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ASU MAT 266 - mat266_ex3_0

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MAT 266 Test3 1 Directions: 1. There are 14 questions worth a total of 60 points. 2. Questions 1 - 10 are Multiple Choice worth 4 points each to be answered on the supplied SCANTRONS. 3. Questions 11 - 14 are Free Responses worth 5 points each and are to be answered in the space provided on the test. 4. Read all the questions carefully. 5. For the Free Response, you must show all work in order to receive credit!! 6. When possible, box your answer, which must be complete, organized, and exact unless otherwise directed. 7. Always indicate how a calculator was used (i.e. sketch graph, etc. …). 8. No calculators with QWERTY keyboards or ones like TI-89 or TI-92 that do symbolic algebra may be used. Honor Statement: By signing below you confirm that you have neither given nor received any unauthorized assistance on this exam. This includes any use of a graphing calculator beyond those uses specifically authorized by the Mathematics Department and your instructor. Furthermore, you agree not to discuss this exam with anyone until the exam testing period is over. In addition, your calculator’s program memory and menus may be checked at any time and cleared by any testing center proctor or Mathematics Department instructor. _______________________________________ _________________ Signature Date MAT 266 TEST 3 ANSWERS/SOLUTIONS SoMSS, ASUMAT 266 Test3 2 1. Find the interval of convergence of the series ∑∞=+−14)1(nnnnx Select the correct answer. a. (0, 2) b. (-1, 2) c. (-1, 1) d. diverges everywhere e. None of these 2. Suppose that the radius of convergence of the power series ∑∞=0nnnxc is 16. What is the radi-us of convergence of the power series ∑∞=02nnnxc? Select the correct answer. a. 256 b. 4 c. 1 d. 16 e. None of these 3. Find the radius of convergence of the series. ∑∞=+12)5(8nnnnx Select the correct answer. a. R = 8 b. R = 1/8 c. R =1 d. R = 5 e. None of these 4. Find the Taylor polynomial 3T for the function ( ) lnf x x= at the number a = 1. Select the correct answer. a. 32)1(21)1(31)1( +++−+ xxx b. 32)1(31)1(21)1( −+−−− xxx c. 32)1(21)1(41)1( −+−+− xxx d. 32)1(21)1(41)1( −+−+− xxx e. None of the above 5. Find the sum of the series: 012 !nnn∞=∑ Select the correct answer. a. 1/e b. e c. 2e d. 2e− e. None of theseMAT 266 Test3 3 6. Evaluate the indefinite integral ∫−dtt )(tan21 as a power series. [The MacLaurin series for 3 5 7 2 110tan ... ( 1)3 5 7 2 1nnnx x x xx xn+∞−== − + − + = −+∑] Select the correct answer. a. ∑∞=+++−+034)34)(12()1(nnnnntC b. ∑∞=++−+034)34()1(nnnntC c. ∑∞=+++−+024)34)(12()1(nnnnntC d. ∑∞=++−+022)12()1(nnnntC e. None of the above 7. Find the first three nonzero terms in the MacLaurin series for the function xexfx4cos)(2−= Select the correct answer. a. 2 41 17 19.17x x− + b. 2 41 9 11.17x x− + c. 2 41 9 19.17x x− + d. 21 9 19.17x x− + e. None of the above. 8. Eliminate the parameter to find the Cartesian equation of the curve cos , sec , 0 / 2x yθ θ θ π= = ≤ < Select the correct answer. a. 2y x= b. y x= c. 2y x= d. 1yx= e. None of these 9. Find the radius of convergence of the binomial series expansion of the 21( )(2 )f xx=+ Select the correct answer. a. 1R= b. 1 / 2R= c. 2R= d. 4R= e. None of these 10. Find parametric equations to represent the line segment from (-1, 2) to (10, -6). Select the correct answer. a. 10,111,810 ≤≤+−=−= ttytx b. 10,82,111 ≤≤−=+−= ttytx c. 10,86,111 ≤≤−−=+−= ttytx d. 10,8,111 ≤≤−=−−= ttytx e. None of the aboveMAT 266 Test3 4 11. Find the polar equation for the curve represented by the given Cartesian equation 4x y+ = Select the correct answer. a. 2cos sinrθ θ=− b. 4cos sinrθ θ=− c. 4(cos sin )rθ θ= + d. 2(cos sin )rθ θ= + e. 4cos sinrθ θ=+ 12. Set up the integral for the length of the curve 2 33 , 2 , 0 1x t y t t= = ≤ ≤. Select the correct answer. a. 12 2 2(6 ) (6 )0t t dt−∫ b. 12 2 2(6 ) (6 )0t t dt+∫ c. 126 60t t dt+∫ d. 12 2(6 6 )0t t dt+∫ e. 12 2 2(36 ) (36 )0t t dt+∫ f. none of these FREE RESPONSE 13. Use series to evaluate: 30(sin )limxx xx→−MAT 266 Test3 5 14. Use the power series representation of ( ) sin( )f x x=to find a power series representation for 2( ) sin( )g x x x= . 15. Test the series for convergence or divergence: 513mmm∞=∑MAT 266 Test3 6 16. Use the Binomial Theorem to find the first 3 terms of the MacLaurin Series for 3( ) 5f x x= + and use it to estimate 308.5. 17. [5 pts] Find the area bounded by the curve 1 2,x t y tt t= − = + and the line y = 4.5 18. [5 pts] Find the slope of the tangent line to the polar curve θ2=r when πθ= 19. [5 pts] Find the area of the region that lies inside the curve 3cosrθ= and outside the curve 1 cos( )rθ= +. 20. [5 pts] Find the exact coordinates of the highest point on the curve 15 , 3t tx te y te−=


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ASU MAT 266 - mat266_ex3_0

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