Today. Logistic equation in applications. Trig review

Transcription

1 Today Logistic equation in applications Trig review

2 Logistic equation in different contexts...

3 Rates of change that are proportional to two things Infectious disease: bsi (S=susceptible, I=infected) Spread of rumour: bnh (N = not heard rumour, H = heard rumour) Spread of new words: bnu (use word or not). Spread of new technologies: bnu (use tech or not). Active oil exploration sites: bud (undiscovered and discovered). Waterlillies in a pond: bsw (waterlillies and space for waterwillies).

4 ...two things that are just different forms of a single thing

5 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X.

6 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. Lose Y: dy dt = bxy and gain X: dx dt = bxy

7 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. dy Lose Y: and gain X: dt = bxy dx dt = bxy X+Y= constant = C so Y=C-X.

8 ...two things that are just different forms of a single thing When X meets Y, there s a chance Y turns into X. dy Lose Y: and gain X: dt = bxy dx dt = bxy X+Y= constant = C so Y=C-X. dx dt = bx(c X)

9 Infectious disease Dr. Erin Mears: Once we know the R0, we'll be able to get a handle on the scale of the epidemic. Minnesota Health #4: So, it's an epidemic now. An epidemic of what? Dave: We sent samples to the CDC. Dr. Erin Mears: In seventy two hours, we'll know what it is, if we're lucky. Minnesota Health #4: Clearly, we're not lucky.

10 Infectious disease

11 Infectious disease N individuals, I of them have a flu, S=N-I do not.

12 Infectious disease N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI.

13 Infectious disease N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease:

14 Infectious disease N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease: (A) di dt = bi(n I) (C) ds dt = bsi (B) di dt = bi(n I) (D) di dt = bsi

15 Infectious disease N individuals, I of them have a flu, S=N-I do not. If everyone interacts, new cases appear at a rate proportional to SI. The DE describing the spread of disease: (A) di dt = bi(n I) (C) ds dt = di di (B) = bi(n I) (D) dt dt = bsi dp dt = rp P Compare this with 1. K bsi

16 Infectious disease What is the carrying capacity? di dt = bi(n I) (A) b/n (C) I (B) N/b (D) N

17 Infectious disease What is the carrying capacity? di dt = bi(n I) (A) b/n (C) I (B) N/b (D) N dp dt = rp 1 P K

18 Infectious disease What is the carrying capacity? di dt = bi(n I) = bni 1 I N (A) b/n (C) I (B) N/b (D) N dp dt = rp 1 P K

19 Infectious disease What is the carrying capacity? di dt = bi(n I) = bni { r 1 I N (A) b/n (C) I (B) N/b (D) N dp dt = rp 1 P K

20 Infectious disease What is the carrying capacity? di = bi(n I) dt = bni 1 { (A) b/n (C) I r I N { K (B) N/b (D) N dp dt = rp 1 P K

21 Infectious disease What is the carrying capacity? di = bi(n I) dt = bni 1 { (A) b/n (C) I r I N { K (B) N/b dp dt = rp 1 (D) N P K Everyone gets sick!

22 Infectious disease

23 Infectious disease Suppose infected people recover at a rate proportional to how many there are.

24 Infectious disease Suppose infected people recover at a rate proportional to how many there are. The DE describing the spread of disease with recovery: (A) di (C) dt = bi(n I) µs di dt = bi(n I)+µI (B) di (D) dt = bi(n I) µi di dt = bi(n I)+µI

25 Infectious disease di dt = bi(n I) µi

26 Infectious disease di dt = bi(n I) µi = bin bi 2 µi

27 Infectious disease di dt = bi(n I) µi = bin bi 2 µi = bi N µ b I

28 Infectious disease di dt = bi(n I) µi = bin bi 2 µi = bi N K µ b I

29 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N µ b I K = N µ b < 0 K

30 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N K µ b I K K = N µ b < 0 I 0 N I

31 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ b >N then = bi N K µ b I K K = N µ b < 0 I 0 N I and the disease dies out.

32 Infectious disease di dt = bi(n I) µi = bin bi 2 µi = bi N K µ b I

33 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ b > 0 = bi N µ b I K

34 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K

35 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic.

36 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1

37 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1

38 Infectious disease di dt = bi(n I) µi = bin bi 2 µi If µ then b <N K = N µ I 0 b > 0 = bi N µ b I K N I K and the disease becomes an epidemic. R 0 = Nb µ > 1

39 Trig review If θ is measured counterclockwise from the positive x axis we define sin and cos so that (A) x=sin(θ), y=tan(θ). (B) x=tan(θ), y=sin(θ). (C) x=sin(θ), y=cos(θ). (D) x=cos(θ), y=sin(θ).

40 Trig review If θ is measured counterclockwise from the positive x axis we define sin and cos so that (A) x=sin(θ), y=tan(θ). (B) x=tan(θ), y=sin(θ). (C) x=sin(θ), y=cos(θ). (D) x=cos(θ), y=sin(θ).

41 Trig review (cosø, sinø) ø Learn special angles in Quad I and modify signs for other Quads.

42 Trig review (cosø, sinø) (cosθ, sinθ) ø θ Learn special angles in Quad I and modify signs for other Quads.

43 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. ø θ

44 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø ø θ

45 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ

46 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ θ (cosθ, sinθ)

47 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ θ ø (cosθ, sinθ)

48 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø θ ø (cosθ, sinθ)

49 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø sinθ = -sinø θ ø (cosθ, sinθ)

50 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø sinθ = -sinø θ ø θ (cosθ, sinθ) (cosθ, sinθ)

51 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø sinθ = -sinø θ ø θ ø (cosθ, sinθ) (cosθ, sinθ)

52 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø sinθ = -sinø (cosθ, sinθ) θ ø cosθ = cosø θ ø (cosθ, sinθ)

53 Trig review (cosø, sinø) (cosθ, sinθ) ø Learn special angles in Quad I and modify signs for other Quads. cosθ = -cosø sinθ = sinø ø θ cosθ = -cosø sinθ = -sinø (cosθ, sinθ) θ ø cosθ = cosø sinθ = -sinø θ ø (cosθ, sinθ)

54 Trig review The other trig functions:

55 Trig review The other trig functions: tanθ = sinθ / cosθ

56 Trig review The other trig functions: tanθ = sinθ / cosθ cscθ = 1 / sinθ

57 Trig review The other trig functions: tanθ = sinθ / cosθ cscθ = 1 / sinθ secθ = 1 / cosθ

58 Trig review The other trig functions: tanθ = sinθ / cosθ cscθ = 1 / sinθ secθ = 1 / cosθ cotθ = 1 / tanθ

59 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2)

60 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) cos(a+b) = cosa cosb - sina sinb

61 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) cos(a+b) = cosa cosb - sina sinb θ 1 cosθ sinθ

62 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ sin 2 θ + cos 2 θ = 1 (B) tan 2 θ + 1 = sec 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) cos(a+b) = cosa cosb - sina sinb θ 1 cosθ sinθ

63 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ sin 2 θ + cos 2 θ = 1 sin 2 θ sin 2 θ sin 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) cos(a+b) = cosa cosb - sina sinb θ 1 cosθ sinθ

64 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ sin 2 θ + cos 2 θ = 1 cos 2 θ cos 2 θ cos 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) cos(a+b) = cosa cosb - sina sinb θ 1 cosθ sinθ

65 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) sin 2 θ + cos 2 θ = 1 cos 2 θ cos 2 θ cos 2 θ <-- Use sin(a+b) (watch today s 2 nd video) 1 sinθ cos(a+b) = cosa cosb - sina sinb θ cosθ

66 Trig review Which of the following is not a trig identity? (A) 1 + cot 2 θ = csc 2 θ (B) tan 2 θ + 1 = sec 2 θ sin 2 θ + cos 2 θ = 1 cos 2 θ cos 2 θ cos 2 θ (C) sin(2θ) = 2 sinθ cosθ (D) cos(θ) = sin(θ-π/2) (E) sin(θ) = cos(θ-π/2) <-- Use sin(a+b) (watch today s 2 nd video) Know graphs, how to shift or use sin(a+b), cos(a+b) cos(a+b) = cosa cosb - sina sinb θ 1 cosθ sinθ

67 Trig review cos(2π/3) = (A) (B) (C) (D) p 3 2 p

68 Trig review cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3

69 Trig review cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3 2π/3

70 Trig review cos(2π/3) = (A) (B) (C) (D) p 3 2 p π/3 2π/3 π/3

71 Trig review cos(2π/3) = (A) p 3 2 p 3 π/3 2π/3 (B) (C) (D) π/3 1/2 3/2

72 Trig review cos(2π/3) = (A) p 3 2 p 3 π/3 2π/3 (B) (C) (D) And 2π/3 is in Quad II so cos(2π/3) < 0. 1 π/3 1/2 3/2

73 Trig review tan ( /4) = 1 (A) p 2 (B) 1 p (C) 2 (D) 1 2

74 Trig review tan ( /4) = 1 (A) p 2 (B) 1 p (C) 2 (D) 1 2