Mobius Answers Assignment 6 PDF

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Q1

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Q6

𝑏

𝑛

𝑛

∫ 𝑓(𝑥)𝑑𝑥 = lim ∑ 𝑓(𝑥𝑖 )∆𝑥 = lim ∑ 𝑎

a) First, transform

4𝑖 4𝑖 (4 + ), 𝑛2 𝑛

n→∞

n→∞

𝑖=0

4𝑖 4𝑖 (4 + ) 𝑛2 𝑛

𝑖=1

so that it has the form 𝑓(𝑥𝑖 )∆𝑥

4𝑖 4𝑖 1 𝑖 𝑖 2 (4 + ) = ( ) + 16 ( ) ) (16 𝑛2 𝑛 𝑛 𝑛 𝑛

Next, write expressions for ∆𝑥, 𝑎, 𝑏, 𝑥𝑖 , 𝑓(𝑥) from the line above ∆𝑥 =

𝑏−𝑎 1 = 𝑛 𝑛

𝑥0 = 𝑎 = 0 ⇒ 𝑏 = 1

𝑥1 = 𝑥0 + ∆𝑥 = 0 +

1 1 = 𝑛 𝑛

𝑥2 = 𝑥0 + 2∆𝑥 = 0 + 2 …

𝑥𝑖 = 𝑥0 + 𝑖

1 2 = 𝑛 𝑛

1 𝑖 = 𝑛 𝑛

b) lim ∑𝑛𝑖=1 4𝑖2 (4 + 4𝑖 ) 𝑛 𝑛 n→∞

𝑛

𝑓(𝑥) = 16𝑥 + 16𝑥 2

𝑛

𝑖 𝑖 2 16 lim ∑ (( ) + ( ) ) 𝑛 𝑛 n→∞ 𝑛 𝑖=1

lim (

n→∞

Q7

𝑛

4𝑖 1 4𝑖 𝑖 𝑖 2 lim ∑ ) = lim ∑ (4 + 𝑛 (16 ( ) + 16 ( ) ) = n→∞ n→∞ 𝑛2 𝑛 𝑛 𝑛 𝑖=1 𝑖=1 𝑛

𝑛

𝑖=1

𝑖=1

16 16 = lim ( 2 ∑(𝑖) + 3 ∑(𝑖)2 ) = n→∞ 𝑛 𝑛

16 𝑛(𝑛 + 1) 16 16 80 40 16 𝑛(𝑛 + 1)(2𝑛 + 1) )) = )+ 3( + = = ( 2 6 2 2 3 6 𝑛 𝑛 3

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