Chapter 0 Preliminaries

Transcription

1 Chapter 0 Preliminaries MA1101 Mathematics 1A Semester I Year 2017/2018 FTMD & FTI International Class Odd NIM (K-46) Lecturer: Dr. Rinovia Simanjuntak

2 0.1 Real Numbers and Logic

3 Real Numbers

4 Repeating and Nonrepeating Decimals Every rational number can be written as a decimal. Irrational numbers can also be expressed as decimals. Every rational number can be written as a repeating decimal. If x is a rational number then x is a repeating decimal. The converse is also true: If x is a repeating decimal then x is a rational number. Example. Show that x = represents a rational number. What about x =

5 In Calculus, the principle numbers are real numbers Be able to calculate with rational numbers (expressed as either repeating or terminating decimals) or irrational numbers (decimals that do NOT terminate or repeat) Be able to ESTIMATE answers before pushing a button on a calculator! Use good mental mathematics. Much done in math must be proven, and different methods of proof can be employed.

6 Denseness Rational numbers and irrational numbers are dense along the real line. Any irrational number can be approximated as closely as we please by a rational number. Example. Find a rational number between 3,14159 and π. Note that π=3,141592

7 A Bit of Logic Important results in mathematics are called theorems. Many theorems are stated in the form If P then Q (P Q). We call P the hypothesis and Q the conclusion of the theorem. The converse of P Q is Q P. The negation of the statement P is ~P. The contrapositive of P Q is ~Q ~P. Example. Write the converse and contrapositive of the following statement: 1. If it rains today, then I will stay at home. 2. If x < y then x 2 < y 2.

8 Order The nonzero real numbers separate nicely into two disjoint sets: the positive real numbers and the negative real numbers. This fact allow us to introduce the order relation <. x < y y x is positive x y y x is positive or zero

9 Quantifiers Many mathematical statements involve a variable x and the truth of the statement depends on the value of x. We will let P(x) denote a statement whose truth depends on the value of x. We say For all x, P(x) or For every x, P(x) when the statement P(x) is true for every value of x. When there is at least one value of x for which P(x) is true, we say There exists an x such that P(x)".

10 Examples Which of the following statements are true? 1. For all x, x 2 > For all x, x < 0 x 2 > For every x, there exists a y such that y > x. 4. There exists a y such that, for every x, y > x.

11 0.2 Inequalities and Absolute Values

12 Equations and Inequalities Solving equations is one of the traditional tasks in mathematics. In Calculus, solving inequalities is also a significant task. To solve an inequality is to find the set of all real numbers that make the inequality true. The solution of an equation normally consists of one or several numbers, but the solution set of an inequality is usually an entire interval of numbers or the union of such intervals.

13 Intervals

14 Solving Inequalities The procedure for solving an inequality consists of transforming the inequality one step of a time until the solution set is obvious. This can be done by comparing the inequality to zero, factor if possible, and solve. We may perform certain operations on both sides of inequality without changing its solution set. 1. We may add the same number to both sides of an equality. 2. We may multiply both sides of an equality by the same positive number. 3. We may multiply both sides of an equality by the same negative number, but then we must reverse the direction of the equality sign.

15 Examples 1. Solve 5 2x + 6 < Solve 3x 2 x 2 > Solve x 1 x Solve 2.9 < 1 x < Solve 3 x+5 < 2

16 Absolute Values The absolute value of a real number x, denoted by x, is defined by One of the best way to think of the absolute value of a number is as an undirected distance. x is the distance between x and the origin. x-a is the distance between x and a.

17 Properties of Absolute Values

18 Inequalities Involving Absolute Values If the distance is greater than a constant, you must get further away in both directions. If the distance is less than a constant, the solution values must be within a certain range of values. Examples. 1. Solve the inequality x 4 < Solve the inequality 3x 5 1.

19 Squares and Square Roots Note that Does the squaring operation preserve inequality? Example. Solve the inequality 3x + 1 < 2 x 6.

20 0.3 The Rectangular Coordinate System

21 Cartesian Coordinates coordinate axes: y-axis and x-axis origin quadrants Cartesian coordinates (a,b): x-coordinate and y-coordinate

22 The Distance Formula Using Phytagorean Theorem we could obtain the following Distance Formula between two points P and Q. Example. Find the distance between P 2, 3 and Q π, π.

23 The Equation of a Circle A circle is the set of points that lie at a fixed distance (the radius) from a fixed point (the center). In general, the circle of radius r and center (h,k) has the equation This is called the standard equation of a circle. Example. Show that the equation x 2 2x + y 2 + 6y = 6 represents a circle, and find its center and radius.

24 The Midpoint Formula Example. Find the equation of the circle having the segment from (1,3) to (7,11) as a diameter.

25 Lines For a line through A x 1, y 1 and B x 2, y 2, where x 1 x 2, we define the slope m of that line by Does the value of the slope depend on which pair of points we use for A and B?

26 The slope of a line The slope is a measure of the steepness of a line.

27 Forms of the Line Equation The line passing through the (fixed) point x 1, y 1 with slope m has equation We call this the point-slope form of the equation of a line. The line intersects the y-axis at 0, b with slope m has equation This is called the slope-intercept form.

28 Equation of a Vertical Line

29 The Form Ax + By + C = 0 This is called the general linear equation and it covers all lines, including vertical lines.

30 Parallel Lines Two lines that have no points in common are said to be parallel. Example. Find the equation on the line through (6,8) that is parallel to the line with equation 3x 5y = 11.

31 Perpendicular Lines Two non-vertical lines are perpendicular if and only if their slopes are negative reciprocal of each other. Example. Find the equation of the line through the point of intersection of the lines 3x + 4y = 8 and 6x + 10y = 7 that is perpendicular to the first of this two lines.

32 0.4 Graphs of Equations

33 The Graphing Procedure To graph an equation, we could follow a simple three-step procedure: Step 1. Obtain the coordinates of a few points that satisfy the equation. Step 2. Plot these points in the plane. Step 3. Connect the points with a smooth curve. Example. Graph the equation y = x 2 3.

34 Symmetry of a Graph A graph is symmetric with respect to the y-axis, if whenever (x,y) is on the graph, (-x,y) is also in the graph. A graph is symmetric with respect to the x-axis, if whenever (x,y) is on the graph, (x,-y) is also in the graph. A graph is symmetric with respect to the origin, if whenever (x,y) is on the graph, (-x,-y) is also in the graph.

35 Symmetric Test Examples. Check the symmetry of the following graphs. 1. x 2 y 2 = 4 2. y = 1 x x + y = 1

36 Intercepts Intercepts are the points where the graph of an equation crosses the two coordinate axes. Example. Find all intercepts of the graph of y 2 x + y 6 = 0.

37 Intersections of Graphs Find the points of intersection of the line y = 2x + 2 and the parabola y = 2x 2 4x 2.

38

39 0.5 Functions and Their Graphs

40 Functions A function f is a rule of correspondence that associates with each object x in one set, called the domain, a single value f(x) from a second set. The set of all values so obtained is called the range of the function.

41 Function Notation A single letter, e.g. f, g, or F, is used to name a function. f(x), read f of x or f at x, is the value that f assigns to x. Example. For f x = x 2 2x, find and simplify a. f 4 b. f 4 + h c. f 4 + h f 4 d. f 4+h f 4 h

42 Domain and Range To specify a function completely, we must state the domain of the function. The rule of correspondence, together with the domain, determine the range of the function. Example. F x = x with domain 1,0,1,2,3. When no domain is specified for a function, we assume that it is the largest of real numbers for which the rule for the function makes sense. This is called the natural domain. Example. Find the natural domains of 1. f x = 1 x 3 2. g t = 9 t 2

43 Graphs of Functions The graph of the function f is the graph of the equation y = f x. Example. Sketch the graphs of 1. f x = x g x = 2 x 1

44 Even and Odd Functions If f x = f(x) for all x, then the graph is symmetric with respect to the y-axis. Such a function is called an even function. If f x = f(x) for all x, then the graph is symmetric with respect to the origin. Such a function is called an odd function. Examples. Is the following functions even, odd, or neither? 1. f x = x3 +3x x 4 3x g x = 2 x 1

45 Two Special Functions The absolute value function The greatest integer function

46 0.6 Operations on Functions

47 Sums, Differences, Products, Quotients, and Powers Consider functions f and g with formulas

48 Example Let F x = 4 x + 1 and G x = 9 x 2. Find formulas for F + G, F G, F G, F G, and F5 and give their natural domains.

49 Composition of Functions If f works on x to produce f(x) and g the works on f(x) to produce g(f(x)), we say that we have composed g with f. The resulting function, called the composition of g with f, is denoted by g f. Example. Let f x = x 3 and g x = x. 2 Find g f and f g.

50 Domain of Composition of Functions Clearly, g f f g (composition of functions is not commutative). The domain of g f is equal to the set of x that satisfy the following properties: 1. x is in the domain of f. 2. f(x) is in the domain of g. Example. Let f x = 6x x 2 9 and g x = Find f g x and give its domain. 3x, with their natural domains.

51 Translations How are the graphs of related to each other?

52 Translations (2) Example. Sketch the graph of f x = x using translation.

53 Partial Catalogue of Functions constant function identity function polynomial function linear function quadratic function rational function

54 0.7 Trigonometric Function

55 Sine and Cosine

56 Basic Properties of Sine and Cosine The domain for both sine and cosine functions is,. The range for both sine and cosine functions is 1,1.

57 Basic Properties of Sine and Cosine (2)

58 Graphs of Sine and Cosine Example. Sketch the graphs of 1. y = cos 2t 2. y = sin 2πt

59 Period of Trigonometric Functions A function f is periodic if there is a positive number p such that f x + p = f x for all real numbers x in the domain of f. The smallest such positive number p is called the period of f. The sine function is periodic because sin x + 2π = sin x. It is also true that. Examples. What are the periods of the following functions. 1. cos(2t) 2. sin(at) 3. sin(2πt)

60 Amplitude of Trigonometric Functions If the periodic function f attains a minimum and a maximum, we define the amplitude A as half the vertical distance between the highest point and the lowest point in the graph. Examples. Find the amplitude of the following periodic functions. 1. sin 2πt cos(2t)

61 Four Other Trigonometric Functions Example. Show that tangent is an odd function.

62 Trigonometric Identities

63 Trigonometric Identities (2)