Learn calculus and applied mathematics using the Symbolic Math Toolbox™. The example shows introductory functions `fplot`

and `diff`

.

To manipulate a symbolic variable, create an object of type `syms`

.

`syms x`

Once a symbolic variable is defined, you can build and visualize functions with `fplot`

.

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

f(x) =$$\frac{1}{4\hspace{0.17em}\mathrm{cos}\left(x\right)+5}$$

fplot(f)

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

f(pi/2)

ans =$$\frac{1}{5}$$

Many functions can work with symbolic variables. For example, `diff`

differentiates a function.

f1 = diff(f)

f1(x) =$$\frac{4\hspace{0.17em}\mathrm{sin}\left(x\right)}{{\left(4\hspace{0.17em}\mathrm{cos}\left(x\right)+5\right)}^{2}}$$

fplot(f1)

`diff`

can also find the $${N}^{th}$$ derivative. Here is the second derivative.

f2 = diff(f,2)

f2(x) =$$\frac{4\hspace{0.17em}\mathrm{cos}\left(x\right)}{{\left(4\hspace{0.17em}\mathrm{cos}\left(x\right)+5\right)}^{2}}+\frac{32\hspace{0.17em}{\mathrm{sin}\left(x\right)}^{2}}{{\left(4\hspace{0.17em}\mathrm{cos}\left(x\right)+5\right)}^{3}}$$

fplot(f2)

`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))

g(x) =$$-\frac{8}{{\mathrm{tan}\left(\frac{x}{2}\right)}^{2}+9}$$

fplot(g)

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

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

e = f - g

e(x) =$$\frac{8}{{\mathrm{tan}\left(\frac{x}{2}\right)}^{2}+9}+\frac{1}{4\hspace{0.17em}\mathrm{cos}\left(x\right)+5}$$

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)

`e(x) = $$1$$`