I want to review some of the material in the first two chapters.

• An even function is one that satisfies the equation f(x) = f(-x). This function is symmetric about the y-axis.
  
  
• An odd function is one that satisfies the equation -f(x) = f(-x). This function is symmetric about the origin.

When looking at a function for the first time, I should always take a moment to ask myself if it is even or odd. Most are neither, but part of what makes mathematics interesting is looking for and noticing such patterns.

• The graph of any polynomial is some form of bumpiness in the middle and then a strong trend up or down at each side, depending on the degree of the polynomial and the sign of the leading coefficient. If the degree is even, then the curve goes up or down at both ends depending on whether the coefficient is + or -. If the degree is odd, then the left side goes up if the coefficient is -ve and down if the coefficient is +, and the right side does the opposite.
  
  
• The graph will cross the y-axis at a value corresponding to f(0).

Once again, any time I see a polynomial expression I should try to visualize the general shape of the graph. This should become second nature to me. It may involve a moment's extra effort, but the goal is always to have a sense of what is happening. That is genuine understanding rather than rote.

• The graphs of reciprocal functions ( 1/x^n) look similar depending on whether n is odd or even. If n is even, then the function is even (i.e. symmetic about the y-axis) and approaches infinity as x approaches zero and approaches 0 as x approaches infinity. If n is odd, the function is odd, and is symmetric about the origin. But the curve does not cross the y-axis (that would imply dividing by 0). Instead the curve approaches infinity as x approaches 0 from the right and approaches 0 as x approaches infinity.

I think that more complicated reciprocal polynomials are unpredictable. This is where Mathematica can be useful. Let's play!

• Here is the link to a brief exploration of the reciprocal function.

This turned out to be much more unpredictable than I would have guessed! I think that when I become more familiar with calculus I will be able to compute derivatives and determine points of inflection and asymptotes for such functions and get a better sense of the curves. I want to move on and have a look at the logarithmic and exponential functions this morning.

• The graph of any exponential function a^x passes through the point (0,1) and rises quickly. The left side is asymptotic to 0.
  
  
• If x is negative, the curve still passes through (0,1) but now the left side rises quickly and the right side is asymptotic to 0.
  
  
• Note that the graph for the exponential function passes the horizontal line test and therefore has an inverse. This inverse is called the logarithm of the function. The curve of an inverse is the mirror image of the function where the mirror is the line y = x.
  
  
• Therefore the graph of log x is asymptotic to the y-axis and grows slowly as x increases.

I should be able to play with these relationships with Mathematica. This will involve learning various ways of having Mathematica evaluate inverses. This might be a good way to spend the next hour.

 

The reciprocal function turned out to be much more complicated that I imagined. Exploring some of the possibiliteis took more time than I anticipated and I thus failed to get deeply into the exponential and logarithmic functions. Total time this morning: about 2 hours.

Tags: mathematics, mathematica