Those with graphing calculators or access to plotting programs may think that this unit is unnecessary. There are several schools of thought on this subject, and for current purposes, this unit is included for completeness. One should not have to resort to mechanical aids to make rough sketches of simple functions. Additionally, the properties of curves, interpreted from the derivatives of the functions that the curves represent, is an important part of knowing calculus.

For a specific example of using a program (Maple) for graphing of functions, see Plotting Functions of a Single Variable on MAPLE, Xdvi or PDF; these notes were written in conjuction with a self-paced class in Differential Equations, and involve functions that are far from simple. However, the commands may be adapted for other functions.

**Curve sketching** uses the first and second
derivatives to locate points of interest on the curve. To sketch the
curve of `y`=`f`(`x`), we first find where
the function reaches maxima and minima. Other than places where the
curve is not "smooth" (i.e., |`x`| is not smooth at
`x`=0), a maximum or minimum occurs only when the first
derivative is zero. (One way to see this: for a maximum, the curve
must stop going up and start going down. Thus either it passes through
a point where it is level, and thus `y´`=0, or the
derivative is not continuous, which is another way of saying the curve
isn't smooth.) To aid the accuracy of our sketch, we also obtain
information from the second derivative. This informs us of the rate at
which the slope of the curve is changing. If this rate is positive,
the slope is becoming steadily larger. Geometrically, the curve is
*concave up* or ``holding water.'' Similarly, if
`y´´<`0 the curve is concave down. Solving for
the points where the second derivative is zero gives us the places
where concavity might change.

A few cautions about curve sketching: Just because a derivative
is zero at a point doesn't necessarily mean that the derivative
changes sign in the vicinity of that point; it could go from positive
to zero and back to positive. (An example is `y`=`x`³
in the vicinity of the
origin.) Also, remember which function `y`,
`y`´, or `y`´´ you
want to use at the moment. If, for example, `y`´´=0 at `x`=5,
graph the point by finding the values of `y` and
`y`´ at `x`=5, and draw the curve through that
point as a striaight line with that slope. Finally,
be careful of discontinuities in the function or its derivatives.
Position, slope and concavity can all change drastically at a
discontinuity.

A second topic that will be covered here that doesn't quite fit anywhere else is the Mean Value Theorem and a useful application of it - L'Hôpital's Rule.

First, the **Mean Value Theorem**. The primary reason to
care about this theorem is that it is used to prove many other results
we care about even more. For the Mean Value Theorem, we assume a
section of a smooth curve (the first derivative is continuous). A
sketch of such a curve is shown:

Aside: For a parabolic arc, the point where the slope is equal to the
ratio `c` is at the midpoint of the `x` - range.

Now that we have the Mean Value Theorem, we can, as promised,
prove something interesting: **L'Hôpital's
Rule**. You remember from
elementary school that 0/0 lacks meaning. If, instead, we
have two functions `f`(`x`) and
`g`(`x`) which both approach zero at
`x`_{0}, sometimes the limit

Remember that `c`/0, for all values of `c` other
than zero, ``is'' infinity. To say this as properly as we can for our
purposes,

- Sketch curves and indicate regions of positive slope, negative slope, positive and negative concavity.
- Understand the Mean Value Theorem.
- Apply L'Ho&pcirc;ital's Rule to appropriate limits.

- Read
*Simmons*4.1-4.2 and 12.1-12.3 - Sorry, there are not yet any World Web Math entries on this topic.
- Again, you need to work lots of problems, especially to get the
hang of graphing.
Simmons has problems on pp. 91, 94. The problems vary widely in
difficulty. Work your way up to the harder ones by solving
*many*of the easier ones. For L'Hopital's rule, do problems from sections 12.2 and 12.3. - Take a
**Sample Unit Test**, Xdvi or PDF. - Ask your instructor to give you a unit test.

watko@mit.edu Last modified July 31, 1998