Mathematics 1

1. Mathematics1 Applied Informatics Štefan BEREŽNÝ

2. 1st lecture

3. Contents • Real Number • Functions • Graphs of Functions • Limit of Function MATHEMATICS 1 Applied Informatics

4. Real Number • The real number system denoted R. • Properties of real numbers: • Commutative laws: • a, bR: a + b = b + a • a, bR: ab = ba • Associative laws: • a, b, cR: (a + b) + c = a + (b + c) • a, b, cR: (ab) c = a (bc) • Distributive laws: • a, b, cR: a (b + c) = ab + ac • a, b, cR: (a + b) c = ac + bc MATHEMATICS 1 Applied Informatics

5. The Real Line Axion: There is a one-to-one correspondence between the points on the line and the system Rof real numbers. • The each real number a there corresponds a point P on the line. • Each point P on the line is the correspondent of a single real number a. • If P is left of Q, then b – a is the distance from P to Q. If P a and Q  b. MATHEMATICS 1 Applied Informatics

6. Order • number a lies to the left of number b „a is less thanb“  ab • number a lies to the left of number b „b is greater than a“ ba • „a is less or equal to b“ ab • „b is greater than or equal to a“  ba • The relations less than and greater than and so on  is called order on the real number system. MATHEMATICS 1 Applied Informatics

7. Order Properties of order: • Reflexivity: aR: aa. • Anti-symetry: a, bR: if ab and ba, then a = b. • Transitivity: a, b, cR: if ab and bc, then ac. MATHEMATICS 1 Applied Informatics

8. Order Properties of order: • Trichotomy: If a and b are real numbers, then exactly one of these three relations holds: aba = bab. • The arithmetic operations +, –,  and  are closely linked to the order relations a, bR: a + bR. a, bR: a – bR. a, bR: abR. a, bR: abR. MATHEMATICS 1 Applied Informatics

9. Order Properties of order: • If ab and if c is any real number, then a + cb + c and a – cb – c. • If ab and c 0, then acbc and a/cb/c. • If ab and c 0, then acbc and a/cb/c. In particular: –1  0, so – b –a. • If 0 ab or ab 0, then 1/b 1/a. • The statement ab is equivalent to the statement b – a 0. • If x is any real number, then x2 0. If x2 = 0, then x = 0. MATHEMATICS 1 Applied Informatics

10. Absolute Value Definition: • The absolute value of real number a, written a, is the distance on the number line from a to 0. MATHEMATICS 1 Applied Informatics

11. Absolute Value Rules for absolute value are: • –a = a • ab = ab • a + ba + b • a – ba + b • a – ba – b • ; (b 0) MATHEMATICS 1 Applied Informatics

12. Intervals The set of all numbers between two fixed numbersis called an interval on the number line. An interval that includes both end points is called a closed interval. We use the notation: a, b, forab. For the closed interval a, b of all points x satisfying: axb. An interval that excludes both end points is called an open interval. We use the notation: (a, b). For the open interval (a, b) of all points x satisfying:axb. Hybrid intervals called half-open or half-closedintervals. The suggestive notation for these intervals is: (a, b for the set of x satisfying axb, a, b) for the set of x satisfying axb. MATHEMATICS 1 Applied Informatics

13. Intervals The intervals a, b, (a, b), (a, b and a, b)are bounded. Unbounded intervals, that is, intervals that go off indefinitely in one direction or the other. We use one of the following notations: (–, b), (–, b, (a, ), a, ). • The inequality x – ar describes the open interval (a – r, a + r). • The inequality x – ar describes the closed interval a – r, a + r. MATHEMATICS 1 Applied Informatics

14. Rational and Irational Number The real number system includes the rational number system. Recall that a rational number is a real number that is the quotient of integers. Each rational number (also called fraction in schol arithmetic) can be written in lowest terms, that is, as a quotient of integers with no common factor larger than 1. A real number that is not rational is called irrational. MATHEMATICS 1 Applied Informatics

15. Functions A functionf: y = f(x) maps its domain (D) onto its range (R). y = f(x) – variable xis called the independent variable – variable yis called the dependent variable. For each number x in the domain of f, we find the corresponding number y = f(x) in the range and then we plot the point x, y. The set of all such points is called the graph of f(x). MATHEMATICS 1 Applied Informatics

16. Functions • IfMR, then mapping f: MR is called a real function ofone real variable. • D(f) = xR: ! yR: y = f(x). • R(f) = yR: xR: y = f(x). • G(f) = x, f(x)R2: xD(f). MATHEMATICS 1 Applied Informatics

17. Functions Function f is said to be one-to-one if x1, x2D(f): x1x2f(x1) f(x2). Inverse function: There are inverse mappings to one-to-one mappings, there are also called inverse function to one-to-one functions. The inverse function to function f will be denoted by f–1. Its domain is R(f), the range is D(f) and xD(f): y = f(x) x = f –1(y). The graphs of the functions f and f –1 are symmetric with respect to the axis of the 1st and 3rd quadrant. MATHEMATICS 1 Applied Informatics

18. Functions Composite function: If f and g are such functions that R(g) D(f), we can define a function h by the equation h(x) = f(g(x)) for xD(g). The function h is called the composite function of functions f and g. We use the notation h = f ◦g. Function f is called the outside function and function g the inside function. MATHEMATICS 1 Applied Informatics

19. Functions Bounded function: Function f is called bounded above (upper bounded) if there exists a real number KR such that xD(f): f(x) K. We can define analogously a function bounded below (lower bounded). Function f is called bounded if fis bounded above and bounded below. LetMD(f). Function f is called upper bounded on the set M if there exists a real number KR such that xM: f(x) K. We can similarly define the notion of a function lower bounded on the set M and the notion of a function bounded on the set M. MATHEMATICS 1 Applied Informatics

20. Functions Extreme values of a function: The function f has its maximum at the point x0D(f) if xD(f): f(x) f(x0). The function f has its minimum at the point x0D(f) if xD(f): f(x) f(x0). The maximum and minimum of function f are both called extreme value of f. Suppose that MD(f). The function f has its maximum on the set M at the point x0M if xM: f(x) f(x0). The minimum of function f on the set M at the pointx0M if xM: f(x) f(x0). MATHEMATICS 1 Applied Informatics

21. Functions Supremum and infimum of a function: The supremum of the set of values of function f is called the supremum of function f and it is the least upper bound of f. The infimum of the set of values of function f is called the infimum of function f and it is the greatest lower bound of f. MATHEMATICS 1 Applied Informatics

22. Functions Monotonic and strictly monotonic function: Let f be a function and MD(f). The function f is called: (a) increasing on M if x1, x2M: x1x2f(x1) f(x2); (b) decreasing on M if x1, x2M: x1x2f(x1) f(x2); (c) non-increasing on M if x1, x2M: x1x2f(x1) f(x2); (d) non-decreasing on M if x1, x2M: x1x2f(x1) f(x2); (e) monotonic on Mif f is non-increasing or non-decreasing on set M; (f) strictly monotonic on Mif f is increasing or decreasing on set M. MATHEMATICS 1 Applied Informatics

23. Functions Elementary functions: The constant function: f(x) = c; where cR, The linear function: f(x) = kx + q; where k, qR and k 0, The quadratic function: f(x) = ax2 + bx + c; where a, b, cR and a 0, The power function: f(x) = xk; where kR, MATHEMATICS 1 Applied Informatics

24. Functions The exponential function with base a: f(x) = ax; where aR and a 0, The logarithmic function with base a: f(x) = logax; where aR and a 0, The function sine: f(x) = sin x; The function cosine: f(x) = cos x; The function tangent: f(x) = tg x; The function cotangent: f(x) = cotg x. MATHEMATICS 1 Applied Informatics

25. Functions The functions sine, cosine, tangent and cotangent are called trigonometric functions. MATHEMATICS 1 Applied Informatics

26. Functions Polynomial function (polynomial): The polynomial of the n-th degree is the function: P(x) = a0 + a1x + a2x2 + a3x3 +  + an–1xn–1 +anxn where a0, a1, a2, a3,  , an are real numbers and an 0. Specially, if n = 0, then P(x) is called constant function, if n = 1, then P(x) is called linear polynomial (linear function), if n = 2, then P(x) is called quadratic polynomial (quadratic function) and if n = 3, then P(x) is cubic polynomial. MATHEMATICS 1 Applied Informatics

27. Limit Of Function The limit as x approaches a of f(x) is L: if for each number  0 there exists a number  0 such that f(x) – L for all x in D(f) that the inequality 0 x – a. MATHEMATICS 1 Applied Informatics

28. Limit Of Function Basic Limit Rules: Suppose both of the limits and exists. Suppose c is a constant. Then the following four limits exist and have the values stated: MATHEMATICS 1 Applied Informatics

29. Limit Of Function if M 0. MATHEMATICS 1 Applied Informatics

30. Limit Of Function Let f(x) be a function whose domain D is an interval, or union of intervals. Let a be a point of D. Then f(x) is continuous at a if Function f(x) is continuous on D if f(x) is continuous at each point D. MATHEMATICS 1 Applied Informatics

31. Thank you for your attention. MATHEMATICS 1 Applied Informatics