Instructor: Hacker Name:Course: AME 302Chapter 8 Homework Set SolutionsNumerical DifferentiationInstructions: 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 donot 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 crediton this assignment, you must show your work on all of the problems. If you fail to show workyou will receive a zero for the problem even if it is correct. There are 5 problems worth 40% (8points 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 notgive 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 definition of “little or no work” will be determined by the instructor, not thestudent.Parts 1-3: Circle your answers here. Do not detach this sheet from the homework.1.T○ F2. TF○3.T○ F4.T○ F5.T○ F6. TF○7. TF○8. TF○9.T○ F10.T○ F11.T○ F12.T○ F13.T○ F14.T○ F15. TF○16. TF○17.T○ F18.T○ F19.T○ F20. TF○21. a b cd○ e22.a○ b c d e23. a b cd○ e24. a b cd○ e25. ab○ c d e26. (10 pts)27. (10 pts)28. (10 pts)Attention: For all problems involving writing a MATLAB program you must turn inyour MATLAB code with the output to receive any credit!AME 302 chapter 8 hw set solns Copyright ©Wayne Hacker 2018. All rights reserved. 2Part 1: True-or-False Concept Questions ComponentNumerical DifferentiationChapter 8.1: BackgroundProblem 1. True or False: Three standard ways to improve the accuracy of derivativeestimates when employing finite differences are: (i) decrease the step size (provided thefunction being differentiated is known), (ii) use a higher-order formula that employsmore points, and (iii) if possible to increase the accuracy of the data by taking bettermeasurements.Solution: True. Some comments are in order. (i) For discrete data sets you cannotdecrease 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 decreasethe step size since it decreases the error regardless of the difference scheme used.Solution: False. The given data for the independent variable sets the step size. Youcan’t just go in and add data points!Problem 3. True or False: Consider the case of being given a discrete set of data forthe 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 takenclose together, then the finite difference approximations should yield accurate results.Solution: True.Problem 4. True or False: Consider the case of being given a discrete set of data for theposition versus time of a particle in rectilinear motion: {(t1, x1), (t2, x2), . . . , (tn, xn)}. Ifthe measurements for the position of a particle as a function of time are known to have lotsof noise in the data, then in general it not a good idea to use low-order finite differences(two-point formulas) to approximate the derivative.Solution: True.Chapter 8.2: Finite difference approximation of the 1st derivativeProblem 5. True or False: Of the standard three broad categories of two-point finitedifference schemes (forward, backward, and centered), the two-point centered differenceis the most accurate for small fixed step size h.Solution: True. In fact, it can often be more accurate even for larger step size, due tothe fact that the it is the secant line is taken oven an interval symmetric with respect tothe point of evaluation.AME 302 chapter 8 hw set solns Copyright ©Wayne Hacker 2018. All rights reserved. 3Chapter 8.3: Finite difference formulas using Taylor series expansionsProblem 6. True or False: There are no formulas for approximating derivatives withfinite differences above third order because there is no way to cancel off the higher-orderterms using Taylor series.Solution: False. There is no theoretical limit as to how high of order you can go, thereis a practical one!Problem 7. True or False: Consider the situation of being given a discrete set of datafor the position versus time of a particle, where the particle’s motion is known to besmooth and the time steps h are large (h > 1). You are asked to compute the velocityof the particle. In this case you must use high-order accuracy finite difference formulasto be guaranteed a small truncation error.Solution: False. When h is large, the error is large, so high accuracy formulas mayeven perform worse! For example, if the step size is h = 100, then an order h3methodhas a much larger error than a first-order method.Chapter 8.4: Summary of Finite difference formulas for numerical differenti-ationProblem 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 datamay or may not be equally spaced. The standard finite difference formulas given in thetables are valid for approximating the derivatives of f.Solution: False. The data must be equally spaced.Chapter 8.5: Differentiation formulas using Lagrange polynomialsProblem 9. True or False: Lagrange polynomials have the advantage over formulasderived from Taylor series of being easier to work with when the points are not equallyspaced.Solution: True.Problem 10. True or False: Lagrange polynomials have the advantage over formulasderived from Taylor series of being easier to work with when you want an approximationfor the derivative at a point that is not one of the given data points.Solution: True.AME 302 chapter 8 hw set solns Copyright ©Wayne Hacker 2018. All rights reserved. 4Chapter 8.6: Differentiation using curve fittingProblem 11. True or False: Suppose we are given a collection of discrete data forthe position of a