Learn Calculus in the Live Editor

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

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

fplot(f1) 

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

f2 = diff(f,2) 

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

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

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

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) = $$1$$