1/7 Quiz 1 The first letter of ___________________________ _____________________________ your LAST name First Name Last Name Q1-1 (25 pts) Q1-2 (20 pts) Q1-3 (30 pts) Q1-4 (25 pts) Total Note: You have 60 min. Be careful about the time allocation. Try not to leave any problems totally blank so that I can give you some partial credit. Good luck! 24-311 NUMERICAL METHOD Fall 02 QUIZ 1 Date and time 9/17 (Tue), 10:30AM-11:30AM (60 min) Weight 10 % of final grade Coverage lectures and reading assignments: (1) – (6) problem sets: PS1, PS2 Format closed book, closed notes Note bring a basic calculator with + - x / keys (no computer allowed in quiz!)2/7 Q1-1 (25pts) (1) To solve an engineering problem, you first establish a mathematical model, or a governing equation, by applying a fundamental law of physics. You can then solve the equation by an analytical method or a numerical method. When you choose to use a numerical method, what are two types of errors that you should be aware of? What are the sources of each of the two types of errors? (2) What is the rate of convergence of the Newton-Raphson method? (Linear, quadratic, cubic, quartic, …?) (3) In what situation does the modified Newton-Raphson method converge faster than the original Newton-Raphson method? (4) Which root finding method converges faster to the solution, false-position or secant?3/7 Q1-2 (20pts) Evaluate the function value at 2.14x = with 3-digit arithmetic with chopping. 2( ) 5.81 0.628 1.08yx x x=++ Show the intermediate steps for full credit (or partial credit, if you make a mistake).4/7 Q1-3 (30pts) (1) Illustrate in the graph below how the false-position method finds a root. Start with an initial bracket, 27x≤≤, and iterate twice. What is the narrowed bracket (a range of x) after the two iterations? (Hint: you do NOT need to calculate x values using the false-position formula—just use a graphical method.)5/7 (2) Illustrate in the graph below how the secant method finds a root. Start with two initial values, 06x = and 18x = , and iterate twice. What are 2x and 3x ? (Hint: you do NOT need to calculate x values using the secant formula—just use a graphical method.)6/7 Q1-4 (25pts) (1) Complete the following Taylor series formula for a function of one variable. The step size is iixxh −=+1. () ()niiRxfxf+++++=+"1, where the remainder term is defined as =nR (2) Find the second order approximation of the following function using the Taylor series expansion with a base point at3xπ= . ()()tanfxx x=+7/7 This is the last page of Quiz 1, and this page is intentionally left blank so that you can use it if you need more space to write your solution or do some