From differen+al equa+ons to trigonometric func+ons. Introducing sine and cosine

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1 From differen+al equa+ons to trigonometric func+ons Introducing sine and cosine

2 Calendar OSH due this Friday by 12:30 in MX 1111 Last quiz next Wedn

3 Office hrs: Today: 11:30-12:30 OSH due by 12:30 Math Annex 1111

4 The mathema+cs of LOVE

5 The mathema+cs of Stormy LOVE hpps://

6 Romeo and Juliet By the famous mathema+cian, physicist, and author of favourites such as: Steven Strogatz

7 Romeo and Juliet Juliet: My passion for Romeo decreases in propor+on to his love. Romeo: My passion for Juliet increases in propor+on to her love.

8 Romeo and Juliet Juliet: My passion for Romeo decreases in propor+on to his love. (The more Romeo loves me, the more I run away from him.. But when he hates me, I start to love him.) Romeo: My passion for Juliet increases in propor+on to her love. (The more Juliet loves me, the more I love her! But when she hates me, my love for her decreases. )

9 Love meter I hate you I love you

10 Romeo and Juliet Let x(t) = Juliet s love for Romeo, y(t) = Romeo s love for Juliet Both x(t) and y(t) will change with +me, as the star-crossed lovers chase each other across the love meter.

11 (1) Juliet s love, x(t) My passion for Romeo decreases at a rate propor+onal to his love. (Assume k 1 >0). (A) dx/dt = k 1 x (C) dy/dt = - k 1 y (E) Not sure (B) dx/dt = - k 1 x (D) dx/dt = - k 1 y

12 (2) Romeo s love, y(t) My passion for Juliet increases at a rate propor+onal to her love (Assume k 2 >0). (A) dx/dt = k 2 x (B) dy/dt = - k 2 x (C) dy/dt = k 2 y (D) dy/dt = - k 2 y (E) dy/dt = k 2 x

13 Love-hate rela+onship Let k 1 =k 2 =1 Juliet: Romeo: dx/dt = - k 1 y dy/dt = k 2 x This is once more a pair of coupled differen+al equa+ons for two func+ons of +me, (x(t), y(t)). What s gonna happen?

14 Direc+on field dx/dt = - k 1 y dy/dt = k 2 x Romeo s love y(t) Juliet s love x(t)

15 (3) What do you think will dx/dt = - k 1 y dy/dt = k 2 x happen? Romeo s love y(t) Juliet s love x(t)

16 Solu+on curve dx/dt = - k 1 y dy/dt = k 2 x Romeo s love y(t) Juliet s love x(t)

17 Solu+on curve dx/dt = - k 1 y dy/dt = k 2 x Romeo s love y(t) Romeo and Juliet chase each other in an endless love-circle! Juliet s love x(t)

18 Shown in 3D with +me axis: Romeo s love y(t) Juliet s love x(t) Time t à

19 Romeo and Juliet Time t à

20 (4) Do you recognize these func+ons? Yes! These are (A) Polynomials (B) Exponen+als (C) Power func+ons (D) Sine and cosine (E) Not sure

21 Romeo and Juliet Time t à These curves are x(t) = cos(t), y(t) = sin(t). Next, aher a short break, we will discuss these trigonometric func+ons!

22 Pause for online evalua+on of Math 102 Please fill in an evalua+on at the following URL: hpps://eval.ctlt.ubc.ca/science These evalua+ons are taken very seriously by the Faculty of Science, the Dept of Mathema+cs, and your instructor. Please par+cipate.

23 Introducing: the trigonometric func+ons sin(t), cos(t) What s special about these func+ons? - Classic periodic func+ons - Describe oscilla+ng systems - Specially nice derivs! - Close rela+ves.. (x(t),y(t))

24 Cosine: Deriva+ves of cosine and sine Sine: (See course notes Sec+on where this is shown using the defini+on of the deriva+ve.)

25 Deriva+ve of sin(x) See course Notes Sec+on

26 Uses Trig iden+ty See course Notes Sec+on

27 Uses two limits: See course Notes Sec+on

28 Uses two limits: See course Notes Sec+on

29 Uses two limits: See course Notes Sec+on

30 Cosine: x(t) = cos(t), y(t) = sin(t) Deriva+ves of sine and cosine sine: dx/dt = - y dy/dt = x (Special rela+onship between these func+ons means that they sa+sfy the Juliet-Romeo equa+ons!)

31 Find the second deriva+ve of the func+on y(t)=sin(t) What differen+al equa+on does this func+on sa+sfy?

32 (5) Sin(t) sa+sfies (A) (B) (C) (D) (E) Not sure

33 Solu+on: 2 nd deriva+ve of func+on = -(original fn)

34 Easy to show that: But these are the same equa+on in two nota+ons: both say that 2 nd deriva+ve of func+on = -(original fn)

35 Both sine and cosine sa+sfy the same kind of differen+al equa+on 2 nd deriva+ve of func+on = -(original fn)

36 Confused??? I thought trig func+ons have to do with angles and triangles Yes they do! Provided we interpret angles in a special way:

37 Angles in radians We define a new measure for angles: 1 revolu+on around a circle 2 π radians Convenient, since it associates angles with the length of an arc subtended by that angle

38 Conven+on Angles increase counterclockwise

39 (6) Convert from degrees to radians: In terms of radians, the angles 30, 45, 60, 90 o are: (A) π /6, π /4, π /3, π /2 (B) π /3, π /2, π /6, π (C) π /30, π /45, π /60, π/90

40 Special angles (See Trig review, Appendix of M102 Course Notes)

41 Connec+on with angle ( ) Now let theta depend on +me, so the point will move around the circle

42 Trig iden+ty: Eqn of circle of radius 1: Point on that circle: Implies special rela+onship:

43 Mo+on around a circle Angle (theta) increases Then we can describe the mo+on by either Or else by

44 Mo+on around a circle So there are two ways of describing the same mo+on, using either Radial coordinates: Rectangular (cartesian) coordinates:

45 Periodic func+ons A func+on is said to be periodic with period T if For Trig func+ons such as sine and cosine, you should be able to iden+fy the period, frequency, amplitude, and mean.

46 Frequency and period The frequency (radians/+me) One cycle (1 full revolu+on) = radians The period (+me to complete 1 cycle:

47 Period (T), amplitude (A) A= Amplitude = 1 T=Period = 2π Frequency:

48 Other trig func+ons (See Trig review, Appendix of M102 Course Notes)

49 (7) Find the deriva+ve of y=tan(x) (A) (B) (C) (D) (E)

50 Important Trigonometric Iden++es Sum of two angles: Circular iden+ty:

51 Law of Cosines Rela+on between side lengths I a triangle and an angle Special case: θ=π/2

52 (8) Law of Cosines In the special case: θ=π/2, the Law of Cosines reduces to which of these? (A) (B) (C) (D)

53 Answers 1 C 2 E 3 C 4 D 5 A 6 A 7 D 8 C

54 Prac+ce Exam problems

55 Mul+ple Choice Q

56 Mul+ple Choice: Q:

57 Mul+ple Choice: Q:

58 Short Answer Problem

59 Solu+ons to Prac+ce Problems from last +me Similar to problems that could appear on an exam

60 Inflec+on points (IPs) Show that the solu+on curves to the Logis+c eq have an IP only if y(0)<0.5: Rewrite: Differen+ate each side with respect to +me: So 2 nd deriva+ve changes sign when (1-2y) changes sign, i.e. at y=1/2=0.5. Chain rule used here

61 Inflec+on Pts cont d At y(0)=0.5, all the slopes are posi+ve, so solu+ons increasing. So we have to start below that to have an IP! y(t) t

62 Inflec+on points, cont d No+ce that only solns star+ng below y(0)=0.5 change concavity! (and it happens precisely at y=0.5) y(t) t

The deriva*ve. Geometric view. UBC Math 102

The deriva*ve. Geometric view. UBC Math 102 The deriva*ve Geometric view Math Learning Center The MLC is a space for undergraduate students to study math together, with friendly support from tutors (grad students in math). Located at LSK310 and