MEng 124n
Advance Mathematics for
Mechanical Engineering
Module 7 and 8
Numerical Differentiation and Integration
Ayrton John V. Bantay
Instructor
Content:
1. Numerical Differentiation
(Finite Divided Difference)
a. Forward
b. Backward
c. Centered
Content:
2. Numerical Integration
a. Trapezoidal Rule
b. Simpson’s 1/3 Rule
c. Simpson’s 3/8 Rule
Numerical Differentiation
Forward
Centered
Backward
Forward finite-divided-difference
Backward finite-divided-difference
Centered finite-divided-difference
Example:
Using the high accuracy formulas for forward,
backward and centered finite difference. Calculate the
first derivative of the function at x = 0.5 with a step size
of h = 0.25.
4
3
2
𝑓 𝑥 = −0.1𝑥 − 0.15𝑥 − 0.5𝑥 − 0.25𝑥 + 1.2
Numerical Integration
Analytical
Trapezoidal
Simpson’s 1/3
Simpson’s 3/8
Trapezoidal Rule
Simpson’s 1/3 Rule
Simpson’s 3/8 Rule
Example:
Using the 3 numerical integration methods, estimate
the integral of the function from a = 0 to b = 0.8
𝑓 𝑥 = 0.2 + 25𝑥 − 200𝑥 2 + 675𝑥 3 − 900𝑥 4 + 400𝑥 5
a. Trapezoidal Rule (Single Application)
b. Trapezoidal Rule (Multiple Application with n = 6)
c. Simpson’s 1/3 Rule (Single Application)
d. Simpson’s 1/3 Rule (Multiple Application with n = 6)
e. Simpson’s 3/8 Rule (Single Application)
f. Simpson’s 3/8 Rule (Multiple Application with n = 6)
Graph:
Thank you.
God bless.
"Courage isn't having the strength to go on – it is going
on when you don't have strength."
― Napoleon Bonapart