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**Unformatted text preview: **MATH 1920, FALL 2009 PRELIM 1 SHOW ALL WORK. NO CALCULATORS. NAME: STUDENT ID #: SECTION #: Who is your TA? PROBLEM 1a 1b 2 3a 3b 4a 4b 4c 5a 5b 6 SCORE TOTAL 1. Find the limit if it exists, or show that it does not exist. (a) (2x − 3y + z ) sin(x − 1) (x,y,z )→(1,−1,−1) (x2 + z 2 )(x − 1) lim (b)
(x,y )→(0,0) lim x2 + y 2 . 2x − y 1 2. Find parametric equations for the line tangent to the curve of intersection of the surfaces 2x2 − 3y + z 2 = 5 and x + y = 3 at the point P (1, 2, 3). 2 3. (a) Consider the function f (x, y, z ) = xy + yz − xz , where x = u − v , y = u + v , and ∂f z = uv . Find as a function of u and v . ∂u (b) The directional derivative of a function f (x, y ) at the point (1, 1) in the direction √ of i + j is 3 2, and the directional derivative in the direction of 3j is −2. Find fx (1, 1) and fy (1, 1). 3 On this problem, you do not need to justify your answers. 4. Consider the domain of the function f (x, y ) = xy + 3 − (x2 + y 2 ) . (x2 + y 2 ) − 1 (a) Give equations for the boundary of the domain of f . (b) Mark the correct statement below. The domain is bounded. The domain is unbounded. The domain is neither bounded nor unbounded. The domain is both bounded and unbounded. (c) Mark the correct statement below. The domain is open. The domain is closed. The domain is neither open nor closed. The domain is both open and closed. 4 5. Consider the plane T with equation 3x − y + z = 13. It contains the point P (2, −3, 4). (a) The line L1 has parametric equations x = −t + 4 y = 2t − 1 z = 5t and lies in the plane T . Find parametric equations for the line L2 that satisﬁes the following conditions: L2 lies in the plane T , L2 is perpendicular to L1 , and L2 goes through the point P . (b) Find the equation of the sphere S through the point (8, −5, 6) and such that the plane T is tangent to S at the point P . 5 6. The ﬁrst octant is the region in space deﬁned by x ≥ 0, y ≥ 0, z ≥ 0. Use Lagrange Multipliers to determine which point in the ﬁrst octant and on the surface √ xy 2 z 2 = 16 2 is closest to the origin. 6 7 ...

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