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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