Numerical Methods – Final work-sheet
Numerical derivatives
Forward finite difference
𝛿(𝑥)=𝑓(𝑥+∆𝑥)−𝑓(𝑥)
∆𝑥
Backward finite difference
𝛿(𝑥)=𝑓(𝑥)−𝑓(𝑥−∆𝑥)
∆𝑥
Centered finite difference
𝛿(𝑥)=𝑓(𝑥+∆𝑥)−𝑓(𝑥−∆𝑥)
2∆𝑥
Numerical Integration
Left endpoint approximation
𝑓(𝑥)𝑑𝑥
≈𝑓(𝑥)∆𝑥
Right endpoint approximation
𝑓(𝑥)𝑑𝑥
≈𝑓(𝑥)∆𝑥
Midpoint approximation
𝑓(𝑥)𝑑𝑥
≈𝑓𝑥+𝑥
2∆𝑥
The Trapezoidal Rule
𝑓(𝑥)𝑑𝑥
=∆𝑥
2(𝑓(𝑥)+2𝑓(𝑥)
+𝑓(𝑥))
Simpson 1/3 Rule
𝑓(𝑥)𝑑𝑥
=∆𝑥
3(𝑓(𝑥)+4𝑓(𝑥)
+2 𝑓(𝑥)
+𝑓(𝑥))
Simpson 3/8 Rule
𝑓(𝑥)𝑑𝑥
=3∆𝑥
8(𝑓(𝑥)+3𝑓(𝑥)
+2𝑓(𝑥)
+𝑓(𝑥))
