Second Order Derivatives ApproximationAna Catalina Torres, Autar KawUniversity of South FloridaUnited States of Americakaw@eng.usf.eduIntroductionThis worksheet demonstrates the use of Maple to illustrate the approximation of the second order derivative of continuous functions. A second order derivative approximation uses a point h ahead and a point h behind of the given value of x at which the second derivative of f (x) is to be found.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 1: InputThe following simulation approximates the second derivative of a function using Second Order Derivatives Approximation. The user inputs are a) function, f(x) b) point at which the derivative is to be found, xv c) starting step size, h d) number of times user wants to halve the step size, nThe outputs include a) approximate value of the second derivative at the point and initial step size given
b) exact value c) true error, absolute relative true error, approximate error and absolute relative approximate error, number of at least correct significant digits in the solution as a function of step size.FunctionLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZHNiVRInhGJ0ZKRk1GL0YvRi8=. 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Value of x at which f ''(x) is desired, xvLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEjeHZGJy8lJWJvbGRHUSZmYWxzZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RKiZjb2xvbmVxO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTi1JI21uR0YkNiVRJDQuMEYnRi9GPC1GOTYuUSI7RidGL0Y8Rj4vRkFGNEZCRkRGRkZIRkovRk1RJjAuMGVtRidGTw==Starting step size, hLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEiaEYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEkMC4yRidGL0Y8LUY5Ni5RIjtGJ0YvRjxGPi9GQUY0RkJGREZGRkhGSi9GTVEmMC4wZW1GJ0ZPNumber of times step size is halvedLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEibkYnLyUlYm9sZEdRJmZhbHNlRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdGMS8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0YxLyUobGFyZ2VvcEdGMS8lLm1vdmFibGVsaW1pdHNHRjEvJSdhY2NlbnRHRjEvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZOLUkjbW5HRiQ2JVEiNkYnRi9GPC1GOTYuUSI7RidGL0Y8Rj4vRkFGNEZCRkRGRkZIRkovRk1RJjAuMGVtRidGTw==This is the end of the user section. All the information must be entered before proceeding to the next section. Re-execute the program.Section 2: ProcedureThe following procedure estimates the solution of second derivate of an equation at a point xv.f (x) = function xv = value at which the solution is desired h = step size value n = number of times step size is 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 3: CalculationThe exact value Ev of the first derivative of the equation:
First, using the diff command the solution is found. In a second step, the exact value of the derivative is shown. 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The next loop calculates the following:Av: Approximate value of the second derivative using Second Order Derivatives Approximation by calling the procedure "SOD"Ev: Exact value of the second derivativeEt: True Erroret: Absolute relative true percentage errorEa: Approximate Errorea: Absolute relative approximate percentage errorSig: Least number of correct significant digits in an 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 loop halves the value of the step size n times. Each time, the approximate value of the second derivative is calculated and saved in a vector. The approximate error is calculated after at least two approximate values of the second derivative have been saved. The number of significant digits is calculated and written as the lowest real number. If the number of significant digits calculated is less than cero, then is shown as cero. Section 4: SpreadsheetThe next table shows the step size value, approximate value, true error, the absolute relative true percentage error, the approximate error, the absolute relative approximate percentage error and the least number of correct significant digits in an approximation as a function of the step size value.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZpcy1JI21pR0YkNihRJXdpdGhGJy8lJ2ZhbWlseUdRK01vbm9zcGFjZWRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lK2ZvcmVncm91bmRHUSpbMjU1LDAsMF1GJy8lMGZvbnRfc3R5bGVfbmFtZUdRLE1hcGxlfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYpLUYjNiUtRiw2KFEnU3ByZWFkRidGL0YyRjVGOEY7LyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvRjxRJ25vcm1hbEYnRi8vJSVib2xkR0Y0RjVGOC9GPFElYm9sZEYnLyUrZm9udHdlaWdodEdGTi1JI21vR0YkNjBRIjpGJ0YvRjVGOEZJLyUmZmVuY2VHRkgvJSpzZXBhcmF0b3JHRkgvJSlzdHJldGNoeUdGSC8lKnN5bW1ldHJpY0dGSC8lKGxhcmdlb3BHRkgvJS5tb3ZhYmxlbGltaXRzR0ZILyUnYWNjZW50R0ZILyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGX28tSSdtc3BhY2VHRiQ2Ji8lJ2hlaWdodEdRJjAuMGV4RicvJSZ3aWR0aEdRJjAuMGVtRicvJSZkZXB0aEdGZ28vJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRlI2MFEifkYnRi9GNUY4RklGVUZXRllGZW5GZ25GaW5GW28vRl5vUSYwLjBlbUYnL0Zhb0ZkcC1GLDYoUSx0YWJsZW91dHB1dEYnRi9GMkY1RjhGOy1GUjYwUSomY29sb25lcTtGJ0YvRjVGOEZJRlVGV0ZZRmVuRmduRmluRltvRl1vRmBvLUYsNihRMkNyZWF0ZVNwcmVhZHNoZWV0RidGL0YyRjVGOEY7LUY/NictRiM2Jy1GLDYjUSFGJy1JI21zR0YkNiNROFNlY29uZH5PcmRlcn5EZXJpdmF0aXZlRidGY3FGRkZJRi9GNUY4RklGUUZib0ZgcC1GLDYoUS9TZXRDZWxsRm9ybXVsYUYnRi9GMkY1RjhGOy1GPzYnLUYjNi1GZnAtRlI2MFEiLEYnRi9GNUY4RklGVS9GWEY0RllGZW5GZ25GaW5GW29GY3AvRmFvUSwwLjMzMzMzMzNlbUYnLUkjbW5HRiQ2J1EiMUYnRi9GNUY4RklGYXItRmhyNidRIjJGJ0YvRjVGOEZJRmFyRmBwLUZncTYjUSpTdGVwflNpemVGJ0ZjcUZGRklGL0Y1RjhGSUZRRmJvRmBwRmpxLUY/NictRiM2LUZmcEZhckZnckZhci1GaHI2J1EiM0YnRi9GNUY4RklGYXJGYHAtRmdxNiNRLUFwcHJveH5WYWx1ZUYnRmNxRkZGSUYvRjVGOEZJRlFGYm9GYHBGanEtRj82Jy1GIzYtRmZwRmFyRmdyRmFyLUZocjYnUSI0RidGL0Y1RjhGSUZhckZgcC1GZ3E2I1ErVHJ1ZX5FcnJvckYnRmNxRkZGSUYvRjVGOEZJRlFGYm9GYHBGanEtRj82Jy1GIzYtRmZwRmFyRmdyRmFyLUZocjYnUSI1RidGL0Y1RjhGSUZhckZgcC1GZ3E2I1EzQWJzflJlbH5UcnVlfkVycm9yRidGY3FGRkZJRi9GNUY4RklGUUZib0ZgcEZqcS1GPzYnLUYjNi1GZnBGYXJGZ3JGYXItRmhyNidRIjZGJ0YvRjVGOEZJRmFyRmBwLUZncTYjUS1BcHByb3h+ZXJyb3JGJ0ZjcUZGRklGL0Y1RjhGSUZRRmJvRmBwRmpxLUY/NictRiM2LUZmcEZhckZnckZhci1GaHI2J1EiN0YnRi9GNUY4RklGYXJGYHAtRmdxNiNRNUFic35SZWx+QXBwcm94fkVycm9yRidGY3FGRkZJRi9GNUY4RklGUUZib0ZgcEZqcS1GPzYnLUYjNi1GZnBGYXJGZ3JGYXItRmhyNidRIjhGJ0YvRjVGOEZJRmFyRmBwLUZncTYjUStTaWd+RGlnaXRzRidGY3FGRkZJRi9GNUY4RkktRlI2MFEiO0YnRi9GNUY4RklGVUZkckZZRmVuRmduRmluRltvRmNwRmBvRmJvRmBwRmBwLUZSNjJRJGZvckYnRi9GS0Y1RjhGTUZPRlVGV0ZZRmVuRmduRmluRltvRmNwRmVwRmBwLUYsNihRImlGJ0YvRjJGNUY4RjtGYHAtRlI2MlElZnJvbUYnRi9GS0Y1RjhGTUZPRlVGV0ZZRmVuRmduRmluRltvRmNwRmVwRmBwLUZocjYnUSIwRidGL0Y1RjhGSUZgcC1GUjYyUSNieUYnRi9GS0Y1RjhGTUZPRlVGV0ZZRmVuRmduRmluRltvRmNwRmVwRmBwRmdyRmBwLUZSNjJRI3RvRidGL0ZLRjVGOEZNRk9GVUZXRllGZW5GZ25GaW5GW29GY3BGZXBGYHAtRiw2KFEibkYnRi9GMkY1RjhGOy1GUjYwUSgmbWludXM7RidGL0Y1RjhGSUZVRldGWUZlbkZnbkZpbkZbby9GXm9RLDAuMjIyMjIyMmVtRicvRmFvRml4RmdyRmBwLUZSNjJRI2RvRidGL0ZLRjVGOEZNRk9GVUZXRllGZW5GZ25GaW5GW29GY3BGZXBGYm9GYHBGYHBGYHBGanEtRj82Jy1GIzYwRmZwRmFyRmBwRmN3LUZSNjBRIitGJ0YvRjVGOEZJRlVGV0ZZRmVuRmduRmluRltvRmh4Rmp4RltzRmFyRmdyRmFyRmN3RmJ5RmdyRkZGSUYvRjVGOEZJRlFGYm9GYHBGYHBGYHBGanEtRj82Jy1GIzYvRmZwRmFyRmBwRmN3RmJ5RltzRmFyRltzRmFyLUYsNihRJmV2YWxmRidGL0YyRjVGOEY7LUY/NictRiM2Ji1GLDYoUSJIRidGL0YyRjVGOEY7LUY/NiktRiM2JUZjd0ZGRklGL0Y1RjhGSS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZGRklGL0Y1RjhGSUZGRklGL0Y1RjhGSUZRRmJvRmBwRmBwRmBwRmpxLUY/NictRiM2L0ZmcEZhckZgcEZjd0ZieUZbc0ZhckZlc0ZhckZpeS1GPzYnLUYjNiYtRiw2KFEjQXZGJ0YvRjJGNUY4RjtGY3pGRkZJRi9GNUY4RklGRkZJRi9GNUY4RklGUUZib0ZgcEZgcEZgcEZqcS1GPzYnLUYjNi9GZnBGYXJGYHBGY3dGYnlGW3NGYXJGX3RGYXJGaXktRj82Jy1GIzYmLUYsNihRI0V0RidGL0YyRjVGOEY7RmN6RkZGSUYvRjVGOEZJRkZGSUYvRjVGOEZJRlFGYm9GYHBGYHBGYHBGanEtRj82Jy1GIzYvRmZwRmFyRmBwRmN3RmJ5RltzRmFyRml0RmFyRml5LUY/NictRiM2Ji1GLDYoUSNldEYnRi9GMkY1RjhGO0ZjekZGRklGL0Y1RjhGSUZGRklGL0Y1RjhGSUZRRmBwRmJvRmBwRmBwRmBwRmpxLUY/NictRiM2L0ZmcEZhckZgcEZjd0ZieUZbc0ZhckZjdUZhckZpeS1GPzYnLUYjNiYtRiw2KFEjRWFGJ0YvRjJGNUY4RjtGY3pGRkZJRi9GNUY4RklGRkZJRi9GNUY4RklGUUZib0ZgcEZgcEZgcEZqcS1GPzYnLUYjNi9GZnBGYXJGYHBGY3dGYnlGW3NGYXJGXXZGYXJGaXktRj82Jy1GIzYmLUYsNihRI2VhRidGL0YyRjVGOEY7RmN6RkZGSUYvRjVGOEZJRkZGSUYvRjVGOEZJRlFGYm9GYHBGYHBGYHBGanEtRj82Jy1GIzYvRmZwRmFyRmBwRmN3RmJ5RltzRmFyRmd2RmFyRml5LUY/NictRiM2Ji1GLDYoUSRTaWdGJ0YvRjJGNUY4RjtGY3pGRkZJRi9GNUY4RklGRkZJRi9GNUY4RklGXXdGYm9GYHBGYHAtRlI2MlEkZW5kRidGL0ZLRjVGOEZNRk9GVUZXRllGZW5GZ25GaW5GW29GY3BGZXBGYHBGW3lGUUZib0ZgcC1GLDYoUTRFdmFsdWF0ZVNwcmVhZHNoZWV0RidGL0YyRjVGOEY7LUY/NictRiM2JUZmcEZGRklGL0Y1RjhGSUZRRmJvLUZjbzYmRmVvRmhvRltwL0ZecFElYXV0b0YnRmNxRkZGSQ==Section 5: GraphsThe following graphs show the approximate solution, absolute relative true error, absolute relative approximate error and least number of significant digits as a function of step size.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=ReferencesNumerical Differentiation of Continuous Functions.
See http://numericalmethods.eng.usf.edu/mws/gen/0.2def/QuestionsThe velocity of a rocket is given 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Use second order derivative approximation method with a step size of 0.25 to find the jerk at t=5s. Compare with the exact answer and study the effect of the step size.Look at the true error vs. step size data for problem 1. Do you see a relationship between the value of the true error and step size ? Is this concidential?ConclusionsTo obtain more accurate values of the second derivative using Second Order Derivative Approximation, the step size needs to be small. As the spreadsheet shows, the smaller the step size value is, the approximation is closest to the exact value. By decreasing the step size, the least number of significant digits that can be trusted increases.
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