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# hw1 - at x = 1(accurate upto at least 8 digits Assuming...

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ENGR 6101: C OMPUTATIONAL E NGINEERING Problem Set 1 (due Monday in my mailbox by 4pm, 8/30) Show all your work on all problems. Correct answers without enough work will be graded as 0/3. Questions: 1. (3 pts.) Write down the Taylor series expansion for f ( x ) = 3 x ln x about 1. Evaluate 3 . 3ln ( 1 . 1 ) by summing the first three non-zero terms in the Taylor series. Compare your result with the actual value of 3 . 3ln ( 1 . 1 ) = 0 . 314523593354. How many accurate (significant) digits do you get? 2. (3 pts) Consider the following function: f ( x ) = x 2 + sin ( x ) (a) Create a table by evaluating this function at x = 0 . 6 , 0 . 8 , 1 , 1 . 2 and 1 . 4, accurate upto at least 8 digits 1 . Then compute the numerical approximation of the derivative at x = 1 using the following methods below: i. Forward, use values f ( 1 ) and f ( 1 . 2). ii. Backward, use values f ( 1 ) and f ( 0 . 8). iii. Central, use values f ( 0 . 8 ) and f ( 1 . 2). iv. New method presented in problem 4, use values f ( 0 . 6 ) , f ( 0 . 8), f ( 1 . 2 ) and f ( 1 . 4). (b) Compute the exact value by taking the derivative of this function and evaluating the derivative
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Unformatted text preview: at x = 1 (accurate upto at least 8 digits). Assuming that this results is “true”, ﬁnd the absolute errors for all the methods above. 3. (3 pts.) Consider the following numerical differentiation formula: f ( x ) ≈ α f ( x + h ) + β f ( x-2 h ) γ h Find the appropriate values for α , β and γ for this formula to work (using Taylor series expansion). What is the order of your approximation? 4. (3 pts) Consider the following numerical differentiation formula: f ( x ) ≈-f ( x + 2 h ) + 8 f ( x + h )-8 f ( x-h ) + f ( x-2 h ) 12 h Prove that this is a valid formula (using Taylor’s expansion) and ﬁnd its order. 1 Note that x is in radians, that is, sin ( 2 π ) = 1. 1...
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