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- 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. 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. 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. 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. Graphically and numerically y x m 4 x 2
- 6. Graphically and numerically y x m 3 5 9 4 x 2 3
- 7. Graphically and numerically y x m 3 5 2.5 4.25 6.25 4 x 2 2.5
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Numerical evidence P(−10 + 0.1) − P(−10) r1990 ≈ ≈ 0.000136 0.1
- 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. 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. 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. Upshot The instantaneous population growth is given by P(t + ∆t) − P(t) lim ∆t ∆t→0
- 28. Marginal costs Problem Given the production cost of a good, ﬁnd the marginal cost of production after having produced a certain quantity.
- 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. 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. 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. Upshot The incremental cost ∆C = C (q + 1) − C (q) is useful, but depends on units.
- 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. 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. The deﬁnition All of these rates of change are found the same way!
- 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. Derivative of the squaring function Example Suppose f (x) = x 2 . Use the deﬁnition of derivative to ﬁnd f (a).
- 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. 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. 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. Diﬀerentiability is super-continuity Theorem If f is diﬀerentiable at a, then f is continuous at a.
- 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. 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. How can a function fail to be diﬀerentiable? Kinks f (x) x
- 45. How can a function fail to be diﬀerentiable? Kinks f (x) f (x) x x
- 46. How can a function fail to be diﬀerentiable? Kinks f (x) f (x) x x
- 47. How can a function fail to be diﬀerentiable? Cusps f (x) x
- 48. How can a function fail to be diﬀerentiable? Cusps f (x) f (x) x x
- 49. How can a function fail to be diﬀerentiable? Cusps f (x) f (x) x x
- 50. How can a function fail to be diﬀerentiable? Vertical Tangents f (x) x
- 51. How can a function fail to be diﬀerentiable? Vertical Tangents f (x) f (x) x x
- 52. How can a function fail to be diﬀerentiable? Vertical Tangents f (x) f (x) x x
- 53. How can a function fail to be diﬀerentiable? Weird, Wild, Stuﬀ f (x) x
- 54. How can a function fail to be diﬀerentiable? Weird, Wild, Stuﬀ f (x) f (x) x x
- 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. 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. Meet the Mathematician: Isaac Newton English, 1643–1727 Professor at Cambridge (England) Philosophiae Naturalis Principia Mathematica published 1687
- 58. Meet the Mathematician: Gottfried Leibniz German, 1646–1716 Eminent philosopher as well as mathematician Contemporarily disgraced by the calculus priority dispute
- 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. 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. 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. function, derivative, second derivative y f (x) = x 2 f (x) = 2x f (x) = 2 x

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