MAT01A1: Functions and Mathematical Models

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1 MAT01A1: Functions and Mathematical Models Dr Craig 21 February 2017

2 Introduction Who: Dr Craig What: Lecturer & course coordinator for MAT01A1 Where: C-Ring 508 Web:

3 Important information Course code: MAT01A1 NOT: MAT1A1E, MAT1A3E, MATE0A1, MAEB0A1, MAA00A1, MAT00A1, MAFT0A1 Learning Guide: available on Blackboard. Please check Blackboard twice a week. Student check this account twice per week or set up forwarding to an address that you check frequently.

4 Important information Lecture times: Tuesday 08h50 10h25 Wednesdays 17h10 18h45 Lecture venues: C-LES 102, C-LES 103 Tutorials: Tuesday afternoons 13h50 15h25: D-LES 104 or D-LES 106 OR 15h30 17h05: C-LES 203 or D1 LAB 408

5 Other announcements No tuts for MAT01A1 on Wednesdays. If you see this on your timetable, it is an error. (To move your Chem. prac., Mr Kgatshe CSC02A2 students: tut on Thu 09h40 to 11h15. Maths Learning Centre in C-Ring 512: 10h30 14h35 Mondays 08h00 15h30 Tuesday to Thursday 08h00 12h55 Fridays

6 Lecturers Consultation Hours Monday: 10h30 11h30 Ms Richardson (C-503) Wednesday: 14h30 16h00 Ms Richardson (C-503) Thursday: 11h00 13h00 Dr Craig (C-508) 13h30 14h00 Ms Richardson (C-503) Friday: 11h30 13h00 Dr Craig (C-508)

7 Note: these lecture slides should be in used in conjunction with the textbook. These slides do not tell the full story. You must read the sections from the textbook and complete the prescribed tutorial exercises in order to fully understand each section.

8 Introduction to functions Four examples of functions: The area of a circle depends on the radius: A = πr 2. Population of the world depends on time: P (1950) = 2, 560, 000, 000. The cost of posting a package depends on the weight: C(w). The vertical ground acceleration during an earthquake: S(t).

9 Domain and range of functions A function f is a rule which assigns to each element x in a set D exactly one element, f(x), in a set E. The set D is the domain of f. The range of f is the set of all possible values of f(x) as x varies through the domain.

10 A symbol representing an arbitrary element of the domain is called an independent variable and a symbol representing an arbitrary element of the range is a dependent variable. In the example of the circle: r is the independent variable while A is the dependent variable.

11 Graphs of functions: a common way of representing a function is by a graph. Formally, the graph of the function f is the set of ordered pairs { ( x, f(x) ) x D } Dom(f)= [1,6] and Ran(f)=[1,7]

12 Examples Sketch the following functions and find their domain and range: (a) f(x) = 3x + 4 (b) g(x) = x 2 2 (c) h(x) = sec x

13 Representations of functions There are four ways to represent a function: Verbally (describe in words) Tables Graphically Algebraically

14 The vertical line test How do we know if a curve is a function? A curve in the xy-plane is a function of x if and only if no vertical line intersects the curve more than once.

15 The curve x 2 + y 2 = 4 is not a function: The curve y = 4 x 2 is a function:

16 Piecewise defined functions Consider the function { x + 2 if x 1 f(x) = x 2 if x > 1 Example: The absolute value function. { x if x 0 f(x) = x = x if x < 0

17 Example: step functions Consider the cost of travelling on the Rea Vaya bus. Here x is the number of km travelled and C(x) is in rands: 6.2 if 0 x 5 C(x) = 8.1 if 5 < x if 10 < x if 15 < x if 25 < x if 35 < x

18 Symmetry in functions If a function f satisfies f( x) = f(x) then f is an even function. Examples: f(x) = x 2 f(x) = cos(x) f(x) = x for all x D Another way of defining an even function is to say that it is a reflection about the y-axis.

19 Symmetry in functions A function is odd if f( x) = f(x) Examples: f(x) = x f(x) = sin(x) f(x) = x 3 for all x D An odd function is a reflection about the origin. Thus, a necessary condition for a function f to be odd is that it must have f(0) = 0.

20 Increasing and decreasing functions A function f is increasing on an interval I if whenever x 1 < x 2, we have f(x 1 ) < f(x 2 ). A function f is decreasing on an interval I if whenever x 1 < x 2 we have f(x 1 ) > f(x 2 ). Example: Is the function f(x) = cos x increasing, decreasing, or neither over the following intervals: (a) x [π, 3π/2] (b) x [0, π/2] (c) x [π/2, 3π/2]

21 Mathematical models A mathematical model is a mathematical description, using a function or an equation, of a real-world problem. If y is a linear function of x then the graph is a straight line: y = mx + c where m is the slope of the graph, and c is the y-intercept.

22 Example: The length of a newly born snake is 10cm and after 3 months the length is 25cm. It grows the same amount each month. (a) Express length as a function of time (in months) where the D = [0, 12]. (b) Draw the graph of the length function. (c) What is the length of the snake after 9 months?

23 Functions from data: If we don t have a function to work from, we can try to determine a function using empirical data. Year Number of registered cars in SA ,500, ,670, ,890, ,100, ,310, ,540, ,770, ,900,000

24 For the last example we can try to find a function which fits the data by calculating the slope of the line which goes through the first and last points. m = Therefore m = 200, , 900, 000 4, 500, To be more accurate, we can use a statistical technique known as linear regression.

25 Polynomials These are functions of the form: f(x) = a n.x n +a n 1.x n a 2.x 2 +a 1.x+a 0 where a 0,..., a n are constants. If the leading coefficient a n 0 then f(x) is a polynomial of degree n. polynomial of degree 1 = linear function polynomial of degree 2 = quadratic function polynomial of degree 3 = cubic function

26 Power functions These are functions of the form f(x) = x a Note that a can be a positive integer a = 1 n where n is a positive integer a negative integer, so f(x) = 1 x a

27 Rational functions A rational function f is a ratio of two polynomials Example: f(x) = P (x) Q(x) f(x) = 2x4 x x 2 4

28 Algebraic functions An algebraic function is one that can be formed by using the algebraic operations of addition, subtraction, multiplication, powers, division and taking roots. Note: any rational function is automatically an algebraic function. Examples: f(x) = x h(x) = x4 16x 2 x + x

29 Trigonometric functions Functions that express the ratio between x, y and r when angles are plotted on the xy-plane. sin x cos x tan x csc x sec x cot x

30 Exponential functions These are functions of the form f(x) = a x where a 0 is constant. Examples: f(x) = 2 x f(x) = 0.5 x f(x) = e x

31 f (x) = 2x

32 f (x) = 0.5x

33 Logarithmic functions f(x) = log a x where a 0 is a constant (also known as the base). Logarithmic functions are the inverse of exponential functions. That is, if y = a x then log a y = x.

34 f(x) = log 2 x

35 Make sure that you are comfortable with the sketches of exponential and logarithmic functions. Know their shape and where the intercepts occur.

MAT01A1. Numbers, Inequalities and Absolute Values. (Appendix A)

MAT01A1. Numbers, Inequalities and Absolute Values. (Appendix A) MAT01A1 Numbers, Inequalities and Absolute Values (Appendix A) Dr Craig 8 February 2017 Leftovers from yesterday: lim n i=1 3 = lim n n 3 = lim n n n 3 i ) 2 ] + 1 n[( n ( n i 2 n n + 2 i=1 i=1 3 = lim