January 2013 - Math and Multimedia

Curve Sketching 4: Identifying Oblique Asymptotes

January 28, 2013 GB High School Calculus, High School Mathematics

In the previous post in this series, we have learned about asymptotes. horizontal asymptote and vertical asymptote. We continue this series by discussing oblique asymptotes in this post.

An oblique asymptote is an asymptote that is not vertical and not horizontal. We need to know these types of asymptotes to sketch graphs especially rational functions. A rational function contains an oblique asymptote if the degree of its numerator is 1 more than that of its denominator. For instance, the function

$y = \displaystyle\frac{x^2-4x-5}{x-3}$

has degree 2 in the numerator and 1 in the denominator. If we divide the expression, we have

$\displaystyle\frac{x^2 - 4x - 5}{x-3} = x - 1 + \displaystyle\frac{8}{x-3}$ .

Notice that as $x$ goes to infinity, the remainder goes to 0. The expression $x - 1$ is the oblique asymptote.

The red dashed line in the graph is the oblique asymptote of the function above. Notice that the function has also a vertical asymptote (see green dashed line) which is $x=3$ .

Note however, that if the degree of the numerator of the rational function is more than the degree of the denominator, but not 1, there are no oblique asymptotes. In addition, there is at most one oblique asymptote or one horizontal asymptote, but not both (Why?).

Reference: Bob Miller’s Calc for the Clueless: Calc I

2 comments curve sketching, curve sketching tutorial, graph sketching, oblique asymtpote, rational function

Ted Talk: Robert Lang on the Mathematics and Magic of Origami

January 27, 2013 GB General Education, Math in Real Life, Resources and Freebies

In the previous post, I have shared to you sites about origami or paper folding. Aside from these resources, I have also posted several examples of using it in the classroom. Several of these examples are introduction to the notion of proof, and getting the square root and cube root of a number. In addition, I have also shared a video using mathematics of origami to fold gift wraps minimizing wastage.

But origami is a lot more than the things you have read above. Today, the art is already used as a model for airbags and telescope in space. There is even a research for using it in devices that will be used in heart surgery. » Read more

One comment magic of origami, mathematics of origami, paper folding, robert lang, ted talk, uses of origami

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January 2013 – The Most Popular Posts

Curve Sketching 4: Identifying Oblique Asymptotes

Ted Talk: Robert Lang on the Mathematics and Magic of Origami