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Sec on 2.1–2.2    The Deriva ve      V63.0121.011: Calculus I    Professor Ma hew Leingang           New York University  ...
Announcements   Quiz this week on   Sec ons 1.1–1.4   No class Monday,   February 21
ObjectivesThe Derivative   Understand and state the defini on of   the deriva ve of a func on at a point.   Given a func on...
ObjectivesThe Derivative as a Function   Given a func on f, use the defini on of   the deriva ve to find the deriva ve   fun...
Outline Rates of Change    Tangent Lines    Velocity    Popula on growth    Marginal costs The deriva ve, defined    Deriva...
The tangent problemA geometric rate of change Problem Given a curve and a point on the curve, find the slope of the line ta...
A tangent problem Example Find the slope of the line tangent to the curve y = x2 at the point (2, 4).
Graphically and numerically       y                    x2 − 22                     x   m=                             x−2 ...
Graphically and numerically       y                      x2 − 22                       x   m=                             ...
Graphically and numerically       y                      x2 − 22                       x   m=                             ...
Graphically and numerically         y                        x2 − 22                         x     m=                     ...
Graphically and numerically         y                         x2 − 22                         x     m=                    ...
Graphically and numerically         y                       x2 − 22                       x     m=                        ...
Graphically and numerically         y                       x2 − 22                       x     m=                        ...
Graphically and numerically         y                         x2 − 22                        x      m=                    ...
Graphically and numerically         y                        x2 − 22                        x    m=                       ...
Graphically and numerically       y                         x2 − 22                       x    m=                         ...
Graphically and numerically       y                         x2 − 22                       x    m=                         ...
Graphically and numerically         y                         x2 − 22                         x    m=                     ...
Graphically and numerically         y                         x2 − 22                         x    m=                     ...
Graphically and numerically         y                       x2 − 22                       x    m=                         ...
Graphically and numerically         y                       x2 − 22                       x    m=                         ...
Graphically and numerically         y                        x2 − 22                        x    m=                       ...
Graphically and numerically         y                        x2 − 22                        x    m=                       ...
Graphically and numerically       y                       x2 − 22                     x    m=                             ...
Graphically and numerically         y                              x2 − 22                              x     m=          ...
Graphically and numerically         y                              x2 − 22                              x     m=          ...
The tangent problemA geometric rate of change Problem Given a curve and a point on the curve, find the slope of the line ta...
The velocity problemKinematics—Physical rates of change Problem Given the posi on func on of a moving object, find the velo...
A velocity problemExampleDrop a ball off the roof of theSilver Center so that its height canbe described by          h(t) =...
Numerical evidence                          h(t) = 50 − 5t2 Fill in the table:                                     h(t) − ...
Numerical evidence                            h(t) = 50 − 5t2 Fill in the table:                                       h(t...
Numerical evidence                            h(t) = 50 − 5t2 Fill in the table:                                       h(t...
Numerical evidence                            h(t) = 50 − 5t2 Fill in the table:                                       h(t...
Numerical evidence                            h(t) = 50 − 5t2 Fill in the table:                                       h(t...
Numerical evidence                             h(t) = 50 − 5t2 Fill in the table:                                        h...
Numerical evidence                             h(t) = 50 − 5t2 Fill in the table:                                        h...
Numerical evidence                          h(t) = 50 − 5t2 Fill in the table:                                       h(t) ...
Numerical evidence                          h(t) = 50 − 5t2 Fill in the table:                                       h(t) ...
A velocity problemExample                                Solu onDrop a ball off the roof of the         The answer is      ...
A velocity problemExample                                Solu onDrop a ball off the roof of the         The answer is      ...
A velocity problemExample                                Solu onDrop a ball off the roof of the         The answer is      ...
A velocity problemExample                                Solu onDrop a ball off the roof of the         The answer is      ...
A velocity problemExample                                Solu onDrop a ball off the roof of the         The answer is      ...
Velocity in general Upshot                                                    y = h(t) If the height func on is given     ...
Population growthBiological Rates of Change Problem Given the popula on func on of a group of organisms, find the rate of g...
Population growth example Example Suppose the popula on of fish in the East River is given by the func on                  ...
Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on:                      ∆P = P(...
Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on:                      ∆P = P(...
Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference         P(t + ∆t) −...
Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference         P(t + ∆t) −...
Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference         P(t + ∆t) −...
Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference         P(t + ∆t) −...
Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference         P(t + ∆t) −...
Solu on (Con nued)   r2010
Solu on (Con nued)             P(10 + 0.1) − P(10)   r2010 ≈                     0.1
Solu on (Con nued)                                         (                       )             P(10 + 0.1) − P(10)    1 ...
Solu on (Con nued)                                         (                       )             P(10 + 0.1) − P(10)    1 ...
Population growth example Example Suppose the popula on of fish in the East River is given by the func on                  ...
Population growthBiological Rates of Change Problem Given the popula on func on of a group of organisms, find the rate of g...
Marginal costsRates of change in economics Problem Given the produc on cost of a good, find the marginal cost of produc on ...
Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is                      ...
Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is                      ...
Comparisons Solu on                      C(q) = q3 − 12q2 + 60q Fill in the table:           q C(q)           4           ...
Comparisons Solu on                      C(q) = q3 − 12q2 + 60q Fill in the table:           q C(q)           4 112       ...
Comparisons Solu on                      C(q) = q3 − 12q2 + 60q Fill in the table:           q C(q)           4 112       ...
Comparisons Solu on                      C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q)           4   112   ...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q     ...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q     ...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q     ...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q     ...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q ∆C =...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q ∆C =...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q ∆C =...
Comparisons Solu on                       C(q) = q3 − 12q2 + 60q Fill in the table:           q   C(q) AC(q) = C(q)/q ∆C =...
Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is                      ...
Marginal costsRates of change in economics Problem Given the produc on cost of a good, find the marginal cost of produc on ...
Outline Rates of Change    Tangent Lines    Velocity    Popula on growth    Marginal costs The deriva ve, defined    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! Defini on Let f be a func on and a a point in the domain...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a).
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on      ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
“Can you do it the other way?”Same limit, different form Solu on           ′           f(2 + h) − f(2)                     ...
“Can you do it the other way?”Same limit, different form Solu on           ′           f(2 + h) − f(2)                     ...
“Can you do it the other way?”Same limit, different form Solu on                                              2+h − 2      ...
“Can you do it the other way?”Same limit, different form Solu on                                              2+h − 2      ...
“Can you do it the other way?”Same limit, different form Solu on                                              2+h − 2      ...
“How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions  Fact         a c  ad ± bc          ± =       ...
“How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions  Fact         a c  ad ± bc          ± =       ...
What does f tell you about f′?   If f is a func on, we can compute the deriva ve f′ (x) at each   point x where f is differ...
What does f tell you about f′?   If f is a func on, we can compute the deriva ve f′ (x) at each   point x where f is differ...
Derivative of the reciprocal Example               1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2).    ...
What does f tell you about f′?   If f is a func on, we can compute the deriva ve f′ (x) at each   point x where f is differ...
Graphically and numerically         y                              x2 − 22                              x     m=          ...
What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b).
What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b). Picture Proof. ...
What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Pro...
What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Pro...
What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Pro...
Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval?
Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval? ...
Outline Rates of Change    Tangent Lines    Velocity    Popula on growth    Marginal costs The deriva ve, defined    Deriva...
Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a.
Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a. Proof. We have          ...
Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a. Proof. We have          ...
Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a. Proof. We have          ...
Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ ....
Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the de...
Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the de...
Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the de...
Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the de...
Differentiability FAILWeird, Wild, Stuff Example              f(x)                .       x This func on is differen able at 0.
Differentiability FAILWeird, Wild, Stuff Example              f(x)              f′ (x)                .       x          .  ...
Differentiability FAILWeird, Wild, Stuff Example              f(x)              f′ (x)                .       x          .  ...
Differentiability FAILWeird, Wild, Stuff Example              f(x)                           f′ (x)                .       x...
Outline Rates of Change    Tangent Lines    Velocity    Popula on growth    Marginal costs The deriva ve, defined    Deriva...
Notation     Newtonian nota on                          f′ (x)     y′ (x)   y′     Leibnizian nota on                     ...
Meet the MathematicianIsaac Newton   English, 1643–1727   Professor at Cambridge   (England)   Philosophiae Naturalis   Pr...
Meet the MathematicianGottfried Leibniz    German, 1646–1716    Eminent philosopher as    well as mathema cian    Contempo...
Outline Rates of Change    Tangent Lines    Velocity    Popula on growth    Marginal costs The deriva ve, defined    Deriva...
The second derivative If f is a func on, so is f′ , and we can seek its deriva ve.                                 f′′ = (...
The second derivative If f is a func on, so is f′ , and we can seek its deriva ve.                                 f′′ = (...
Function, derivative, second derivative                 y                             f(x) = x2                           ...
SummaryWhat have we learned today?    The deriva ve measures instantaneous rate of change    The deriva ve has many interp...
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Lesson 7: The Derivative (slides)

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Many functions in nature are described as the rate of change of another function. The concept is called the derivative. Algebraically, the process of finding the derivative involves a limit of difference quotients.

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  1. 1. Sec on 2.1–2.2 The Deriva ve V63.0121.011: Calculus I Professor Ma hew Leingang New York University February 14, 2011. NYUMathematics
  2. 2. Announcements Quiz this week on Sec ons 1.1–1.4 No class Monday, February 21
  3. 3. ObjectivesThe Derivative Understand and state the defini on of the deriva ve of a func on at a point. Given a func on and a point in its domain, decide if the func on is differen able at the point and find the value of the deriva ve at that point. Understand and give several examples of deriva ves modeling rates of change in science.
  4. 4. ObjectivesThe Derivative as a Function Given a func on f, use the defini on of the deriva ve to find the deriva ve func on f’. Given a func on, find its second deriva ve. Given the graph of a func on, sketch the graph of its deriva ve.
  5. 5. Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve
  6. 6. The tangent problemA geometric rate of change Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point.
  7. 7. A tangent problem Example Find the slope of the line tangent to the curve y = x2 at the point (2, 4).
  8. 8. Graphically and numerically y x2 − 22 x m= x−2 4 . x 2
  9. 9. Graphically and numerically y x2 − 22 x m= x−2 3 9 4 . x 2 3
  10. 10. Graphically and numerically y x2 − 22 x m= x−2 3 5 9 4 . x 2 3
  11. 11. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 6.25 4 . x 2 2.5
  12. 12. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 6.25 4 . x 2 2.5
  13. 13. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.41 4 . x 2.1 2
  14. 14. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 4.41 4 . x 2.1 2
  15. 15. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.014.0401 4 . x 2.01 2
  16. 16. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.014.0401 4 . x 2.01 2
  17. 17. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 1 . x 1 1 2
  18. 18. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 1 . x 1 3 1 2
  19. 19. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 2.25 . 1.5 x 1 3 1.5 2
  20. 20. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 2.25 . 1.5 3.5 x 1 3 1.5 2
  21. 21. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 3.61 1.9 . 1.5 3.5 x 1 3 1.9 2
  22. 22. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 3.61 1.9 3.9 . 1.5 3.5 x 1 3 1.9 2
  23. 23. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 43.9601 1.99 1.9 3.9 . 1.5 3.5 x 1 3 1.99 2
  24. 24. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 43.9601 1.99 3.99 1.9 3.9 . 1.5 3.5 x 1 3 1.99 2
  25. 25. Graphically and numerically y x2 − 22 x m= x−2 3 5 2.5 4.5 2.1 4.1 2.01 4.01 4 1.99 3.99 1.9 3.9 . 1.5 3.5 x 1 3 2
  26. 26. Graphically and numerically y x2 − 22 x m= x−2 3 5 9 2.5 4.5 2.1 4.1 6.25 2.01 4.01 limit 4.414.0401 43.9601 3.61 1.99 3.99 2.25 1.9 3.9 1 1.5 3.5 . x 1 3 1 1.52.1 3 1.99 2.01 1.92.5 2
  27. 27. Graphically and numerically y x2 − 22 x m= x−2 3 5 9 2.5 4.5 2.1 4.1 6.25 2.01 4.01 limit 4 4.414.0401 43.9601 3.61 1.99 3.99 2.25 1.9 3.9 1 1.5 3.5 . x 1 3 1 1.52.1 3 1.99 2.01 1.92.5 2
  28. 28. The tangent problemA geometric rate of change Problem Given a curve and a point on the curve, find the slope of the line tangent to the curve at that point. Solu on 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
  29. 29. The velocity problemKinematics—Physical rates of change Problem Given the posi on func on of a moving object, find the velocity of the object at a certain instant in me.
  30. 30. A velocity problemExampleDrop a ball off the roof of theSilver Center so that its height canbe described by h(t) = 50 − 5t2where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
  31. 31. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15
  32. 32. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5
  33. 33. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5
  34. 34. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1
  35. 35. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1 − 10.5
  36. 36. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01
  37. 37. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05
  38. 38. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05 1.001
  39. 39. Numerical evidence h(t) = 50 − 5t2 Fill in the table: h(t) − h(1) t vave = t−1 2 − 15 1.5 − 12.5 1.1 − 10.5 1.01 − 10.05 1.001 − 10.005
  40. 40. A velocity problemExample Solu onDrop a ball off the roof of the The answer is (50 − 5t2 ) − 45Silver Center so that its height can v = limbe described by t→1 t−1 h(t) = 50 − 5t2where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
  41. 41. A velocity problemExample Solu onDrop a ball off the roof of the The answer is (50 − 5t2 ) − 45Silver Center so that its height can v = limbe described by t→1 t−1 5 − 5t2 = lim h(t) = 50 − 5t2 t→1 t − 1where t is seconds a er droppingit and h is meters above theground. How fast is it falling onesecond a er we drop it?
  42. 42. A velocity problemExample Solu onDrop a ball off the roof of the The answer is (50 − 5t2 ) − 45Silver Center so that its height can v = limbe described by t→1 t−1 5 − 5t2 = lim h(t) = 50 − 5t2 t→1 t − 1 5(1 − t)(1 + t)where t is seconds a er dropping = lim t→1 t−1it and h is meters above theground. How fast is it falling onesecond a er we drop it?
  43. 43. A velocity problemExample Solu onDrop a ball off the roof of the The answer is (50 − 5t2 ) − 45Silver Center so that its height can v = limbe described by t→1 t−1 5 − 5t2 = lim h(t) = 50 − 5t2 t→1 t − 1 5(1 − t)(1 + t)where t is seconds a er dropping = lim t→1 t−1it and h is meters above the = (−5) lim(1 + t) t→1ground. How fast is it falling onesecond a er we drop it?
  44. 44. A velocity problemExample Solu onDrop a ball off the roof of the The answer is (50 − 5t2 ) − 45Silver Center so that its height can v = limbe described by t→1 t−1 5 − 5t2 = lim h(t) = 50 − 5t2 t→1 t − 1 5(1 − t)(1 + t)where t is seconds a er dropping = lim t→1 t−1it and h is meters above the = (−5) lim(1 + t) t→1ground. How fast is it falling onesecond a er we drop it? = −5 · 2 = −10
  45. 45. Velocity in general Upshot y = h(t) If the height func on is given h(t0 ) by h(t), the instantaneous ∆h velocity at me t0 is given by h(t0 + ∆t) h(t) − h(t0 ) v = lim t→t0 t − t0 h(t0 + ∆t) − h(t0 ) = lim ∆t→0 ∆t . ∆t t t0 t
  46. 46. Population growthBiological Rates of Change Problem Given the popula on func on of a group of organisms, find the rate of growth of the popula on at a par cular instant.
  47. 47. Population growth example Example Suppose the popula on of fish in the East River is given by the func on 3et P(t) = 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish popula on growing fastest in 1990, 2000, or 2010? (Es mate numerically)
  48. 48. Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so ideally we would want ( ) ∆P 1 3et+∆t 3et lim = lim − ∆t→0 ∆t ∆t→0 ∆t 1 + et+∆t 1 + et
  49. 49. Derivation Solu on Let ∆t be an increment in me and ∆P the corresponding change in popula on: ∆P = P(t + ∆t) − P(t) This depends on ∆t, so ideally we would want ( ) ∆P 1 3et+∆t 3et lim = lim − ∆t→0 ∆t ∆t→0 ∆t 1 + et+∆t 1 + et But rather than compute a complicated limit analy cally, let us approximate numerically. We will try a small ∆t, for instance 0.1.
  50. 50. Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t r1990 r2000
  51. 51. Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t P(−10 + 0.1) − P(−10) r1990 ≈ 0.1 P(0.1) − P(0) r2000 ≈ 0.1
  52. 52. Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t ( ) P(−10 + 0.1) − P(−10) 1 3e−9.9 3e−10 r1990 ≈ = − 0.1 0.1 1 + e−9.9 1 + e−10 ( ) P(0.1) − P(0) 1 3e0.1 3e0 r2000 ≈ = − 0.1 0.1 1 + e0.1 1 + e0
  53. 53. Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t ( ) P(−10 + 0.1) − P(−10) 1 3e−9.9 3e−10 r1990 ≈ = − 0.1 0.1 1 + e−9.9 1 + e−10 = 0.000143229 ( ) P(0.1) − P(0) 1 3e0.1 3e0 r2000 ≈ = − 0.1 0.1 1 + e0.1 1 + e0
  54. 54. Numerical evidence Solu on (Con nued) To approximate the popula on change in year n, use the difference P(t + ∆t) − P(t) quo ent , where ∆t = 0.1 and t = n − 2000. ∆t ( ) P(−10 + 0.1) − P(−10) 1 3e−9.9 3e−10 r1990 ≈ = − 0.1 0.1 1 + e−9.9 1 + e−10 = 0.000143229 ( ) P(0.1) − P(0) 1 3e0.1 3e0 r2000 ≈ = − 0.1 0.1 1 + e0.1 1 + e0 = 0.749376
  55. 55. Solu on (Con nued) r2010
  56. 56. Solu on (Con nued) P(10 + 0.1) − P(10) r2010 ≈ 0.1
  57. 57. Solu on (Con nued) ( ) P(10 + 0.1) − P(10) 1 3e10.1 3e10 r2010 ≈ = − 0.1 0.1 1 + e10.1 1 + e10
  58. 58. Solu on (Con nued) ( ) P(10 + 0.1) − P(10) 1 3e10.1 3e10 r2010 ≈ = − 0.1 0.1 1 + e10.1 1 + e10 = 0.0001296
  59. 59. Population growth example Example Suppose the popula on of fish in the East River is given by the func on 3et P(t) = 1 + et where t is in years since 2000 and P is in millions of fish. Is the fish popula on growing fastest in 1990, 2000, or 2010? (Es mate numerically) Answer We es mate the rates of growth to be 0.000143229, 0.749376, and 0.0001296. So the popula on is growing fastest in 2000.
  60. 60. Population growthBiological Rates of Change Problem Given the popula on func on of a group of organisms, find the rate of growth of the popula on at a par cular instant. Solu on The instantaneous popula on growth is given by P(t + ∆t) − P(t) lim ∆t→0 ∆t
  61. 61. Marginal costsRates of change in economics Problem Given the produc on cost of a good, find the marginal cost of produc on a er having produced a certain quan ty.
  62. 62. Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that?
  63. 63. Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that? Answer If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should produce more to lower average costs.
  64. 64. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) 4 5 6
  65. 65. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) 4 112 5 6
  66. 66. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) 4 112 5 125 6
  67. 67. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) 4 112 5 125 6 144
  68. 68. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q 4 112 5 125 6 144
  69. 69. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q 4 112 28 5 125 6 144
  70. 70. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q 4 112 28 5 125 25 6 144
  71. 71. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q 4 112 28 5 125 25 6 144 24
  72. 72. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 5 125 25 6 144 24
  73. 73. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: q C(q) AC(q) = C(q)/q ∆C = C(q + 1) − C(q) 4 112 28 13 5 125 25 6 144 24
  74. 74. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: 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
  75. 75. Comparisons Solu on C(q) = q3 − 12q2 + 60q Fill in the table: 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
  76. 76. Marginal Cost Example Example Suppose the cost of producing q tons of rice on our paddy in a year is C(q) = q3 − 12q2 + 60q We are currently producing 5 tons a year. Should we change that? Answer If q = 5, then C = 125, ∆C = 19, while AC = 25. So we should produce more to lower average costs.
  77. 77. Marginal costsRates of change in economics Problem Given the produc on cost of a good, find the marginal cost of produc on a er having produced a certain quan ty. Solu on The marginal cost a er producing q is given by C(q + ∆q) − C(q) MC = lim ∆q→0 ∆q
  78. 78. Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve
  79. 79. The definition All of these rates of change are found the same way!
  80. 80. The definition All of these rates of change are found the same way! Defini on Let f be a func on and a a point in the domain of f. If the limit f(a + h) − f(a) f(x) − f(a) f′ (a) = lim = lim h→0 h x→a x−a exists, the func on is said to be differen able at a and f′ (a) is the deriva ve of f at a.
  81. 81. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a).
  82. 82. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on f(a + h) − f(a) f′ (a) = lim h→0 h
  83. 83. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on ′ f(a + h) − f(a) (a + h)2 − a2 f (a) = lim = lim h→0 h h→0 h
  84. 84. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on ′ f(a + h) − f(a) (a + h)2 − a2 f (a) = lim = lim h→0 h h→0 h (a + 2ah + h ) − a 2 2 2 = lim h→0 h
  85. 85. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on ′ f(a + h) − f(a) (a + h)2 − a2 f (a) = lim = lim h→0 h h→0 h (a + 2ah + h ) − a 2 2 2 2ah + h2 = lim = lim h→0 h h→0 h
  86. 86. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on ′ f(a + h) − f(a) (a + h)2 − a2 f (a) = lim = lim h→0 h h→0 h (a + 2ah + h ) − a 2 2 2 2ah + h2 = lim = lim h→0 h h→0 h = lim (2a + h) h→0
  87. 87. Derivative of the squaring function Example Suppose f(x) = x2 . Use the defini on of deriva ve to find f′ (a). Solu on ′ f(a + h) − f(a) (a + h)2 − a2 f (a) = lim = lim h→0 h h→0 h (a + 2ah + h ) − a 2 2 2 2ah + h2 = lim = lim h→0 h h→0 h = lim (2a + h) = 2a h→0
  88. 88. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x
  89. 89. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y 1/x − 1/2 f′ (2) = lim x→2 x−2 . x
  90. 90. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y 1/x − 1/2 2−x f′ (2) = lim = lim x→2 x−2 x→2 2x(x − 2) . x
  91. 91. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y 1/x − 1/2 2−x f′ (2) = lim = lim x→2 x−2 x→2 2x(x − 2) −1 = lim x→2 2x . x
  92. 92. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y 1/x − 1/2 2−x f′ (2) = lim = lim x→2 x−2 x→2 2x(x − 2) −1 1 = lim =− x→2 2x 4 . x
  93. 93. “Can you do it the other way?”Same limit, different form Solu on ′ f(2 + h) − f(2) 1 2+h− 1 2 f (2) = lim = lim h→0 h h→0 h
  94. 94. “Can you do it the other way?”Same limit, different form Solu on ′ f(2 + h) − f(2) 1 2+h− 1 2 f (2) = lim = lim h→0 h h→0 h 2 − (2 + h) = lim h→0 2h(2 + h)
  95. 95. “Can you do it the other way?”Same limit, different form Solu on 2+h − 2 1 1 ′ f(2 + h) − f(2) f (2) = lim = lim h→0 h h→0 h 2 − (2 + h) −h = lim = lim h→0 2h(2 + h) h→0 2h(2 + h)
  96. 96. “Can you do it the other way?”Same limit, different form Solu on 2+h − 2 1 1 ′ f(2 + h) − f(2) f (2) = lim = lim h→0 h h→0 h 2 − (2 + h) −h = lim = lim h→0 2h(2 + h) h→0 2h(2 + h) −1 = lim h→0 2(2 + h)
  97. 97. “Can you do it the other way?”Same limit, different form Solu on 2+h − 2 1 1 ′ f(2 + h) − f(2) f (2) = lim = lim h→0 h h→0 h 2 − (2 + h) −h = lim = lim h→0 2h(2 + h) h→0 2h(2 + h) −1 1 = lim =− h→0 2(2 + h) 4
  98. 98. “How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions Fact a c ad ± bc ± = b d bd 1 1 2−x − x 2 = 2x = 2 − x x−2 x−2 2x(x − 2)
  99. 99. “How did you get that?”The Sure-Fire Sally Rule (SFSR) for adding fractions Fact a c ad ± bc ± = b d bd 1 1 2−x − x 2 = 2x = 2 − x x−2 x−2 2x(x − 2) Paul Sally
  100. 100. What does f tell you about f′? If f is a func on, we can compute the deriva ve f′ (x) at each point x where f is differen able, and come up with another func on, the deriva ve func on. What can we say about this func on f′ ?
  101. 101. What does f tell you about f′? If f is a func on, we can compute the deriva ve f′ (x) at each point x where f is differen able, and come up with another func on, the deriva ve func on. What can we say about this func on f′ ? If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve) on that interval
  102. 102. Derivative of the reciprocal Example 1 Suppose f(x) = . Use the defini on of the deriva ve to find f′ (2). x Solu on y 1/x − 1/2 2−x f′ (2) = lim = lim x→2 x−2 x→2 2x(x − 2) −1 1 = lim =− x→2 2x 4 . x
  103. 103. What does f tell you about f′? If f is a func on, we can compute the deriva ve f′ (x) at each point x where f is differen able, and come up with another func on, the deriva ve func on. What can we say about this func on f′ ? If f is decreasing on an interval, f′ is nega ve (technically, nonposi ve) on that interval If f is increasing on an interval, f′ is posi ve (technically, nonnega ve) on that interval
  104. 104. Graphically and numerically y x2 − 22 x m= x−2 3 5 9 2.5 4.5 2.1 4.1 6.25 2.01 4.01 limit 4 4.414.0401 43.9601 3.61 1.99 3.99 2.25 1.9 3.9 1 1.5 3.5 . x 1 3 1 1.52.1 3 1.99 2.01 1.92.5 2
  105. 105. What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b).
  106. 106. What does f tell you about f′? Fact If f is decreasing on the open interval (a, b), then f′ ≤ 0 on (a, b). Picture Proof. If f is decreasing, then all secant lines point downward, hence have y nega ve slope. The deriva ve is a limit of slopes of secant lines, which are all nega ve, so the limit must be ≤ 0. . x
  107. 107. What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. If ∆x > 0, then f(x + ∆x) − f(x) f(x + ∆x) < f(x) =⇒ <0 ∆x
  108. 108. What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. If ∆x > 0, then f(x + ∆x) − f(x) f(x + ∆x) < f(x) =⇒ <0 ∆x If ∆x < 0, then x + ∆x < x, and f(x + ∆x) − f(x) f(x + ∆x) > f(x) =⇒ <0 ∆x
  109. 109. What does f tell you about f′? Fact If f is decreasing on on the open interval (a, b), then f′ ≤ 0 on (a, b). The Real Proof. f(x + ∆x) − f(x) Either way, < 0, so ∆x f(x + ∆x) − f(x) f′ (x) = lim ≤0 ∆x→0 ∆x
  110. 110. Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval?
  111. 111. Going the Other Way? Ques on If a func on has a nega ve deriva ve on an interval, must it be decreasing on that interval? Answer Maybe.
  112. 112. Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve
  113. 113. Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous at a.
  114. 114. Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous 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 · lim (x − a) x→a x−a x→a
  115. 115. Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous 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 · lim (x − a) x→a x−a x→a ′ = f (a) · 0
  116. 116. Differentiability is super-continuity Theorem If f is differen able at a, then f is con nuous 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 · lim (x − a) x→a x−a x→a ′ = f (a) · 0 = 0
  117. 117. Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) . x
  118. 118. Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  119. 119. Differentiability FAILKinks Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  120. 120. Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) . x
  121. 121. Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  122. 122. Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  123. 123. Differentiability FAILCusps Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  124. 124. Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) . x
  125. 125. Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  126. 126. Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  127. 127. Differentiability FAILVertical Tangents Example Let f have the graph on the le -hand side below. Sketch the graph of the deriva ve f′ . f(x) f′ (x) . x . x
  128. 128. Differentiability FAILWeird, Wild, Stuff Example f(x) . x This func on is differen able at 0.
  129. 129. Differentiability FAILWeird, Wild, Stuff Example f(x) f′ (x) . x . x This func on is differen able at 0.
  130. 130. Differentiability FAILWeird, Wild, Stuff Example f(x) f′ (x) . x . x This func on is differen able at 0.
  131. 131. Differentiability FAILWeird, Wild, Stuff Example f(x) f′ (x) . x . x This func on is differen able But the deriva ve is not at 0. con nuous at 0!
  132. 132. Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve
  133. 133. Notation Newtonian nota on f′ (x) y′ (x) y′ Leibnizian nota on dy d df f(x) dx dx dx These all mean the same thing.
  134. 134. Meet the MathematicianIsaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathema ca published 1687
  135. 135. Meet the MathematicianGottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathema cian Contemporarily disgraced by the calculus priority dispute
  136. 136. Outline Rates of Change Tangent Lines Velocity Popula on growth Marginal costs The deriva ve, defined Deriva ves of (some) power func ons What does f tell you about f′ ? How can a func on fail to be differen able? Other nota ons The second deriva ve
  137. 137. The second derivative If f is a func on, so is f′ , and we can seek its deriva ve. f′′ = (f′ )′ It measures the rate of change of the rate of change!
  138. 138. The second derivative If f is a func on, so is f′ , and we can seek its deriva ve. f′′ = (f′ )′ It measures the rate of change of the rate of change! Leibnizian nota on: d2 y d2 d2 f f(x) dx2 dx2 dx2
  139. 139. Function, derivative, second derivative y f(x) = x2 f′ (x) = 2x f′′ (x) = 2 . x
  140. 140. SummaryWhat have we learned today? The deriva ve measures instantaneous rate of change The deriva ve has many interpreta ons: slope of the tangent line, velocity, marginal quan es, etc. The deriva ve reflects the monotonicity (increasing-ness or decreasing-ness) of the graph
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