A Simple Calculus Example

This demo shows how to use the Symbolic Math Toolbox with a simple example from calculus.

Copyright 1984-2002 The MathWorks, Inc. $Revision: 1.10 $ $Date: 2002/03/31 23:38:47 $

To manipulate a symbolic variable, create an object of type SYM.

x = sym('x')

 
x =
 
x
 

Once a symbolic variable is defined, you can use it to build functions. EZPLOT makes it easy to plot symbolic expressions.

f = 1/(5+4*cos(x))
ezplot(f)

 
f =
 
1/(5+4*cos(x))
 

Many functions can work with symbolic variables. For example, DIFF differentiates a function.

f1 = diff(f)
ezplot(f1)

 
f1 =
 
4/(5+4*cos(x))^2*sin(x)
 

DIFF can also find the Nth derivative. Here is the second derivative.

f2 = diff(f,2)
ezplot(f2)

 
f2 =
 
32/(5+4*cos(x))^3*sin(x)^2+4/(5+4*cos(x))^2*cos(x)
 

INT integrates functions of symbolic variables. The following is an attempt to retrieve the original function by integrating the second derivative twice.

g = int(int(f2))
ezplot(g)

 
g =
 
-8/(tan(1/2*x)^2+9)
 

At first glance, the plots for f and g look the same. Look carefully, however, at their formulas and their ranges on the y-axis.

subplot(1,2,1)
ezplot(f)
subplot(1,2,2)
ezplot(g)

e is the difference between f and g. It has a complicated formula, but its graph looks like a constant.

e = f - g
subplot(1,1,1)
ezplot(e)

 
e =
 
1/(5+4*cos(x))+8/(tan(1/2*x)^2+9)
 

To show that the difference really is a constant, simplify the equation. This comfirms that the difference between them really is a constant.

e = simple(e)
ezplot(e)

 
e =
 
1