Tutorial-9
1. Using Newton forward divided difference formula, compute first derivative for the
tabulated function:
𝑥
𝑓(𝑥)
1
2
2
5
3
7
4
10
2. Given the values of empirical function 𝑓(𝑥) for certain value of 𝑥
1.0
1.2
1.4
1.6
1.8
2.0
𝑥
𝑓(𝑥)
0.0000
0.1280
0.5440
1.2960
2.4320
4.0000
Find the first and second derivatives of the function 𝑓(𝑥) at the point 𝑥 = 1.1.
3. Derive the rectangular rule with the error term
𝑏
∫𝑎 𝑓(𝑥) 𝑑𝑥 = (𝑏 − 𝑎) 𝑓(𝑎) +
(𝑏−𝑎)2
2
𝑓 ′ (𝜉), 𝜉 ∈ (𝑎, 𝑏)
and hence write the composite rule with error terms.
4. Find 𝑥1, 𝐴0 , 𝐴1 , 𝛼 so that the following integration rule is exact for a
polynomial of degree as high as possible
1
1
′′′
∫0 √𝑥 𝑓(𝑥)𝑑𝑥 = 𝐴0 𝑓 (3) + 𝐴1 𝑓(𝑥1 ) + 𝛼 𝑓 (𝜉), 𝜉 ∈ (0,1).
5. Find 𝑥0, , 𝑥1, 𝐴0 , 𝐴1 , and 𝛼 so that the following rule is exact for all polynomials
of degree as high as possible
1
∫−1 𝑓(𝑥)𝑑𝑥 = 𝐴0 𝑓(𝑥0 ) + 𝐴1 𝑓(𝑥1 ) + 𝛼 𝑓 (4) (𝜉), for some 𝜉 ∈ (−1,1).
0
