1. Numerical Differentiation and Integration in MATLAB; Function M-files
259 Lecture 16
Numerical Differentiation and Integration in MATLAB;
Function M-files

2. Derivatives and Integrals
We can use MATLAB to numerically differentiate or integrate functions!
The key is to remember the definitions of derivative and definite integral: 2

3. Numerical Differentiation
For small x, we have the following estimate for f’(x):
Using the MATLAB commands “linspace” and “diff”, we can find reasonable approximations to f’(x), provided x is small and f is differentiable!

4. Numerical Differentiation
Example 1:
Use MATLAB to find the numerical derivative of y = sin(x) on the interval [0, 2].
Compare this estimate for dy/dx to the actual derivative of y.

5. Numerical Differentiation
Example 1 (cont.):
Try each of the following commands with a = 0, b = 2*pi, and n = 50.
linspace(a, b, n)
a:(b-a)/(n-1):b
What do you notice?
Try “linspace(a,b)”.
What happens in this case?

6. Numerical Differentiation
Example 1 (cont.):
Next, try these commands:
x = linspace(1, 10, 10)
diff(x)
What happens?
In general, for x = [a, b, c, d], diff(x) = [b-a, c-b, d-c].
Now we are ready to estimate dy/dx!

7. Numerical Differentiation
Example 1 (cont.):
Enter the following commands to create an estimate for dy/dx, which we’ll call yprime:
x = linspace(0, 2*pi, 1000);
y = sin(x);
deltax = diff(x);
deltay = diff(y);
yprime = deltay./deltax;

8. Numerical Differentiation
Example 1 (cont.)
To compare our estimate to the actual derivative, let’s look at a table of the first 10 values of yprime and cos(x), via concatenation:
[yprime(1:10); cos(x(1:10))]’.
Note the use of the colon (:) and transpose (‘) commands!

9. Numerical Differentiation
Example 1 (cont.)
Let’s also compare graphically!
One way to do this is with the “subplot” command.
Try the following commands:
subplot(1,2,1)
plot(x(1:999), yprime, 'r‘)
title(‘yprime = \Deltay/\Deltax’)
subplot(1,2,2)
plot(x(1:999), cos(x(1:999)),‘b')
title(‘dy/dx = cos(x)’)
Why can’t we just use “x, yprime”, etc. in our plot commands?

10. Numerical Integration
For integrable function f(x), choosing xi = (b-a)/n and xi* to be the right endpoint of the ith subinterval, i.e. xi*=a+i*(b-a)/n, we get the following estimate for large n:
Using the “sum” command, we can find an estimate for definite integrals via Riemann sums!

11. Numerical Integration
Example 2:
Use MATLAB to numerically estimate the definite integral
Compare this estimate as n gets larger to the actual value for the integral.

12. Numerical Integration
Example 2 (cont.)
Enter the following commands to compute Riemann sums for our integral!
a = 0;
b = 1;
n = 50;
deltax = (b-a)/n;
xstar = a+deltax:deltax:b;
Rn = sum((xstar.*xstar) *deltax)

13. Function M-Files
In addition to using M-files to run scripts, we can use them to create functions!
Function M-files can accept input and produce output. One example of a function defined by a function M-file is “linspace”.
Let’s make a function via an M-file!

14. Function M-Files Within the M-file script editor, type the following:
function y = Sample1(x)
%Here is where you put in information about how the function is used.
%The syntax and variables can be outlined here as well.
y = x + x.^2 – x.^4;
Save the file as Sample1.m on the Desktop and make sure the Path is set to see the file!

15. Function M-Files
To use the new function, for example to find the value of Sample1(x) at x = 3, type:
Sample1(3)
Find the numerical derivative of Sample1(x) and plot both y = Sample1(x) and its numerical derivative on [-1,1].

16. References
Using MATLAB in Calculus by Gary Jenson 16