Symbolic Math Toolbox

Calculus Example

This section describes how to analyze a simple function to find its asymptotes, maximum, minimum, and inflection point. The section covers the following topics:

• Defining the Function
• Finding the Asymptotes
• Finding the Maximum and Minimum
• Finding the Inflection Point

Defining the Function

The function in this example is

To create the function, enter the following commands:

• syms x
  num = 3*x^2 + 6*x -1;
  denom = x^2 + x - 3;
  f = num/denom
  

This returns

• f = 
  (3*x^2+6*x-1)/(x^2+x-3)
  

You can plot the graph of f by entering

• ezplot(f)
  

This displays the following plot.

Finding the Asymptotes

To find the horizontal asymptote of the graph of f, take the limit of f as x approaches positive infinity:

• limit(f, inf)
  ans = 
  3
  

The limit as x approaches negative infinity is also 3. This tells you that the line
y = 3 is a horizontal asymptote to the graph.

To find the vertical asymptotes of f, set the denominator equal to 0 and solve by entering the following command:

• roots = solve(denom)
  

This returns to solutions to :

• roots =
  [ -1/2+1/2*13^(1/2)]
  [ -1/2-1/2*13^(1/2)]
  

This tells you that vertical asymptotes are the lines

and

You can plot the horizontal and vertical asymptotes with the following commands:

• ezplot(f)
  hold on % Keep the graph of f in the figure
  % Plot horizontal asymptote
  plot([-2*pi 2*pi], [3 3],'g')
  % Plot vertical asymptotes 
  plot(double(roots(1))*[1 1], [-5 10],'r') 
  plot(double(roots(2))*[1 1], [-5 10],'r')
  title('Horizontal and Vertical Asymptotes')
  hold off
  

Note that roots must be converted to double to use the plot command.

The preceding commands display the following figure.

To recover the graph of f without the asymptotes, enter

• ezplot(f)
  

Finding the Maximum and Minimum

You can see from the graph that f has a local maximum somewhere between the points x = 2 and x = 3, and might have a local minimum between x = -4 and x = -2. To find the x-coordinates of the maximum and minimum, first take the derivative of f:

• f1 = diff(f)
  

This returns

• f1 = (6*x+6)/(x^2+x-3)-(3*x^2+6*x-1)/(x^2+x-3)^2*(2*x+1)
  

To simplify this expression, enter

• f1 = simplify(f1)
  

which returns

• f1 = -(3*x^2+16*x+17)/(x^2+x-3)^2
  

You can display f1 in a more readable form by entering

• pretty(f1)
  

which returns

•                                    2
                                  3 x  + 16 x + 17
                                - ----------------
                                     2         2
                                   (x  + x - 3)
  

Next, set the derivative equal to 0 and solve for the critical points:

• crit_pts = solve(f1)
  

This returns

• ans =
  [ -8/3-1/3*13^(1/2)]
  [ -8/3+1/3*13^(1/2)]
  

It is clear from the graph of f that it has a local minimum at

and a local maximum at

Note    MATLAB does not always return the roots to an equation in the same order.

You can plot the maximum and minimum of f with the following commands:

• ezplot(f)
  hold on
  plot(double(crit_pts), double(subs(f,crit_pts)),'ro')
  title('Maximum and Minimum of f')
  text(-5.5,3.2,'Local minimum')
  text(-2.5,2,'Local maximum')
  hold off
  

This displays the following figure.

Finding the Inflection Point

To find the inflection point of f, set the second derivative equal to 0 and solve.

• f2 = diff(f1);
  inflec_pt = solve(f2);
  double(inflec_pt)
  

This returns

• ans =
    -5.2635          
    -1.3682 - 0.8511i
    -1.3682 + 0.8511i
  

In this example, only the first entry is a real number, so this is the only inflection point. (Note that in other examples, the real solutions might not be the first entries of the answer.) Since you are only interested in the real solutions, you can discard the last two entries, which are complex numbers.

• inflec_pt = inflec_pt(1)
  

To see the symbolic expression for the inflection point, enter

• pretty(simplify(inflec_pt))
  

This returns

•                    1/2 2/3                        1/2 1/3
        (676 + 156 13   )    + 52 + 16 (676 + 156 13   )
  - 1/6 ---------------------------------------------------
                                    1/2 1/3
                       (676 + 156 13   )
  

To plot the inflection point, enter

• ezplot(f, [-9 6])
  hold on
  plot(double(inflec_pt), double(subs(f,inflec_pt)),'ro')
  title('Inflection Point of f')
  text(-7,2,'Inflection point')
  hold off
  

The extra argument, [-9 6], in ezplot extends the range of x values in the plot so that you see the inflection point more clearly, as shown in the following figure.

Taylor Series

Extended Calculus Example