Instructor Hacker Course AME 302 Chapter 8 Homework Set Solutions Numerical Di erentiation Name Instructions This homework is broke into three components Part 1 true false concept questions Part 2 basic computational skills and Part 3 derivation analysis and problem solving questions The rules for each component are listed below Part 1 There is no partial credit given for true false problems Since these are concept questions you do not need to show any work for these problems There are 20 problems worth 60 3 points each Part 2 There is no partial credit given for multiple choice problems Although there is no partial credit on this assignment you must show your work on all of the problems If you fail to show work you will receive a zero for the problem even if it is correct There are 5 problems worth 40 8 points each Part 3 On these problems you must show all of your work to receive any credit If in doubt write it out Show your work as clearly as you can if I can t understand how you got an answer I will not give you credit for it Remember I know how to solve the problem and to make matters worse I have a lot of training in following logical arguments Warning The de nition of little or no work will be determined by the instructor not the student Parts 1 3 Circle your answers here Do not detach this sheet from the homework 1 2 3 4 5 6 7 8 9 T F T F T F T F T F T F T F T F T F 11 T F 12 T F 13 T F 14 T F 15 T F 16 T F 17 T F 18 T F 19 T F 10 T F 20 T F 21 a b 22 a b b b a a a 23 24 25 26 27 28 c c c c d e d e d e d e b c d e 10 pts 10 pts 10 pts Attention For all problems involving writing a MATLAB program you must turn in your MATLAB code with the output to receive any credit AME 302 chapter 8 hw set solns Copyright Wayne Hacker 2018 All rights reserved 2 Part 1 True or False Concept Questions Component Numerical Di erentiation Chapter 8 1 Background Problem 1 True or False Three standard ways to improve the accuracy of derivative estimates when employing nite di erences are i decrease the step size provided the function being di erentiated is known ii use a higher order formula that employs more points and iii if possible to increase the accuracy of the data by taking better measurements Solution True Some comments are in order i For discrete data sets you cannot decrease the step size it is set by the data for the independent variable ii If h 1 then going to higher order formula with more points may or may not be more successful iii More accurate data is always good Problem 2 True or False Given a discrete set of data you should always decrease the step size since it decreases the error regardless of the di erence scheme used Solution False The given data for the independent variable sets the step size You can t just go in and add data points Problem 3 True or False Consider the case of being given a discrete set of data for the position versus time of a particle in rectilinear motion t1 x1 t2 x2 tn xn If the particle s motion is known to be smooth and the time measurements are taken close together then the nite di erence approximations should yield accurate results Solution True Solution True Problem 4 True or False Consider the case of being given a discrete set of data for the position versus time of a particle in rectilinear motion t1 x1 t2 x2 tn xn If the measurements for the position of a particle as a function of time are known to have lots of noise in the data then in general it not a good idea to use low order nite di erences two point formulas to approximate the derivative Chapter 8 2 Finite di erence approximation of the 1st derivative Problem 5 True or False Of the standard three broad categories of two point nite di erence schemes forward backward and centered the two point centered di erence is the most accurate for small xed step size h Solution True In fact it can often be more accurate even for larger step size due to the fact that the it is the secant line is taken oven an interval symmetric with respect to the point of evaluation AME 302 chapter 8 hw set solns Copyright Wayne Hacker 2018 All rights reserved 3 Chapter 8 3 Finite di erence formulas using Taylor series expansions Problem 6 True or False There are no formulas for approximating derivatives with nite di erences above third order because there is no way to cancel o the higher order terms using Taylor series Solution False There is no theoretical limit as to how high of order you can go there is a practical one Problem 7 True or False Consider the situation of being given a discrete set of data for the position versus time of a particle where the particle s motion is known to be smooth and the time steps h are large h 1 You are asked to compute the velocity of the particle In this case you must use high order accuracy nite di erence formulas to be guaranteed a small truncation error Solution False When h is large the error is large so high accuracy formulas may even perform worse For example if the step size is h 100 then an order h3 method has a much larger error than a rst order method Chapter 8 4 Summary of Finite di erence formulas for numerical di erenti ation Problem 8 True or False Consider the situation of being given a set of data x1 f x1 x2 f x2 xn f xn where the function f x dependent variable may be known or unknown and the data may or may not be equally spaced The standard nite di erence formulas given in the tables are valid for approximating the derivatives of f Solution False The data must be equally spaced Chapter 8 5 Di erentiation formulas using Lagrange polynomials Problem 9 True or False Lagrange polynomials have the advantage over formulas derived from Taylor series of being easier to work with when the points are not equally spaced Solution True Solution True Problem 10 True or False Lagrange polynomials have the advantage over formulas derived from Taylor series of being easier to work with when you want an approximation for the derivative at a point that is not one of the given data points AME 302 chapter 8 hw set solns Copyright Wayne Hacker 2018 All rights reserved 4 Chapter 8 6 Di erentiation using curve tting Problem 11 True or False Suppose we are given a …