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Sp00ex1 - x r f x l f x r |e A | 0.0000 2.0000 c Find the...

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Sp 2000 EGN 3420 Exam I Name_____________ SHOW ALL WORK! Problem 1 (25 pts) For the function f x x ( ) sin = π a) Find the second order truncated Taylor Series expansion of f ( x ) about some point x 0 . The expression for f 2 ( x ) should be in terms of x , x 0 and the appropriate derivatives of the function sin π x evaluated at x 0 . b) Simplify the expression for f 2 ( x ) when x 0 = 0.5 c) Find the true error E T when using f 2 ( x ) to approximate f ( x ) at x = 0.6
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Sp 2000 EGN 3420 Exam I Name_____________ SHOW ALL WORK! Problem 2 (25 pts) Consider the function f x x ( ) cos = . a) Find the fourth order truncated series expansion of f ( x ) about the point x 0 = 0. b) Use the fourth order truncated series f 4 ( x ) to estimate the root of f ( x ) = 0 located between 0 and 2. In other words, use the Bisection Method to locate the root of f x 4 0 ( ) = Fill in the table below and stop when the magnitude of the approximate relative error is less than 5 % or after 5 iterations, whichever comes first. Express x l , x u , and x r to 4 places after the decimal point and the remaining columns to 2 places after the decimal point
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Unformatted text preview: x r f( x l ) f( x r ) |e A |, % 0.0000 2.0000 c) Find the true root R of f x 4 ( ) = by letting y = x 2 and solving the resulting quadratic in y for its two roots and then using x y = . d) Find the true error E T = R - x r . Sp 2000 EGN 3420 Exam I Name_____________ SHOW ALL WORK! Problem 3 (25 pts) Solve the system of equations A x = b below by the Gauss-Jordan Method, i.e. start with the augmented matrix (A | b ) and transform it to (I | b ') by a sequence of elementary row operations so that the solution is x = b '. 2 4 3 4 2 8 3 2 10 u x y z u x y z u x y z u x y z-+-=-+ + + = + +-= + + + = Sp 2000 EGN 3420 Exam I Name_____________ SHOW ALL WORK! Problem 4 (25 pts) The system of equations A x = b below is inconsistent, i.e. there is no solution. Find the value(s) of K . x x x x x x x x x x x x Kx x x x 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 7 4 1 2 4 3 2 4 2 3 2 3 4 + +-=-+-=-+ + + = + +-= Hint: The matrix A must be singular....
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