Multiple Choice.(5 pts.) Find the sum of the geometric series n=0 ( ) n. (c) (d).(5 pts.) Find the 5 th Maclaurin polynomial for the function f(x) = sin x. (Recall that Maclaurin polynomial is another name for Taylor polynomial centered at 0.) x x + 5 x5 x + x x + 5 x5 (c) x 6 x + 0 x5 (d) x x + x 5 x x 8 x5.(5 pts.) Calculate lim x 0 cos(x) cos(x) x. 5 (c) 0 (d).(5 pts.) Find the indefinite integral x x + x dx. ln x ln x + + C ln x + ln x + + C (c) arctan (x + x ) ln x + + C (d) ln x + + ln x + C (x + ) (x ) + C
5.(5 pts.) Which of the following is the form of the partial fraction decomposition of x + x + x + x (x? + ) x + B x + Cx + D x + x + Bx x + Cx + D x + (c) x + Bx + C x + D x + (d) x + B x + C x + x + B x + C x + D x + 6.(5 pts.) Consider the sequence {a n } given by a n = n sin n = sin(/n) /n Which of the following statements describes the behavior of this sequence? It has limit. It has limit 0. (c) It has limit. (d) It has limit π. It does not converge.. 7.(5 pts.) Consider the series (I) statements is true? n= n! n n and (II) n= ( ) n n. Which one of the following Both series converge. (I) converges and (II) diverges. (c) (I) diverges and (II) converges. (d) Both series diverge. Both series are geometric series. 8.(5 pts.) Find the sum of the convergent series n= (n + )(n + ). (c) (d) 0x 0 + 9x 9 + 8x 8 + 7x 7 + 6x 6 + 5x 5 + x + x + x + x 9.(5 pts.) Calculate lim x e x. 0 0! (c) (d) e 0
0.(5 pts.) Let P (x) be the second Maclaurin polynomial for the function f(x) = Find the value of P (). (c) 0 (d) x. 6x.(5 pts.) Determine whether the improper integral (x dx converges or diverges, and evaluate the integral if it converges. + ) Suggestion: you can make use of the indefinite integral formula 6x (x + ) dx = x + x + + C. It converges to. It diverges. (c) It converges to 0. (d) It converges to. It converges to π..(5 pts.) Find the radius of convergence of the power series n= n ( x ) n. (n + )! (c) (d) e.(5 pts.) What is the area of the region above the x-axis, below the graph of y = e x, and to the right of the line x =? e (c) e (d) ln.(5 pts.) There is a number L such that the series and diverges if k > L. What is L? n= ( ) n kn converges if k < L n + (c) (d) e Partial Credit You must show your work on the partial credit problems to receive credit!
5.(0 pts.) Compute the indefinite integral x 5 x dx by using a trigonometric substitution. 6.(0 pts.) person marooned on a desert island decides to compute cos() by using the 5 th Maclaurin polynomial P 5 (x) of f(x) = cos x. Use Taylor s theorem to give an upper bound for the maximum error that can occur in the castaway s approximation. You don t have to express your answer in decimal notation; giving a formula in terms of specific numbers is sufficient. 7.(0 pts.) In a small midwestern city, the annual spending by tourists is about $,000,000. bout 60% (/5) of this revenue is again spent in the city, and of that amount approximately 60% is again spent in the city, and so on. Write down the geometric series that gives the total amount of spending generated by the $,000,000 and find the sum of the series. (s long as you give a formula for the sum in terms of specific numbers, you don t have to carry out the arithmetic. If you do end up with complicated arithmetic, though, you are probably on the wrong track.) 5
Name: Instructor: NSWERS NSWERS Math 0 Exam II March 8, 00 The Honor Code is in effect for this examination. ll work is to be your own. No calculators. The exam lasts for 75 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all 5 pages of the test. The backs of pages may be used if you need additional room to work on a problem. Good Luck! PLESE MRK YOUR NSWERS WITH N X, not a circle!. ( ) (c) (d). ( ) (c) (d). ( ) (c) (d). ( ) (c) (d) 5. ( ) (c) (d) 6. ( ) (c) (d) 9. ( ) (c) (d) 0. ( ) (c) (d). ( ) (c) (d). ( ) (c) (d). ( ) (c) (d). ( ) (c) (d) 7. ( ) (c) (d) 8. ( ) (c) (d) DO NOT WRITE IN THIS BOX! Total multiple choice: 5. 6. 7. Total:
Name: Instructor: Math 0 Exam II March 8, 00 The Honor Code is in effect for this examination. ll work is to be your own. No calculators. The exam lasts for 75 minutes. Be sure that your name is on every page in case pages become detached. Be sure that you have all 5 pages of the test. The backs of pages may be used if you need additional room to work on a problem. Good Luck! PLESE MRK YOUR NSWERS WITH N X, not a circle!. (c) (d). (c) (d). (c) (d). (c) (d) 5. (c) (d) 6. (c) (d) 9. (c) (d) 0. (c) (d). (c) (d). (c) (d). (c) (d). (c) (d) 7. (c) (d) 8. (c) (d) DO NOT WRITE IN THIS BOX! Total multiple choice: 5. 6. 7. Total: