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02.16.08
Posted in Math 1a, Spring 2008 at 6:08 pm by leingang
In the past week we’ve defined the derivative as the limit of a difference quotient. Graphically, it’s the slope of the line tangent to a curve at a point. But it has quite a few interpretations in various models. If the function is position, the derivative is velocity. If the function is cost, the derivative is marginal cost. If the function is mass, the derivative is density. The number of different uses for the derivative are what make it so important to study.
We looked at various interplays between a function and its derivative. For instance, when a function is increasing on an interval, the derivative is nonnegative (positive or zero) on that interval. If the function is decreasing, the derivative is negative or zero. And if the graph has a horizontal tangent line, the derivative is zero. It turns out–and we will be showing this when we get to The Most Important Theorem in Calculus–that the converses of these statements are pretty much true as well. That is, if a function has a positive derivative on an interval, it’s increasing on that interval, and so on.
The derivative can’t tell you everything, however. For one, it can’t tell you the function value at a specific point without any other information. This is true even of constant functions, which all have the derivative zero, no matter what the constant is. It also can’t tell you about the concavity of the graph of a function. But that can be computed from the second derivative.
Although we define the derivative in terms of a limit, eventually we will develop a set of rules to make calculating derivatives easier. That’s the subject of Chapter 3. However, it’s important not to lose the foundational ideas behind what the derivative actually is and what is for.
Slides for this week are posted.
Tags: math, math1a, function, derivative, calculus
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02.07.08
Posted in Math 1a, Spring 2008 at 2:14 pm by leingang
Continuity is the property that the limit of a function near a point is the value of the function near that point. It’s one of many “nice” properties a function can have. Functions can fail to be continuous in a number of different ways, including jumping from one value to another, having a simple “removable” discontinuity, or blowing up to infinity near a point.
Lots of times we want to model a situation with a function, and we assume the function is continuous to make our lives easier. For instance, the population of a bacteria culture in a dish. Clearly there are only whole numbers of organisms in the culture, but we pretend the population function of time is continuous because relative to the population size, an error of one by rounding isn’t that much.
An important consequence of continuity is the intermediate value theorem, which says that a continuous function on an interval assumes all the values in between the values at the endpoints. This theorem has important consequences, for instance:
- the square root of two exists
- at some point in your life your height in inches equaled your weight in pounds
- A table on an uneven floor can always be turned so that it won’t wobble
- Right now there are points on opposite sides of the world with the same temperature!
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Tags: math, math1a, function, continuity, limit
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02.05.08
Posted in Math 1a, Spring 2008 at 11:57 am by leingang
Today (yesterday, actually) we introduced the concept of limit: The limit of a function near a point is the number to which function values get arbitrarily close to by making the points sufficiently close to a given point. Or,
if values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a.
There are several ways a function can fail to have a limit. One is that limits “from the right” and “from the left” exist but do not agree. In this case there are two good candidates for the limit, but neither satisfies the definition completely. Another possibility is for the function to be unbounded near a. Then we cannot get arbitrarily close to any finite value. A third possibility is the kind of oscillation you see in the graph of y = sin(1/x): the frequency of oscillation increases near zero.
But limits do behave well with respect to arithmetic: for instance, the limit of a sum of functions at a point is the sum of the limits of those functions at that point (provided these limits exist). The only caveat here being that we still can’t divide by zero, so for a limit of a quotient where the denominator has a limit of zero, no limit law applies.
In many cases, the limit of a function at a point is the value of the function near that point. We call this property the Direct Substitution Property, and we will talk much more about it Wednesday.
Another useful method of computing limits is the Squeeze Theorem. It allows to “pinch” a function between two functions that are known to have the same limit. It follows that the middle function has the same limit.
Slides are now on the website.
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Tags: math, math1a, function, limit
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02.01.08
Posted in Math 1a, Spring 2008 at 3:50 pm by leingang
Today we had a rousing game of Name Bingo, and then we settled down to set up the one of the principal ideas of the course.
Finding the velocity of a moving object involves finding a distance traveled and dividing by the time it takes to move that distance. Smaller intervals are better, but there’s no smallest positive interval so what can we do? For now we take as small an interval as we can and approximate.
Finding the slope of a line tangent to a curve at a point is another problem. We need its slope, but how do we find that when we only know one point on the line? We approximate the tangent line with a secant line, measure the slope of the secant line (knowable since it goes through two points on the curve), and use secants to get the best possible approximation. Again, there’s no smallest distance between two points on the curve, so we have to decide what the successive approximations are actually approximating.
But the idea of taking something easy and well-understood (the slope of a line defined by two points) and using it to get at something complicated (the slope of a line defined by a point and a tangency condition) is pervasive. And it turns out that our two calculations–velocities and tangents–are formally very similar. In fact, we can find the instantaneous rate of change for any function using a limiting process like this. Formalizing the process is the matter of the next few sections.
Slides are on the web now.
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Tags: mathmath1a function limit derivative
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01.30.08
Posted in Math 1a, Spring 2008 at 11:43 am by leingang
It’s shopping period, so my Math 1a class today was basically an advertisement for the course. I clicked through the Introductory presentation and handed out copies of the syllabus.
Here’s one of my favorite slides.
I want students to know that my goal is not to prevent them from succeeding, but to enable them to succeed. Even though a college-level math course is a lot of work, we try to provide support with office hours, problem sessions, and homework help in the Math Question Center
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Tags: mathmath1a slideshow humor
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Posted in Math 1a, Spring 2008, Math 21a, Spring 2008 at 6:50 am by leingang
Math 21a meets in sections, so it won’t start officially until Monday the 4th. But Math 1a has its first meetings today and Friday.
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01.29.08
Posted in Math 1a, Spring 2008 at 3:56 pm by leingang
I was the course head for Math 1a (Introduction to Calculus) last fall, and I’m happy to be continuing it in the Spring. The material will be the same, but in the spring there will only be a single section. So I’ll have less administrative overhead and more time to concentrate on a smaller number of students.
I’m also trying a new blogging idea. Rather than keep separate blogs on each of my course websites, I’m putting as much on my office blog and filtering it to the other sites with RSS feeds. We’ll see how that goes.
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Tags: math1a
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05.21.07
Posted in Math, Math 1b, Math 1a, Education at 7:50 am by leingang
Last week an Ontario-based programmer named Antonio Cangiano started writing his Math Blog - Mathematics is wonderful! (I agree, BTW). Only two articles so far, but one of them rose up the digg ranks pretty quickly and crashed his server. So maybe he’s doing something right. 
Refresh your High School Math skills is a post containing precalculus math problems. I’d agree with him that these are the kinds of faculties we’d like our students to have going into calculus–algebra, trigonometry, inequalities, familiarity with exponentials and logarithms, etc. I wish we could assume more of the conic sections material was taught but it doesn’t seem that way anymore.
His other post is called “The most enlightening Calculus books” and is about his favorite books. There is massive debate among college math teachers about how best to teach calculus: reform, IBL, “Harvard Calculus” (which I do not teach), the list goes on. And as someone who has perused dozens of free calculus books from publishing companies, I can say that I still haven’t found the perfect book for wide university appeal.
What I want in a freshman calculus book is:
Tell no lies
I don’t insist on epsilons and deltas in a book, but I think we can get within epsilon of it (sorry). The concept that f(x) can be made arbitrarily close to L by taking x sufficiently close to a is precisely the definition without the greek letters, absolute value bars, and the dreaded less-than sign.
I think the derviative should be defined as a limit of difference quotients, and the integral should be defined as a limit of Riemann sums. I don’t think we need to prove that all continuous functions are integrable (that requires uniform continuity, which requires compactness of closed intervals, which I think is a little much), but the Fundamental Theorem of Calculus needs to be proved.
There is a tightrope to walk here. If you get too technical, students’ eyes will glaze over. I just don’t think everyone needs to know about epsilons and deltas. But if you get too hand-wavy, you lose the faculty to speak in any rigorous fashion about any limit, and suddenly every theorem becomes an article of faith.
Relevance
I think today’s students are interested in putting everything together rather than following many subjects down their separate paths. So I’d like a book that includes as many applications as possible. Calculus is the universal language of science, and I want my students to think of it as something that continues to be relevant. Of course there are the myriad physics applications that mathematicians are most familiar with, but the majority of our students are concentrating in (a) economics or (b) some sort of life sciences or pre-med. So give me problems in comparative statics, theory of the firm, population systems, rates of absorbing medicine, etc.
Problems
Many of our students get discouraged about the difference between homework and test problems. I really believe that for a student to demonstrate mastery of calculus, they need to be able to solve “new” problems. I don’t think the students are well served if each exam problem is similar to a homework problem. Again, calculus is not something that has been solved and put in books to be memorized; it is a tool which can be used ad infinitum.
So I also want conceptual problems that are unique, and enough of them to give the impression that this is what calculus “is.” I like drill-type problems for practicing the techniques (after all, the word “calculus” means a set of rules for deriving something), but limiting calculus to that is like saying all you need to know to be a carpenter is how to saw a board in half.
Antonio’s a big fan of Calculus by Michael Spivak. Indeed, it is a beautiful book; it changed my life in my first year of college at the University of Chicago. It has excellent prose, wonderful, challenging problems, and the kind of snarkiness that appeals to smart math students and their teachers. I still pick it up about once a month. Yet, as someone in charge of teaching calculus to hundreds of college students, I can’t imagine using it. I don’t think every single student is going to be receptive to that kind of book.
So the quest continues.
technorati tags:math, books, calculus, spivak
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01.06.06
Posted in Math, News, Math 1a at 6:56 am by leingang
Here’s the story of a guy who’s got some of his priorities straight. Farris Hassan, 16, of Ft. Lauderdale, traveled secretly to Iraq to practice what he learned in his high school about “immersion journalism.” He’s promising to talk about what he saw there, but has to catch up on his schoolwork:
“I have a big calculus test on relative rates of change that I need to study for, so I have a lot of things going on right now,” Hassan said.
Good luck on the test, Farris, and we’re looking forward to hearing about your experience.
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