giải tích 2 lê xuân đại functions of single variable sinhvienzone com

C o e on ELECTRONIC VERSION OF LECTURE nZ Dr Lê Xuân Đại hV ie HoChiMinh City University of Technology Faculty of Applied Science, Department of Applied Mathematics Email: ytkadai@hcmut.edu.vn in m FUNCTIONS OF SINGLE VARIABLE Dr Lê Xuân Đại (HCMUT-OISP) HCMC — 2016 https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o e FUNCTIONS OF SINGLE VARIABLE BASIC PROPERTIES OF FUNCTIONS ELEMENTARY FUNCTIONS MATL AB hV ie nZ on in m OUTLINE Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o Function and its graph on e DEFINITION 1.1 A function f is a rule that assigns to each element x in a set X ⊂ R exactly one element y , called f (x) in a set E ⊂ R Denoted by: nZ f : X −→ E y = f (x), x − independent, y − dependent hV ie The set X = {x ∈ R : f (x)is defined} is called the domain of the function f and is denoted by D(f ) The set f (X ) = {y = f (x) ∈ R : x ∈ X } is called the range of the function f and is denoted by E(f ) f (x) is the value of f at x and is read " f of x " in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o Function and its graph hV ie nZ on e DEFINITION 1.2 The set consists of all points (x, f (x)), x ∈ X in the coordinate plane Oxy is called the graph of the function f in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o Function and its graph on e EXAMPLE 1.1 Find the domain and range of function f (x) = x + hV ie nZ SOLUTION The domain of f consists of all values of x such that x + ⇔ x −2, so the domain is the interval [−2, +∞) The range of f consists of all values of y such that y = x + 0, so the range is the interval [0, +∞) in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o Piecewise defined functions nZ on e DEFINITION 1.3 The functions which are defined by different formulas in different parts of their domain, are called piecewise defined functions ie EXAMPLE 1.2 A function f is defined by hV f (x) = in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) − x, if x −1 x2 , if x > −1 https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o The composite function e Suppose that y = f (u), where u is a function of x : u = g(x) We compute this by substitution on y = f (g(x)) nZ The procedure is called composition because the new function is composed of the given functions f and g hV ie DEFINITION 1.4 Given functions f and g, the composite function f ◦ g (read: f circle g ) is defined by in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) (f ◦ g)(x) = f (g(x)) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 ie nZ on e C o The composite function hV in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o The composite function nZ SOLUTION We have on e EXAMPLE 1.3 If f (x) = x2 , and g(x) = x3 − 7, find the composite function f ◦ g and g ◦ f ie (f ◦ g)(x) = f (g(x)) = f (x3 − 7) = (x3 − 7)2 hV (g ◦ f )(x) = g(f (x)) = g(x2 ) = (x2 )3 − = x6 − in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 / 52 C o One-to-one functions on e DEFINITION 1.5 A function f is called a one-to-one function if it never takes on the same value twice; that is, nZ f (x1 ) = f (x2 ) whenever x1 = x2 hV ie EXAMPLE 1.4 The function f (x) = x3 is one-to-one because if x1 = x2 then x13 = x23 The function g(x) = x2 is not one-to-one because, for instance g(1) = = g(−1), and so and −1 have the same output in m Functions of single variable Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 10 / 52 C o Trigonometric functions Function cosine y = cos x Domain: D = R Function is periodic of period 2π : Range: E = [−1, 1] e nZ Function is increasing on the interval (−π, 0) , and decreasing on the interval (0, π) , Function is even, the graph is symmetric with respect to the y−axis hV ie on cos(x) = cos(x + 2π) = cos(x − 2π) in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 38 / 52 C o Trigonometric functions on e sin x Function tangent y = tan x = cos x Domain: D = R \ π2 + kπ, k ∈ Z Range: E = R Function is periodic of period π : tan(x) = tan(x + π) = tan(x − π) hV ie Function is increasing on the interval − π2 , π2 Function is odd, the graph is symmetric about the origin O(0, 0) nZ in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 39 / 52 C o Trigonometric functions on e x Function cotangent y = cot x = cos sin x Domain: D = R \ {kπ, k ∈ Z} Range: E = R Function is periodic of period π : cot(x) = cot(x + π) = cot(x − π) hV ie Function is decreasing on the interval (0, π) Function is odd, the graph is symmetric about the origin O(0, 0) nZ in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 40 / 52 C o Trigonometric functions Some basic formulas sin x cos x tan(π − x) = tan(−x) = − tan x tan x = hV ie nZ on sin x + cos x = sin 2x = sin x cos x sin 3x = sin x − sin3 x − cos 2x sin2 x = π sin = 1; sin(kπ) = cos 2x = cos2 x − sin2 x cos 2x = cos2 x − = − sin2 x + cos 2x cos2 x = cos = 1; cos π = −1 π cos ± = e in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) tan(π + x) = tan(x) π tan = 0, tan is undefined cos x sin x cot(π−x) = cot(−x) = − cot x cot x = cot(π + x) = cot x π cot = 0, cot is undefined https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 41 / 52 C o Inverse trigonometric functions Function arcsine y = arcsin x e y = arcsin x ⇐⇒ x = sin y π π y −1 x 1, − 2 ie nZ on (8) hV in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 42 / 52 C o Inverse trigonometric functions Function arcosine y = arccos x e y = arccos x ⇐⇒ x = cos y −1 x y π ie nZ on (9) hV in m Elementary functions Dr Lê Xuân Đại (HCMUT-OISP) https://fb.com/sinhvienzonevn FUNCTIONS OF SINGLE VARIABLE HCMC — 2016 43 / 52 e ⇐⇒ x = tan y π π −∞ < x < ∞ −