University:
East Georgia State College
Course:
MATH 1540 | Calculus I
Academic year:

2024
Views:

236

Pages:

2
Author:

somospumasl3o

Question 1 Given the recursive formula, write the explicit formula: 𝑎 1 = 5 a 1 =5 𝑎 𝑛 = 𝑎 𝑛 − 1 + 3 a n =a n−1 +3 Given the explicit formula, write the recursive formula: 2) 𝑎 𝑛 = 15 + 4 𝑛 a n =15+4n Answer: 1) 𝑎 1 = 5 a 1 =5, 𝑎 𝑛 = 𝑎 𝑛 − 1 + 3 a n =a n−1 +3 d = a_n - a_{n-1} = 3 ] a_n = a_1 + (n - 1)d ] 𝑎 𝑛 = 5 + ( 𝑛 − 1 ) 3 a n =5+(n−1)3 𝑎 𝑛 = 5 + 3 𝑛 − 3 a n =5+3n−3 𝑎 𝑛 = 2 + 3 𝑛 a n =2+3n 2) 𝑎 𝑛 = 15 + 4 𝑛 a n =15+4n General explicit formula: 𝑎 𝑛 = ( 𝑎 𝑛 − 1 − 𝑑 ) + 𝑛 𝑑 a n =(a n−1 −d)+nd Here, 𝑑 = 4 d=4, and 𝑎 1 = 15 a 1 =15: 𝑎 𝑛 = 𝑎 𝑛 − 1 + 𝑑 a n =a n−1 +d 𝑎 𝑛 = 𝑎 𝑛 − 1 + 4 a n =a n−1 +4 Question 2 a) Evaluate the polynomial: 𝑦 = 𝑥 3 − 7 𝑥 2 + 8 𝑥 − 0.35 y=x 3 −7x 2 +8x−0.35 at 𝑥 = 1.37 x=1.37. Use 3-digit arithmetic with chopping. Evaluate the percent relative error. b) Repeat (a) but express 𝑦 y as: 𝑦 = ( ( 𝑥 − 7 ) 𝑥 + 8 ) 𝑥 − 0.35 y=((x−7)x+8)x−0.35 Evaluate the error and compare with part (a). Answer 1: (a) First, determine the powers of 𝑥 x needed for evaluating the polynomial: 𝑥 = 1.37 , 𝑥 2 = 1.882 , 𝑥 3 = 2.579 x=1.37,x 2 =1.882,x 3 =2.579 Chopping to 3-digit arithmetic: 𝑥 2 = 1.3 7 2 = 1.88 , 𝑥 3 = 1.88 ⋅ 1.37 = 2.57 x 2 =1.37 2 =1.88,x 3 =1.88⋅1.37=2.57 Substitute into the polynomial: 𝑝 ( 1.37 ) = 𝑥 3 − 7 𝑥 2 + 8 𝑥 − 0.35 p(1.37)=x 3 −7x 2 +8x−0.35 𝑝 ( 1.37 ) = 2.57 − 7 ( 1.88 ) + 8 ( 1.37 ) − 0.35 p(1.37)=2.57−7(1.88)+8(1.37)−0.35 𝑝 ( 1.37 ) = 2.57 − 13.16 + 10.96 − 0.35 p(1.37)=2.57−13.16+10.96−0.35 𝑝 ( 1.37 ) = 0.02 p(1.37)=0.02 True value: 𝑝 ( 1.37 ) exact = 0.043053 p(1.37) exact =0.043053 Percent relative error: 𝜀 = ∣ 0.043053 − 0.02 ∣ ∣ 0.043053 ∣ × 100 = 53.54 % ε= ∣0.043053∣ ∣0.043053−0.02∣ ×100=53.54% (b) Expressing 𝑝 ( 𝑥 ) p(x) in the second form: 𝑝 ( 𝑥 ) = ( ( 𝑥 − 7 ) 𝑥 + 8 ) 𝑥 − 0.35 p(x)=((x−7)x+8)x−0.35 Substituting: ( 1.37 − 7 ) ⋅ 1.37 + 8 ⋅ 1.37 − 0.35 (1.37−7)⋅1.37+8⋅1.37−0.35 = − 5.63 ⋅ 1.37 + 10.96 − 0.35 =−5.63⋅1.37+10.96−0.35 = − 7.71 + 10.96 − 0.35 = 0.047 =−7.71+10.96−0.35=0.047 Percent relative error: 𝜀 = ∣ 0.043053 − 0.047 ∣ ∣ 0.043053 ∣ × 100 = 9.16 % ε= ∣0.043053∣ ∣0.043053−0.047∣ ×100=9.16% Comparison: The second form is more accurate, with a lower relative error.