The solutions to Practice Exam #3 are now available.

Believe it or not, we know most of what there is to know about differentiation. It is useful in calculus to be able to reverse the differentation process. Given a function f, can we find a function F such that ? (For instance, knowing the velocity of an automobile may help us find the overall position function of the car if we know where the car was at one particular moment.)

Such a function F is called an antiderivative (or indefinite integral) for f. Because of the rule we learned at the end of the previous section, we can write the most general antiderivative of f as



The number C is any real number and is given the name the constant of integration. The above is called an integral equation.

For example, we have that since .

Any derivative equation we can write can be rewritten in integral form as above. There is a small integral table on the hard card in the back of your book. We learned about how to reverse the Power Rule (when our power is not -1), and about finding antiderivatives for the natural logarithm, exponential, and basic trigonometric functions.

One type of reversal we learned was how to reverse the General Power Rule and the Chain Rule. The idea is to take a difficult integral and write it as one of the ones on our basic integral table. For example, if we let u = sin x, then du = cos x dx and we could transform the following integral.



One thing to remember here is not to proceed with the integration until the entire integrand is expressed in terms of u. (No x's can remain.)

We will see the usefulness of this in Chapter 5 and a connection to computing area of a region in the xy-plane. We'll work on it after Friday's exam.

Exam #3 happens this Friday, November 7th. It covers Sections 4.1 through 4.6.

Here are practice problems for the exam. The usual 5 point offer is in effect.

The solutions to Quiz #7 are available from the Test and Quiz Solutions page. You will get your graded quizzes back on Wednesday.

I'm behind in my lecture blogging, so here's a quick attempt to catch up before the exam on Wednesday.

The section on the Mean Value Theorem is transitional for us. On the one hand, it continues in the vein of figuring out what the derivative says about the graph of a function f. On the other hand, we are getting towards the point of wondering what we can say about an unknown function if we know some things about its derivative.

The Mean Value Theorem (MVT) is simply Rolle's Theorem with the restriction on the function values at a and b removed. In that case, then there is a number such that



(Informally, the Mean Value Theorem is a "rotated" version of Rolle's Theorem. See Section 4.6.) That is, there is a point c where the tangent line is parallel to the secant line connecting the endpoints. An application comes with the fact that a car that has traveled 120 miles in 3 hours must have been going precisely 40 mph at some point in its journey.

Two applications that pave the way to the future for us are:

• If the derivative of a function on an interval (a,b) is zero everywhere on (a,b), then the function is constant on (a,b).
• Two functions with the same derivative must differ by only an additive constant (the "+C").

We will use this last fact to introduce antidifferentiation.