%% 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))
fplot(f)

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

f(pi/2)

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

f1 = diff(f) 
fplot(f1) 

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

f2 = diff(f,2) 
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)) 
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 

% 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) 

% Copyright 2016 The MathWorks, Inc.