2 1 x 2 f x x 2 1 x 3 f x x 1 x 1 x 1 x 1 f x e x ln

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2 1+ x 2 f ( x ) = ( x 2 - 1)( x - 3) f ( x ) = x - 1 x +1 - x +1 x - 1 f ( x ) = e x ln x f ( x ) = ln( x ) - x - 1 f ( x ) = e - x 2 f ( x ) = 3 x 4 + x ln x 2 13. Find the first and second partial derivatives of the following functions: (a) f ( x, y ) = e 3 x +2 y (b) f ( x, y ) = ln(5 x + 9 y ) 14. Find the following integrals: R e x (1+ e x ) 2 dx R x +3 x +2 dx R ( e x - 3 x ) dx R x ( x 2 - 1) 99 dx R x 2 2+ x 3 dx R xe x dx R x 2 e x dx R (4 x 3 + 1) ln xdx R x 2 ln xdx 15. Are A - B and A + B defined here? A = 5 - 1 2 0 3 1 6 2 - 1 B = 4 0 3 - 1 2 5 16. Calculate A + B + C + D A = - 3 2 1 0 B = 3 5 1 2 C = - 6 5 1 - 3 D = 0 2 - 1 3 17. Multiply the following matrices A = 4 2 1 - 3 - 1 3 B = 2 1 3 0 1 2 - 4 7 1 18. Show that B = B - 1 B = - 1 2 - 2 4 - 3 4 4 - 4 5 19. For which values of x is the following matrix diagonal? 1 ln 2 x - 1 ln 2 x - ln x 2 20. If A is invertible, prove that kA ( k R * ) is also invertible and that ( kA ) - 1 = (1 /k ) A - 1 21. Assume A = 1 2 2 4 , find the values of x and y for which A 2 = xA + yI . 22. If A = 2 - 1 3 4 , prove that: 2
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(a) A 2 - 6 A + 11 I = O (b) Show that A is invertible and calculate its inverse matrix 23. Find the eigenvalues and eigenvectors of the following matrices: A 1 = 6 3 3 - 2 A 2 = 1 - 1 - 1 1 A 3 = - 1 3 2 0 A 4 = 1 1 1 1 A 5 = 2 4 1 2 A 6 = 0 . 5 0 . 7 0 . 5 0 . 3 A 7 = 1 0 2 0 5 0 3 0 2 A 8 = 0 1 1 1 0 1 1 1 0 A 9 = 2 0 0 0 3 0 0 0 5 24. Solve the following linear systems (a) 5 x - 2 y = - 4 - x + 3 y = - 7 (b) 5 x - 2 y + 3 z = 16 2 x + 3 y - 5 z = 2 4 x - 5 y + 6 z = 7 3
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  • Spring '10
  • ddfs
  • following statements, Invertible matrix, following functions, ln xdx

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