The Unknown Thing

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Here’s why x is so ubiquitous in mathematics.

Now one more unknown thing is known!

This TED Talk can be found here.

 

 

 

 

Dominance

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When considering functions made up of the sums, differences, products or quotients of different sorts of functions (polynomials, exponentials and logarithms), or different powers of the same sort of function we  say that one function dominates the other. This means that as x approaches infinity or negative infinity, the graph will eventually look like the dominating function.

  • Exponentials dominate polynomials,
  • Polynomials dominate logarithms,
  • Among exponentials, larger bases dominate smaller,
  • Among polynomials, higher powers dominate lower,

For example, consider the function x{{e}^{x}}. The exponential function dominates the polynomial. As x\to \infty , the graph looks like an exponential approaching infinity; that is, \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,x{{e}^{x}}=\infty . As  x\to -\infty the graph looks like an exponential with very small negative values (i.e. small in absolute value); so, \displaystyle \underset{x\to -\,\infty }{\mathop{\lim }}\,x{{e}^{x}}=0.

Another example, consider a rational function (the quotient of two polynomials). If the numerator is of higher degree than the denominator, as x\to \pm \infty the numerator dominates, and the limit is infinite. If the denominator is of higher degree, the denominator dominates, and the limit is zero. (And if they are of the same degree, then the limit is the ratio of the leading coefficients. Dominance does not apply.)

Dominance works in other ways as well. Consider the graphs of y=3{{x}^{2}} and y={{2}^{x}}. In a standard graphing window, the graphs appear to intersect twice. But on the right side the exponential function is lower than the polynomial. Look farther out and farther up, the exponential dominates and will eventually lie above the polynomial   (after x = 7.334).

Here’s an example that pretty much has to be done using the dominance approach.

\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{\ln \left( {{x}^{5}} \right)}{{{x}^{0.02}}}=0

The polynomial function in the denominator, even with the very small exponent, will dominate the logarithm function. The denominator will eventually get larger than the numerator and drive the quotient towards zero. We will return to this function when we know about finding maximums and points of inflection and find where it starts decreasing. For more on this see my post Far Out!

A Note on Notation

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For quite a while I’ve been writing sin(x), ln(x) and the like with parentheses instead of the usual sin x or ln x .

The main reason is that I want to emphasize that sin(x), ln(x), etc. are the same level and type of notation as f(x). The only difference is that sin(x) and ln(x) always represent the same function, while things like f(x) represent different functions from problem to problem. I hope this makes things just a little clearer to the students.

I also favor using (sin(x))² instead of sin²(x), again to make clearer just what is getting squared. Notation can be inconsistent: I don’t think I’ve ever seen ln²(x) or even f ²(x).  So this helps in that regard as well.

Of course, when entering functions in calculators or computers you almost always must use the “extra” parentheses in both cases. (Except for the new Casio PRIZM which will understand sin x and ln x, but not sin²(x).)

Now we can use that spot in the notation exclusively for inverse functions, as in {{\sin }^{-1}}\left( x \right) and {{f}^{-1}}\left( x \right). Maybe that will lessen the confusion there.

Another possible inconsistency is trying to write sin′(x)  for the derivative as you do with {f}'\left( x \right). Although, if I saw it I would understand it. (LaTex won’t even parse  sin′(x).)

Finding Limits

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Ways to find limits (summary):

  1. If the function is continuous at the value x approaches, then substitute that value and the number you get will be the limit.
  2. If the function is not continuous at the value x approaches, then
    1. If you get something that is not zero divided by zero, the limit does not exist (DNE) or equals infinity (see below).
    2. If you get \frac{0}{0} or \frac{\infty }{\infty } the limit may exist. Simplify by factoring, or using different trig functions. Later in the year a method called L’Hôpital’s Rule can often be used in this situation.
  3. Dominance is a quick way of finding many limits. Exponentials dominate, polynomials, polynomials dominate logarithms, higher powers dominate lower powers. The next post will give some hints about dominance.

Infinity is not a number, but it often is used as if it were. When we say a limit is infinity, what we mean is that the function increases without bound, or there is some x-value that will make the expression larger than any number you choose. Writing things like \infty -\infty =0,\frac{\infty }{\infty }=1,\infty +\infty =2\infty  are common mistakes.

DNE or Infinity?  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{{{\left( x-3 \right)}^{2}}} does not exist, and DNE is a correct answer. However, it is a bit better to say the limit is (equals) infinity, indicating that the expression gets larger without bound as x approaches 3. Both answers will get credit on an AP exam.  \displaystyle \underset{x\to 3}{\mathop{\lim }}\,\frac{1}{x-3} DNE since the one-sided limits (from the left and from the right) are different.  Only DNE gets credit here.

Take a look at this AP question 1998 AB-2: In (a) students found that \displaystyle \underset{x\to \infty }{\mathop{\lim }}\,2x{{e}^{2x}}=\infty \text{ or }DNE, in (b) they found the minimum value of 2x{{e}^{2x}}  is -{{e}^{-1}} and in (c) they had to state the range of the function is [-{{e}^{-1}},\infty )\text{ or }x>-{{e}^{-1}}. Thus making the students show they knew that this kind of DNE is the kind where the value increases without bound.

Deltas and Epsilons

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The Formal Definition of Limit

Teachers are often concerned that the Advanced Placement Calculus tests do not test the formal definition of limit and the delta-epsilon idea. I think there are two reasons for this.

  1. For linear functions the relationship is always \delta =\frac{\varepsilon }{\left| m \right|} where m is the slope of the line. Anyone can memorize that, so testing it does not provide any evidence that the student knows anything. For any other function (polynomials of higher degree, trigonometric functions, etc.) some special one-time “trick” is required, often different for each type of function. Thus, the question becomes unreasonably difficult even for one who understands the concept.
  2. As a multiple choice question, since “none of these” is never a choice on AP tests, the smallest answer must be correct. If \delta =\tfrac{1}{2}\varepsilon  is correct, then so is \delta =\tfrac{1}{3}\varepsilon ,\ \delta =\tfrac{1}{10}\varepsilon  etc. Again no real knowledge of the definition is required – just choose the smallest delta.

Of course, the definition of limit is important. You may teach it in your class. It just will not appear on the AP Calculus exams.

Here are some notes The Definition of Limit contributed by Paul A . Foerster of San Antonio, Texas, that will help your students understand the concept.

The First Week

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After reading my post on “The First and Second Day of School” Paul A. Foerster,  was nice enough to share  these problems from a recent presentation. They give a taste of derivatives and integrals in the first week of school and get the kids into calculus right off the bat.

The First Week of AP Calculus

Paul who recently retired after 50 (!) years of teaching, is Teacher Emeritus of Mathematics of Alamo High Heights School in San Antonio, Texas. He is the author of several textbooks including Calculus: Concepts and Applications, Second Edition, 2005, published by Kendall Hunt Publishing Company, www.kendallhunt.com (Formerly published by Key Curriculum Press).

Many years ago (almost 50) I remember teaching from his fine Trigonometry book . 

Thank you, Paul!

Why Limits?

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There are four important things before calculus and in beginning calculus for which we need the concept of limit.

    1. The first is continuity. Most of the time in pre-calculus mathematics and in the calculus we deal with nice continuous functions or functions that are not continuous at just a few points. Limits give us the vocabulary and the mathematics necessary to describe and deal with discontinuities of functions. Historically, the modern (delta-epsilon) definition of limit comes out of Weierstrass’ definition of continuity.
    2. Asymptotes: A vertical asymptote is the graphical feature of function at a point where its limit equals positive or negative infinity. A horizontal asymptote is the (finite) limit of a function as x approaches positive or negative infinity.

Ideally, one would hope that students have seen these phenomena and have used the terms limit and continuity informally before they study calculus. This is where the study of calculus starts. The next two items are studied in calculus and are based heavily on limit.

3. The tangent line problem. The definition of the derivative as the limit of the slope of a secant line to a graph is the first of the two basic ideas of the calculus. This single idea is the basis for all the concepts and applications of differential calculus.

4. The area problem. Using limits it is possible to find the area of a region with a curved side, even if the curve is not something simple like a semi-circle. The definite integral is defined as the limit of a Riemann sum and gives the area regions with a curved side. This then can be extended to huge number of very practical applications many having nothing to do with area.

So these are the main ways that limits are used in beginning calculus. Students need a good visual  understanding, what the graph looks like,  of the first two situations listed above and how limits describe and define them. This is also necessary later when third and fourth come up.