The most common use of calculus (the one that motivated our discussions of the previous chapter) are those that involve change in some quantity over time. While can still be used in this case (x would represent time), we often will change the variables—say, for change in height over time, or for change in velocity over time…
Note that if the variables have positive association (they increase and decrease together), then the derivative will be positive. If one variable increases when the other decreases, the derivative will be negative.
A common (introductory) application of calculus is motion on a straight line. The original function represents the position (x-coordinate) of a moving object. Typically, this function is given as .
The first derivative then gives the instantaneous velocity—positive value indicating motion to the right, and negative values indicating motion to the left. To find average velocity, you must be given an initial an a final position. Note that a velocity of zero indicates that the object is at rest.
Speed and velocity are not the same thing! Velocity is magnitude and direction—speed is magnitude only. Thus, the instantaneous speed of a moving object is .
The second derivative gives instantaneous acceleration—positive values indicating increasing velocity, and negative values indicating decreasing velocity. Be careful—"increasing velocity" does not necessarily mean "going faster!" Consider the following graph of velocity versus time:
Figure 1 - Velocity vs. Time
Between times zero and three, the velocity of the object is increasing. Since the velocity is non-zero, the object is not at rest. When , the velocity is zero, and the object is at rest. How would you describe the motion of an object that is moving, then comes to rest?
You'd say it was decelerating! Even though the velocity is increasing!
So—be careful. When velocity is positive, increasing and decreasing velocity have their usual meanings (accelerate and decelerate); when velocity is negative, the usual meanings are reversed…
To find average acceleration, you must be given an initial and final velocity.
You can also determine the distance traveled by the object. The difference between the initial position and the final position can be called the net distance traveled—but is usually called the displacement. The total distance traveled (or gross distance traveled) is the sum of the distances traveled in each direction.
Here—consider this graph of position versus time:
Figure 2 - Position vs. Time
The position at is zero; the position at is zero. Thus, the displacement for is zero. The total distance traveled is different—the object first moves about 3 units to the right; then changes direction and moves about 6 units to the left (for a total of 9 units); then turns once more and moves about 3 units to the right, for a total of 12 units.
For displacement, we add the movements with the signs intact; for total distance traveled, we add the absolute values of the movements.
A function is monotonic on the interval if does not change sign (from positive to negative, or vice versa) on that interval.
For example, if it is rising or flat, but doesn't fall—that's monotonic. If the function changes direction (on that interval), then it's not monotonic.
A function is increasing on the interval means that for any pair of x values in the interval, if , then .
Another way to say this is that the function is increasing on the interval if the first derivative is positive for all values in the interval.
Those who actually bother to read the text will note a discrepancy—some of us insist that increasing (and decreasing) are properties of open intervals, while others insist that closed intervals are fine. I belong to the former group.
There are similar definitions for intervals where the function is decreasing—I'll let you figure them out.
A sign diagram (of the first derivative) is a great way to determine intervals where a function is increasing and decreasing.
If a function is increasing, then switches to decreasing, what happens in-between?
If a function is decreasing, then switches to increasing, what happens in-between?
The answers actually depend on the function, and whether or not it is a nice function—a continuous function. For the moment, let us only consider functions that are continuous on the set of real numbers.
In this case, an increasing function (positive derivative) changing to a decreasing function (negative derivative) must pass through some place where it is neither increasing nor decreasing—a place where the derivative is zero! Since the function was rising, then started falling, it must have achieved a local maximum (singular; plural is maxima).
has a local maximum at if there exists some open interval so that every x value in that interval causes . In other words, there is some (small) region where is bigger than any other .
has a global maximum at if for all values of x in the domain of f.
Again, there are similar definitions for local and global minima…
Note that a maximum or minimum must (for now) be a location where the derivative equals zero; however, just because the derivative equals zero, that does not mean that there is an extreme value (maximum or minimum).
Sometimes, the function levels off (derivative equals zero), but it doesn't change direction. The text calls this a horizontal (or stationary) inflection.
A rational function is one that can be written as a ratio of polynomials. Many of the details involved in graphing rational functions involve (or touch on topics from) calculus. Thus, here are some steps you should follow…
That leaves just one item—which the text breaks into a bunch of cases, even though there is one guiding principle! Specifically, the final item of consideration is end behavior—what happens to the function for values of x that are (very) far from zero?
The guiding principle is this—divide the rational function (polynomial long or synthetic short—take your pick) out into a quotient and remainder. For very large (far from zero) values of x, the rational function will behave/look like the quotient from the division.
When dividing the rational function, there are three (or four) possible results, depending on the degrees of the numerator and denominator.
Let me emphasize that a horizontal asymptote is a very different creature from a vertical asymptote (it is unfortunate that they are both called asymptotes). A vertical asymptote is a domain issue—there is a value that simply cannot be used; thus, we get a line that cannot be touched, crossed, etc. However, a horizontal asymptote is end behavior—it is an indication of the tendency that the function has. In this case, the asymptote can be crossed at will—the only requirement is that the function approach that line as x moves away from zero.
Back to the possible cases from the division…
It is this last case that some break into two cases—specifically, some will separately consider the case where the degree of the numerator is exactly one greater than the denominator. That would result in a linear quotient, which is often called an oblique asymptote (or, if you are a bit backwards, a slant asymptote). My opinion: why (at this point) treat the lines any differently from the other polynomials?
I said that this relates to calculus…and it does! I just didn't make the connections for you…
There are two types of curve: concave up, and concave down.
Figure 3 - Concavity
A curve is concave up at those locations where its second derivative is positive; it is concave down where its second derivative is negative.
A Point of Inflection is a location where the concavity of the curve (and thus, the sign of the second derivative) changes.
This is the process of finding and verifying locations where a function has an extreme value. If you've got a graphing calculator, then you could just look at the graph…so I will focus my comments on the algebraic methods.
First of all, we have to determine where the extreme values might occur. Begin by taking the derivative. Any points where the tangent is horizontal (the first derivative equals zero) are candidates. Also (technically), any points where the first derivative is undefined are candidates. This is because of a type of feature called a cusp.
Figure 4 - Cusps
These should be pretty unusual—better to be safe and check for them, even if you never run across one.
Also—any domain restrictions will create additional candidates. If the function is restricted to a closed interval , then and are also candidates.
In summary: potential locations of extrema are…
Having identified all candidates for extrema, now you've got to check and see which ones are, in fact, extreme values.
If the domain is restricted to a closed interval, then just plug the candidates back into the original function, and compare the results—the largest is the maximum, and the smallest is the minimum.
If you're not working on a closed interval, then you have some options.
This term is a really fancy way of saying "word problem." This means that you may have to write the function on your own!
Some details about some specific types of calculus problem (of major interest to business majors)…
A Cost Function gives the cost (y) of producing a certain number (x) of some item. This should be an increasing function—it is best when this function is concave down (the rate of change in cost is decreasing).
The Average Cost Function is .
The Marginal Cost is the instantaneous rate of change of cost with respect to the number of items produced (how fast the cost is changing as you increase production; ).
The Demand Function gives the price (per item) if x items are sold. Typically, a business will charge less per item as the number of items increases—thus, the demand function should be a decreasing function.
The Revenue Function gives the revenue produced by selling x items. This is the product of x and the demand function. This should be an increasing function! Its derivative is the marginal revenue function.
The Profit Function is the difference in the revenue and cost functions. Its derivative is the marginal profit function. Of prime interest is maximum profit—which will occur when the marginal profit is zero.
So far, we have exclusively dealt with explicit functions—functions where the y-variable can be separated from x completely; hence, the form y = for our functions.
Of course, this is not the only way to do it…what about the equation of a circle? This equation is implicit—x and y are mixed together.
Okay, okay—that's not a function. It is a perfectly fine curve, and it makes perfect sense to discuss the slope of the tangent at a point! How do you use the definition in order to take limits and find the slope of the tangent line?
The answer is to do everything the same, but treat y as another function (and don't forget the chain rule!).
For example, is really a function raised to a power—which, by the general power rule, has a derivative of (the is from the chain rule).
We'll do some examples of this in class…
Page last validated 2010-08-15