quiz1sol - 1. (a) The volume of the parallelepiped spanned...

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Unformatted text preview: 1. (a) The volume of the parallelepiped spanned by u, v and w is the absolute value of determinant of the matrix spanned by u, v and w . Calculate 123 149 1 8 27 = 123 026 0 6 24 =1· 26 6 24 123 149 1 8 27 = 12. = 12. (b) Note that w′ = w − v . Therefore, 123 149 0 4 18 = Therefore, the volume of the parallelepiped spanned by u, v and w′ is also 12. 2. We denote log as ln. Let f (x, y ) = ex log(1 + y ). Then ∂f = ex log(1 + y ), ∂x ∂f ex = . ∂y 1+y At the base point (x0 , y0 ) = (0, 0), we have ∂f (0, 0) = e0 log(1 + 0) = 0, ∂x ∂f e0 (0, 0) = = 1. ∂y 1+0 Therefore, by linear approximation, we have · e0.02 log 1.01 = f (0.02, 0.01) = f (0, 0) + 1 ∂f ∂f (0, 0) · 0.02 + (0, 0) · 0.01 = 0.01. ∂x ∂y ...
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