## Symbolic Math Toolbox |

This example shows how to perform simple calculus operations using Symbolic Math Toolbox™.

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

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

Evaluate the function at x=pi/2 using math notation.

f(pi/2)

ans = 1/5

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

f1 = diff(f) ezplot(f1)

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

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

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

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

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

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

e = simplify(e) ezplot(e)

e(x) = 1