Math380_Summer08_Exam1

# Math380_Summer08_Exam1 - Name Math 380 Exam 1 50 points...

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Name: Math 380 – Exam 1 June 26, 2008 50 points possible 1. Let T ( x ) : R 3 R 3 be deﬁned by the matrix A = 3 - 2 0 1 3 2 1 0 5 and let S ( y ) : R 3 R be deﬁned by the matrix B = h 4 0 - 2 i (a) (1pt) Is S a vector-valued function? Brieﬂy explain your answer. (b) (2pts) Find the matrix corresponding to the linear mapping C = S T . (c) (2pts) Why is the composition T S undeﬁned? (d) (3pts) Find the volume of the parallelepiped spanned by the position vectors u = (3 , - 2 , 0), v = (1 , 3 , 2), and w = (1 , 0 , 5). (e) (2pts) Describe the image of the unit cube spanned by e 1 , e 2 and e 3 under the transfor- mation T . What is the volume of this image? 2. Let f ( x,y ) = 3 p x 4 + y 4 . (a) (3pts) Using the deﬁnition of a partial derivative, show that f x (0 , 0) = 0. (Note: The exact same computation will show f y (0 , 0) = 0 as well.) (b) (3pts) Find the derivative of f ( x,y ). (c) (2pts) On what set is f ( x,y ) a diﬀerentiable function. (d) (2pts) Do your answers to part (a) and (c) reconcile? Brieﬂy explain.

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Math380_Summer08_Exam1 - Name Math 380 Exam 1 50 points...

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