Announcements. Topics: Homework: - sec0ons 1.2, 1.3, and 2.1 * Read these sec0ons and study solved examples in your textbook!

Transcription

1 Announcements Topics: - sec0ons 1.2, 1.3, and 2.1 * Read these sec0ons and study solved examples in your textbook! Homework: - review lecture notes thoroughly - work on prac0ce problems from the textbook and assignments from the coursepack as assigned on the course web page (under the SCHEDULE + HOMEWORK link)

2 Models A mathema&cal model is a descrip0on of a biological pavern, observa0on, or rule using mathema0cal concepts and language (such as func0ons and equa0ons). When we have a model, we can apply tools of calculus to study how a living system changes.

3 Dynamical Systems Discrete-&me dynamical systems describe a sequence of measurements made at equally spaced intervals Con&nuous-&me dynamical systems, usually known as differen0al equa0ons, describe measurements that change con0nuously

4 Conversions To be studied independently look at tables in sec0on 1.2, do conversion, etc.

5 Rela0ons and Func0ons A rela&on between two variables is the set of all pairs of values that occur. A func0on is a special type of rela0on.

6 Func0ons A func&on f is a rule that assigns to each real number x in some set D (called the domain) a unique real number f(x) in a set R (called the range).

7 Most (almost all) data collected in life sciences cons0tutes a RELATION and not a FUNCTION Example: the graph on the next slide shows the cranial capacity (i.e., the brain volume) calculated from the skulls of early humans and modern humans, between 3 million years in the past and today

8 four skulls, roughly the same age but with different cranial capacity cranial capacity in millilitres 2,100 1,700 1, M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) This diagram shows a rela0on, and not a func0on

9 Using sta0s0cal methods such as regression (these methods are covered in sta0s0cs courses in levels 2 and above), we can iden0fy a func0on which approximates the data cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1,

10 And then we work with the func0on we obtained. Why? Because we have no choice. It is not possible to work with rela0ons and obtain quan0ta0ve results desired in our research in the life sciences. cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1,

11 cranial capacity in millilitres cranial capacity in millilitres 3 M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1, M 2.5 M 2 M 1.5 M 1 M 0.5 M NOW years ago (M=million) 2,100 1,700 1, Of course, we can say something for example, the data on the right suggests some kind of exponen0al growth. But in order to quan0fy that growth, and further work with it, we need to have a func0on

12 Domain The domain of a func0on f is the largest set of real numbers (possible x-values) for which the func0on is defined (as a real number). Example: Find the domain of the following func0ons.

13 Graphs The graph of a func0on f is a curve that consists of all points (x,y) where x is in the domain of f and y=f(x). Example: Sketch the graph and find the domain and range of f (x) = x 2 +8x 17. y x

14 Piecewise Func0ons A piecewise func0on f(x) is a func0on whose defini0on changes depending on the value of x. Example: Absolute Value Func4on The absolute value of a number x, denoted by x, is the distance between x and 0 on the real number line. $ f (x) = x = % x if x 0 & x if x < 0

15 Piecewise Func0ons Example: Sketch the graph of f(x). f (x) = # 1 % x, % $ % 2, % & x 1, x <1 x =1 x >1 y x

16 Variables and Parameters A variable represents a measurement that can change during the course of an experiment. A parameter represents a measurement that remains constant during an experiment but can change between different experiments.

17 Variables and Parameters Example: Body Mass Index (BMI) BMI = m h 2 where and h m is a person s mass in kilograms is their height in metres. BMI is the dependent variable; and h are the two independent variables. m

18 Variables and Parameters We can study how a func0on depends on one of its variables at a 0me by holding all other variables constant. For example, to study how BMI depends on mass, we fix height to be constant (i.e., collect data from all people of the same height).

19 Body Mass Index Height as a Parameter BMI = 0.416m m

20 Propor0onal and Inversely Propor0onal Rela0onships Example: Body Mass Index (BMI) BMI = m h 2 Note: BMI is propor4onal to mass. If a person s mass changes (and their height remains the same), then their BMI will change by the same amount.

21 Propor0onal and Inversely Propor0onal Rela0onships If m new =1.10 m old Then BMI new = m new h 2 = 1.10 m old h 2 =1.10 BMI old So a 10% increase in body mass results in a 10% increase in BMI.

22 Propor0onal and Inversely Propor0onal Rela0onships Example: Body Mass Index (BMI) BMI = m h 2 Note: BMI is inversely propor4onal to height squared. So an increase in height (with mass held constant), will result in a decrease in BMI.

23 Propor0onal and Inversely Propor0onal Rela0onships If h new =1.10 h old Then BMI new = m h new 2 = m (1.10h old ) 2 = BMI old 0.83 BMI old So a 10% increase in height results in a 17% decrease in BMI.

24 Linear Func0ons For a linear func0on, the change in output ( Δy ) is propor4onal to the change in input ( Δx ) Δy Δx Δy = m Δx If the change in input is scaled by some factor, then the change in output is scaled by the same factor. We call this constant m the slope of the line

25 Linear Func0ons slope: m = Δy Δx = y 2 y 1 x 2 x 1 y 2 y P 2 point-slope equa0on: y 2 y 1 y y 1 = m( x x ) 1 y 1 P 1 slope-y-intercept equa0on: x 2 x 1 x 1 x 2 x y = mx + b

26 Linear Model for the Popula0on of Canada Data: Year Time, t Popula&on, P(t) (in thousands) P t

27 Linear Model for the Popula0on of Canada P Create a linear model for the popula0on of Canada as a func0on of 0me using the first two data points t

28 Power Func0ons A power func0on is a func0on of the form where a f (x) = x a is a constant. Note: Although a can be any real number, we usually omit the case when a = 0.

29 Power Func0ons Some special cases: a=2: f (x) = x 2 a=3: f (x) = x

30 Power Func0ons Some special cases: a=1/2: f (x) = x 1 2 = x a=1/3: square root func0on f (x) = x 1 3 = 3 x cube root func0on

31 Power Func0ons Some special cases: a=-1: f (x) = x 1 = 1 x ra0onal func0on a=-2: f (x) = x 2 = 1 x

32 Models Involving Power Func0ons Example: Blood Circula0on Time in Mammals Blood circula0on 0me is the average 0me needed for the blood to reach a site in the body and come back to the heart. It has been determined that, for mammals, the blood circula0on 0me is propor4onal to the fourth root of the body mass.

33 Models Involving Power Func0ons Example: Blood Circula0on Time in Mammals Model: where T is the blood circula0on, in seconds, B is the body mass, in kilograms, and a is some propor0onality constant.

34 Example: Blood Circula0on Time in Mammals Graph: 4 T(B) = a B, B 0 T 152 T(B) B

35 Models Involving Power Func0ons If the body mass increases 10-fold, how does the blood circula0on 0me change?

36 Models Involving Power Func0ons This means that the blood circula0on 0me of an elephant weighing 5400 kg is about mes longer than the blood circula0on 0me of a cow that weights 540 kg.

37 How to read and understand math in journal ar0cles, books about science, and other sources? Math in math textbooks does not look exactly the same as math found elsewhere (we will see examples soon). Although the differences are mathema&cally insignificant (such as different nota&on), it takes some &me to get used to math wrioen in a non-textbook format What do we do? We learn concepts, formulas and algorithms using math textbooks, because it is easier that way Then we use our knowledge to apply to contexts takes from various disciplines

38 Example Journal Ar0cle What is in it?

39 Our focus is on math parts How do we make ourselves understand this formula?

40 Exercise - Equa0on Analysis C = A rw βt If C is a func0on and t is independent variable: (1) Which quan00es are parameters (2) Replace all parameters by numbers do you recognize the equa0on? (3) Keep the parameters (do not give them numeric values); what is the graph of C (you will need to make assump0ons, for instance β could be posi0ve, nega0ve, or zero) (4) If C is a func0on of A, what is its graph? (5) If C is a func0on of r, what is its graph? (6) If C is a func0on of W, what is its graph?