University:
Amherst College
Course:
MATH 121 | Intermediate Calculus
Academic year:

2021
Views:

406

Pages:

4
Author:

do3rang8n

Review Packet for Exam #1 Fall 2021, Math 121 Derivatives: Compute the derivative for each of the following functions. Do not worry about simplifying your answers: 1. f (x) = arcsin x · arctan x + arctan(sin(ln x)) sinh(x2 − 2) 2. ///////////////////////// f (x) = skip this one x + sin−1 x x 3. f (x) = sin earcsin e 7 4. f (x) = ln 1 − arcsin x 5. Compute f ′′ (x) where f (x) = ln(1 − x). 6. Compute f ′′ (x) where f (x) = arctan(3x). 7. Compute f ′′ (x) where f (x) = arcsin(4x). Proofs: 8. Prove that d 1 arcsin x = √ dx 1 − x2 9. Prove that d 1 arctan x = dx 1 + x2 d 1 10. /////// Prove////// that////////////////////////////// sinh−1 x = √ dx 1 + x2 skip this one 11. Prove that d 5 arcsin(5x) = √ dx 1 − 25x2 12. Prove that d 1 ln x = dx x 13. Prove that d 3 arctan(3x) = dx 1 + 9x2 Z x 1 1 dx = arctan +C 9 + x2 3 3 Z x 1 √ 15. Prove, using Integration, that dx = arcsin +C 4 16 − x2 14. Prove, using Integration, that 1 Limits: Compute each of the following limit. 1 x 16. lim 1 + x→∞ x 7xex − arctan(7x) x→0 sin x + ln(1 − x) 17. lim 18. lim x→0 cos(4x) − 1 − arctan(4x) + 4x ln(1 − x) + arcsin x 1 − e−4x − arctan(4x) x→0 x2 19. lim 20. lim x3 ln x x→0+ 21. lim x ln x→0+ 1 x 1 22. lim x x2 x→∞ 1 23. lim (1 − 2x) x x→0+ 24. lim x3 + 1 1 ln x x→∞ 25. lim x2 sin x→∞ 26. lim x→0+ √ 1 x2 x ln x 3 4x 27. lim 1 − x→∞ x x2 6 28. lim 1 + ln 1 − 2 x→∞ x 3x4 5 29. lim 1 − arctan x→∞ x4 r 30. CHALLENGE: lim x→∞ 5 1 − 2 − sin x 1 x2 !5x2 x8 1 8 31. lim e x8 − 8 x→∞ x x x 32. lim x→∞ x + 3 2 Integrals: Compute each of the following integrals. Z 33. (ex + x)2 dx Z x sin2 x dx Z 1 √ dx 25 − x2 34. 35. Z 36. x2 Z 37. 1 dx + 25 1 p dx x 9 − ln2 x Z 38. x arcsin x dx Z 39. 1 3 (4 − x2 ) 2 Z dx e 40. ln x dx 1 Z ln(2x5 ) dx x2 Z cos5 x dx Z ln2 (x20 ) dx Z sin5 x cos2 x dx Z sin2 x cos3 x dx 41. 42. 43. 44. 45. Z 46. e3 1 dx x(9 + (ln x)2 ) √ e 3 Z e 47. 1 3 x(1 + (ln x)2 ) 2 1 Z 48. π 2 cos x 7 (1 + sin2 x) 2 0 √ Z 49. 3 √ 1 Z 50. (x2 dx dx 1 dx 4 − x2 1 dx + 4)2 3 Z 51. 1 (x2 Z 52. 7 dx + 4) 2 x4 arcsin x dx Z 53. x arctan x dx Z 1 x tan−1 (x2 ) dx 54. 0 Z 55. Z x2 dx x6 + 1 e2 56. √ x ln x dx 1 x2 Z 57. 3 (1 − x2 ) 2 Z dx e (ln x)2 dx 58. 1 √ Z 3 √ 59. 0 1 1 + 2 dx 2 x +9 4−x x2 √ dx 16 − x2 Z p 61. x3 9 − x2 dx Z 60. Z 62. Z x2 dx x2 + 3 3 63. p 9 − x2 dx −3 Z 64. e√ x ln x dx 1 Z 65. x+3 √ dx 4 − x2 Z 66. CHALLENGE: (arcsin x)2 dx 4

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