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# X2 T08 03 inequalities & graphs

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### X2 T08 03 inequalities & graphs

1. 1. Inequalities & Graphs
2. 2. Inequalities & Graphs x2 e.g. i  Solve 1 x2
3. 3. Inequalities & Graphs x2 e.g. i  Solve 1 x2
4. 4. Inequalities & Graphs x2 e.g. i  Solve 1 x2
5. 5. Inequalities & Graphs x2 e.g. i  Solve 1 x2
6. 6. Inequalities & Graphs x2 e.g. i  Solve 1 x2 Oblique asymptote:
7. 7. Inequalities & Graphs x2 e.g. i  Solve 1 x2  x2 4 x2 Oblique asymptote: x2 x2
8. 8. Inequalities & Graphs x2 e.g. i  Solve 1 x2  x2 4 x2 Oblique asymptote: x2 x2
9. 9. Inequalities & Graphs x2 e.g. i  Solve 1 x2  x2 4 x2 Oblique asymptote: x2 x2
10. 10. Inequalities & Graphs x2 e.g. i  Solve 1 x2
11. 11. Inequalities & Graphs x2 e.g. i  Solve 1 x2
12. 12. Inequalities & Graphs x2 e.g. i  Solve 1 x2 x2 1 x2 x2  x  2 x2  x  2  0  x  2 x  1  0 x  2 or x  1
13. 13. Inequalities & Graphs x2 e.g. i  Solve 1 x2 x2 1 x2 x2  x  2 x2  x  2  0  x  2 x  1  0 x  2 or x  1 x2 1 x2 x  2 or  1  x  2
14. 14. (ii) (1990) Consider the graph y  x
15. 15. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0
16. 16. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx
17. 17. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx y x dy 1  dx 2 x
18. 18. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx y x dy 1  dx 2 x dy   0 for x  0 dx
19. 19. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx y x at x  0, y  0 dy 1  dx 2 x dy   0 for x  0 dx
20. 20. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx y x at x  0, y  0 dy 1 when x  0, y  0  dx 2 x dy   0 for x  0 dx
21. 21. (ii) (1990) Consider the graph y  x a) Show that the graph is increasing for all x  0 dy Curve is increasing when 0 dx y x at x  0, y  0 dy 1 when x  0, y  0  dx 2 x dy   0 for x  0  curve is increasing for x  0 dx
22. 22. b) Hence show that; n 2 1  2    n   xdx  n n 0 3
23. 23. b) Hence show that; n 2 1  2    n   xdx  n n 0 3
24. 24. b) Hence show that; n 2 1  2    n   xdx  n n 0 3 As x is increasing; Area outer rectangles  Area under curve
25. 25. b) Hence show that; n 2 1  2    n   xdx  n n 0 3 As x is increasing; Area outer rectangles  Area under curve n 1  2    n   xdx 0
26. 26. b) Hence show that; n 2 1  2    n   xdx  n n 0 3 As x is increasing; Area outer rectangles  Area under curve n 1  2    n   xdx 0 n 2 x x   3 0  2  n n 3
27. 27. b) Hence show that; n 2 1  2    n   xdx  n n 0 3 As x is increasing; Area outer rectangles  Area under curve n 1  2    n   xdx 0 n 2 x x   3 0  2  n n 3 n 2  1  2    n   xdx  n n 0 3
28. 28. c) Use mathematical induction to show that; 4n  3 1  2  n  n for all integers n  1 6
29. 29. c) Use mathematical induction to show that; 4n  3 1  2  n  n for all integers n  1 6 Test: n = 1
30. 30. c) Use mathematical induction to show that; 4n  3 1  2  n  n for all integers n  1 6 Test: n = 1 L.H .S  1 1
31. 31. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6
32. 32. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S
33. 33. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S Hence the result is true for n = 1
34. 34. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S Hence the result is true for n = 1 Assume the result is true for n  k where k is a positive integer
35. 35. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S Hence the result is true for n = 1 Assume the result is true for n  k where k is a positive integer 4k  3 i.e. 1  2    k  k 6
36. 36. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S Hence the result is true for n = 1 Assume the result is true for n  k where k is a positive integer 4k  3 i.e. 1  2    k  k 6 Prove the result is true for n  k  1
37. 37. c) Use mathematical induction to show that; 4n  3 1 2  n  n for all integers n  1 6 Test: n = 1 41  3 L.H .S  1 R.H .S  1 6 1 7  6  L.H .S  R.H .S Hence the result is true for n = 1 Assume the result is true for n  k where k is a positive integer 4k  3 i.e. 1  2    k  k 6 Prove the result is true for n  k  1 4k  7 i.e. Prove 1  2    k  1  k 1 6
38. 38. Proof:
39. 39. Proof: 1  2  k 1  1  2  k  k 1
40. 40. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6
41. 41. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6
42. 42. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6 16k 3  24k 2  9k  6 k  1  6
43. 43. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6 16k 3  24k 2  9k  6 k  1  6 k  116k 2  8k  1  1  6 k  1  6
44. 44. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6 16k 3  24k 2  9k  6 k  1  6 k  116k 2  8k  1  1  6 k  1  6 k  14k  12  1  6 k  1  6
45. 45. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6 16k 3  24k 2  9k  6 k  1  6 k  116k 2  8k  1  1  6 k  1  6 k  14k  12  1  6 k  1  6 k  14k  1  6 k  1 2  6
46. 46. Proof: 1  2  k 1  1  2  k  k 1 4k  3  k  k 1 6 4k  3 k  6 k  1 2  6 16k 3  24k 2  9k  6 k  1  6 k  116k 2  8k  1  1  6 k  1  6 k  14k  12  1  6 k  1  6 k  14k  1  6 k  1 2  6 4k  1 k  1  6 k  1  6  4k  7  k  1 6
47. 47. Hence the result is true for n = k +1 if it is also true for n =k
48. 48. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
49. 49. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred
50. 50. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred 2 4n  3 n n  1 2  n  n 3 6
51. 51. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred 2 4n  3 n n  1 2  n  n 3 6 2 410000   3 10000 10000  1  2    10000  10000 3 6
52. 52. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred 2 4n  3 n n  1 2  n  n 3 6 2 410000   3 10000 10000  1  2    10000  10000 3 6 666700  1  2    10000  666700
53. 53. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred 2 4n  3 n n  1 2  n  n 3 6 2 410000   3 10000 10000  1  2    10000  10000 3 6 666700  1  2    10000  666700  1  2    10000  666700 to the nearest hundred
54. 54. Hence the result is true for n = k +1 if it is also true for n =k Since the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n d) Use b) and c) to estimate; 1  2    10000 to the nearest hundred 2 4n  3 n n  1 2  n  n 3 6 2 410000   3 10000 10000  1  2    10000  10000 3 6 666700  1  2    10000  666700  1  2    10000  666700 to the nearest hundred Exercise 10F