Sections 2.1–2.2
         Derivatives and Rates of Changes
          The Derivative as a Function

                    V63...
Outline

   Rates of Change
      Tangent Lines
      Velocity
      Population growth
      Marginal costs

   The deriva...
The tangent problem

   Problem
   Given a curve and a point on the curve, find the slope of the line
   tangent to the cur...
The tangent problem

   Problem
   Given a curve and a point on the curve, find the slope of the line
   tangent to the cur...
Graphically and numerically

        y

                              x   m




    4




                       x
       ...
Graphically and numerically

        y

                              x   m
                              3   5
    9




...
Graphically and numerically

         y

                              x     m
                              3     5
     ...
Graphically and numerically

         y

                              x     m
                              3     5
     ...
Graphically and numerically

         y

                              x      m
                              3      5
   ...
Graphically and numerically

        y

                              x      m
                              3      5
    ...
Graphically and numerically

         y

                              x      m
                              3      5
   ...
Graphically and numerically

         y

                              x      m
                              3      5
   ...
Graphically and numerically

         y

                              x      m
                              3      5
   ...
Graphically and numerically

         y

                                x       m
                                3      ...
The tangent problem

   Problem
   Given a curve and a point on the curve, find the slope of the line
   tangent to the cur...
Velocity

   Problem
   Given the position function of a moving object, find the velocity of
   the object at a certain ins...
Velocity

   Problem
   Given the position function of a moving object, find the velocity of
   the object at a certain ins...
Numerical evidence




                                  h(t) − h(1)
                 t       vave =
                     ...
Velocity

   Problem
   Given the position function of a moving object, find the velocity of
   the object at a certain ins...
Upshot
If the height function is given by h(t), the instantaneous velocity
at time t is given by

                        ...
Population growth

   Problem
   Given the population function of a group of organisms, find the
   rate of growth of the p...
Population growth

   Problem
   Given the population function of a group of organisms, find the
   rate of growth of the p...
Numerical evidence




                     P(−10 + 0.1) − P(−10)
           r1990 ≈                         ≈ 0.000136
  ...
Numerical evidence




                     P(−10 + 0.1) − P(−10)
           r1990 ≈                         ≈ 0.000136
  ...
Numerical evidence




                     P(−10 + 0.1) − P(−10)
           r1990 ≈                         ≈ 0.000136
  ...
Population growth

   Problem
   Given the population function of a group of organisms, find the
   rate of growth of the p...
Upshot
The instantaneous population growth is given by

                         P(t + ∆t) − P(t)
                     lim...
Marginal costs


   Problem
   Given the production cost of a good, find the marginal cost of
   production after having pr...
Marginal costs


   Problem
   Given the production cost of a good, find the marginal cost of
   production after having pr...
Comparisons




       q C (q) AC (q) = C (q)/q ∆C = C (q + 1) − C (q)
       4 112          28                 13
       ...
Marginal costs


   Problem
   Given the production cost of a good, find the marginal cost of
   production after having pr...
Upshot
   The incremental cost

                      ∆C = C (q + 1) − C (q)

   is useful, but depends on units.
Upshot
   The incremental cost

                      ∆C = C (q + 1) − C (q)

   is useful, but depends on units.
   The m...
Outline

   Rates of Change
      Tangent Lines
      Velocity
      Population growth
      Marginal costs

   The deriva...
The definition



   All of these rates of change are found the same way!
The definition



   All of these rates of change are found the same way!
   Definition
   Let f be a function and a a point...
Derivative of the squaring function


   Example
   Suppose f (x) = x 2 . Use the definition of derivative to find f (a).
Derivative of the squaring function


   Example
   Suppose f (x) = x 2 . Use the definition of derivative to find f (a).

 ...
What does f tell you about f ?



      If f is a function, we can compute the derivative f (x) at each
      point x wher...
Outline

   Rates of Change
      Tangent Lines
      Velocity
      Population growth
      Marginal costs

   The deriva...
Differentiability is super-continuity

   Theorem
   If f is differentiable at a, then f is continuous at a.
Differentiability is super-continuity

   Theorem
   If f is differentiable at a, then f is continuous at a.

   Proof.
   W...
Differentiability is super-continuity

   Theorem
   If f is differentiable at a, then f is continuous at a.

   Proof.
   W...
How can a function fail to be differentiable?
Kinks




            f (x)




                      x
How can a function fail to be differentiable?
Kinks




            f (x)                         f (x)




               ...
How can a function fail to be differentiable?
Kinks




            f (x)                         f (x)




               ...
How can a function fail to be differentiable?
Cusps




            f (x)




                      x
How can a function fail to be differentiable?
Cusps




            f (x)                         f (x)




               ...
How can a function fail to be differentiable?
Cusps




            f (x)                         f (x)




               ...
How can a function fail to be differentiable?
Vertical Tangents




                    f (x)




                         ...
How can a function fail to be differentiable?
Vertical Tangents




                    f (x)                 f (x)




   ...
How can a function fail to be differentiable?
Vertical Tangents




                    f (x)                 f (x)




   ...
How can a function fail to be differentiable?
Weird, Wild, Stuff




                    f (x)




                         ...
How can a function fail to be differentiable?
Weird, Wild, Stuff




                    f (x)                 f (x)




   ...
Outline

   Rates of Change
      Tangent Lines
      Velocity
      Population growth
      Marginal costs

   The deriva...
Notation



       Newtonian notation

                             f (x)     y (x)    y

       Leibnizian notation
     ...
Meet the Mathematician: Isaac Newton




     English, 1643–1727
     Professor at Cambridge
     (England)
     Philosoph...
Meet the Mathematician: Gottfried Leibniz




     German, 1646–1716
     Eminent philosopher as
     well as mathematicia...
Outline

   Rates of Change
      Tangent Lines
      Velocity
      Population growth
      Marginal costs

   The deriva...
The second derivative



   If f is a function, so is f , and we can seek its derivative.

                               ...
The second derivative



   If f is a function, so is f , and we can seek its derivative.

                               ...
function, derivative, second derivative

                         y
                                          f (x) = x 2
...
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Lesson 7-8: Derivatives and Rates of Change, The Derivative as a function

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The derivative is one of the fundamental quantities in calculus, partly because it is ubiquitous in nature. We give examples of it coming about, a few calculations, and ways information about the function an imply information about the derivative

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Lesson 7-8: Derivatives and Rates of Change, The Derivative as a function

  1. 1. Sections 2.1–2.2 Derivatives and Rates of Changes The Derivative as a Function V63.0121, Calculus I February 9–12, 2009 Announcements Quiz 2 is next week: Covers up through 1.6 Midterm is March 4/5: Covers up to 2.4 (next T/W)
  2. 2. Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be differentiable? Other notations The second derivative
  3. 3. The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point.
  4. 4. The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Example Find the slope of the line tangent to the curve y = x 2 at the point (2, 4).
  5. 5. Graphically and numerically y x m 4 x 2
  6. 6. Graphically and numerically y x m 3 5 9 4 x 2 3
  7. 7. Graphically and numerically y x m 3 5 2.5 4.25 6.25 4 x 2 2.5
  8. 8. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 4.41 4 x 2.1 2
  9. 9. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 4.0401 4 x 2.01 2
  10. 10. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 4 1 3 1 x 1 2
  11. 11. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 4 1.5 3.5 2.25 1 3 x 1.5 2
  12. 12. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1.9 3.9 4 3.61 1.5 3.5 1 3 x 1.9 2
  13. 13. Graphically and numerically y x m 3 5 2.5 4.25 2.1 4.1 2.01 4.01 1.99 3.99 1.9 3.9 4 3.9601 1.5 3.5 1 3 x 1.99 2
  14. 14. Graphically and numerically y x m 3 5 2.5 4.25 9 2.1 4.1 2.01 4.01 6.25 limit 4 1.99 3.99 4.41 1.9 3.9 4.0401 4 3.9601 3.61 1.5 3.5 2.25 1 3 1 x 1 1.5 2.1 3 1.99 1.9 2.5 2.01 2
  15. 15. The tangent problem Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Example Find the slope of the line tangent to the curve y = x 2 at the point (2, 4). Upshot If the curve is given by y = f (x), and the point on the curve is (a, f (a)), then the slope of the tangent line is given by f (x) − f (a) mtangent = lim x −a x→a
  16. 16. Velocity Problem Given the position function of a moving object, find the velocity of the object at a certain instant in time. Example Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 10t 2 where t is seconds after dropping it and h is meters above the ground. How fast is it falling one second after we drop it?
  17. 17. Velocity Problem Given the position function of a moving object, find the velocity of the object at a certain instant in time. Example Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 10t 2 where t is seconds after dropping it and h is meters above the ground. How fast is it falling one second after we drop it? Solution The answer is (50 − 10t 2 ) − 40 = −20. lim t −1 t→1
  18. 18. Numerical evidence h(t) − h(1) t vave = t −1 −30 2 −25 1.5 −21 1.1 −20.01 1.01 −20.001 1.001
  19. 19. Velocity Problem Given the position function of a moving object, find the velocity of the object at a certain instant in time. Example Drop a ball off the roof of the Silver Center so that its height can be described by h(t) = 50 − 10t 2 where t is seconds after dropping it and h is meters above the ground. How fast is it falling one second after we drop it? Solution The answer is (50 − 10t 2 ) − 40 = −20. lim t −1 t→1
  20. 20. Upshot If the height function is given by h(t), the instantaneous velocity at time t is given by h(t + ∆t) − h(t) v = lim ∆t ∆t→0
  21. 21. Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant.
  22. 22. Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant. Example Suppose the population of fish in the East River is given by the function 3e t P(t) = 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish population growing fastest in 1990, 2000, or 2010? (Estimate numerically)?
  23. 23. Numerical evidence P(−10 + 0.1) − P(−10) r1990 ≈ ≈ 0.000136 0.1
  24. 24. Numerical evidence P(−10 + 0.1) − P(−10) r1990 ≈ ≈ 0.000136 0.1 P(0.1) − P(0) r2000 ≈ ≈ 0.75 0.1
  25. 25. Numerical evidence P(−10 + 0.1) − P(−10) r1990 ≈ ≈ 0.000136 0.1 P(0.1) − P(0) r2000 ≈ ≈ 0.75 0.1 P(10 + 0.1) − P(10) r2010 ≈ ≈ 0.000136 0.1
  26. 26. Population growth Problem Given the population function of a group of organisms, find the rate of growth of the population at a particular instant. Example Suppose the population of fish in the East River is given by the function 3e t P(t) = 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish population growing fastest in 1990, 2000, or 2010? (Estimate numerically)? Solution The estimated rates of growth are 0.000136, 0.75, and 0.000136.
  27. 27. Upshot The instantaneous population growth is given by P(t + ∆t) − P(t) lim ∆t ∆t→0
  28. 28. Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity.
  29. 29. Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity. Example Suppose the cost of producing q tons of rice on our paddy in a year is C (q) = q 3 − 12q 2 + 60q We are currently producing 5 tons a year. Should we change that?
  30. 30. Comparisons q C (q) AC (q) = C (q)/q ∆C = C (q + 1) − C (q) 4 112 28 13 5 125 25 19 6 144 24 31
  31. 31. Marginal costs Problem Given the production cost of a good, find the marginal cost of production after having produced a certain quantity. Example Suppose the cost of producing q tons of rice on our paddy in a year is C (q) = q 3 − 12q 2 + 60q We are currently producing 5 tons a year. Should we change that? Example If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should produce more to lower average costs.
  32. 32. Upshot The incremental cost ∆C = C (q + 1) − C (q) is useful, but depends on units.
  33. 33. Upshot The incremental cost ∆C = C (q + 1) − C (q) is useful, but depends on units. The marginal cost after producing q given by C (q + ∆q) − C (q) MC = lim ∆q ∆q→0 is more useful since it’s unit-independent.
  34. 34. Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be differentiable? Other notations The second derivative
  35. 35. The definition All of these rates of change are found the same way!
  36. 36. The definition All of these rates of change are found the same way! Definition Let f be a function and a a point in the domain of f . If the limit f (a + h) − f (a) f (a) = lim h h→0 exists, the function is said to be differentiable at a and f (a) is the derivative of f at a.
  37. 37. Derivative of the squaring function Example Suppose f (x) = x 2 . Use the definition of derivative to find f (a).
  38. 38. Derivative of the squaring function Example Suppose f (x) = x 2 . Use the definition of derivative to find f (a). Solution (a + h)2 − a2 f (a + h) − f (a) f (a) = lim = lim h h h→0 h→0 2 + 2ah + h2 ) − a2 2ah + h2 (a = lim = lim h h h→0 h→0 = lim (2a + h) = 2a. h→0
  39. 39. What does f tell you about f ? If f is a function, we can compute the derivative f (x) at each point x where f is differentiable, and come up with another function, the derivative function. What can we say about this function f ? If f is decreasing on an interval, f is negative (well, nonpositive) on that interval If f is increasing on an interval, f is positive (well, nonnegative) on that interval
  40. 40. Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be differentiable? Other notations The second derivative
  41. 41. Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a.
  42. 42. Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have f (x) − f (a) lim (f (x) − f (a)) = lim · (x − a) x −a x→a x→a f (x) − f (a) · lim (x − a) = lim x −a x→a x→a = f (a) · 0 = 0
  43. 43. Differentiability is super-continuity Theorem If f is differentiable at a, then f is continuous at a. Proof. We have f (x) − f (a) lim (f (x) − f (a)) = lim · (x − a) x −a x→a x→a f (x) − f (a) · lim (x − a) = lim x −a x→a x→a = f (a) · 0 = 0 Note the proper use of the limit law: if the factors each have a limit at a, the limit of the product is the product of the limits.
  44. 44. How can a function fail to be differentiable? Kinks f (x) x
  45. 45. How can a function fail to be differentiable? Kinks f (x) f (x) x x
  46. 46. How can a function fail to be differentiable? Kinks f (x) f (x) x x
  47. 47. How can a function fail to be differentiable? Cusps f (x) x
  48. 48. How can a function fail to be differentiable? Cusps f (x) f (x) x x
  49. 49. How can a function fail to be differentiable? Cusps f (x) f (x) x x
  50. 50. How can a function fail to be differentiable? Vertical Tangents f (x) x
  51. 51. How can a function fail to be differentiable? Vertical Tangents f (x) f (x) x x
  52. 52. How can a function fail to be differentiable? Vertical Tangents f (x) f (x) x x
  53. 53. How can a function fail to be differentiable? Weird, Wild, Stuff f (x) x
  54. 54. How can a function fail to be differentiable? Weird, Wild, Stuff f (x) f (x) x x
  55. 55. Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be differentiable? Other notations The second derivative
  56. 56. Notation Newtonian notation f (x) y (x) y Leibnizian notation dy d df f (x) dx dx dx These all mean the same thing.
  57. 57. Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687
  58. 58. Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute
  59. 59. Outline Rates of Change Tangent Lines Velocity Population growth Marginal costs The derivative, defined Derivatives of (some) power functions What does f tell you about f ? How can a function fail to be differentiable? Other notations The second derivative
  60. 60. The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change!
  61. 61. The second derivative If f is a function, so is f , and we can seek its derivative. f = (f ) It measures the rate of change of the rate of change! Leibnizian notation: d 2y d2 d 2f f (x) dx 2 dx 2 dx 2
  62. 62. function, derivative, second derivative y f (x) = x 2 f (x) = 2x f (x) = 2 x
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