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Lesson 1: Functions

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Functions defined, ways to represent functions, properties of functions …

Functions defined, ways to represent functions, properties of functions

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  • 1. Section 1.1 Functions and their Representations V63.0121.021/041, Calculus I New York University September 8, 2010Announcements First WebAssign-ments are due September 13 First written assignment is due September 15 Do the Get-to-Know-You survey for extra credit!
  • 2. Announcements First WebAssign-ments are due September 13 First written assignment is due September 15 Do the Get-to-Know-You survey for extra credit!V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 2 / 33
  • 3. Function
  • 4. Objectives: Functions and their Representations Understand the definition of function. Work with functions represented in different ways Work with functions defined piecewise over several intervals. Understand and apply the definition of increasing and decreasing function.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 3 / 33
  • 5. What is a function?DefinitionA function f is a relation which assigns to to every element x in a set D asingle 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 .V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 4 / 33
  • 6. OutlineModelingExamples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verballyProperties of functions Monotonicity SymmetryV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 5 / 33
  • 7. The Modeling Process Real-world model Mathematical Problems Model solve test Real-world interpret Mathematical Predictions ConclusionsV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 6 / 33
  • 8. Plato’s CaveV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 7 / 33
  • 9. The Modeling Process Real-world model Mathematical Problems Model solve test Real-world interpret Mathematical Predictions Conclusions Shadows FormsV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 8 / 33
  • 10. OutlineModelingExamples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verballyProperties of functions Monotonicity SymmetryV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 9 / 33
  • 11. Functions expressed by formulasAny expression in a single variable x defines a function. In this case, thedomain is understood to be the largest set of x which after substitution,give a real number.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 10 / 33
  • 12. Formula function exampleExample x +1Let f (x) = . Find the domain and range of f . x −2V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
  • 13. Formula function exampleExample x +1Let f (x) = . Find the domain and range of f . x −2SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
  • 14. Formula function exampleExample x +1Let f (x) = . Find the domain and range of f . x −2SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve x +1 2y + 1 y= =⇒ x = x −2 y −1So as long as y = 1, there is an x associated to y .V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
  • 15. Formula function exampleExample x +1Let f (x) = . Find the domain and range of f . x −2SolutionThe denominator is zero when x = 2, so the domain is all real numbersexcept 2. As for the range, we can solve x +1 2y + 1 y= =⇒ x = x −2 y −1So as long as y = 1, there is an x associated to y . Therefore domain(f ) = { x | x = 2 } range(f ) = { y | y = 1 }V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 11 / 33
  • 16. How did you get that? x +1 start y= x −2 cross-multiply 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 −1V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 12 / 33
  • 17. No-no’s for expressions Cannot have zero in the denominator of an expression Cannot have a negative number under an even root (e.g., square root) Cannot have the logarithm of a negative numberV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 13 / 33
  • 18. Piecewise-defined functionsExampleLet x2 0 ≤ x ≤ 1; f (x) = 3−x 1 < x ≤ 2.Find the domain and range of f and graph the function.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
  • 19. Piecewise-defined functionsExampleLet x2 0 ≤ x ≤ 1; f (x) = 3−x 1 < x ≤ 2.Find the domain and range of f and graph the function.SolutionThe domain is [0, 2]. The range is [0, 2). The graph is piecewise. 2 1 0 1 2V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 14 / 33
  • 20. Functions described numericallyWe can just describe a function by a table of values, or a diagram.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 15 / 33
  • 21. ExampleIs this a function? If so, what is the range? x f (x) 1 4 2 5 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
  • 22. ExampleIs this a function? If so, what is the range? 1 4 x f (x) 1 4 2 5 2 5 3 6 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
  • 23. ExampleIs this a function? If so, what is the range? 1 4 x f (x) 1 4 2 5 2 5 3 6 3 6Yes, the range is {4, 5, 6}.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 16 / 33
  • 24. ExampleIs this a function? If so, what is the range? x f (x) 1 4 2 4 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
  • 25. ExampleIs this a function? If so, what is the range? 1 4 x f (x) 1 4 2 5 2 4 3 6 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
  • 26. ExampleIs this a function? If so, what is the range? 1 4 x f (x) 1 4 2 5 2 4 3 6 3 6Yes, the range is {4, 6}.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 17 / 33
  • 27. ExampleHow about this one? x f (x) 1 4 1 5 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
  • 28. ExampleHow about this one? 1 4 x f (x) 1 4 2 5 1 5 3 6 3 6V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
  • 29. ExampleHow about this one? 1 4 x f (x) 1 4 2 5 1 5 3 6 3 6No, that one’s not “deterministic.”V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 18 / 33
  • 30. An ideal functionV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
  • 31. An ideal function Domain is the buttonsV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
  • 32. An ideal function Domain is the buttons Range is the kinds of soda that come outV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
  • 33. An ideal function Domain is the buttons Range is the kinds of soda that come out You can press more than one button to get some brandsV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
  • 34. An ideal function Domain is the buttons Range is the kinds of soda that come out You can press more than one button to get some brands But each button will only give one brandV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 19 / 33
  • 35. Why numerical functions matterIn science, functions are often defined by data. Or, we observe data andassume that it’s close to some nice continuous function.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 20 / 33
  • 36. Numerical Function ExampleHere is the temperature in Boise, Idaho measured in 15-minute intervalsover 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 8/29V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 21 / 33
  • 37. Functions described graphicallySometimes all we have is the “picture” of a function, by which we mean,its graph.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
  • 38. Functions described graphicallySometimes all we have is the “picture” of a function, by which we mean,its graph.The one on the right is a relation but not a function.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 22 / 33
  • 39. Functions described verballyOftentimes our functions come out of nature and have verbal descriptions: The temperature T (t) in this room at time t. The elevation h(θ) of the point on the equator at longitude θ. The utility u(x) I derive by consuming x burritos.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 23 / 33
  • 40. OutlineModelingExamples of functions Functions expressed by formulas Functions described numerically Functions described graphically Functions described verballyProperties of functions Monotonicity SymmetryV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 24 / 33
  • 41. MonotonicityExampleLet P(x) be the probability that my income was at least $x last year.What might a graph of P(x) look like?V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
  • 42. MonotonicityExampleLet P(x) be the probability that my income was at least $x last year.What might a graph of P(x) look like? 1 0.5 $0 $52,115 $100KV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 25 / 33
  • 43. MonotonicityDefinition A function f is decreasing if f (x1 ) > f (x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f . A function f is increasing if f (x1 ) < f (x2 ) whenever x1 < x2 for any two points x1 and x2 in the domain of f .V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 26 / 33
  • 44. ExamplesExampleGoing back to the burrito function, would you call it increasing?V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
  • 45. ExamplesExampleGoing back to the burrito function, would you call it increasing?ExampleObviously, the temperature in Boise is neither increasing nor decreasing.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 27 / 33
  • 46. SymmetryExampleLet I (x) be the intensity of light x distance from a point.ExampleLet F (x) be the gravitational force at a point x distance from a black hole.V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 28 / 33
  • 47. Possible Intensity Graph y = I (x) xV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 29 / 33
  • 48. Possible Gravity Graph y = F (x) xV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 30 / 33
  • 49. DefinitionsDefinition A function f is called even if f (−x) = f (x) for all x in the domain of f. A function f is called odd if f (−x) = −f (x) for all x in the domain of f .V63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 31 / 33
  • 50. Examples Even: constants, even powers, cosine Odd: odd powers, sine, tangent Neither: exp, logV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 32 / 33
  • 51. Summary The fundamental unit of investigation in calculus is the function. Functions can have many representationsV63.0121.021/041, Calculus I (NYU) Section 1.1 Functions September 8, 2010 33 / 33