x = 0. Use this information to determine the constant in the relationship between f and g.

5. Confirm your calculation in the preceding step by graphing two functions that should be the same. If they are, you will see only one graph.

6. Further confirm the relationship in step 4 symbolically: Ask your computer algebra system for a symbolic formula for g(x), and compare the result to the formula for f(x).

7. Now let's abstract the general principle from our example. Suppose
   

   but we don't know anything about f(x) other than the fact that it is continuously differentiable -- i.e., f ' exists and is a continuous function. Write a formula for the relationship between g and f. Use the fact that both of these functions are antiderivatives of f '. Since you don't know values for f, your formula will have to contain the unknown value f(0).

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