S09mt2an

# S09mt2an - f at the point P(1 3 2 in the direction of the...

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Sample Second Midterm Answer all seven questions. You must show your working to obtain full credit. Points may be deducted if you do not justify your Fnal answer. 1. Let f ( x, y )=2 x 2 + x 3 y + x 2 y 4 8 y . (i) ±ind all the second partial derivatives of f . (ii) The function f has a critical point at ( 2 , 1). [You need NOT prove this.] Determine whether the critical point is a local maximum or a local minimum or a saddle. 2. ±or the function z = xy + x 2 /y suppose that x is measured as 5 with a possible error of ± 0 . 02 and y is measured as 2 with a possible error of ± 0 . 04. Use di²erentials to estimate the maximum possible error in the value of z . 3. Suppose that z = xy + xw + yw where x = e st , y = s 2 , and w = s + t . Use the chain rule to evaluate ∂z ∂s and ∂z ∂t when s = 1 and t =2 . 4. ±or the function f ( x, y, z )= x 2 yz 3 + xz Fnd (i) the gradient of f ;
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Unformatted text preview: f at the point P (1 , 3 , 2) in the direction of the vector ° 2 , − 1 , 1 ² ; (iii) the equation of the tangent plane at P to the level surface of f through P . 5. Use the method of Lagrange multipliers to Fnd the maximum and minimum values of f ( x, y, z ) = 2 x + 3 y − z on the ellipsoid 4 x 2 + y 2 + 2 z 2 = 42. ±ind also the points where the maximum and minimum values are obtained. 6. ±ind the volume of the solid under the surface z = 6 + x + y 2 and above the region bounded by y = x and x = 2 − y 2 . 7. Use polar coordinates to evaluate ° ° D y dA where D is the region in the Frst quadrant inside the circle x 2 + y 2 = 4 and outside the circle x 2 + y 2 = 2 x ....
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