MRQ1 - UCSC MATH 21 FALL 2009 Review For Midterm 1(1...

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UCSC MATH 21 FALL 2009 Review For Midterm 1 (1) Polynomial interpolation . The polynomial f ( x ) = c n x n + c n - 1 x n - 1 + · · · + c 1 x + c 0 interpolates the points ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x k , y k ) if f ( x j ) = y j for j = 1 , 2 , . . . , k . In other words, f ( x ) interpolates those points if its graph passes through them. Find the cubic (degree 3) polynomial that interpolates the points ( - 1 , 1) , (1 , 2) , (2 , 4) and (3 , 1). Is the solution unique? Why? Hint: The equations f ( x j ) = ( y j ) give linear equations in the coefficients c 0 , c 1 , . . . c n of the polynomial f ( x ). (2) Use Gauss-Jordan elimination to find the reduced-row-echelon form of the matrices A and B , below. A = 2 3 - 1 3 1 - 2 3 5 1 4 - 3 - 1 and B = 2 3 7 0 8 2 4 8 2 4 2 1 4 2 3 1 - 1 2 1 3 . (3) Write down the systems of equations for which the matrices A and B , in problem (2) above, are the augmented matrices. Based on your work in the problem (2), determine the solution sets for these two systems.
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