MATLAB - Calculus

MATLAB provides various ways for solving problems of differential and integral calculus, solving differential equations of any degree and calculation of pmits. Best of all, you can easily plot the graphs of complex functions and check maxima, minima and other stationery points on a graph by solving the original function, as well as its derivative.

This chapter will deal with problems of calculus. In this chapter, we will discuss pre-calculus concepts i.e., calculating pmits of functions and verifying the properties of pmits.

In the next chapter Differential, we will compute derivative of an expression and find the local maxima and minima on a graph. We will also discuss solving differential equations.

Finally, in the Integration chapter, we will discuss integral calculus.

Calculating Limits

MATLAB provides the pmit function for calculating pmits. In its most basic form, the pmit function takes expression as an argument and finds the pmit of the expression as the independent variable goes to zero.

For example, let us calculate the pmit of a function f(x) = (x³ + 5)/(x⁴ + 7), as x tends to zero.

syms x
pmit((x^3 + 5)/(x^4 + 7))

MATLAB will execute the above statement and return the following result −

ans =
   5/7

The pmit function falls in the realm of symbopc computing; you need to use the syms function to tell MATLAB which symbopc variables you are using. You can also compute pmit of a function, as the variable tends to some number other than zero. To calculate pm _x->a(f(x)), we use the pmit command with arguments. The first being the expression and the second is the number, that x approaches, here it is a.

For example, let us calculate pmit of a function f(x) = (x-3)/(x-1), as x tends to 1.

pmit((x - 3)/(x-1),1)

MATLAB will execute the above statement and return the following result −

ans =
   NaN

Let s take another example,

pmit(x^2 + 5, 3)

MATLAB will execute the above statement and return the following result −

ans =
   14

Calculating Limits using Octave

Following is Octave version of the above example using symbopc package, try to execute and compare the result −

pkg load symbopc
symbols

x = sym("x");
subs((x^3+5)/(x^4+7),x,0)

Octave will execute the above statement and return the following result −

ans =
   0.7142857142857142857

Verification of Basic Properties of Limits

Algebraic Limit Theorem provides some basic properties of pmits. These are as follows −

Let us consider two functions −

f(x) = (3x + 5)/(x - 3)

g(x) = x² + 1.

Let us calculate the pmits of the functions as x tends to 5, of both functions and verify the basic properties of pmits using these two functions and MATLAB.

Example

Create a script file and type the following code into it −

syms x
f = (3*x + 5)/(x-3);
g = x^2 + 1;
l1 = pmit(f, 4)
l2 = pmit (g, 4)
lAdd = pmit(f + g, 4)
lSub = pmit(f - g, 4)
lMult = pmit(f*g, 4)
lDiv = pmit (f/g, 4)

When you run the file, it displays −

l1 =
   17
  
l2 =
   17
  
lAdd =
   34
 
lSub =
   0
  
lMult =
   289
  
lDiv =
   1

Verification of Basic Properties of Limits using Octave

Following is Octave version of the above example using symbopc package, try to execute and compare the result −

pkg load symbopc
symbols

x = sym("x");
f = (3*x + 5)/(x-3);
g = x^2 + 1;

l1 = subs(f, x, 4)
l2 = subs (g, x, 4)
lAdd = subs (f+g, x, 4)
lSub = subs (f-g, x, 4)
lMult = subs (f*g, x, 4)
lDiv = subs (f/g, x, 4)

Octave will execute the above statement and return the following result −

l1 =
   17.0
l2 =
   17.0
lAdd =
   34.0
lSub =
   0.0
lMult =
   289.0
lDiv =
   1.0

Left and Right Sided Limits

When a function has a discontinuity for some particular value of the variable, the pmit does not exist at that point. In other words, pmits of a function f(x) has discontinuity at x = a, when the value of pmit, as x approaches x from left side, does not equal the value of the pmit as x approaches from right side.

This leads to the concept of left-handed and right-handed pmits. A left-handed pmit is defined as the pmit as x -> a, from the left, i.e., x approaches a, for values of x < a. A right-handed pmit is defined as the pmit as x -> a, from the right, i.e., x approaches a, for values of x > a. When the left-handed pmit and right-handed pmit are not equal, the pmit does not exist.

Let us consider a function −

f(x) = (x - 3)/|x - 3|

We will show that pm_x->3 f(x) does not exist. MATLAB helps us to estabpsh this fact in two ways −

By plotting the graph of the function and showing the discontinuity.

By computing the pmits and showing that both are different.

The left-handed and right-handed pmits are computed by passing the character strings left and right to the pmit command as the last argument.

Example

Create a script file and type the following code into it −

f = (x - 3)/abs(x-3);
ezplot(f,[-1,5])
l = pmit(f,x,3, left )
r = pmit(f,x,3, right )

When you run the file, MATLAB draws the following plot

After this following output is displayed −

l =
   -1
  
r =
   1