JC TUTORIALS
MAT2611
ASSIGNMENT 10
SEMESTER 1
2022
, Problem 35:
Solution:
𝐷𝑒𝑓𝑖𝑛𝑒 𝑎𝑛 𝑖𝑛𝑛𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑛 𝑃3 (𝑅) 𝑏𝑦
1
〈𝑓, 𝑔〉 = ∫ 𝑓(𝑥) ∙ 𝑔(𝑥) 𝑑𝑥
0
𝐿𝑒𝑡 𝑞(𝑥) = 2 + 3𝑥 𝑎𝑛𝑑 𝑙𝑒𝑡
𝑈 = {𝑝 ∈ 𝑃3 (𝑅): 𝑝(0) = 0 , 𝑝′ (0) = 0}
𝐿𝑒𝑡𝑠 𝑓𝑖𝑛𝑑 𝑎𝑛 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 𝑏𝑎𝑠𝑖𝑠 𝑜𝑓 𝑈
𝐴 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑝(0) = 0 , 𝑝′ (0) = 0 ℎ𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚 0 𝑎𝑛𝑑 𝑓𝑖𝑟𝑠𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 𝑡𝑒𝑟𝑚
𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0. 𝑇ℎ𝑢𝑠 𝑎 𝑏𝑎𝑠𝑖𝑠 𝑜𝑓 𝑈 𝑖𝑠
(𝑥 2 , 𝑥 3 )
𝐴𝑝𝑝𝑙𝑦 𝐺𝑟𝑖𝑚 − 𝑆𝑐ℎ𝑚𝑖𝑑𝑡 𝑝𝑟𝑜𝑐𝑒𝑑𝑢𝑟𝑒 𝑡𝑜 𝑡ℎ𝑖𝑠 𝑏𝑎𝑠𝑖𝑠
𝐿𝑒𝑡 𝑓 = 𝑥 2 𝑎𝑛𝑑 𝑔 = 𝑥 3
𝑓
𝑒1 =
‖𝑓‖
MAT2611
ASSIGNMENT 10
SEMESTER 1
2022
, Problem 35:
Solution:
𝐷𝑒𝑓𝑖𝑛𝑒 𝑎𝑛 𝑖𝑛𝑛𝑒𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑛 𝑃3 (𝑅) 𝑏𝑦
1
〈𝑓, 𝑔〉 = ∫ 𝑓(𝑥) ∙ 𝑔(𝑥) 𝑑𝑥
0
𝐿𝑒𝑡 𝑞(𝑥) = 2 + 3𝑥 𝑎𝑛𝑑 𝑙𝑒𝑡
𝑈 = {𝑝 ∈ 𝑃3 (𝑅): 𝑝(0) = 0 , 𝑝′ (0) = 0}
𝐿𝑒𝑡𝑠 𝑓𝑖𝑛𝑑 𝑎𝑛 𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙 𝑏𝑎𝑠𝑖𝑠 𝑜𝑓 𝑈
𝐴 𝑝𝑜𝑙𝑦𝑛𝑜𝑚𝑖𝑎𝑙 𝑝 𝑠𝑎𝑡𝑖𝑠𝑓𝑦𝑖𝑛𝑔 𝑝(0) = 0 , 𝑝′ (0) = 0 ℎ𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑡𝑒𝑟𝑚 0 𝑎𝑛𝑑 𝑓𝑖𝑟𝑠𝑡 𝑑𝑒𝑔𝑟𝑒𝑒 𝑡𝑒𝑟𝑚
𝑎𝑙𝑠𝑜 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 0. 𝑇ℎ𝑢𝑠 𝑎 𝑏𝑎𝑠𝑖𝑠 𝑜𝑓 𝑈 𝑖𝑠
(𝑥 2 , 𝑥 3 )
𝐴𝑝𝑝𝑙𝑦 𝐺𝑟𝑖𝑚 − 𝑆𝑐ℎ𝑚𝑖𝑑𝑡 𝑝𝑟𝑜𝑐𝑒𝑑𝑢𝑟𝑒 𝑡𝑜 𝑡ℎ𝑖𝑠 𝑏𝑎𝑠𝑖𝑠
𝐿𝑒𝑡 𝑓 = 𝑥 2 𝑎𝑛𝑑 𝑔 = 𝑥 3
𝑓
𝑒1 =
‖𝑓‖
