ODE Background: Differential (1A) Young Won Lim 12/29/15

Transcription

1 ODE Background: Differential (1A

2 Copyright (c Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.

3 Differentials Background : Differential (1A 3

4 A triangle and its slope y = f x, f f h f h f ( +h f ( +h h, f h f Background : Differential (1A 4

5 Many smaller triangles and their slopes f ( + h f ( h ( +h, f ( +h f ( + h 1 f ( h 1 f ( + h 2 f ( h 2 f ( +h f ( +h 1 f ( +h 2 f ( (, f ( +h 2 +h 1 +h h 2 h 1 lim h 0 f ( + h f ( h h ( +h, f ( +h (, f ( Background : Differential (1A 5

6 The limit of triangles and their slopes y = f x The derivative of the function f at The derivative function of the function f f ' ( = lim h 0 f ( + h f ( h f '(x = lim h 0 f (x + h f (x h y ' = f ' (x = df dx = d dx f (x Background : Differential (1A 6

7 The derivative as a function f x y = f x x 2 x 3 f f x 2 f x 3 Derivative Function y ' = f ' (x f ' x f ' = lim h 0 f (x + h f (x h x 2 x 3 f ' x 2 f ' x 3 Background : Differential (1A 7

8 The notations of derivative functions Largrange's Notation y ' = f ' x f ' (x f ' Leibniz's Notation x 2 f ' x 2 dy dx = d dx f (x x 3 f ' x 3 Newton's Notation ẏ = ḟ x not a ratio. slope of a tangent line lim h 0 f (x + h f (x h Euler's Notation D x y = D x f x derivative with respect to x x is an independent variable Background : Differential (1A 8

9 Another kind of triangles and their slope y = f x y ' = f ' x given x1 f the slope of a tangent line Background : Differential (1A 9

10 Differential in calculus Differential: dx, dy, infinitesimals a change in the linearization of a function of, or relating to differentiation function f dx the linearization of a function f dy f dx dx dx f Background : Differential (1A 10

11 Approximation Differential: dx, dy, f ( + dx f ( + dy = f ( + f '( dx f '( = lim h 0 f ( + h f ( h function the linearization of a function f dx f dy f dx dx dx f Background : Differential (1A 11

12 Differential as a function Line equation in the new coordinate. dy dy = f ' ( dx slope = f ' ( = dy 1 d f ( dy 1 dy 2 dy 3 d dx 2 dx 3 dx = dy 2 dx 2 = dy 3 dx 3 Background : Differential (1A 12

13 Differentials and Derivatives (1 dy = f ' (x dx dy dy = f ' ( dx dy = df dx dx slope = f ' ( = dy 1 d differentials derivative f ( dy 1 dy 2 dy 3 d dx 2 dx 3 dx = dy 2 dx 2 = dy 3 dx 3 dy dx = f ' (x ratio not a ratio Background : Differential (1A 13

14 Differentials and Derivatives (2 f ( + dx f ( + dy = f ( + f '( dx dy dy = f ' ( dx slope = f ' ( for small enough dx f ( + dx = f ( + dy lim dx 0 f ( dy 1 dy 2 dy 3 d dx 2 dx 3 dx = dy 1 d = dy 2 dx 2 = dy 3 dx 3 = f ( + f '( dx lim dx 0 f ( + dx f ( dx = f ' ( Background : Differential (1A 14

15 Differentials and Derivatives (3 dy = f ' (x dx dy = df dx dx dy = ḟ dx dy = D x f dx dy = f '(x dx dy = df dx dx y = f (x dy = 1dy = y Background : Differential (1A 15

16 Integration Constant C place a constant place another constant differs by a constant place only one constant from the beginning dy = f '(x dx dy = f '(x dx + C dy = df dx dx dy = df dx dx + C y + C 1 = f (x + C 2 y = f (x + C y = f (x + C Background : Differential (1A 16

17 Differential as a function The differential of a function ƒ(x of a single real variable x is the function of two independent real variables x and dx given by dy = f ' (x dx Line equation in the new coordinate. dy dy = f ' dx slope = f ' (x, dx dy f dx Background : Differential (1A 17

18 Applications of Differentials (1 Substitution Rule f ( g(x g'(x dx = f ( u du (I u = g( x du = g ' (xdx du = dg dx dx (II f (g dg dx dx = f ( g dg Background : Differential (1A 18

19 Applications of Differentials (2 Integration by parts f (xg ' (x dx = f (xg(x f ' (xg(x dx u = f (x v = g(x du = f ' (x dx dv = g'( xdx du = df dx dx dv = dg dx dx f (xg ' (x dx = f (xg(x f ' (xg(x dx u dv = u v v du Background : Differential (1A 19

20 Derivatives and Differentials (large dx dx = 0.6; m = f ' ( tangent at f ( + f ' ( dx f ' ( dx f (x 2 + f ' (x 2 dx f ( = sin( dx f (x = sin(x x 2 Background : Differential (1A 20

21 Euler Method of Approximation (large dx dx = 0.6; m = f ' ( tangent at f (x 2 f ( + f ' ( dx f (x 3 f (x 2 + f ' (x 2 dx (f (x 2 + f ' (x 2 dx + f '(x 2 dx f ( = sin( Euler Method dx f (x = sin(x Initial Value x 2 x 3 Background : Differential (1A 21

22 Derivatives and Differentials (large dx = 0.6 dx = 0.6; dy = f ' (x dx dy = df dx dx dy = f '(x dx dy = df dx dx y = f (x Background : Differential (1A 22

23 Derivatives and Differentials (small dx = 0.2 dx = 0.2; dy = f ' (x dx dy = df dx dx dy = f '(x dx dy = df dx dx y = f (x Background : Differential (1A 23

24 Euler's Method of Approximation Background : Differential (1A 24

25 Octave Code clf; hold off; dx = 0.2; x = 0 : dx : 8; y = sin(x; plot(x, y; t = sin(x + cos(x*dx ; y1 = [y(1, t(1:length(y-1]; y2 = [0]; y2(1 = y(1; for i=1:length(y-1 y2(i+1 = y2(i + cos((i*dx*dx; endfor hold on t = 0:0.01:8; plot(t, sin(t, "color", "blue"; plot(x, y, "color", 'blue', "marker", 'o'; plot(x, y1, "color", 'red', "marker", '+'; plot(x, y2, "color", 'green', "marker", '*'; Background : Differential (1A 25

26 Anti-derivatives Background : Differential (1A 26

27 Anti-derivative? differentiation derivative of? f (x? Anti-derivative of f(x Anti-differentiation f (x Background : Differential (1A 27

28 Anti-derivative and Indefinite Integral F '(x = f (x F(x Anti-derivative without constant the most simple anti-derivative F( x + C the most general anti-derivative f (xdx Indefinite Integral : a function of x f (xdx = F (x + C Background : Differential (1A 28

29 Anti-derivative Examples F 1 (x= 1 3 x3 differentiation All are Anti-derivative of f(x F 2 (x= 1 3 x Anti-differentiation f (x=x 2 F 3 (x= 1 3 x3 49 the most general anti-derivative of f(x 1 3 x3 + C indefinite Integral of f(x x 2 dx Background : Differential (1A 29

30 Indefinite Integrals a 1 dx x a 1 dx dx dy a x a x + C y + C given a variable x indefinite integral a d f d x dx x c d f d x dx d f d x dx f ( f (a f (x f (a f (x + C given a variable x indefinite integral Background : Differential (1A 30

31 Indefinite Integrals via the Definite Integral x a f (t dt definite integral x a f (t dt anti-derivative f (x indefinite integral f (xdx anti-derivative f (x f (x d x = F (x + C x a f (t d t = F (x F(a a common reference point : arbitrary Background : Differential (1A 31

32 Definite Integrals via the Definite Integral x a f (t dt definite integral x a f (t dt anti-derivative f (x indefinite integral f ( xdx anti-derivative f (x x 2 f (t dt x = 1 a x f (t dt + 2 a f (t dt a common reference point : arbitrary [ F (x + c ] x1 x 2 = F( x 2 F ( [ F (x ] x1 x 2 = F ( F (x 2 Anti-derivative without constant Background : Differential (1A 32

33 Indefinite Integral Examples x 0 f (x dx = [ 1 3 x3]0 x = 1 3 x3 f (x=x 2 x a f (x dx = [ 1 3 x3]a x = 1 3 x3 1 3 a3 x a f (t dt = [ 1 3 t3]a x = 1 3 x3 1 3 a3 anti-derivative by the definite integral of f(x x a t 2 dt = 1 3 x3 1 3 a2 d x dx a f (t dt = f (x = x 2 indefinite integral of f(x x 2 dx = 1 3 x3 + C Background : Differential (1A 33

34 Definite Integrals on [a, ] 1 dx f ' (x dx f '(x = 1 a a view (I 1 dx a a g(x dx g(x = 1 view (II view (I view (II f ' (x dx [ f (x ] a = f ( f (a a g(x dx [G(x] a = G( G(a a Background : Differential (1A 34

35 Definite Integrals over an interval [, x 2 ] view (I view (II length G( x = x 1 area f '( x = 1 x g(x = 1 a x 2 a x 2 arbitrary reference point (a, f(a arbitrary reference point (a, G(a x 2 x 2 f ' (x dx = g(x dx = [ f (x ] x1 x 2 = f (x 2 f ( [G(x] x1 x 2 = G(x 2 G( Background : Differential (1A 35

36 Definite Integrals on [a, ] and [a, x 2 ] f ' (x G(x area G(x 2 G( length a x 2 a x 2 G(x area G(x 2 G( length a x 2 a x 2 c x 2 f '(xdx c f ' ( xdx c x 2 g(xdx c g(xdx Background : Differential (1A 36

37 Derivative Function and Indefinite Integrals f ' ( lim h 0 f ( + h f ( h x 2 f ( x dx f ' (x 2 lim h 0 f (x 2 + h f (x 2 h x 4 x 3 f ( x dx f ' (x 3 lim h 0 f (x 3 + h f (x 3 h x 6 x 5 f ( x dx, x 2, x 3 [, x 2 ],[ x 3, x 4 ], [x 5, x 6 ] f ' (x = lim h 0 f ( x + h f (x h x F (x + C = a f ( x dx f ' (, f ' (x 2, f '( x 3 function of x [ F( x ] x 2, [ F( x ]x 3 x 4 x, [ F (x 6 ]x5 function of x Background : Differential (1A 37

38 a Background : Differential (1A 38

39 Differential Equation f (x = e 3 x f (x? f ' (x = 3e 3 x f ' (x 3 f (x = 3 e 3x 3 e 3 x = 0 f ' (x 3 f (x = 0 Background : Differential (1A 39

40 First Order Examples (y=f(x f ' (x = f (x y ' = y f (x? y? An Example of A First Order Differential Equation f (x = c e x for all x y = c e x I : (, + f ' (x = f (x y ' = y f (x? y? f (0 = 3 f (0 = 3 An Example of A First Order Initial Value Problem f (x = 3e x for all x y = 3e x I : (, + Background : Differential (1A 40

41 Second Order Examples (y=f(x f ' ' (x = f (x y ' ' = y f (x = y = An Example of A Second Order Differential Equation c 1 e +x + c 2 e x for all x c 1 e +x + c 2 e x I : (, + f ' ' (x = f (x y ' ' = y f (x = y = f ' (0 = 0 y ' (0 = 0 f (0 = 1 f (0 = 1 +1 e +x 1 e x +1 e +x 1 e x An Example of A Second Order Initial Value Problem for all x I : (, + Guess the possible solution. Background : Differential (1A 41

42 General First & Second Order IVPs (y=f(x First Order Initial Value Problem f ' (x = f (x y ' = y f (0 = 3 f (0 = 3 d y = g(x, y y ' = g(x, y d x y(x 0 = y 0 y(x 0 = y 0 Second Order Initial Value Problem f ' ' (x = f (x y ' ' = y f ' (0 = 0 y ' (0 = 0 f (0 = 1 f (0 = 1 d 2 y d x 2 = g(x, y, y ' y(x 0 = y 0 y ' (x 0 = y 1 y ' ' = g(x, y, y ' y(x 0 = y 0 y ' (x 0 = y 1 Guess the possible solution. Background : Differential (1A 42

43 References [1] [2] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics