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48 In spherical coordinates, the graph of the equation r = 0 is (a) an infinitely tall vertical cylinder (b) an infinitely long vertical line (c) a sphere (d) a point (e) undefined 49 How can we add two polar-coordinate vectors (a) Convert them to standard form in Cartesian coordinates, add the Cartesian vectors, and then convert the Cartesian sum back to polar form (b) Convert them to standard form in cylindrical coordinates, add the cylindrical vectors, and then convert the cylindrical sum back to polar form (c) Convert them to standard form in spherical coordinates, add the spherical vectors, and then convert the spherical sum back to polar form (d) Add the direction angles of the addend vectors to get the direction angle of the sum vector, and add the magnitudes of the addend vectors to get the magnitude of the sum vector (e) We can t! Addition of polar vectors is not defined 50 In a system of spherical coordinates, the constant-radius increments appear as (a) concentric circles (b) concentric cylinders (c) concentric spheres (d) straight lines passing through the origin (e) parallel planes 51 Consider the sum of the tangent and the cotangent Let f (q) = tan q + cot q What s the positive peak amplitude in the graph of f (a) 1 (b) 2 (c) p (d) 2p (e) It s not defined 52 Consider the following system of parametric equations representing a straight line in Cartesian xyz space: x = 4 + 7t y = 3 5t z = 2 + 6t
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Which of the following is a valid expression of the line s direction numbers (a) ( 4,3,2) (b) (3, 2,8) (c) ( 11,8, 4) (d) (11, 8,4) (e) ( 14,10, 12) 53 When we add, subtract, or multiply one function by another function of the same variable, the domain of the resultant function is (a) the intersection of the ranges of the two functions (b) the union of the ranges of the two functions (c) the intersection of the domains of the two functions (d) the union of the domains of the two functions (e) None of the above 54 Consider the following equation in three variables x, y, and z: (x + 2)2/4 + (x + 3)2/9 + (x + 4)2/16 = 1 In Cartesian xyz-space, this equation represents (a) an elliptic cone (b) an oblate ellipsoid (c) a cylinder (d) a hyperboloid (e) a paraboloid 55 The solid black curve in Fig FE-8 shows the graph of a relation f in the Cartesian xy plane Which, if any, of the four dashed gray curves portrays the graph of f 1 (a) Curve A (b) Curve B (c) Curve C (d) Curve D (e) None of the above 56 Which, if any, of the curves in Fig FE-8 represents a relation that s not a true function of either x or y (a) Curve A (b) Curve B (c) Curve C (d) Curve D (e) The solid black curve 57 In the Cartesian xy plane, the unit hyperbola intersects the line x = 3 at (a) no points (b) one point
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y 6 4 2 x 6 4 2 2 4 Graph of relation f 6 2 4 6 Graph of relation f
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Curve A Curve B Curve C Curve D
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Figure FE-8 (c) two points (d) four points (e) infinitely many points
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58 Which of the following is an arithmetic sequence and also a geometric sequence (a) 4, 4, 4, 4, 4, 4, (b) 4, 2, 4, 2, 4, 2, (c) 1, 1, 1, 1, 1, 1, (d) 3, 2, 1, 0, 1, 2, (e) 1, 1/2, 1/3, 1/4, 1/5, 1/6, 59 What does the following expression represent
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2 i
i =0
(a) A divergent harmonic series (b) A divergent geometric series
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(c) A convergent harmonic series (d) A convergent geometric series (e) An undefined series 60 What s the value of the following limit
x 0
Lim ln ( x )
(a) (b) (c) (d) (e)
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