Review of Basic Results and Concepts

While Maple can easily create a plot of the graph of a function from the formula, important information about the graph of a function often requires tools from calculus and algebra. Such information includes analysis of asymptotes (both both horixzontal, vertical or linear), critical points, intervals of increase or decrease, concavity and points of inflection. Critical points are points on the graph of f ( x ) where the derivative is either zero or undefined. A function is increasing  on any interval on which the derivative is positive  and decreasing  on any interval on which the derivative is negative . A function is concave up  on any interval where the second derivative is positive and concave down on any interval where the second derivative is negative. A point on the graph of f ( x ) where the concavity changes is called a point of inflection .

Remember that the goal of this project is not to obtain the graph of the function f ( x ) but rather to obtain a complete analysis of the various properties of the graph. We now discuss some tools from Maple that are useful for obtaining such an analysis.

Some Useful Maple Tools

When the formula for a function becomes complex, it may be impractical to calculate the derivatives and their roots by hand. Instead we turn to Maple to compute the derivatives and the numerical solver ( fsolve ) to find good numerical approximations to the roots. In this situation the fsolve  command is very important.

In analyzing the graph of a function one often wants information about the sign of either the function or one of its derivatives. One can get qualitative information about the sign of a function by having Maple plot the graph of . In order to avoid "vertical lines" at points where the function changes sign, add the option "discont=true" to the plot command.

>	plot([sin(x),sin(x)/abs(sin(x))],x=-2*Pi..2*Pi,color=[red,blue],discont=true);

>	plot((x^2-4)/abs(x^2-4),x=-5..5,color=blue,discont=true);

Maple has several tools that are useful in dealing with asymptotes of a graph. One such tool is the limit  command. Sometimes one wants to calculate one-sided limits when verifying vertical asymptotes. Suppose .

>	plot(x^2/((x+2)*(2*x-3)),x=-5..5,y=-10..10,discont=true);

>	limit(x^2/((x+2)*(2*x-3)),x=-2,right);

>	limit(x^2/((x+2)*(2*x-3)),x=-2,left);

• Note that when using the limit command for a vertical asymptote, if you only have a numerical approximation for the x -value, the limit command may not give the correct response. In such cases you can verify the apparent behavior indicated by your plot by using the fsolve  command. For example if your plot seems to indicate that your function approaches  as x  approaches a particular number c  from one side, you can use the fsolve  command to solve f(x) = b where b is a large positive number such as 1000 or 10000.

For horizontal asymptotes, the limit  command with x  = infinity or x  = -infinity can be used. Recall that a rational function  has a horizontal asymptote as x  approaches plus or minus infinity if  and  are of the same degree.

>	limit(x^2/((x+2)*(2*x-3)),x=infinity);

>	limit(x^2/((x+2)*(2*x-3)),x=-infinity);

Recall that a rational function  has a linear  or oblique  asymptote when the degree of  is exactly one more than the degree of . Determining such an asymptote often requires long division . Maple has a slightly different tool for accomplishing this called a partial fraction decomposition . This tool in fact does more than long division.

>	convert(x^2/((x+2)*(2*x-3)),parfrac,x);

This command gives us an equivalent way of expressing . Notice that the terms with x  in the denominator approach 0 as  approaches . Thus the entire expression approaches  as  approaches . In other words,  is a horizontal asymptote. An example with a linear  asymptote is

>	convert((x^3-5*x^2+1)/(x^2-1),parfrac,x);

Again the terms with x  in the denominator approach 0 as  approaches , and so the values of our function approach values of the linear function x  - 5. In other words,  is a linear  or oblique  asymptote for the function . We plot the linear asymptote along with the function

>	plot([(x^3-5*x^2+1)/(x^2-1),x-5],x=-5..5,y=-10..10,color=[red,blue],discont=true);

If a rational function has been expanded into partial fractions, the simplify command applied to the expanded form will return to the standard form .

>	simplify(x-5-3/2*1/(x-1)+5/2/(x+1));

Project

Your instructor will either assign you one of the following functions or let you choose one yourself. Plot the graph of your function and do a complete analysis of the graph. Be sure to discuss the following properties

• Critical points
• Intervals of increase or decrease
• Concavity
• Points of Inflection
• Asymptotes
• Any other interesting features of the graph

Remember that you may need to give several different plots in different windows in order to illustrate all the interesting features of your function.

a)

b)

c)

Extra Credit

Construct a rational function f(x) having all of the following properties:

     1) The line  is an oblique asymptote

     2) f(x) has vertical asymptotes at x  = 2 and x  = -1.

     3) The graph of f(x) has a local minimum at (1,1)

     4) The graph of f(x) approaches  on both sides of x  = -1.

Try to construct your rational function similar to the output from a partial fraction expansion. Use functions of the form   where A is a constant and n  is a small integer as building blocks. Make a plot of your function together with the oblique asymptote to verify that it has all of the required properties. How many times does your function cross the oblique asymptote?