Student Print Section Last First Middle Student Sign Student ID Instructor MATH 152 Exam 3 Spring 2000 Test Form A Part I is multiple choice There is no partial credit Part II is work out Show all your work Partial credit will be given 1 10 50 11 10 12 15 13 15 14 10 You may not use a calculator TOTAL 1 Part I Multiple Choice 5 points each There is no partial credit You may not use a calculator 1 Find the values of x such that the vectors x 1 3 and 2 5 x are orthogonal a 1 only b 0 only c 1 only d 0 and 1 only e 1 and 1 only x 3 lim e 16 x xv0 x HINT The series for e x may be helpful a 0 b 1 3 c 1 3 d 1 2 3 2 Compute e 3 Consider the parametric curve r t t sin t t 3 Find parametric equations for the line tangent to the curve at t a x 1 t y t z 3 2 3 t b x 1 t y 1 z 3 2 3 t c x t y 1 t cos t z 3 3t 3 d x t y t cos t z 3 3t 3 e x t y t z 3 3 2 t 4 Find the Taylor series for f x x 2 3 about x 2 a 7 4 x 2 x 2 2 b 7 4 x 2 2 x 2 2 4 x 2 3 c 7 4 x 2 x 2 2 2 x 2 d 7 4 x 2 2 x 2 2 e 7 4 x 2 2 x 2 2 4 C 3 4 x 2 3 2 x 2 4 C 2 x 2 3 4 x 2 4 3 3 2 5 The vectors a b and c b a all lie in the same plane as shown in the diagram Which of the following statements is TRUE a a b 0 b a b points into the page a b c c 0 d b a c points in the direction of a e None of These 6 Find a power series centered at x 0 for the function f x of convergence a 1 n n 3n 1 R 1 8 1 n n 3n 1 R 8 8 x n 0 b 8 x x and determine its radius 1 8x 3 n 0 c 8 n x 3n 1 n n 0 d R 2 8 n x 3n 1 R 1 8 8 n x 3n 1 R 1 2 n 0 e n 0 7 Find the distance from the point 3 2 4 to the center of the sphere x 1 a b c d e 2 y 1 2 z 2 2 4 2 3 9 61 61 3 8 Let f x sin x 2 Compute f 14 0 the 14 th derivative of f x evaluated at 0 HINT Use a series for sin x 2 1 a 14 7 b 7 14 c 14 7 d 14 7 e 14 7 9 Find the angle between the vectors u 1 1 0 and v 1 2 1 a 0 b 30 c 45 d 60 e 90 a 1 n 0 b n 0 c n 0 d n 0 e n 0 1 3n 1 1 n 3n 1 1 n 1 3n 1 2 n 3n n 3n 1 1 2 1 2 1 1 2 1 2 1 dx as an infinite series 1 x3 1 13 16 19 C 2 2 2 10 Evaluate the integral 0 3n 1 3n 1 1 2 1 1 1 C 4 24 7 27 10 2 10 1 1 1 1 C 4 24 10 2 10 7 27 2 1 2 3n 1 3n 1 2 22 55 88 C 2 2 2 2 1 1 1 C 2 22 5 25 8 28 4 Part II Work Out points indicated below Show all your work Partial credit will be given You may not use a calculator 11 10 points Consider the planes P1 2x y z 1 P2 x y 3z 2 a 2 pts Fill in the blanks A normal to the plane P 1 is N 1 A normal to the plane P 2 is N 2 b 3 pts Find a vector parallel to the line of intersection of the two planes c 3 pts Find a point on the line of intersection of the two planes d 2 pts Find parametric equations for the line of intersection of the two planes 5 12 15 points Let f x lnx a 10 pts Find the 3 rd degree Taylor polynomial T 3 for f x about x 2 b 5 pts If this polynomial T 3 is used to approximate f x on the interval 1 t x t 3 estimate the maximum error R 3 in this approximation using Taylor s Inequality M x 2 n 1 where M u f n 1 x for 1 t x t 3 R n x n 1 6 13 15 points Consider the points P 1 0 1 Q 2 3 1 and R 0 4 1 a 5 pts Find a vector orthogonal to the plane determined by P Q and R b 5 pts Find the area of the triangle with vertices P Q and R c 5 pts Find the equation of the plane determined by P Q and R 7 14 10 points Find the radius of convergence and the interval of convergence of the series n 0 n 1 x 2 3n n 1 8
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