III CALCULUS FALL 2002 ASSIGNMENT 15
Exercises
Due Wed. Dec. 4
SIMPLIFY ALL ANSWERS. Unless the contrary is stated GIVE EXACT ANSWERSNOT DECIMAL APPROXIMATIONS. (1) The equations of the line containing the points (5, 2, 1) and (9, 4, 1) are y2 zk xh = = a 3 c where a = ,c= ,h= , and k = . (2) Find the symmetric equations of the line containing the point (1, 0, 1) which is parallel to x4 2y 3 3z 7 the line = = . 2 5 6 xh yj z+1 Answer: = = where a = ,b= ,h= , and j = . a b 4 (3) The equation of the plane which passes through the points (0, 1, 1), (5, 0, 1), and (4, 1, 0) is ax + by + cz = 1 where a = ,b= , and c = . a (4) The angle between the planes 4x + 4z 16 = 0 and 2x + 2y 13 = 0 is where a = b and b = . (5) The equation of the plane which passes through the point (4, 5, 1) and is parallel to the vectors (2, 0, 1) and (0, 1, 4) is x + by + cz = d where b = ,c= , and d = . (6) The point of intersection of the plane 2x 2y z + 1 = 0 and the line which passes through the points (3, 0, 4) and (2, 1, 1) is ( , , ). (7) The length of the curve r(t) = et i + et j + 2 t k between the points where t = 0 and t = 1 1 is a where a = . a (8) The length of the curve r(t) = (arctan t, 1 ln(1 + t2 ), 5) between the points where t = 0 and 2 t = 1 is ln(a + b) where a = and b = . (9) The length of the curve r(t) = (8t, 6t2 , 3t3 ) between the points where t = 0 and t = 2 is . (10) The length of the curve forming the intersection of the surfaces y = 2 2x and z = ln x between the points (1, 2 2, 0) and (2, 4, ln 2) is a + ln b where a = and b = . a (11) The curvature of the ellipse x2 + 4y 2 = 8 at the point (2, 1) is where a = and b b b= . a (12) The curvature of the spiral of Archimedes r = at the point where = 2 2 is where b a= and b = . 1 (13) The curvature of the curve r(t) = (t sin t) i + (1 cos j t) at the point 23 i + 1 j is 2 3 a where a = .
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CALCULUS III
FALL 2002 ASSIGNMENT 15
(14) The curvature of the curve r(t) = t2 i + (t4 t5 ) j at the point r(0) is a (15) The curvature of the curve y = ln x is greatest at the point x = where a = b b b= . At that point the curvature is where c = . c c (16) At the point where the curve y = 3x x3 attains a local maximum its curvature is a (17) The curvature of the cardioid r = 2(1 cos ) is rp where a = , b = b p= .
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(18) A particle P moves along a curve C. The position of P at time t is given by r(t) = 6t i + 2 t2 j + 5 k. a (a) When t = 6 the tangential component of the acceleration of P is where a = . 3 a a (b) When t = 6 the normal component of the acceleration of P is where a = b and b = . 2 (c) At the point r(6) the curvature of C is where a = . a (19) A particle P moves along a curve C. The position of P at time t is given by r(t) = 3t i + 1 t2 j 7 k. 2 (a) At the point r(3) the equation of the tangent line to the curve C is R(u) = (a + 3u) i + 9 ( 2 + bu) j + (c + du) k where a = ,b= ,c= , an...