Month: April 2010

Maxima Tutorial 3 – Trigonometric Functions and Calculus

Maxima, Software Tutorials, Web 2.0 Technology

In this tutorial, we are going to discuss some of the most elementary commands used in Maxima in Trigonometry and Calculus. Note that in maxima, most constants are written with the percent sign before it. For instance, the most common constants are π, e and i are written as %pi, %e and %i.

Maxima can perform the following trigonometric and calculus operations:

  • Get the value of trigonometric functions
  1. sin(%pi/2);
  2. cos(%pi/4);
  3. tan(%pi/6);
  4. [sin(%pi/3),cos(%pi/3)];

As you can see, in 4, we can get the values simultaneously by separating the functions with comma and enclosing them with bracket.

  • Get the Limits of Functions
  1. computes the limit of 2x3 + 5x2– 2x as x approaches -1: limit(2*x^3 + 5*x^2 – 2*x – 1, x, -1);
  2. computes the limit of (x4 + 4x2)/(2x4 – 2) as x approaches negative infinity: limit((x^4 + 4*x^2)/(2*x^4 – 2), x, minf);
  3. computes the limit of (x4 + 4x2)/(2x4 – 2) as x approaches infinity: limit((x^4 + 4*x^2)/(2*x^4 – 2), x, inf);
  • Get the Derivatives and Integrals of Functions
  1. Get the derivative of derivative of (x4 + 2x3 + 5x2): diff(x^4 + 2*x^3 + 5*x^2, x);
  2. Perform indefinite integration: integrate(x^3 + 4*x^2 – 2*x, x);
  3. Perform definite integration: integrate(x^3 – 2*x^2 + x, x, -1, 2)
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Applets For Visualizing Isometric and Three Dimensional Views

Resources and Freebies, Teaching Tools, Web 2.0 Technology

I came across with two applets I found very useful. They can be used for practicing visualizing the relationship between isometric and three-dimensional views. The first applet is called “Rotating houses”.  The applet requires the user to rotate an object until the desired isometric view is copied.

Note: Click the pictures below to load the applets.

The second applet is called “Building houses with side views“. Given the isometric views (front, top, side), the applet lets you construct the actual object in three dimensions.

The highest point is given if you can construct the “houses” with the least number of blocks.

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The Definition of “Undefined”

High School Calculus, High School Mathematics, Questions and Quandaries

In learning mathematics, we often encounter terms that are not always clear. Example of such term is the word undefined. What do we mean by undefined?

The word undefined may slightly differ in meaning depending on the context. In plain language, it means something which has no sensible meaning. For instance, during the time when the negative numbers were not yet invented, the numerical expression 5 – 8 has no meaning. In our time, we can say that 5 – 8 is undefined in the set of positive integers.

Below are some examples of the different contexts where the different meanings of “undefined” can be drawn.

Numbers

Square Root of Negative Numbers. The \sqrt{-1} is undefined in the set of real numbers. This means that no real number exists that when multiply it by itself, the product is equal to -1. Note, however, that some operations may be undefined under some sets, but defined in other sets. We know from high school mathematics that square root of – 1 equals i in the set of complex numbers.

Algebra

Division by 0. Since we do not know the answer if a number divided by 0, let us examine the quotient of numbers when divided by numbers close to 0. To make it simple, let us try 1/0.

Figure 1 – The value of the 1/x as x approaches 0.

As we can see, as x approaches 0 from the right, the quotient of 1/x is getting larger and larger. On the other hand, as x approaches 0 from the left, 1/x is getting smaller and smaller. As a consequence, there is no single number that 1/x approaches as x approaches 0.   Therefore we can say that 1/0 is undefined.

Geometry

Intersection of two lines. In Euclidean Geometry, if we talk about the intersection of two lines, we can have three cases: intersecting lines have one intersection, coinciding lines have infinitely many intersections, and parallel lines have no intersection.

Figure 2 – The three cases, in terms of intersection, two lines can be places on a plane.

We can say that if two lines are parallel, no intersection exists. Algebraically, the solution to the system of equations of the two lines is their intersection.  Hence, the solution of the two systems of linear equations of parallel lines as graphs is undefined.

 

Matrices

Matrices with Different Sizes. If A is a 2 by 2 matrix and B is a 3 x 3 matrix, then A + B has no meaning since five of the entries of matrix B have no corresponding entries in matrix A. We can say that the sum of matrix A and matrix B is not well-defined.

Sets

Intersections of Sets. If E is the set of even integers and O be the set of odd inteers, then there is no common value to both sets.

Figure 3 – The Venn Diagram of the intersection of even and odd numbers.

In set theory, we call the common values the intersection, and in this example, the intersection is the empty set.  As a consequence, we can say that the intersection of set E and O is undefined.

***

Although the word “undefined” has different meanings depending on the context, by now, you would have realized that the phrases “does not exist”, “without sensible meaning” and “cannot be determined” are somewhat synonymous to it. If the result of an operation yields no value  at all (note that 0 is a value and is not the same to no value), then it is more likely that it is undefined.

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