1 / 42

FUNCTIONS Reference: Croft & Davision, Chapter 6 p.125 math.utep/sosmath

Learn about functions, domains, ranges, graphs, compositions, and inverses. Explore trigonometric functions and equations, including amplitudes and conversions. Explanations and exercises provided.

refugioa
Télécharger la présentation

FUNCTIONS Reference: Croft & Davision, Chapter 6 p.125 math.utep/sosmath

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. FUNCTIONS • Reference: Croft & Davision, Chapter 6 p.125 • http://www.math.utep.edu/sosmath • Basic Concepts of Functions A function is a rule which operates on an input and produces a single output from that input. Consider the function given by the rule: 'double the input'. Page 1

  2. e.g.1 Given f (x) = 2x + 1 find:(a) f (3) (b) f (0) (c) f (–1 ) (d) f (a) (e) f (2a) (f ) f (t) (g) f ( t + 1 ) • a) 2(3)+1=7 b) 2(0)+1=1 c) 2(-1) +1=-1 • d) 2(a)+1=2a+1 e) 2(2a)+1=4a+1 f) 2(t)+1=2t+1 • g) 2(t+1)+1=2t+3 • End of Block Exercise • p.129 Page 2

  3. The Graph of a Function A function may be represented in graphical form. The function f (x) = 2x is shown in the figure. We can write: • . Page 3

  4. In the function y = f (x), x is the independent variable and y is the dependent variable. • The set of x values used as input to the function is called the domain of the function • The set of values thaty takes is called the range of the function. Page 4

  5. e.g.2 The figure shows the graph of the function • f (t) given by • (a) State the domain of the function. • (b) State the range of the function by inspecting the graph. • End of Block Exercise • p.135 [-3, 3] [0, 9] e.g.3 Explain why the value t = 0 must be excluded from the domain of the function f (t) = 1/t. ∵ 1/0 is undefined Page 5

  6. Determine the domain of each of the following functions: All real number All real number except 0 For S≧2 All real number except 5 & -0.5 • (a) • (b) • (c) • (d) Page 6

  7. Composition of Functions Reference URL: • http://archives.math.utk.edu/visual.calculus/0/compositions.5/ When the output from one function is used as the input to another function - Composite Function • Consider • End of Block Exercise • p.141 Page 7

  8. One-to-many rules • Note: • e.g. • One-to-many rule is not a function. • But functions can be one-to-one or many-to-one. • f (x) = 5x +1 is an example of one-to-one function. • is an example of many-to-one function. Page 8

  9. Inverse of a Function • is the notation used to denote the inverse function of f (x). The inverse function, if exists, reverse the process in f (x). e.g.4 Find the inverse function of • End of Block Exercise • p.148 Page 9

  10. Solution of e.g.4 : The inverse function, g-1, must take an input 4x – 3 and give an output x. That is, g-1(4x-3) = x Let Z = 4x-3, and transpose this to give x = (Z+3)/4 Then, g-1(Z) = (Z+3)/4 Writing with x as its argument instead of Z gives g-1(x) = (x+3)/4 Page 10

  11. f-1(3x-8)=x --------------------------------- Step 1 Let Z=3x-8, then x=(Z+8)/3 ------------Step 2 And then, f-1(Z) =(Z+8)/3 ----------------Step 3 Writing with x instead of Z, then, f-1(x)=(x+8)/3 ----------------------------Step 4 Class Exercises Find the inverse functions for the following functions. • 1. f(x) = 3x – 8 • 2. g(x) = 8 – 7x • 3. f(x) = (3x – 2)/x Let Z=8-7x, then x=(8-Z)/7 Writing with x instead of Z, then, g-1(x)=(8-x)/7 Let Z=(3x-2)/x, then x=-2/(Z-3) Writing with x instead of Z, then, f-1(x)=-2/(x-3) Page 11

  12. L r r TRIGONOMETRIC FUNCTIONS • Reference: Croft & Davision, Chapter 9 • http://www.math.utep.edu/sosmath Angles Two main units of angle measures: • degree 90o, 180o • radian  , 1/2  Unit Conversion  radian = 180o e.g.1. Convert 127o in radians. Page 12

  13. Trigonometric functions • Reference URL: http://home.netvigator.com/~leeleung/sinBox.html y • P(x,y) r  y x x Page 13

  14. y sin’+’ve S All ‘+’ve A x C cos ‘+’ve T tan’+’ve • The sign of a trigonometric ratio depends on the quadrants in which  lies. • The sign chart will help you to remember this. Page 14

  15. y II I y     x x O O    III IV y y   x x O  O  • Reference Angle:  Page 15

  16. Reduction Principle • Where the sign depends on A S • e.g.2 Without using a calculator, find T C 1 45○ 1 30○ 2 =sin (1800-300 ) =sin300 =1/2 or 0.5 =tan(3600-450) =-tan450 =-1 =cos(1800+300) =-cos300 = 60○ 1 Page 16

  17. Negative Angles • Negative angles are angles generated by clockwise rotations. • Therefore • e.g.3 Find (a) sin(-30o) (b) cos (-300o) x  =cos(360o- 60o) =cos 60o =1/2 =-sin 30o =-1/2 Page 17

  18. Trigonometric graphs Consider the function y = A sin x, where A is a positive constant. The number A is called the amplitude. Page 18

  19. Example • State the amplitude of each of the following functions: • 1. y = 2 sin x • 2.y = 4.7cos x • 3. y = (2 sin x) / 3 • 4. y = 0.8cos x • -2≦y ≦2 • -4.7≦y ≦4.7 • -2/3≦y ≦2/3 • -0.8≦y≦0.8 Page 19

  20. Simple trigonometric equations • Notation : • If sin  = k then  = sin-1k ( sin-1is written as inv sin or arcsin). Similar scheme is applied to cos and tan. • e.g.4 Without using a calculator, solve sin  =  0.5, • where 0o    360o • e.g.5 Solve cos 2 =  0.4 , where 0   2 sin-1 0.5 = 210o, 330o cos-1(-0.4)= 2 113.58o= 2 or 246.4o=2 = 56.8o or =123.2o <--- WRONG UNIT Page 20

  21.  /2, … , 3, … 0, 2 … 3 /2, … cos-1(-0.4) = 2 = 113.580 = 0.631 rad • e.g.5 Solve cos2 =  0.4 , where 0   2 Thus: 2 =  - 1.16;  + 1.16; 3 - 1.16;3 + 1.16, ….. = 1.98, 4.3, 8.26, 10.58, …… Thus:  = 0.99, 2.15, 4.13 or 5.29 rad A S 2 =113.580 1.16 rad 1.16 rad 1.16 rad 1.16 rad T C Page 21

  22. TRIGONOMTRIC EQUATIONS • Reference: Croft & Davision, Chapter 9, Blocks 5, 6, 7 • Some Common Trigonometric Identities A trigonometric identity is an equality which contains one or more trigonometric functions and is valid for all values of the angles involved. • e.g. (1) • (2) • (3) Page 22

  23. Exercise: Derive (2) and (3) from (1) Page 23

  24. 2x -1 x 1 2x2 + (x)(-1) + 2x + 1(-1) = 2x2 + x - 1 • (2x -1) (x +1) = 0 • x = ½ or x = -1 • e.g.1 (a) Solve • (b) Using (a), or otherwise, solve 2(1-sin2θ) - sinθ-1 = 0 2 - 2sin2θ- sinθ-1 = 0 2sin2θ+ sinθ-1 = 0 sinθ= 0.5 or sinθ= -1 θ= 30°, 150°, 270° = π/6, 5π/6, 3π/2 Page 24

  25. 4tanθ 1 tanθ -1 4tan2θ – 4tanθ+ tanθ - 1 = 4tan2θ – 3tanθ - 1 • e.g.2 Solve (4tanθ+ 1) (tanθ- 1) = 0 tanθ= 1 or tanθ= - 0.25 θ= 45°, 225°, 166° or 346° = π/4, 5π/4, 0.92π or 1.92π End of Block Exercise: p.336 Page 25

  26. Solving equations with given identities • e.g.3 Using the compound angle formula • find the acute angle  such that 1 45○ 1 30○ 2 60○ 1 Page 26

  27. e.g.4 Using the double-angle formula • solve sin2 = sin  , where 0º <360 º To make the answer to be 0, either sinθ=0 or 2cosθ-1=0 cosθ=0.5 θ= 60° or 300° θ= 0° or 180° Page 27

  28. 2sinθ -1 sinθ 1 2sin2θ – sinθ+ 2sinθ - 1 = 2sin2θ + sinθ - 1 • e.g.5 Using the double-angle formula • solve cos2 = sin  , where 0   2. sinθ= 0.5 or sinθ= -1 θ= 30°, 150° or 270° = π/6, 5π/6 or 3 π/2 Page 28

  29. Engineering waves Reference: Croft &Davison , pp 348 Often voltages and currents vary with time and may be represented in the form • where A: Amplitude of the combined wave •  : Angular frequency (rad/sec) of the combined wave (Affect wave width) •  : Phase angle (left and right movement) t : time in second • Example • State (i) the amplitude and (ii) the angular frequency of the following waves: • (a) y = 2 sin 5t • (b) y = sin (t/2) • 2 ii) 5 phase angle=0 • i) 1 ii) ½ phase angle =0 Page 29

  30. Page 30

  31. The period, T, of both y = A sin ωt and y = A cos ω t is given by T = (2π)/ω • Example • State the period of each of the following functions: • 1. y = 3 sin 6t • 2. y = 5.6 cosπ t 2π/6 =π/3 2π/π = 2 • The frequency, f, of a wave is the number of cycles completed in 1 second. It is measured in hertz (Hz). • T = 1 / f • Example • State the period and frequency of the following waves: • 1. y = 3 sin 6 t • 2. y = 5.6cosπ t T=π/3, f = 3/π T=2, f = ½ Page 31

  32. ∵ max sin θ=1 and min sin θ=-1 ∴ max = 5 and min = -5 • E.g. 6 (a) Find the maximum and minimum value of • 5 sin ( t + 0.93 ) • (b) Solve 5 sin ( t + 0.93) = 3.8, where 0  t  2 What ? –ve rad ??? • End of Chapter Exercise: p.360 Page 32

  33. Formula for Reference(Given in the Exam) Page 33

  34. EXPONENTIAL AND LOGARITHMIC FUNCTIONS Reference: Croft & Davision, Chapter 8 p.253 http://www.math.utep.edu/sosmath The exponential function is where e = 2.71828182….. Properties y 30 25 20 15 10 5 1 x -3 0 -2 -1 1 2 3 Page 34

  35. e.g. 1 Simplify • Exercise: p.259 Page 35

  36. Applications : Laws of growth and decay (A)Growth curve e.g. Change of electrical resistance (R) with temp.  0 y A x Page 36

  37. y • (B) Decay Curve • e.g. Discharge of a capacitor • Exercise: p.259 A x 0 Page 37

  38. Class Exercise • Q=50, C=0.25 and R=2 • a) When t=1, q(t) = ? • b) When R is double, q(1) = ? q(1) = 6.77 q(1) = 18.39 Page 38

  39. Logarithmic Functions • The number a is called the base of the logarithm. • In particular, • Exercise: p.271 Page 39

  40. Properties of • Exercise: p.275 y 1 1 2 3 4 5 x 0 -1 -2 -3 Page 40

  41. Solving equations • e.g.2 Solve Page 41

  42. e.g.3 The decay of current in an inductive circuit is given by • Find (a) the current when t=0; • (b) the value of the current when t=3; • (c) the time when the value of the current is 15. Page 42

More Related