.                 Sec on 1.1      Func ons and their Representa ons                    V63.0121.001, Calculus I           ...
Section 1.1Functions and their Representations   V63.0121.001, Calculus I Professor Ma hew Leingang       New York Univers...
Announcements   First WebAssign-ments   are due January 31   Do the Get-to-Know-You   survey for extra credit!
Objectives  Understand the defini on of  func on.  Work with func ons  represented in different ways  Work with func ons defi...
What is a function? Defini on A func on f is a rela on which assigns to to every element x in a set D a single element f(x)...
Outline Modeling Examples of func ons    Func ons expressed by formulas    Func ons described numerically    Func ons desc...
The Modeling Process      Real-world           .           .        model      Mathema cal                                ...
Plato’s Cave               .
The Modeling Process             Real-world                  .                  .        model      Mathema cal           ...
Outline Modeling Examples of func ons    Func ons expressed by formulas    Func ons described numerically    Func ons desc...
Functions expressed by formulas Any expression in a single variable x defines a func on. In this case, the domain is unders...
Formula function example Example              x+1 Let f(x) =       . Find the domain and range of f.              x−2
Formula function example Example              x+1 Let f(x) =       . Find the domain and range of f.              x−2 Solu...
Formula function example Example              x+1 Let f(x) =       . Find the domain and range of f.              x−2 Solu...
How did you get that?                             x+1          start         y=                             x−2
How did you get that?                                 x+1              start         y=                                 x−...
How did you get that?                                  x+1               start         y=                                 ...
How did you get that?                                   x+1                start         y=                               ...
How did you get that?                                    x+1                start          y=                             ...
How did you get that?                                    x+1                start          y=                             ...
No-no’s for expressions   Cannot have zero in the   denominator of an   expression   Cannot have a nega ve   number under ...
Piecewise-defined functionsExampleLet             {              x2    0 ≤ x ≤ 1;      f(x) =              3−x   1 < x ≤ 2....
Piecewise-defined functionsExample                          Solu onLet                              The domain is [0, 2]. T...
Piecewise-defined functionsExample                          Solu onLet                              The domain is [0, 2]. T...
Piecewise-defined functionsExample                          Solu onLet                              The domain is [0, 2]. T...
Piecewise-defined functionsExample                          Solu onLet                              The domain is [0, 2]. T...
Functions described numerically We can just describe a func on by a table of values, or a diagram.
Functions defined by tables IExampleIs this a func on? If so, what isthe range?            x f(x)            1 4           ...
Functions defined by tables IExample                             Solu onIs this a func on? If so, what isthe range?        ...
Functions defined by tables IExample                             Solu onIs this a func on? If so, what isthe range?        ...
Functions defined by tables IExample                             Solu onIs this a func on? If so, what isthe range?        ...
Functions defined by tables IExample                             Solu onIs this a func on? If so, what isthe range?        ...
Functions defined by tables IExample                             Solu onIs this a func on? If so, what isthe range?        ...
Functions defined by tables IIExampleIs this a func on? If so, what isthe range?            x f(x)            1 4          ...
Functions defined by tables IIExample                             Solu onIs this a func on? If so, what isthe range?       ...
Functions defined by tables IIExample                             Solu onIs this a func on? If so, what isthe range?       ...
Functions defined by tables IIExample                             Solu onIs this a func on? If so, what isthe range?       ...
Functions defined by tables IIExample                             Solu onIs this a func on? If so, what isthe range?       ...
Functions defined by tables IIExample                             Solu onIs this a func on? If so, what isthe range?       ...
Functions defined by tables IIIExampleIs this a func on? If so, what isthe range?            x f(x)            1 4         ...
Functions defined by tables IIIExample                             Solu onIs this a func on? If so, what isthe range?      ...
Functions defined by tables IIIExample                             Solu onIs this a func on? If so, what isthe range?      ...
Functions defined by tables IIIExample                             Solu onIs this a func on? If so, what isthe range?      ...
Functions defined by tables IIIExample                             Solu onIs this a func on? If so, what isthe range?      ...
Functions defined by tables IIIExample                             Solu onIs this a func on? If so, what isthe range?      ...
An ideal function
An ideal function   Domain is the bu ons
An ideal function   Domain is the bu ons   Range is the kinds of soda   that come out
An ideal function   Domain is the bu ons   Range is the kinds of soda   that come out   You can press more than   one bu o...
An ideal function   Domain is the bu ons   Range is the kinds of soda   that come out   You can press more than   one bu o...
Why numerical functions matter Ques on Why use numerical func ons at all? Formula func ons are so much easier to work with.
Why numerical functions matter Ques on Why use numerical func ons at all? Formula func ons are so much easier to work with...
Numerical Function Example Example Here is the temperature in Boise, Idaho measured in 15-minute intervals over the period...
Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. The graph...
Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph.          ...
Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph.          ...
Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph.          ...
Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph.          ...
Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons:     The temperature T(t...
Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons:     The temperature T(t...
Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons:     The temperature T(t...
Outline Modeling Examples of func ons    Func ons expressed by formulas    Func ons described numerically    Func ons desc...
MonotonicityExampleLet P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?
MonotonicityExample            Solu onLet P(x) be theprobability that              1my income wasat least $x lastyear. Wha...
Monotonicity Defini on    A func on f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for    any two points x1 and x2 in ...
Examples Example Going back to the burrito func on, would you call it increasing?
Examples Example Going back to the burrito func on, would you call it increasing? Answer Not if they are all consumed at o...
Examples Example Going back to the burrito func on, would you call it increasing? Answer Not if they are all consumed at o...
Symmetry Consider the following func ons described as words Example Let I(x) be the intensity of light x distance from a p...
Possible Intensity Graph Example                     Solu on Let I(x) be the intensity of light x distance from           ...
Possible Gravity Graph Example                    Solu on Let F(x) be the gravita onal force at a              y = F(x) po...
Definitions Defini on    A func on f is called even if f(−x) = f(x) for all x in the domain    of f.    A func on f is calle...
Examples Example    Even: constants, even powers, cosine    Odd: odd powers, sine, tangent    Neither: exp, log
Summary  The fundamental unit of inves ga on in calculus is the func on.  Func ons can have many representa ons
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Lesson 1: Functions and their representations (slides)

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The function is the fundamental unit in calculus. There are many ways to describe functions: with words, pictures, symbols, or numbers.

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Lesson 1: Functions and their representations (slides)

  1. 1. . Sec on 1.1 Func ons and their Representa ons V63.0121.001, Calculus I Professor Ma hew Leingang New York UniversityAnnouncements First WebAssign-ments are due January 31 Do the Get-to-Know-You survey for extra credit!
  2. 2. Section 1.1Functions and their Representations V63.0121.001, Calculus I Professor Ma hew Leingang New York University
  3. 3. Announcements First WebAssign-ments are due January 31 Do the Get-to-Know-You survey for extra credit!
  4. 4. Objectives Understand the defini on of func on. Work with func ons represented in different ways Work with func ons defined piecewise over several intervals. Understand and apply the defini on of increasing and decreasing func on.
  5. 5. What is a function? Defini on A func on f is a rela on which assigns to to every element x in a set D a single element f(x) in a set E. The set D is called the domain of f. The set E is called the target of f. The set { y | y = f(x) for some x } is called the range of f.
  6. 6. Outline Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry
  7. 7. The Modeling Process Real-world . . model Mathema cal . Problems Model solve test Real-world interpret Mathema cal . . Predic ons Conclusions
  8. 8. Plato’s Cave .
  9. 9. The Modeling Process Real-world . . model Mathema cal . Problems Model Shadows Forms solve test Real-world interpret Mathema cal . . Predic ons Conclusions
  10. 10. Outline Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry
  11. 11. Functions expressed by formulas Any expression in a single variable x defines a func on. In this case, the domain is understood to be the largest set of x which a er subs tu on, give a real number.
  12. 12. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2
  13. 13. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solu on The denominator is zero when x = 2, so the domain is all real numbers except 2. We write: domain(f) = { x | x ̸= 2 }
  14. 14. Formula function example Example x+1 Let f(x) = . Find the domain and range of f. x−2 Solu on x+1 2y + 1 As for the range, we can solve y = =⇒ x = . So as x−2 y−1 long as y ̸= 1, there is an x associated to y. range(f) = { y | y ̸= 1 }
  15. 15. How did you get that? x+1 start y= x−2
  16. 16. How did you get that? x+1 start y= x−2 cross-mul ply y(x − 2) = x + 1
  17. 17. How did you get that? x+1 start y= x−2 cross-mul ply y(x − 2) = x + 1 distribute xy − 2y = x + 1
  18. 18. How did you get that? x+1 start y= x−2 cross-mul ply y(x − 2) = x + 1 distribute xy − 2y = x + 1 collect x terms xy − x = 2y + 1
  19. 19. How did you get that? x+1 start y= x−2 cross-mul ply y(x − 2) = x + 1 distribute xy − 2y = x + 1 collect x terms xy − x = 2y + 1 factor x(y − 1) = 2y + 1
  20. 20. How did you get that? x+1 start y= x−2 cross-mul ply y(x − 2) = x + 1 distribute xy − 2y = x + 1 collect x terms xy − x = 2y + 1 factor x(y − 1) = 2y + 1 2y + 1 divide x= y−1
  21. 21. No-no’s for expressions Cannot have zero in the denominator of an expression Cannot have a nega ve number under an even root (e.g., square root) Cannot have the logarithm of a nega ve number
  22. 22. Piecewise-defined functionsExampleLet { x2 0 ≤ x ≤ 1; f(x) = 3−x 1 < x ≤ 2.Find the domain and range of fand graph the func on.
  23. 23. Piecewise-defined functionsExample Solu onLet The domain is [0, 2]. The graph { can be drawn piecewise. x2 0 ≤ x ≤ 1; f(x) = 2 3−x 1 < x ≤ 2. 1Find the domain and range of fand graph the func on. . 0 1 2
  24. 24. Piecewise-defined functionsExample Solu onLet The domain is [0, 2]. The graph { can be drawn piecewise. x2 0 ≤ x ≤ 1; f(x) = 2 3−x 1 < x ≤ 2. 1Find the domain and range of fand graph the func on. . 0 1 2
  25. 25. Piecewise-defined functionsExample Solu onLet The domain is [0, 2]. The graph { can be drawn piecewise. x2 0 ≤ x ≤ 1; f(x) = 2 3−x 1 < x ≤ 2. 1Find the domain and range of fand graph the func on. . 0 1 2
  26. 26. Piecewise-defined functionsExample Solu onLet The domain is [0, 2]. The graph { can be drawn piecewise. x2 0 ≤ x ≤ 1; f(x) = 2 3−x 1 < x ≤ 2. 1Find the domain and range of fand graph the func on. . 0 1 2 The range is [0, 2).
  27. 27. Functions described numerically We can just describe a func on by a table of values, or a diagram.
  28. 28. Functions defined by tables IExampleIs this a func on? If so, what isthe range? x f(x) 1 4 2 5 3 6
  29. 29. Functions defined by tables IExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6
  30. 30. Functions defined by tables IExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6
  31. 31. Functions defined by tables IExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6
  32. 32. Functions defined by tables IExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6
  33. 33. Functions defined by tables IExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 5 3 6 3 6 Yes, the range is {4, 5, 6}.
  34. 34. Functions defined by tables IIExampleIs this a func on? If so, what isthe range? x f(x) 1 4 2 4 3 6
  35. 35. Functions defined by tables IIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6
  36. 36. Functions defined by tables IIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6
  37. 37. Functions defined by tables IIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6
  38. 38. Functions defined by tables IIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6
  39. 39. Functions defined by tables IIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 2 4 3 6 3 6 Yes, the range is {4, 6}.
  40. 40. Functions defined by tables IIIExampleIs this a func on? If so, what isthe range? x f(x) 1 4 1 5 3 6
  41. 41. Functions defined by tables IIIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6
  42. 42. Functions defined by tables IIIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6
  43. 43. Functions defined by tables IIIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6
  44. 44. Functions defined by tables IIIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6
  45. 45. Functions defined by tables IIIExample Solu onIs this a func on? If so, what isthe range? 1 . 4 x f(x) 2 5 1 4 1 5 3 6 3 6 This is not a func on.
  46. 46. An ideal function
  47. 47. An ideal function Domain is the bu ons
  48. 48. An ideal function Domain is the bu ons Range is the kinds of soda that come out
  49. 49. An ideal function Domain is the bu ons Range is the kinds of soda that come out You can press more than one bu on to get some brands
  50. 50. An ideal function Domain is the bu ons Range is the kinds of soda that come out You can press more than one bu on to get some brands But each bu on will only give one brand
  51. 51. Why numerical functions matter Ques on Why use numerical func ons at all? Formula func ons are so much easier to work with.
  52. 52. Why numerical functions matter Ques on Why use numerical func ons at all? Formula func ons are so much easier to work with. Answer In science, func ons are o en defined by data. Or, we observe data and assume that it’s close to some nice con nuous func on.
  53. 53. Numerical Function Example Example Here is the temperature in Boise, Idaho measured in 15-minute intervals over the period August 22–29, 2008. 100 90 80 70 60 50 40 30 20 10 . 8/22 8/23 8/24 8/25 8/26 8/27 8/28 8/29
  54. 54. Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. The graph on the right represents a rela on but not a func on.
  55. 55. Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. . The graph on the right represents a rela on but not a func on.
  56. 56. Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. . . The graph on the right represents a rela on but not a func on.
  57. 57. Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. . . The graph on the right represents a rela on but not a func on.
  58. 58. Functions described graphically Some mes all we have is the “picture” of a func on, by which we mean, its graph. . . The graph on the right represents a rela on but not a func on.
  59. 59. Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons: The temperature T(t) in this room at me t.
  60. 60. Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons: The temperature T(t) in this room at me t. The eleva on h(θ) of the point on the equator at longitude θ.
  61. 61. Functions described verbally O en mes our func ons come out of nature and have verbal descrip ons: The temperature T(t) in this room at me t. The eleva on h(θ) of the point on the equator at longitude θ. The u lity u(x) I derive by consuming x burritos.
  62. 62. Outline Modeling Examples of func ons Func ons expressed by formulas Func ons described numerically Func ons described graphically Func ons described verbally Proper es of func ons Monotonicity Symmetry
  63. 63. MonotonicityExampleLet P(x) be theprobability thatmy income wasat least $x lastyear. Whatmight a graph ofP(x) look like?
  64. 64. MonotonicityExample Solu onLet P(x) be theprobability that 1my income wasat least $x lastyear. What 0.5might a graph ofP(x) look like? . $0 $52,115 $100K
  65. 65. Monotonicity Defini on A func on f is decreasing if f(x1 ) > f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f. A func on f is increasing if f(x1 ) < f(x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f.
  66. 66. Examples Example Going back to the burrito func on, would you call it increasing?
  67. 67. Examples Example Going back to the burrito func on, would you call it increasing? Answer Not if they are all consumed at once! Strictly speaking, the insa ability principle in economics means that u li es are always increasing func ons.
  68. 68. Examples Example Going back to the burrito func on, would you call it increasing? Answer Not if they are all consumed at once! Strictly speaking, the insa ability principle in economics means that u li es are always increasing func ons. Example Obviously, the temperature in Boise is neither increasing nor decreasing.
  69. 69. Symmetry Consider the following func ons described as words Example Let I(x) be the intensity of light x distance from a point. Example Let F(x) be the gravita onal force at a point x distance from a black hole. What might their graphs look like?
  70. 70. Possible Intensity Graph Example Solu on Let I(x) be the intensity of light x distance from y = I(x) a point. Sketch a possible graph for I(x). . x
  71. 71. Possible Gravity Graph Example Solu on Let F(x) be the gravita onal force at a y = F(x) point x distance from a black hole. Sketch a possible graph for F(x). . x
  72. 72. Definitions Defini on A func on f is called even if f(−x) = f(x) for all x in the domain of f. A func on f is called odd if f(−x) = −f(x) for all x in the domain of f.
  73. 73. Examples Example Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, log
  74. 74. Summary The fundamental unit of inves ga on in calculus is the func on. Func ons can have many representa ons

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