Exponen'al growth and differen'al equa'ons

Transcription

1 Exponen'al growth and differen'al equa'ons

2 But first.. Thanks for the feedback!

3 Feedback about M102 Which of the following do you find useful?

4 How many resources students typically use 25 Frequency I use every resource that M102 provides I only talk to friends for help

5 Common sugges'ons Do more harder problems just like the ones on the midterms and the exam (40-50 responses).

6 Really? Let us clear up a few misconcep'ons first..

7 Comparison of some problems we did IN CLASS How to outrun a cheeta Gazelle has constant velocity: find its posi'on Determine when they meet by equa'ng posi'ons Examtype ques'on ON MIDTERM Hare and Fox problem Fox has constant velocity; find its posi'on Determine when they meet by equa'ng posi'ons

8 Comparison of some problems we did IN CLASS Aphid laydybug problem Solve an equa'on of the form ON MIDTERM Hare and Fox problem Solve an equa'on of the form Examtype ques'on

9 Comparison of some problems we did IN CLASS Rodent popula'on problem (Worksheet Oct 10) Find minimum of the func'on ON MIDTERM Hare and Fox problem Find where the following func'on is increasing/decreasing

10 Comparison of some problems IN CLASS Find cylinder of max volume that fits inside sphere (WS Oct 10): Write down Volume V Formulate constraint (constant sphere R) Eliminate on variable Op'mize V(r) Check the type of CP ON MIDTERM E coli volume op'miza'on problem Write down Volume V Formulate constraint (constant surface S) Eliminate on variable Op'mize V(r) Check the type of CP

11 Comparison of some problems IN BOOK A cylindrical cell with minimal surface area ON MIDTERM E coli volume op'miza'on problem See P 148 Sec 7.2

12 Comparison of some problems IN BOOK E coli volume op'miza'on problem ON MIDTERM E coli volume op'miza'on problem See Problem 16.15

13 More comparison IN CLASS Clicker problems on cri'cal points, Inflec'on points, deriva'ves, etc: - MathGame Q # 25 - Lect 5.1 ON MIDTERM MC # 6 the hard one

14 Really hard ques'ons.. Yes there were a few tricky bits.. OK..

15 Feedback, cont d Do more step-by-step solu'ons More detail on problems/solu'ons More guidance through worksheets More 'me on each ques'on (ABOUT 13 such responses) good ideas, and easy to fix!

16 Feedback Display clicker answers immediately aoer a clicker ques'on good idea, and easy to fix! Solving problems on the board or document camera, rather than just showing solu'ons on the slides good idea, and easy to fix!

17 Miscellanious Feedback I want [the instructor] to show me how to solve problems.. I don t want to try to figure it out on my own

18 That s a bit like this: Coach, if you don t mind, I d rather just watch you teach me how to swim, instead of gerng into the water myself

19 Some expecta'ons.. Mastery of a subject requires thinking, analysing, prac'cing, ques'oning, and persistence (not just rote, not just WebWork) You must learn how to learn.. In class we can demo tools and ideas, but you have to take responsibility for owning them, using them crea'vely in new ways.. We can not teach you how to solve every problem..

20 Feedback Talk louder It is hard to focus in this class

21 Feedback Talk louder please be quieter and more respecsul It is hard to focus in this class

22 My idea for a fix Time to discuss Time to focus and be quiet

23 Observa'on about exponen'al func'ons

24 Exponen'al func'on Last 'me: The deriva've of e x is e x (by the chain rule) The deriva've of e kx is k e kx Let us see what this implies

25 A new kind of equa'on The func'on y= f(x)= e x has the same func'on as its deriva've. Hence the func'on f(x) sa'sfies the equa'on dy/dx = y The deriva've is the same as the original func'on

26 What if it depends on 'me? The func'on y= f(t)= e t has the same func'on as its deriva've. Hence the func'on f(t) sa'sfies the equa'on dy/dt = y The deriva've is the same as the original func'on

27 One more slight change The func'on y= f(t)= e kt has the deriva've ke kt. Hence the func'on f(t) sa'sfies the equa'on dy/dt = k y The deriva've is some constant 'mes the original func'on

28 Differen'al equa'on An equa'on that involves (one or more) deriva've of a func'on (and possibly the func'on itself) is called a differen'al equa'on. For example: dy/dt = k y is a differen'al equa'on (DE). The func'on y = f(t) is a solu'on if it sa'sfies this equa'on. We have just seen that y = e kt sa'sfies this DE.

29 Solu'on to a differen'al equa'on y=e kt is a solu'on to the differen'al equa'on: dy/dt = ky Is that the only solu'on that works?

30 Solu'on to a differen'al equa'on Which of the following func'ons sa'sfy the differen'al equa'on

31 Solu'on to a differen'al equa'on Which of the following func'ons sa'sfy the differen'al equa'on

32 Solu'ons to a differen'al equa'on The func'on y=ce kt is a solu'on to the differen'al equa'on dy/dt = ky for any constant C!

33 Solu'ons to a differen'al equa'on The func'on y=ce kt is a solu'on to the differen'al equa'on dy/dt = ky for any constant C!

34 Ini'al condi'on A func'on sa'sfies the the differen'al equa'on We are told that at 'me t = 0 the value of y is y(0) = 3. Then the func'on is

35 Ini'al condi'on A func'on sa'sfies the the differen'al equa'on We are told that at 'me t = 0 the value of y is y(0) = 3. Then the func'on is

36 Solu'ons to a differen'al equa'on The func'on y=ce kt is a solu'on to the differen'al equa'on dy/dt = ky for any constant C! We need more informa'on (such as the state at 'me t=0) to specify the value of the constant C.

37 Solu'on to ini'al value problem (diff l eqn + IC) The func'on y=y 0 e kt is the solu'on to the differen'al equa'on dy/dt = ky and y(0)=y 0 y 0 t=0

38 Differen'al equa'ons in Exponen'al Popula'on growth

39 Balance equa'ons Previously, we asked when two processes exactly balance E.g.: nutrient absorp'on rate = nutrient consump'on rate Aphid birth rate = rate of aphid mortality due to preda'on

40 (0) But.. What if.. What is the two things DON T BALANCE?? In that case we get change e.g. if aphid birth rate > mortality rate then (A) The popula'on increases (B) The popula'on decreases (C) The problem is not defined (D) Not sure what happens

41 (0) But.. What if.. What is the two things DON T BALANCE?? In that case we get change e.g. if aphid birth rate > mortality rate then (A) The popula'on increases (B) The popula'on decreases (C) The problem is not defined (D) Not sure what happens

42 A differen'al equa'on is.. A statement that allows us to track those changes rate flow in rate flow out

43 A differen'al equa'on: Rate of change of amount = rate flow in rate flow out rate flow in rate flow out

44 A differen'al equa'on Describes the rate of change of some state variable Example: t = 'me N(t) = popula'on at 'me t Rate of change of N(t) (ΔN/Δ'me) Rate of = births - deaths + (num/'me) Rate of (num/'me) Rate of immigra'on (num/'me)

45 Units No'ce: units of each term are the same Example: t = 'me Rate of change of N(t) (number/.me) Rate of births (num/.me) Rate of deaths (num/.me) = - + Rate of immigra'on (num/.me)

46 (1) What is this term? (A) N(t) (B) N/t (C) dn/dt (D) None of the above (E) Not sure Rate of change of N(t) (number/.me)

47 (1) What is this term? (A) N(t) (B) N/t (C) dn/dt (D) None of the above (E) Not sure Rate of change of N(t) (number/.me)

48 Simple Example Per capita birth rate = r (per unit 'me) Per capita mortality = m (per unit 'me) Immigra'on rate = I (number per unit 'me) Assume r, m, I constants > 0

49 (2) We obtain the differen'al equa'on for popula'on growth: (A) dn/dt = r m + I (B) dn/dt = r + m + I (C) dn/dt = (r-m) N + I (D) dn/dt = (m-r) N (E) dn/dt = (r-m) I

50 (2) We obtain the differen'al equa'on for popula'on growth: (A) dn/dt = r m + I (B) dn/dt = r + m + I (C) dn/dt = (r-m) N + I (D) dn/dt = (m-r) N (E) dn/dt = (r-m) I

51 The differen'al equa'on: We obtain Example: t = 'me Rate of change of N(t) (number/.me) Rate of births (num/.me) Rate of deaths (num/.me) = - + Rate of immigra'on (num/.me) dn/dt = r N - m N + I

52 What does this equa'on predict? Case 1: r>0, m>0, I = 0 (No immigra'on) Let k = (r-m) ß net growth rate Note: k could be 0, posi've or nega've.

53 (3) What does this equa'on predict? Case 1: r>0, m>0, I = 0 (No immigra'on) dn/dt = k N where k = (r-m) Then if k>0 we expect the popula'on would: (A) Increase (B) decrease (C) no change (D) Not sure

54 (3) What does this equa'on predict? Case 1: r>0, m>0, I = 0 (No immigra'on) dn/dt = k N where k = (r-m) Then if k>0 we expect the popula'on would: (A) Increase (B) decrease (C) no change (D) Not sure

55 Intui'vely: k>0 If r>m à birth rate > mortality rate N é Also k=0 (r = m) à birth = mortality k<0 (r < m) à birth < mortality Nê But we can do even bezer! We can say exactly what the popula'on level will be at any later 'me, if we know its size at 'me t=0

56 What func'on sa'sfies The Diffl Eqn: dn/dt = kn And Ini'al value: N(0)=N 0 Looking for a func'on of 'me N(t) whose deriva've is propor'onal to itself!

57 (4) What func'on sa'sfies The Diffl Eqn: dn/dt = kn And Ini'al value: N(0)=N 0 (A) N(t)=kt (B) N(t)= (1/2)kt 2 (C) (1/2) k N 2 (D) N(t)= e kt (E) N(t)= N 0 e kt

58 (4) What func'on sa'sfies The Diffl Eqn: dn/dt = kn And Ini'al value: N(0)=N 0 (A) N(t)=kt (B) N(t)= (1/2)kt 2 (C) (1/2) k N 2 (D) N(t)= e kt (E) N(t)= N 0 e kt

59 We know how to solve this ini'al value problem! Differen'al equa'on: dn/dt =k N k= (r-m) Some ini'al condi'on: N(0)=N 0 The above pair of equa'ons (a differen'al equa'on together with an ini'al condi'on) is called an Ini'al Value Problem (IVP) à Solu'on: N(t)= N 0 e kt

60 Check: Note: given a func'on, we can always test whether it sa'sfies a differen'al equa'on! Check that N(t)= N 0 e kt is a solu'on to the diff l eqn (DE) and ini'al condi'on(ic) dn/dt =k N N(0)=N 0

61 Check: Differen'ate the func'on N(t)= N 0 e kt We get: So indeed, this func'on sa'sfies the DE and IC

62 Sketch our result Draw a rough sketch of the func'on N(t)= N 0 e kt (where k=r-m) for (1) r>m, (2)r<m, (3) r=m.

63 Sketch our result Draw a rough sketch of the func'on N(t)= N 0 e kt (where k=r-m) for (1) k>0, (2)k<0, (3) k=0.

64 Sketch our result Draw a rough sketch of the func'on N(t)= N 0 e kt (where k=r-m) for (1) k>0, (2)k<0, (3) k=0. expon.al growth expon decay no change

65 What s this all about??? Early part of term: given a func'on, we learned how to find its deriva've (and used that to help graph, find max, mins, etc) Now: Given some informa'on about the deriva've, we are trying to find the func'on! (Not as simple as just finding an an'deriva've because the func'on and its deriva've are all mixed up together.)

66 Case 2: Immigra'on in Europe In Europe, birth rates are lower than mortality, but there is a constant rate of immigra'on, I dn/dt = I μ N where μ=m-r>0

67 Case 2: Immigra'on (I) dn/dt = I μ N I, μ=m-r>0 Belongs to a group of diff l eqns such as dy/dt = a by a, b >0

68 What it says: dn/dt = rate immigr rate mortality dn/dt = I - μ N rate immigra'on rate of mortality

69 When is there a balance between immigra'on and mortality? dn/dt = I - μ N Is there a constant popula'on level N that sa'sfies this equa'on?

70 When is there a balance between immigra'on and mortality? dn/dt = I - μ N When I = μ N then the two balance and then N = I /μ (and also dn/dt = 0)

71 When is there a balance between immigra'on and mortality? dn/dt = I - μ N When I = μ N then the two balance and then N = I /μ (and also dn/dt = 0) We refer to that situa'on as a Steady State (there is no change overall, even though both processes con'nue)

72 What if immigra'on and mortality Then N(t) will change! don t balance? dn/dt = I - μ N Popula'on will either increase or decrease. It turns out that it will move towards the steady state N = I /μ

73 Now back to human popula'on explosion:

74 Overall human popula'on on Planet Earth: Simple assump'on about fer'lity Simple assump'on about mortality rate See P233 Sec'on 11.2

75 Rough es'mates for r and m: Per capita birth rate r per year Per capita mortality rate m per year à k=(r-m) = /year

76 Doubling 'me: dn/dt = k N, with k = (r-m) = /yr à N(t) = N 0 e t How long 'll the popula'on DOUBLES?

77 (5) Doubling 'me: dn/dt = k N, with k = (r-m) = /yr à N(t) = N 0 e t How long 'll the popula'on DOUBLES? (A) years (B) e years (C) 10 years (D) ln(2)/ years (E) ln(0.0125)/2

78 (5) Doubling 'me: dn/dt = k N, with k = (r-m) = /yr à N(t) = N 0 e t How long 'll the popula'on DOUBLES? (A) years (B) e years (C) 10 years (D) ln(2)/ years (E) ln(0.0125)/2

79 (5) Doubling 'me: N(t) = N 0 e t Doubling 'me = ln(2)/ = 55.5 years

80 Predicted human popula'on growth: dn/dt = k N, with k = (r-m) = /yr N(t) = N 0 e t If we start with 6 billion now, how many in 100 ys?

81 (6) Rough es'mate According to our model ( Malthusian growth) in 100 years there will be how many humans? (A) 7 billion (B) 10 billion (C) 15 billion (D) 20 billion (E) 40 billion

82 (6) Rough es'mate According to our model ( Malthusian growth) in 100 years there will be how many humans? (A) 7 billion (B) 10 billion (C) 15 billion (D) 20 billion (E) 40 billion

83 Predicted human popula'on growth: N(t) = 6 e t billion If we keep growing, at t= 100 ys we will be around 20 billion strong!

84 Problems to test your skills

85 Final exam Q

86 Final Exam Q:

87 Solu'ons from last 'me

88 Exam ques'on:

89 Solu'on: The point (0,1) is on the graph of f so (1,0) is the corresponding point on the graph of f -1.

90 A midterm problem puzzler Compute the deriva've of y=f(x) = x x Hint: take ln of both sides

91 The deriva've of y=f(x) = x x

92 From last worksheet

93 The plan Circle circular orbit Parabola Parabolic orbit

94 Solu'on: Set up the problem Label the point(s) at which the orbits intersect

95 Equate slopes of tangent lines Find dy/dx for both the curves and get the equa'on that results by serng them equal at the points (x 0,y 0 ). What are possible solu'ons?

96 Find points of intersec'on parabola circle Get quadra'c eqn in y 0 à This is the y coordinate of that point. We can get the x coordinate from the above

97 Get slopes of tangents to each curve Implicit differen'a'on used for slope of circle

98 Equate slopes at mee'ng point There are two possible cases: or

99 Case 1 This case is less interes'ng it turns out to have one single solu'on at the 'p or the base of the circle: Only one tangent point, at the bozom of the parabola

100 Case 2 We have two simultaneous equa'ons for y 0, which both have to be sa'sfied. Using both equa'ons we get

101 Requirement for such point to exist The above gives us a rela'on between a and b (parabola parameters) that must be sa'sfied, that we can solve for b in terms of a:

102 Designing a parabolic orbit We should set b= This is a requirement for the parabola to intersect the circle at two tangent points. We can test this in Desmos.

103 Try this yourself! We use the value of b that we just computed and a slider for a

104 Are there any special constraints on a? Explora'on on desmos suggests that we have to be a lizle careful! If a is too small, the two curves fail to meet at all! Check it out. So what else did we have to no'ce?

105 Conclusion For the given circular orbit of the supply crao, we need to select a parabolic orbit of the space sta'on of the form: For some sufficiently large constant a. (There are other considera'ons, such as speeds and 'me of contact that we will leave for NASA rocket scien'sts).

106 Differen'al equa'ons for exponen'al growth and decay (preview of next 'me)

107 Differen'al equa'on The func'on sa'sfies the equa'on: A differen'al equa'on is an equa'on linking a func'on and its deriva'ves.

108 Solu'on to a differen'al equa'on We say that y=e x is a solu.on to the differen'al equa'on: dy/dx = y

109 Solu'on to a differen'al equa'on We want to use these facts in 'me-dependent systems. Hence the independent variable will usually be t for 'me rather than x.

110 Solu'on to a differen'al equa'on We say that y=e kt is a solu'on to the differen'al equa'on: dy/dt = ky No'ce: independent variable t ('me)

111 Solu'on from last 'me:

112 Solu'on, cont d

113 Solu'on from last 'me:

114 Solu'on cont d

115 Miscellanious feedback Hmm OK, which one wold you prefer?