Exam_1_Solns - Sorting Number: _ INTRODUCTION TO NUMERICAL...

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Sorting Number: ______________________ Fall 2009 1 INTRODUCTION TO NUMERICAL METHODS OF ENGINEERING ANALYSIS EGM 3344 EXAM 1 HANDWRITTEN (50 points total) General instructions: This exam is closed book and closed notes except for one page, front and back. The only allowable calculator functions for the in-class portion are +, -, *, /, sin, cos, and ^, and sqrt. All work on this exam should be done individually, and you are reminded that the University of Florida has an honor code. The instructor has a zero-tolerance policy for cheating on exams. Any suspected violation of the honor code will be reported to Student Judicial Affairs. If you are found guilty, you may receive a permanent mark on your UF academic transcript and will receive an automatic failing grade in this course. Please show all your work clearly and neatly, as partial credit will be given if it is clear that you understood the concepts required to solve the problem. All three problems should be completed. No credit will be given without a signature and date on the honor code statement at the bottom of the last page of this exam .
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Sorting Number: ______________________ Fall 2009 2 Problem 1 – General Concepts. (12 points) a) (3 points) Up through chapter 12 of the Chapra book, we have had three key concepts for the entire course. List these concepts in the spaces provided below: Key concept 1: Taylor series Key concept 2: Iterate Key concept 3: Specialize b) (3 points) Represent the base-10 integer 39 in base-4. Report your answer in the space provided below: (Suggestion: Use a table like in the lecture notes to solve the problem.) k 4 k Number Value Difference 39 3 64 0 0 39 2 16 2 32 9 1 4 1 4 3 0 1 3 3 0 0 2 1 3 c) (3 points) Given the function 3 () 5 4 fx x x = −+ , write out a Taylor series expansion about 1 x = that can be used to predict (3) f exactly . Provide mathematical expressions for any derivatives that you need. Solution: 23 ''(1) '''(1) (3) (1) '(1)(3 1) (3 1) (3 1) 2! 3! ff f =+ + + where 2 1 5 4 0 '( ) 3 5, '(1) 2 ''( ) 6 , ''(1) 6 '''( ) 6, '''(1) 6 f fx x f f f =−+ = =− = == d) (3 points) Derive a central (as opposed to forward or backward) difference approximation for the second derivative ''( ) f x of a function f x using Taylor series concepts. (Hint: Your answer should be a function of 1 i x , i x , and 1 i x + .) Solution: First write a Taylor series for one point in front of the current point:
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Sorting Number: ______________________ Fall 2009 3 2 11 1 ( ) ( ) '( )( ) ( ''( ) / 2!)( ) ... ii i i i i i i fx f x x x f x x x ++ + =+ + + Next write a Taylor series for one point behind the current point: 2 1 ( ) ( ) '( )( ) ( ''( ) / 2!)( ) ... i i i i i i x x x −− + + Write differences in x values as 1 1 hx x + =− −= and substitute into our two Taylor series expressions: 2 1 ( ) () ' () (' ' () / 2 ! ) . . . i i f xh f x h + + + 2 1 ' / 2 ! ) . . . i i h + + Now add (rather than subtract, as for the first derivative) the second equation and the first equation: 2 2 ( ) 2 ( ' ' ( ) / 2 ! ) i i f xf x f x f x h +− +=+ Rearrange to solve for '( ) i f x : 2 2 ( ) '( ) i i f x f x h −+ =
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Sorting Number: ______________________ Fall 2009 4 Problem 2 – Rootfinding Concepts. (13 points) a) (7 points) Assuming ) sin( 3 1 . 0 ) ( 2 x x x f = , use the false position method to locate a root of ) ( x f starting from initial guesses of 0 . 2 = l x and 0 . 10 = u x . Iterate until the approximate
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.

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Exam_1_Solns - Sorting Number: _ INTRODUCTION TO NUMERICAL...

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