M09E1 - MATH 102 MAKE-UP OF EXAM 1 TIME 90 MINUTES...

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Unformatted text preview: MATH 102 MAKE-UP OF EXAM 1 TIME: 90 MINUTES 02.04.2009 NAME: . . . . . . . . . . . . . . . . ..................................................................................................................................................................................................................................................................................................................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................................................................................................................................................................................................................................. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .................................................................................................................................................................................................................................................................................................................. . . . . . . . . . . . 1. Find the limits: n2 + 2 = n!1 (n + 1)(2n2 + 5) (a) (5 pts) lim (b) (5 pts) lim cos n!1 3n2 + 2 2n2 + n + 1 = n5 + 3n = n!1 3n + n2 (c) (5 pts) lim 2. (10 pts) Find the sum of the geometric series 4 3 8 9 + 16 27 2 3. (10 pts) (a) Find the radius of convergence R of the power series 1 X n=1 1 xn : (n + 1)3n (b) (5 pts) If R is your answer to (a), does the series converge for x = R and x = R? 4. (10 pts) For the function f (x) = x 3x write the second degree Taylor polynomial about x = 1: 3 5. (10 pts) (a) Write the …rst 3 nonzero terms (including the constant one) of the Taylor series expansions about x = 0 of the functions f (x) = cos(2x) and g (x) = (1 + 4x2 ) 1=2 (b) (5 pts) Use your result in (a) to decide which function takes larger values for x close to 0. 6. (10 pts) Sketch the level curves (contour lines) of the function z = values 1; 4: Indicate clearly near each curve the corresponding value of z: x2 1+y 2 for the 4 2 7. (10 pts) Find the Taylor series of the function h(x) = (1 + 2x)ex about 0: Write the …rst four terms and the terms of degree 99 and 100. 8. (15 pts) Explain why the statement is true or give a counterexample if it is false. (a) The series (b) The series P1 n=1 1 X sin n 2 n=1 (c) The series an converges if an ! 0 as n ! 1: 1 X n=2 n converges. 1 converges. (Hint: Use the Integral Test.) n ln n (d) The cross– section of the graph of the function z = circle. p x2 + y 2 with yz – plane is a ( cos(x2 + 2xy ) (x; y ) 6= (0; 0) (e) The function f (x; y ) = 2 (x; y ) = (0; 0) is continuous at (0,0). ...
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