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Symbolic Math Toolbox

Calculus

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

```