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#### Carnegie Mellon University Department of Mathematical Sciences 21256, Spring 2021 Final Exam Ver1 1

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Carnegie Mellon University Department of Mathematical Sciences 21256, Spring 2021 Final Exam Ver1 1. Use an augmented matrix to solve the system of equations, expressing your result in vector form: x1 2x1 3x1 + x2 2x2 x2 + x3 + 2x3 + 3x3 + + x4 2x4 x4 + x5 + x5 x5 = = = 7 1 9 2. Let f (x; y) = 16x4 32x2 y 4 + 8y 2 : Find all critical points and classify them as local maximum, local minimum, or saddle points. 3. Use the Lagrangian function and the Bordered Hessian matrix to determine any local maximum or minimum values of f (x1 ; x2 ; x3 ; x4 ) = x21 + x22 + x23 + x24 subject to the constraints x1 + x2 + x3 + x4 = 2 and x1 + 2x2 + 3x3 + 4x4 = 10 4. Suppose X is a continuous random variable with normal distribution, mean 70; and standard deviation = 4: Determine P (70 X 77) : 2 2 5. Let f (x; y) = ke x y : Determine k such that f (x; y) is a probability density function on R2 : And then …nd a such that P X 2 + Y 2 a2 = 21 : 6. Let E be the solid tetrahedron with vertices A ( 1; 1; 1) ; B ( 1; 2; 1) ; C ( 3; 1; 1) ; and D ( 1; 1; 1) : Evaluate the triple integral ZZZ z dV (x; y; z) E 7. Let E be the solid in the …rst octant which p lies above the cone 3z 2 = x2 + y 2 ; below the sphere 2 2 2 x + y + z = 4; and between the planes x = 3y and x = y : Find the volume of E: 8. Suppose X; Y; and Z are continuous random variables with probability density function 2 2 2 e x y z f (x; y; z) = k p x2 + y 2 + z 2 for all points except (0; 0; 0) : Determine k such that this is a probability density function, and …nd the probability P X 2 + Y 2 + Z 2 2 :

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