Math380_Summer07_Exam1

# Math380_Summer07_Exam1 - a in the set A ⊂ R n State the...

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Name: Math 125 – Exam 1 June 26, 2007 50 points possible 1. (a) (3pts) Clearly state the Cauchy-Schwarz Inequality (b) (2pts) Deﬁne the angle between the vectors a and b in R n . (c) (2pts) Explain why Cauchy-Schwarz is necessary to deﬁne angles in R n when n > 3. 2. Let T be a mapping from R n to R m . (a) (3pts) Deﬁne what it means for T to be linear. (b) (5pts) Let T : R 2 R 3 be the linear transformation T ( x,y ) = ( x - 2 y, - x +3 y, 3 x - 2 y ). Find the unique matrix A such that T ( x,y ) = A x y for all ( x,y ) in R 2 . 3. Let F : X R n R m and let x 0 be a point in X such that lim x x 0 F ( x ) = L. (a) (3pts) Using the concept of neighborhoods, geometrically describe what it means for the above limit to exist. (b) (5pts) Show that lim ( x,y ) (0 , 0) x 2 y x 4 + y 2 does not exist. 4. Let F ( x,y ) = ( e y/x ,x ln y ). (a) (2pts) State the domain of F . (b) (4pts) Find the derivative of F . (c) (2pts) Explain why F is diﬀerentiable on its domain. 5. (a) (3pts) Let G : R n R m and F : R m R p . Assume F G is diﬀerentiable at all points

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Unformatted text preview: a in the set A ⊂ R n . State the Chain Rule for D ( F ◦ G )( a ). (b) (7pts) Let F ( x,y,z ) = ( x + y + z,x 3-e yz ) and G ( s,t,u ) = ( st,tu,su ). Find D ( F ◦ G ) at the point (1 , 1 , 0) using the Chain Rule. 6. Let F ( x,y,z ) = xy + z + 3 xz 5 . (a) (2pts) What theorem is used to determine if the equation F ( x,y,z ) = c is solvable for z as a diﬀerentiable function of ( x,y ) near a point? (b) (3pts) Show that if F ( x,y,z ) = 4, that this equation is solvable for z as a diﬀerentiable function of ( x,y ) near the point (1 , , 1). (c) (2pts) Compute ∂z ∂x at (1 , 0). (d) (2pts) Are there any points of the surface F ( x,y,z ) = 4 where you can not solve for z as a diﬀerentiable function of ( x,y )? Clearly explain your answer....
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Math380_Summer07_Exam1 - a in the set A ⊂ R n State the...

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