While I think it’s important to focus on the math, it’d be an inaccurate post if I didn’t dedicate some of it to our Scribe Post discussion, since we spent about half the class debating this issue.
Benefits:
•The posts are useful for review later in the course, after we’ve moved on to other units.
•The Scribe can (in some cases) absorb the information better when writing.
•Writing about mathematics is a useful and important part of any math course.
•Note taking prepares us for college.
Challenges:
•Posts take more than 1 hr.
•It’s sometimes more difficult for the scribe to understand the lesson.
•People prefer their own notes and therefore don’t read the scribe post.
•The post would be easier to read with bullet points
•If the Scribe doesn’t understand the material, then the notes are obsolete.
Andy suggested that maybe someone could just write down what we did, and maybe post a link or two to help us understand it, instead of writing so much. Mr. O’Brien expressed the importance of thorough scribe posts, but reminded us that we will not be compared to previous scribes. We will be judged by the rubric.
http://math-ob.wikispaces.com/
Also, if you guys want to make it easier, Mr. O’Brien suggested Diigo, which allows you to highlight as you read, then go back and just read the important text when it’s test time. Check it out: http://www.diigo.com/
Now, to the math:
Today’s topic was Continuity, a concept that can apply to both a point, or a whole function. For point c to be continuous, three things must be true of the function, f:
1.
2. *
3.
*Note that at the endpoints of a function, the function needs only to be one-sided.
Now, a function is called continuous if all points in the function’s domain are continuous. Therefore, linear functions, parabolas, cube-root functions, exponential and square root functions are all continuous. In fact, even reciprocal functions are continuous, since the asymptotes are not technically part of the functions’ domains. In fact, all of the functions we encountered in the first half of high school were continuous, except the Step Function. For those of you who've forgotten about the step function, this should be a (very) quick reminder:
http://www.mathwords.com/f/floor_function.htm
The step function shares something with the function we encountered last week: A non-removable discontinuity. Unlike the removable variety, the unfriendly non-removable discontinuity is a huge, vertical gap. We can't just simply imagine it was a connecting piece in the function, because it'd be a vertical line (Vertical lines ruin functions. Always.)
Mr O’Brien then set some rules for continuous functions:
Continuous + Continuous = Continuous
Continuous - Continuous = Continuous
Continuous * Continuous = Continuous
Continuous / Continuous = Continuous
Continuous(Continuous) = Continuous
Let's now take a look an interesting function:
We know this function is a continuous function. How? Both the numerator and the denominator are continuos functions. Now, let's take a look at this function numerically, graphically, and algebraically.
When we plug this function into our calculator, and substitute values approaching 1, we find that when approaching from above and below, the limit= -1.359. Look familiar? Not unless you're used to dividing the irrational number "e" by two, and then negating it, because that's exactly what's going on.
So, we know that . Now, to take a look graphically. In class we found it was useful to break the function in two, and then take a look at both continuous functions on one set of axes. Here, the red function is 2x-2 and the green function is
.
As we learned a few weeks back, if you zoom in close enough on a curve, you find a section that's close enough to a straight line to be considered one. This concept, called local linearity, can be used to estimate the slopes of the two lines at their intersection point. Here's what happens as we zoom in on 1.
well, we know the slope of the 2x-2 is 2, and the but the slope of for this segment is more difficult to tell. At this point in class, we used the nDeriv function on our graphing calculators (push the math button to find it). The set up here is nDeriv(the function, the variable, the number at which we are evaluating, how close we want to look). In this case nDeriv(
, x, 1, .0000001). The answer is -e, which makes sense when we look back at the graph. The slope is negative, and slightly steeper than 2. Now, we can see the that slope at f(1) is -e for the numerator function, and 2 for the denominator function. Remember, the slope at a particluar point is the derivative. When we put the function back together as a fraction, we can see one derivative over the other:
That's how we get the limit of the function graphically.
While both of these approaches are successful, it's the algebraic approach that's truly enlightening. Let's look at the two functions separately:
Numerator:
Denominator:
Even without the graph, we know that both of these functions equal 0 at f(1)/g(1). We can take this shared point and the derivatives at (1,0) and plug it all into the point-slope formula (y-y1=m(x-x1)) to get the equations very quickly.
--> y=-e(x-1)
--> y=2(x-1)
Again, let's put the function back together as one fraction,. Now, we still have the function of numerator over denominator, but by using local linearity to find the derivative, we have transformed the numerator function from a equation of a curve to the equation of a line. Does
? Of course. As x approaches one, the numerator function BECOMES the linear function -e(x-1). At this point, we can cancel and rewrite the limit as
. Now, e has numerical value, and two does as well, so as x approaches one, the limit is
. As we can see, the slope of the numerator over the slope of the denominator is the limit.
We then revisited a previously difficult limit question: . Using local linearity, we can look at the relationship of slope to slope. Let's factor an (x-5) out of the numerator and denominator leaving us with
. We have the slope over the slope, now, so the limit equals 3/2.
So, we now see why . At zero, the slope of sin(x) is one, and at zero (and everywhere else) the slope of x is one. Therefore, asking for the
is the same as asking for the
, which is clearly one. The great thing is, we can use this trick to simplify limits to the point where we don't need graphical interpretation or tables.
Here's a great link that really explains the basics of continuity, and then goes a bit more in depth: http://tutorial.math.lamar.edu/Classes/CalcI/Continuity.aspx
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