Rutgers University Numerical Analysis I Math 373:640– Exam Practice Midterm 1 Instructor: Agnieszka Hejna 4 MARCH 2024 Name: This exam contains 12 pages (including this cover page) and 5 questions. Total of points is 50. During the exams, you are allowed to use a simple calculator. No other resources and electronic devices like the textbook, lecture notes, graphing calculators, cellphones, dictionaries, other electronic devices, formula sheets, or any other material not authorized by the instructor is allowed during an exam. You have 60 minutes to solve the problems. Good luck! Distribution of Marks Question Points Score 1 5 2 16 3 9 4 10 5 10 Total: 50 1 Math 373:640 Exam 4 MARCH 2024 1. Determine which statements are true. You do not have to justify your answers. (a) (1 point) There is a differentiable function g on interval [0, 1] such that g([0, 1]) ⊆ [0, 1], |g ′ (x)| ≤ 0.2, and there are two distinct points x1 , x2 in [0, 1] such that g(x1 ) = x1 and g(x2 ) = x2 . (b) (1 point) In order to apply secant method with the initial approximations p0 , p1 of the zero of the function f , we have to check first if f (p0 )f (p1 ) < 0. (c) (1 point) If zero x0 of the function f has the multiplicity greater than 1, then Newton’s method cannot be applied to approximate this zero. (d) (1 point) There is a function f and nodes x0 , x1 , x2 such that Lagrange interpolating polynomial P2 and Hermite’s polynomial H5 are the same for this function. (e) (1 point) Lagrange interpolating polynomial P3 for the nodes x0 , x1 , x2 , x4 can have degree 4. Page 2 of 12 Math 373:640 Exam Solution to Question 1 Page 3 of 12 4 MARCH 2024 Math 373:640 Exam 4 MARCH 2024 2. In this question, our task will be to approximate the only zero x0 of the function f (x) = sin(x)/2 x − 0.25 It can be done in several ways. (a) (5 points) Perform the first step of Newton’s method with p0 = 0, i.e. find p1 . (b) (5 points) Let us consider g(x) = 0.25 − sin(x)/2. Justify that x0 is a fixed point of g and perform the first step of fixed point iteration for g with p0 = 0, i.e. find p1 . (c) (5 points) Perform the first step of the secant method with p0 = 0 and p1 = π, i.e. find p2 . (d) (1 point) Actually, 0.1669247050256201247600505181189463976022. Which method from (a), (b), (c) gave the best approximation? Page 4 of 12 Math 373:640 Exam Solution to Question 2 Page 5 of 12 4 MARCH 2024 Math 373:640 Exam 4 MARCH 2024 3. Let sin(cos(x)) x2 7. 800 100 (a) (1 point) Define the fixed point of g. g(x) = (b) (8 points) Prove that g has an unique fixed point in the interval [6.5, 7.5]. If you would like to use some theorems from lecture or from the textbook, you have to (briefly) state these theorems. Hint: please be aware that cos, sin functions can reach negative values! Page 6 of 12 Math 373:640 Exam Solution to Question 3 Page 7 of 12 4 MARCH 2024 Math 373:640 Exam 4 MARCH 2024 4. Let f (x) = √1x . (a) (3 points) Find the Lagrange Polynomial P2 using nodes x0 = 1, x1 = 1.21 = (1.1)2 , x2 = 1.69 = (1.3)2 , (b) (2 points) Use your result to find an approximation of the number √11.5 . (c) (5 points) Prove that 1 |√ − P2 (1.5)| ≤ 0.1 1.5 Page 8 of 12 Math 373:640 Exam Solution to Question 4 Page 9 of 12 4 MARCH 2024 Math 373:640 Exam 5. (10 points) Let us consider Hermite interpolating polynomial 3 H3 (x) = A 2x − x2 (x − 2) 2 for the data from the table for some A, B, C, D ∈ R. Find A, B, C, D. x f (x) 0 -1 2 C f ′ (x) B D Page 10 of 12 4 MARCH 2024 Math 373:640 Exam Solution to Question 5 Page 11 of 12 4 MARCH 2024 Math 373:640 Exam 4 MARCH 2024 This page is intentionally left blank to accommodate work that wouldn’t fit elsewhere and/or scratch work. Page 12 of 12