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finalexamreview - Math 222 Final Review Packet Disclaimer 1...

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Math 222: Final Review Packet Disclaimer 1. This review packet should only be viewed as supplementary to (and not a replacement for) studying the notes, old homeworks, quizzes, exams, etc. For example, note that there is no spring-mass problem in these notes even though these problems are fair game for the exam. This was prepared without any knowledge of what will be emphasized on the final exam; however, if you can master all of the problems in this packet (and be able to solve them without help), in your homeworks, and in past quizzes and exams, then you should have almost no trouble on the final exam. Integration Techniques 1. integraldisplay xdx x 2 + 2 x + 2 2. integraldisplay dx ( x 1)( x + 1)( x + 2) 3. integraldisplay dx cos 4 ( x ) 4. integraldisplay (3 x + 1) dx 3 x 2 + 2 x + 1 5. integraldisplay x 2 e x 3 dx 6. integraldisplay sin 2 (2 x ) dx 7. integraldisplay x 4 ln( x ) dx 8. integraldisplay sin 3 ( x ) dx 9. integraldisplay sin( x ) dx radicalbig 1 + cos( x ) 10. integraldisplay parenleftbigg 1 + x x parenrightbigg 2 dx 11. integraldisplay tan 3 ( x ) sec( x ) dx 12. integraldisplay 4 1 e x dx 13. integraldisplay π 0 sin 2 parenleftBig x 3 parenrightBig dx 14. integraldisplay xdx x 2 + 5 x + 6 15. integraldisplay e 2 1 ln( x ) dx 16. integraldisplay π/ 2 0 sin( x ) dx (3 2 cos( x )) 2 17. integraldisplay x 5 dx x 3 + 1 18. integraldisplay x 2 e x dx 19. integraldisplay sec 6 ( x ) dx 20. integraldisplay x 7 + x 3 x 4 1 dx 21. integraldisplay (2 x + 3) dx 4 x 2 + 4 x + 5 22. integraldisplay cos( x ) dx sin 2 ( x ) 3 sin( x ) + 2 23. integraldisplay x sin(ln( x )) dx 24. integraldisplay 1 0 x + 1 x 2 + 2 x 4 dx 25. integraldisplay x sec 2 ( x ) dx 26. integraldisplay π 2 0 radicalbig 1 cos( x ) dx 27. integraldisplay π 0 sin 3 parenleftBig x 3 parenrightBig dx 28. integraldisplay x 3 dx x 2 + 5 29. integraldisplay xdx x 2 + 4 x + 4 30. integraldisplay 1 0 x + 1 x 2 + 2 x + 4 dx 31. integraldisplay x csc 2 ( x ) dx 32. integraldisplay tan 2 parenleftBig x 4 parenrightBig dx 33. integraldisplay 4 2 x 1 x dx 34. integraldisplay x 2 ln( x ) dx 35. integraldisplay 4 x 4 + 1 4 x 3 x dx 36. integraldisplay π 2 π 2 sin 3 ( x ) dx cos( x ) + x 2 + 1 37. integraldisplay 1 0 x 3 e x 2 dx 38. integraldisplay 1 0 x 2 dx 4 x 3 39. integraldisplay x + 2 x 3 x 2 2 x dx 40. integraldisplay dt e 3 t 1 41. integraldisplay e 2 e dx x ln( x ) 42. integraldisplay x 3 + 2 x 3 x dx 43. integraldisplay x 3 + 13 x 2 + 4 x + 5 dx 44. integraldisplay sin 2 ( x ) sin(2 x ) dx 45. integraldisplay cos( x ) dx sin 3 ( x ) sin( x )
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Math 222: Final Review Packet 46. integraldisplay 4 x 3 20 x x 4 10 x 2 + 9 dx 47. integraldisplay e 1 dx x + x ln( x ) 48. integraldisplay ln( x ) x 2 dx 49. integraldisplay radicalBig 1 + tdt 50. integraldisplay π 2 0 radicalbig sin( x ) dx radicalbig sin( x ) + radicalbig cos( x ) 51. integraldisplay sin 3 (4 x ) cos(6 x ) dx Taylor’s Series 1. We wish to evaluate the function f ( x ) = ln(1 + x ) for values of x with 1 10 x 1 10 . We decide to approximate f ( x ) by an expression of the form x + Ax 2 for some constant A . What would be a good choice for A , and what would be a reasonable estimate for the maximum error in this approximation?
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