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Practice Final

# Practice Final - x = y and x = y 2 Write the integrals for...

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Final 70 min 1)(30 points) Let z = f ( x, y ) = x 2 + y 2 - x 2 y . Find all critical points of f and use the second derivative test to classify them, if possible. 2)(30 points) A particle moves along the line ax + by = 1, where ab negationslash = 0. a) What point Q ( x, y ) is closest to the origin P (0 , 0)? b) What is the distance of Q from P ? 3)(40 points) Let R be the region bounded by the curves x = - 1 , x = 1 - y 2 . a) Write the limits of integration for integraltext integraltext R f dx dy b) Write the limits of integration for integraltext integraltext R f dy dx c) Compute integraltext integraltext R y dA

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FINAL EXAM NINETY MINUTES 1)(30 points) Let z = f ( x, y ) = 1 3 y 3 + x 4 - x 2 y . Find all critical points of f . 2)(30 points) Use the method of Lagrange multipliers to find all extremes of f ( x, y ) = x 2 + y 2 - x 2 y , along the curve x 2 + y 2 = 1. 3)(40 points) Let
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Unformatted text preview: x = y and x = y 2 . Write the integrals for i i R fdA in two diﬀerent ways a) As a type one region. b) As a type two region. FINAL EXAM NINETY MINUTES 1)(30 points) Let z = f ( x, y ) = x 3 + y 2-y 2 x . Find all critical points of f . 2)(30 points) Use the method of Lagrange multipliers to ﬁnd the points on the curve 1 4 x 2 + y 2 = 1, that are closest and furthest from P (1 , 0). 3)(40 points) Let R be the region bounded by the curves x = y 1 3 and y = x 2 . Write the integrals for i i R fdA in two diﬀerent ways a) As a type one region. b) As a type two region....
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Practice Final - x = y and x = y 2 Write the integrals for...

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