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- 1. Section 1.2–1.3 A Catalog of Essential Functions The Limit of a Function V63.0121.027, Calculus I September 10, 2009 Announcements Syllabus is on the common Blackboard Ofﬁce Hours MTWR 3–4pm Read Sections 1.1–1.3 of the textbook this week. . . . . . .
- 2. Outline Classes of Functions Linear functions Quadratic functions Cubic functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions Limits Heuristics Errors and tolerances Examples Pathologies . . . . . .
- 3. Classes of Functions linear functions, deﬁned by slope an intercept, point and point, or point and slope. quadratic functions, cubic functions, power functions, polynomials rational functions trigonometric functions exponential/logarithmic functions . . . . . .
- 4. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. . . . . . .
- 5. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. Example In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a function of distance x traveled. . . . . . .
- 6. Linear functions Linear functions have a constant rate of growth and are of the form f(x) = mx + b. Example In New York City taxis cost $2.50 to get in and $0.40 per 1/5 mile. Write the fare f(x) as a function of distance x traveled. Answer If x is in miles and f(x) in dollars, f(x) = 2.5 + 2x . . . . . .
- 7. Quadratic functions These take the form f(x) = ax2 + bx + c . . . . . .
- 8. Quadratic functions These take the form f(x) = ax2 + bx + c The graph is a parabola which opens upward if a > 0, downward if a < 0. . . . . . .
- 9. Cubic functions These take the form f(x) = ax3 + bx2 + cx + d . . . . . .
- 10. Other power functions Whole number powers: f(x) = xn . 1 negative powers are reciprocals: x−3 = 3 . x √ fractional powers are roots: x1/3 = 3 x. . . . . . .
- 11. Rational functions Deﬁnition A rational function is a quotient of polynomials. Example x 3 (x + 3 ) The function f(x) = is rational. (x + 2)(x − 1) . . . . . .
- 12. Trigonometric Functions Sine and cosine Tangent and cotangent Secant and cosecant . . . . . .
- 13. Exponential and Logarithmic functions exponential functions (for example f(x) = 2x ) logarithmic functions are their inverses (for example f(x) = log2 (x)) . . . . . .
- 14. Outline Classes of Functions Linear functions Quadratic functions Cubic functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions Limits Heuristics Errors and tolerances Examples Pathologies . . . . . .
- 15. Transformations of Functions Take the sine function and graph these transformations: ( π) sin x + ( 2 π) sin x − 2 π sin (x) + 2 π sin (x) − 2 . . . . . .
- 16. Transformations of Functions Take the sine function and graph these transformations: ( π) sin x + ( 2 π) sin x − 2 π sin (x) + 2 π sin (x) − 2 Observe that if the ﬁddling occurs within the function, a transformation is applied on the x-axis. After the function, to the y-axis. . . . . . .
- 17. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .
- 18. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .
- 19. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .
- 20. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units . . . . . .
- 21. Vertical and Horizontal Shifts Suppose c > 0. To obtain the graph of y = f(x) + c, shift the graph of y = f(x) a distance c units upward y = f(x) − c, shift the graph of y = f(x) a distance c units downward y = f(x − c), shift the graph of y = f(x) a distance c units to the right y = f(x + c), shift the graph of y = f(x) a distance c units to the left . . . . . .
- 22. Outline Classes of Functions Linear functions Quadratic functions Cubic functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions Limits Heuristics Errors and tolerances Examples Pathologies . . . . . .
- 23. Composition is a compounding of functions in succession g . ◦f . x . f . . g . . g ◦ f)(x) ( f .(x) . . . . . .
- 24. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. . . . . . .
- 25. Composing Example Let f(x) = x2 and g(x) = sin x. Compute f ◦ g and g ◦ f. Solution f ◦ g(x) = sin2 x while g ◦ f(x) = sin(x2 ). Note they are not the same. . . . . . .
- 26. Decomposing Example √ Express x2 − 4 as a composition of two functions. What is its domain? Solution √ We can write the expression as f ◦ g, where f(u) = u and g(x) = x2 − 4. The range of g needs to be within the domain of f. To insure that x2 − 4 ≥ 0, we must have x ≤ −2 or x ≥ 2. . . . . . .
- 27. The Far Side . . . . . .
- 28. Outline Classes of Functions Linear functions Quadratic functions Cubic functions Other power functions Rational functions Trigonometric Functions Exponential and Logarithmic functions Transformations of Functions Compositions of Functions Limits Heuristics Errors and tolerances Examples Pathologies . . . . . .
- 29. Limit . . . . . .
- 30. Zeno’s Paradox That which is in locomotion must arrive at the half-way stage before it arrives at the goal. (Aristotle Physics VI:9, 239b10) . . . . . .
- 31. Heuristic Deﬁnition of a Limit Deﬁnition We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a (on either side of a) but not equal to a. . . . . . .
- 32. The error-tolerance game A game between two players to decide if a limit lim f(x) exists. x→a Player 1: Choose L to be the limit. Player 2: Propose an “error” level around L. Player 1: Choose a “tolerance” level around a so that x-points within that tolerance level are taken to y-values within the error level. If Player 1 can always win, lim f(x) = L. x→a . . . . . .
- 33. The error-tolerance game L . . a . . . . . . .
- 34. The error-tolerance game L . . a . . . . . . .
- 35. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 36. The error-tolerance game T . his tolerance is too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 37. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 38. The error-tolerance game S . till too big L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 39. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 40. The error-tolerance game T . his looks good L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 41. The error-tolerance game S . o does this L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. . . . . . .
- 42. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . .
- 43. The error-tolerance game L . . a . To be legit, the part of the graph inside the blue (vertical) strip must also be inside the green (horizontal) strip. If Player 2 shrinks the error, Player 1 can still win. . . . . . .
- 44. Example Find lim x2 if it exists. x→0 . . . . . .
- 45. Example Find lim x2 if it exists. x→0 Solution I claim the limit is zero. . . . . . .
- 46. Example Find lim x2 if it exists. x→0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. . . . . . .
- 47. Example Find lim x2 if it exists. x→0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round. . . . . . .
- 48. Example Find lim x2 if it exists. x→0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round. What should the tolerance be if the error is 0.0001? . . . . . .
- 49. Example Find lim x2 if it exists. x→0 Solution I claim the limit is zero. If the error level is 0.01, I need to guarantee that −0.01 < x2 < 0.01 for all x sufﬁciently close to zero. If −0.1 < x < 0.1, then 0 ≤ x2 ≤ 0.01, so I win that round. What should the tolerance be if the error is 0.0001? By setting tolerance equal to the square root of the error, we can guarantee to be within any error. . . . . . .
- 50. Example |x| Find lim if it exists. x→0 x . . . . . .
- 51. Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? . . . . . .
- 52. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 53. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 54. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 55. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 56. The error-tolerance game y . . . 1 . x . . Part of graph in- . 1. − side blue is not inside green . . . . . .
- 57. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 58. The error-tolerance game y . . . 1 . x . . 1. − . . . . . .
- 59. The error-tolerance game y . . Part of graph in- side blue is not . . 1 inside green . x . . 1. − . . . . . .
- 60. The error-tolerance game y . . Part of graph in- side blue is not . . 1 inside green . x . . 1. − These are the only good choices; the limit does not exist. . . . . . .
- 61. One-sided limits Deﬁnition We write lim f(x) = L x→a+ and say “the limit of f(x), as x approaches a from the right, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a (on either side of a) and greater than a. . . . . . .
- 62. One-sided limits Deﬁnition We write lim f(x) = L x→a− and say “the limit of f(x), as x approaches a from the left, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufﬁciently close to a (on either side of a) and less than a. . . . . . .
- 63. Example |x| Find lim if it exists. x→0 x Solution The function can also be written as { |x| 1 if x > 0; = x −1 if x < 0 What would be the limit? The error-tolerance game fails, but lim f(x) = 1 lim f(x) = −1 x→0+ x→0− . . . . . .
- 64. Example 1 Find lim if it exists. x→0+ x . . . . . .
- 65. The error-tolerance game y . .? . L . x . 0 . . . . . . .
- 66. The error-tolerance game y . .? . L . x . 0 . . . . . . .
- 67. The error-tolerance game y . .? . L . x . 0 . . . . . . .
- 68. The error-tolerance game y . . The graph escapes the green, so no good .? . L . x . 0 . . . . . . .
- 69. The error-tolerance game y . .? . L . x . 0 . . . . . . .
- 70. The error-tolerance game y . E . ven worse! .? . L . x . 0 . . . . . . .
- 71. The error-tolerance game y . . The limit does not exist because the function is unbounded near 0 .? . L . x . 0 . . . . . . .
- 72. Example 1 Find lim if it exists. x→0+ x Solution The limit does not exist because the function is unbounded near 0. Next week we will understand the statement that 1 lim = +∞ x→0+ x . . . . . .
- 73. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x . . . . . .
- 74. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x f(x) = 0 when x = f(x) = 1 when x = f(x) = −1 when x = . . . . . .
- 75. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k f(x) = 1 when x = f(x) = −1 when x = . . . . . .
- 76. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k 1 f(x) = 1 when x = for any integer k 2k + 1/2 f(x) = −1 when x = . . . . . .
- 77. Weird, wild stuff Example (π ) Find lim sin if it exists. x→0 x 1 f(x) = 0 when x = for any integer k k 1 f(x) = 1 when x = for any integer k 2k + 1/2 1 f(x) = −1 when x = for any integer k 2k − 1/2 . . . . . .
- 78. Weird, wild stuff continued Here is a graph of the function: y . . . 1 . x . . 1. − There are inﬁnitely many points arbitrarily close to zero where f(x) is 0, or 1, or −1. So the limit cannot exist. . . . . . .
- 79. What could go wrong? Summary of Limit Pathologies How could a function fail to have a limit? Some possibilities: left- and right- hand limits exist but are not equal The function is unbounded near a Oscillation with increasingly high frequency near a . . . . . .
- 80. Meet the Mathematician: Augustin Louis Cauchy French, 1789–1857 Royalist and Catholic made contributions in geometry, calculus, complex analysis, number theory created the deﬁnition of limit we use today but didn’t understand it . . . . . .
- 81. Precise Deﬁnition of a Limit No, this is not going to be on the test Let f be a function deﬁned on an some open interval that contains the number a, except possibly at a itself. Then we say that the limit of f(x) as x approaches a is L, and we write lim f(x) = L, x→a if for every ε > 0 there is a corresponding δ > 0 such that if 0 < |x − a| < δ , then |f(x) − L| < ε. . . . . . .
- 82. The error-tolerance game = ε, δ L . . a . . . . . . .
- 83. The error-tolerance game = ε, δ L . +ε L . . −ε L . a . . . . . . .
- 84. The error-tolerance game = ε, δ L . +ε L . . −ε L . . −δ a a . a . +δ . . . . . .
- 85. The error-tolerance game = ε, δ T . his δ is too big L . +ε L . . −ε L . . −δ a a . a . +δ . . . . . .
- 86. The error-tolerance game = ε, δ L . +ε L . . −ε L . a a . −δ . . +δ a . . . . . .
- 87. The error-tolerance game = ε, δ T . his δ looks good L . +ε L . . −ε L . a a . −δ . . +δ a . . . . . .
- 88. The error-tolerance game = ε, δ S . o does this δ L . +ε L . . −ε L . aa . −δ . . +δ a . . . . . .

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