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Final Exam Questions for Introduction to Numerical Analysis | MATH 170C, Exams of Mathematics

University of California - San DiegoMathematics

Material Type: Exam; Class: Intro Numerical Analy/Ord Diff; Subject: Mathematics; University: University of California - San Diego; Term: Unknown 2005;

Typology: Exams

2009/2010

Uploaded on 03/28/2010

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Math 170C Final
June 9, 2005
•Please put your name, ID number, and sign and date.
•There are 6 problems worth a total of 150 points.
•Calculators are allowed but you must show your work to receive credit.
Print Name:
Student ID:
Signature and Date:
Problem Score
1 /25
2 /25
3 /25
4 /25
5 /25
6 /25
Total /150
1
pf3
pf4
pf5

Partial preview of the text

Download Final Exam Questions for Introduction to Numerical Analysis | MATH 170C and more Exams Mathematics in PDF only on Docsity!

Math 170C Final

June 9, 2005

  • Please put your name, ID number, and sign and date.
  • There are 6 problems worth a total of 150 points.
  • Calculators are allowed but you must show your work to receive credit. Print Name: Student ID: Signature and Date: Problem Score 1 / 2 / 3 / 4 / 5 / 6 / Total /
  1. (25 pts) Show the finite difference approximation f ′(x) ≈ f^ (x^ + 2h)^ h−^ f^ (x^ +^ h) is first order accurate.
  1. (25 pts) Consider adaptive Simpson’s rule applied to ∫ (^9) 1 √x dx for an error < 4 × 10 −^4. Determine if S(3, 4) + S(4, 5), where S(a, b) = h 3 [f (x 0 ) + 4f (x 1 ) + f (x 2 )], is an adequate approximation to (^) ∫ (^5) or whether more subdivision is needed.^3 √x dx (Use |S(c, d) − S(c, (c + d)/2) − S((c + d)/ 2 , d)|/10)
  1. (25 pts) Determine if f (t, x) = 2|y| + t^4 satisfies a Lipschitz condition in D = {(t, y)| 1 ≤ t ≤ 2 , −∞ < y < ∞}.
  1. (25 pts) Consider the initial value problem { y y′(1) = 2^ = 1 + y/t, 1 ≤ t ≤ 2 with exact solutiongenerate the approximation 3 y(t) = t ln.2710703 at t + 2t. Suppose we have used an explicit method to t = 1.4. Use this as a predicted value in the implicit method wi+1 = wi− 1 + h 3 [f (ti+1, wi+1) + 4f (ti, wi) + f (ti− 1 , wi− 1 )] with stepsizeat t = 1.4 (One iteration of predictor-corrector). h = 0.1 and exact initial data to generate the corrected approximation