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# finalexam-practice3 - Math 21a 1 Third Old Final Exam...

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Unformatted text preview: Math 21a 1 Third Old Final Exam Spring, 2009 True-False questions. Circle the correct letter. No justiﬁcations are required. T F The curve r(t) = (1 − t)A + tB , t ∈ [0, 1] connects the point A with the point B . T F For every c, the function u(x, t) = (2 cos(ct) + 3 sin(ct)) sin(x) is a solution to the wave equation utt = c2 uxx . T F Let (x0 , y0 ) be the maximum of f (x, y ) under the constraint g (x, y ) = 1. Then fxx (x0 , y0 ) < 0. √ T F The function f (x, y, z ) = x2 − y 2 − z 2 decreases in the direction 2, −2, −2 / 12 at the point (1, 1, 1). T F Assume F is a vector ﬁeld satisfying |F(x, y, z )| ≤ 1 everywhere. For every curve C given by r(t), 0 ≤ t ≤ 1, the line integral C F · dr is less or equal than the arc length of C . T F Let F be a vector ﬁeld which coincides with the unit normal vector n for each point on a curve C . Then C F · dr = 0. T F If for two vector ﬁelds F and G one has curl F = curl G, then F = G + a, b, c , where a, b, c are constants. T F For every vector ﬁeld F the identity grad(div F) = 0 holds. T F If div F(x, y, z ) = 0 for all (x, y, z ), then curl F = 0, 0, 0 for all (x, y, z ). T F For every function f (x, y, z ), there exists a vector ﬁeld F such that div F = f . 2 Indicate with a check in the column below “conservative” if a vector ﬁelds is conservative (that is if curl F(x, y, z ) = 0, 0, 0 for all points (x, y, z )). Similarly, mark the ﬁelds which are incompressible (that is if div F(x, y, z ) = 0 for all (x, y, z )). No justiﬁcations are needed. Vector Field F(x, y, z ) = −5, 5, 3 F(x, y, z ) = x, y, z F(x, y, z ) = −y, x, z F(x, y, z ) = x2 + y 2 , xyz, x − y + z F(x, y, z ) = x − 2yz, y − 2zx, z − 2xy 3 conservative curl F = 0 incompressible div F = 0 Let E be a parallelogram in three dimensional space deﬁned by two vectors u and v. (a) Express the diagonals of the parallelogram as vectors in terms of u and v. (b) What is the relation between the length of the cross product of the diagonals and the area of the parallelogram? (c) Assume that the diagonals are perpendicular. What is the relation between the lengths of the sides of the parallelogram? 4 Find the volume of the wedge shaped solid that lies above the xy -plane, below the plane z = x, and within the cylinder x2 + y 2 = 4. Math 21a 5 Third Old Final Exam Spring, 2009 Match the equations with the objects. No justiﬁcations are needed. .................. ..................... .... ...... ..... .... ... .... .... ... ... ... ... ... ... ... ... . ... ... ... ... .. . .. .. .. .. .. . .. .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . .. .. .. .. .. .. .. .. ... .. ... ... ... ... ... ... ... ... ... ... .... ... .... .... ..... .... ..... ..................... ..................... 15 10 5 0 0 5 10 15 I 15 10 5 0 II 14 III IV 12 15 10 10 8 5 6 0 4 0 5 10 15 2 4 6 8 10 12 14 V Equations: VI VII .. ... .. . .. .. . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. .. .. .. .. .. . .. . .. . ... ... ... ... VIII (a) g (x, y, z ) = cos(x) + sin(y ) = 1 (c) r(t) = cos(t), sin(t) (e) F(x, y, z ) = cos(x), sin(x), 1 (g) g (x, y ) = cos(x) − sin(y ) = 1 6 Consider the surface parameterized by x = uv cos v (b) y = cos(x) − sin(x) (d) r(u, v ) = cos(u), sin(v ), cos(u) sin(v ) (f) z = f (x, y ) = cos(x) + sin(y ) (h) F(x, y ) = cos(x), sin(x) y = uv sin v z = v, where 0 ≤ u ≤ 1 and 0 ≤ v ≤ π Which of the following integrals is an integral for the surface area of this surface? π 1 π 1 (a) 0 π 0 1 u2 + u2 v 4 du dv (b) 0 π 0 1 v 2 + u4 v 2 du dv (c) 0 0 v 2 + u2 v 4 du dv (d) 0 0 u2 + u4 v 4 du dv (e) None of the above. Math 21a 7 8 Third Old Final Exam Spring, 2009 Let the curve C be parameterized by r(t) = t, sin t, t2 cos t for 0 ≤ t ≤ π . Let f (x, y, z ) = z 2 ex+2y + x2 and F = f . Find C F · dr. Evaluate the line integral of the vector ﬁeld F(x, y ) = y 2 , x2 in the clockwise direction around the triangle in the xy -plane deﬁned by the points (0, 0), (1, 0) and (1, 1) in two ways: (a) By evaluating the three line integrals. (b) Using Green’s theorem. 8 2 y 1/3 9 Evaluate 0 y 2 ex dx dy . x8 2 10 Match each of the following iterated integrals with its domain of integration. 1 1 y 1 1 y 1 y 0 x 0 y 0 x y 1 y 0 1 y 0 x y 1 1 (a) 0 f (x, y, z ) dz dx dy (b) 0 f (x, y, z ) dz dx dy (c) 0 f (x, y, z ) dz dx dy (d) 0 f (x, y, z ) dz dx dy (e) 0 f (x, y, z ) dz dx dy 1 1 1 z 0.5 1 0.5 0 0 0 z 0.5 z0.5 0.5 1 1 y0.5 0 0 0 0.5 y 0 0 0.5 0 x 1 x0.5 1 y x 1 Region (i) 1 Region (ii) Region (iii) 1 z 0.5 0.5 1 z 0 0 0 0.5 x 0.5 y 1 1 0.5 00 0 0.5 x y 1 Region (iv) Region (v) Math 21a Third Old Final Exam Spring, 2009 11 Find the volume of the largest rectangular box with sides parallel to the coordinate planes that can 2 2 z2 be inscribed in the ellipsoid x + y9 + 25 = 1. 4 12 (a) Find all the critical points of the function f (x, y ) = −(x4 − 8x2 + y 2 + 1). (b) Classify the critical points. (c) Locate the local and absolute maxima of f . (d) Find the equation for the tangent plane to the graph of f at each absolute maximum. 13 Use Stokes’ theorem to evaluate the line integral of F(x, y, z ) = −y 3 , x3 , −z 3 along the curve r(t) = cos(t), sin(t), 1 − cos(t) − sin(t) with 0 ≤ t ≤ 2π . 14 Let S be the graph of the function f (x, y ) = 2 − x2 − y 2 which lies above the disk {(x, y ) : x2 + y 2 ≤ 1} in the xy -plane. The surface S is oriented so that the normal vector points upwards. Compute the ﬂux S F · dS of the vector ﬁeld F= −4x + x2 + y 2 − 1 2xz , 3y, 7 − z − 2 1 + 3y 1 + 3y 2 through S using the divergence theorem. ...
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## This note was uploaded on 04/26/2010 for the course SCI 35254 taught by Professor George during the Spring '10 term at Aarhus Universitet.

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