Introduction to ODE's (0A) Young Won Lim 3/9/15
|
|
- Alison Chandler
- 6 years ago
- Views:
Transcription
1 Introduction to ODE's (0A)
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 A triangle and its slope y = f x, f f h f h f ( +h) f ( ) +h h, f h f Intro to ODEs (0A) 3
4 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 ( )) Intro to ODEs (0A) 4
5 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) Intro to ODEs (0A) 5
6 The derivative as a function f x y = f x x 3 f f f x 3 Derivative Function y ' = f ' (x) f ' x f ' = lim h 0 f (x + h) f (x) h x 3 f ' f ' x 3 Intro to ODEs (0A) 6
7 The notations of derivative functions Largrange's Notation y ' = f ' x f ' (x) f ' Leibniz's Notation f ' dy dx = d dx f (x) x 3 f ' x 3 not a ratio. Newton's Notation ẏ = ḟ x 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 Intro to ODEs (0A) 7
8 Another kind of triangles and their slopes y = f x y ' = f ' x given x1 f the slope of a tangent line Intro to ODEs (0A) 8
9 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 Intro to ODEs (0A) 9
10 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 Intro to ODEs (0A) 10
11 Differential as a function Line equation in the new coordinate. dy dy = f ' ( ) dx slope = f ' ( ) f ( ) dx Intro to ODEs (0A) 11
12 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 Intro to ODEs (0A) 12
13 Differentials and Derivatives (1) differentials ratio dy = f ' (x) dx dy = df dx dx derivative dy dx = f ' (x) not a ratio f ( + dx) f ( ) + dy for small enough dx = f ( ) + f '( )dx f ( + dx) = f ( ) + dy f ( + dx) f ( ) dx = f ( ) + f '( )dx = f '( ) Intro to ODEs (0A) 13
14 Differentials and Derivatives (2) dy = f ' (x) dx dy = df dx dx dy = ḟ dx dy = f '(x) dx dy = df dx dx dy = 1dy = y dy = D x f dx Intro to ODEs (0A) 14
15 Differentials and Derivatives (3) dy = f '(x) dx dy = f '(x) dx + C dy = df dx dx dy = df dx dx + C a constant is included another constant is included differ by a constant y + C 1 = f (x) + C 2 y = f (x) + C y = f (x) + C Intro to ODEs (0A) 15
16 Applications of Differentials (1) Substitution Rule f ( g(x) ) g'(x) dx = f ( u ) du (I) u = g( x) du = g ' (x)dx du = dg dx dx (II) f (g) dg dx dx = f ( g ) dg Intro to ODEs (0A) 16
17 Applications of Differentials (2) Integration by parts f (x)g ' (x) dx = f (x)g(x) f ' (x)g(x) dx u = f (x) v = g(x) du = f ' (x) dx dv = g'( x)dx du = df dx dx dv = dg dx dx f (x)g ' (x) dx = f (x)g(x) f ' (x)g(x) dx u dv = u v v du Intro to ODEs (0A) 17
18 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 (x)dx Indefinite Integral : a function of x f (x)dx = F (x) + C Intro to ODEs (0A) 18
19 Anti-derivative Examples (1)? Anti-derivative of f(x) differentiation Anti-differentiation derivative of f (x)? F 1 (x)= 1 3 x3 f (x)= All are Anti-derivative of f(x) F 2 (x)= 1 3 x F 3 (x)= 1 3 x3 49 the most general anti-derivative of f(x) 1 3 x3 + C dx indefinite Integral of f(x) Intro to ODEs (0A) 19
20 Anti-derivative Examples (2)? Anti-derivative of f(x) differentiation Anti-differentiation derivative of? f (x) x 0 f (x) dx = [ 1 3 x3]0 x = 1 3 x3 f (x)= 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 + C d x dx a f (t ) dt = f (x) = the Indefinite Integral of f(x) dx = 1 3 x3 + C Intro to ODEs (0A) 20
21 Definite Integrals on [a, ] 1 y = 1 1 a a a f ' (x) = 1 f (x) = 1 a 1 dx a 1 dx [ x ] a = a dx dy = dy dx dx view (I) view (II) a a f ' (x) dx [ f (x) ] a = f ( ) f (a) f (x) dx [ F (x)] a = F ( ) F (a) Intro to ODEs (0A) 21
22 Definite Integrals on [, ] view (I) view (II) f (x) F (x) : anti-derivative of f '(x) : anti-derivative of f ( x) 1 dx Anti-derivative x 1 y = 1 area Anti-derivative without constant a [ x + c ] x1 = = [ x ] x1 length view (I) [ f (x) ] x1 = f ( ) f ( ) x view (II) [ F (x)] x1 = F ( ) F ( ) a Intro to ODEs (0A) 22
23 Definite Integrals on [, ] through F(x)+C view (I) Anti-derivative view (II) Anti-derivative 1 dx f (x) = x 1 dx F (x) = x f '(x) f (x) = [ f (x)] x1 = [ F (x)] x1 = {f ( ) f (a)} {f ( ) f (a)} = {F ( ) F(a)} {F ( ) F(a)} = [ f (x) f (a)] x1 = [ F (x) F (a) ] x1 = [ f (x) + C ] x1 arbitrary reference point (a, f(a)) = [ F (x) + C ] x1 arbitrary reference point (a, F(a)) = [ f ' (x)dx ] x1 = [ f (x)dx ] x1 Intro to ODEs (0A) 23
24 Indefinite Integrals through Definite Integrals view (I) view (II) 1 dx x a 1 dx 1 dx x a 1 dx f ' (x) dx a f ' (x) dx f (x) dx a f (x) dx = f (x) f (a) = x a = f (x) + C = F(x) + C = F(x) F (a) = x a a a f (x) = x a F (x) = x a a a a a Intro to ODEs (0A) 24
25 Definite Integrals on [a, ] and [a, ] (I) a f '(x) dx (II) a f (x) dx a (I) a f '(x) dx (II) a f (x) dx a Intro to ODEs (0A) 25
26 Definite Integrals on [, ] through F(x) d f d x dx Anti-derivative F (x) = f (x) y = f ' (x) area a f ( ) f ( ) = [ F (x)] x1 F (x) = f (x) F ( ) F ( ) length a Intro to ODEs (0A) 26
27 Definite Integrals on [, ] through F(x)+C d f d x dx F (x) = f (x) y = f ' (x) a common reference point a f ( ) f ( ) = [ F (x)] x1 = {F ( ) F(a)} {F ( ) F(a)} = [ F (x) F (a)] x1 = [ F (x) C ] x1 = [ f (x) f (a)] x1 arbitrary reference point (a, F(a)) Intro to ODEs (0A) 27
28 Indefinite Integrals through F(x)+C d f d x dx = F ( x) + C x a d f d x dx y = f ' (x) a common reference point a F (x) F (a) = F (x) f (a) f ( ) f (a) y = f (x) f (a) a Intro to ODEs (0A) 28
29 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 Intro to ODEs (0A) 29
30 Derivative Function and Indefinite Integrals f ' ( ) lim h 0 f ( + h) f ( ) h f ( x) dx f ' ( ) lim h 0 f ( + h) f ( ) 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 3 [, ],[ 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 ' ( ), f '( x 3 ) [ F( x) ], [ F( x) ]x 3 x 4 x, [ F (x) 6 ]x5 Intro to ODEs (0A) 30
31 a Intro to ODEs (0A) 31
32 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 Intro to ODEs (0A) 32
33 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 : (, + ) Intro to ODEs (0A) 33
34 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. Intro to ODEs (0A) 34
35 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 = 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. Intro to ODEs (0A) 35
36 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
ODE Background: Differential (1A) Young Won Lim 12/29/15
ODE Background: Differential (1A Copyright (c 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationSeparable Equations (1A) Young Won Lim 3/24/15
Separable Equations (1A) Copyright (c) 2011-2015 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
More informationIntroduction to ODE's (0A) Young Won Lim 3/12/15
Introduction to ODE's (0A) Copyright (c) 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or
More informationHigher Order ODE's (3A) Young Won Lim 7/7/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 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
More informationHigher Order ODE's (3A) Young Won Lim 12/27/15
Higher Order ODE's (3A) Copyright (c) 2011-2015 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
More informationHigher Order ODE's (3A) Young Won Lim 7/8/14
Higher Order ODE's (3A) Copyright (c) 2011-2014 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
More informationHigher Order ODE's, (3A)
Higher Order ODE's, (3A) Initial Value Problems, and Boundary Value Problems Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms
More informationIntroduction to ODE's (0P) Young Won Lim 12/27/14
Introuction to ODE's (0P) Copyright (c) 2011-2014 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationSecond Order ODE's (2A) Young Won Lim 5/5/15
Second Order ODE's (2A) Copyright (c) 2011-2015 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
More informationBackground Trigonmetry (2A) Young Won Lim 5/5/15
Background Trigonmetry (A) Copyright (c) 014 015 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. or
More informationDifferentiation Rules (2A) Young Won Lim 1/30/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationLinear Equations with Constant Coefficients (2A) Young Won Lim 4/13/15
Linear Equations with Constant Coefficients (2A) Copyright (c) 2011-2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 1/31/15
Definitions of the Laplace Transform (A) Copyright (c) 24 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationDefinite Integrals. Young Won Lim 6/25/15
Definite Integrls Copyright (c 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version
More informationGeneral Vector Space (2A) Young Won Lim 11/4/12
General (2A Copyright (c 2012 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
More informationMatrix Transformation (2A) Young Won Lim 11/9/12
Matrix (A Copyright (c 01 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. or any later version published
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 10/22/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationDifferentiation Rules (2A) Young Won Lim 2/22/16
Differentiation Rules (2A) Copyright (c) 2011-2016 Young W. Lim. Permission is grante to copy, istribute an/or moify this ocument uner the terms of the GNU Free Documentation License, Version 1.2 or any
More informationRelations (3A) Young Won Lim 3/27/18
Relations (3A) Copyright (c) 2015 2018 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
More informationSurface Integrals (6A)
Surface Integrals (6A) Surface Integral Stokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation
More informationLine Integrals (4A) Line Integral Path Independence. Young Won Lim 11/2/12
Line Integrals (4A Line Integral Path Independence Copyright (c 2012 Young W. Lim. Permission is granted to copy, distriute and/or modify this document under the terms of the GNU Free Documentation License,
More informationExpected Value (10D) Young Won Lim 6/12/17
Expected Value (10D) Copyright (c) 2017 Young W. Lim. Permissios granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationComplex Series (3A) Young Won Lim 8/17/13
Complex Series (3A) 8/7/3 Copyright (c) 202, 203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or
More informationCT Rectangular Function Pairs (5B)
C Rectangular Function Pairs (5B) Continuous ime Rect Function Pairs Copyright (c) 009-013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU
More informationDFT Frequency (9A) Each Row of the DFT Matrix. Young Won Lim 7/31/10
DFT Frequency (9A) Each ow of the DFT Matrix Copyright (c) 2009, 2010 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GU Free Documentation License,
More informationMatrix Transformation (2A) Young Won Lim 11/10/12
Matrix (A Copyright (c 0 Young W. im. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation icense, Version. or any later version published
More informationBackground ODEs (2A) Young Won Lim 3/7/15
Background ODEs (2A) Copyright (c) 2014-2015 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
More informationGeneral CORDIC Description (1A)
General CORDIC Description (1A) Copyright (c) 2010, 2011, 2012 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationComplex Functions (1A) Young Won Lim 2/22/14
Complex Functions (1A) Copyright (c) 2011-2014 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
More informationIntegrals. Young Won Lim 12/29/15
Integrls Copyright (c) 2011-2015 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version 1.2 or ny lter version published
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationSurface Integrals (6A)
urface Integrals (6A) urface Integral tokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License,
More informationRoot Locus (2A) Young Won Lim 10/15/14
Root Locus (2A Copyright (c 2014 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
More informationFourier Analysis Overview (0A)
CTFS: Fourier Series CTFT: Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2011-2016 Young W. Lim. Permission is granted to copy, distribute
More informationCapacitor Young Won Lim 06/22/2017
Capacitor Copyright (c) 2011 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
More informationBackground Complex Analysis (1A) Young Won Lim 9/2/14
Background Complex Analsis (1A) Copright (c) 2014 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License, Version 1.2 or
More informationThe Growth of Functions (2A) Young Won Lim 4/6/18
Copyright (c) 2015-2018 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
More informationDetect Sensor (6B) Eddy Current Sensor. Young Won Lim 11/19/09
Detect Sensor (6B) Eddy Current Sensor Copyright (c) 2009 Young W. Lim. Permission is granteo copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 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
More informationComplex Trigonometric and Hyperbolic Functions (7A)
Complex Trigonometric and Hyperbolic Functions (7A) 07/08/015 Copyright (c) 011-015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationPhasor Young Won Lim 05/19/2015
Phasor Copyright (c) 2009-2015 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
More informationSignal Functions (0B)
Signal Functions (0B) Signal Functions Copyright (c) 2009-203 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationFourier Analysis Overview (0B)
CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission
More informationBackground LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 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
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 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
More informationCapacitor and Inductor
Capacitor and Inductor Copyright (c) 2015 2017 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
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-213 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationCLTI System Response (4A) Young Won Lim 4/11/15
CLTI System Response (4A) Copyright (c) 2011-2015 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
More informationMagnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect. Young Won Lim 9/23/09
Magnetic Sensor (3B) Magnetism Hall Effect AMR Effect GMR Effect Copyright (c) 2009 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationDown-Sampling (4B) Young Won Lim 10/25/12
Down-Sampling (4B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More informationAudio Signal Generation. Young Won Lim 1/29/18
Generation Copyright (c) 2016-2018 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
More informationDiscrete Time Rect Function(4B)
Discrete Time Rect Function(4B) Discrete Time Rect Functions Copyright (c) 29-23 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationHamiltonian Cycle (3A) Young Won Lim 5/5/18
Hamiltonian Cycle (3A) Copyright (c) 015 018 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. or any
More informationDispersion (3A) 1-D Dispersion. Young W. Lim 10/15/13
1-D Dispersion Copyright (c) 013. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1. or any later version published
More informationBayes Theorem (10B) Young Won Lim 6/3/17
Bayes Theorem (10B) Copyright (c) 2017 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
More informationDefinitions of the Laplace Transform (1A) Young Won Lim 2/9/15
Definition of the aplace Tranform (A) 2/9/5 Copyright (c) 24 Young W. im. Permiion i granted to copy, ditriute and/or modify thi document under the term of the GNU Free Documentation icene, Verion.2 or
More informationDown-Sampling (4B) Young Won Lim 11/15/12
Down-Sampling (B) /5/ Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later
More informationGroup & Phase Velocities (2A)
(2A) 1-D Copyright (c) 2011 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
More informationUp-Sampling (5B) Young Won Lim 11/15/12
Up-Sampling (5B) Copyright (c) 9,, Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
More informationSection 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point
Section 2.1, Section 3.1 Rate of change, Tangents and Derivatives at a point Line through P and Q approaches to the tangent line at P as Q approaches P. That is as a + h a = h gets smaller. Slope of the
More informationPropagating Wave (1B)
Wave (1B) 3-D Wave Copyright (c) 2011 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
More informationSignals and Spectra (1A) Young Won Lim 11/26/12
Signals and Spectra (A) Copyright (c) 202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationDouble Integrals (5A)
Doule Integrls (5A) Doule Integrl Doule Integrls in Polr oordintes Green's Theorem opyright (c) 2012 Young W. Lim. Permission is grnted to copy, distriute nd/or modify this document under the terms of
More informationChapter 2: Differentiation
Chapter 2: Differentiation Spring 2018 Department of Mathematics Hong Kong Baptist University 1 / 82 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationBilateral Laplace Transform (6A) Young Won Lim 2/16/15
Bilterl Lplce Trnsform (6A) 2/6/5 Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny
More informationFSM Examples. Young Won Lim 11/6/15
/6/5 Copyright (c) 2 25 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version published
More informationCLTI Differential Equations (3A) Young Won Lim 6/4/15
CLTI Differential Equations (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationBilateral Laplace Transform (6A) Young Won Lim 2/23/15
Bilterl Lplce Trnsform (6A) Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny lter
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationHilbert Inner Product Space (2B) Young Won Lim 2/7/12
Hilbert nner Product Space (2B) Copyright (c) 2009-2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationImplication (6A) Young Won Lim 3/17/18
Implication (6A) 3/17/18 Copyright (c) 2015 2018 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
More informationAudio Signal Generation. Young Won Lim 2/2/18
Generation Copyright (c) 2016-2018 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
More information2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems
2.12: Derivatives of Exp/Log (cont d) and 2.15: Antiderivatives and Initial Value Problems Mathematics 3 Lecture 14 Dartmouth College February 03, 2010 Derivatives of the Exponential and Logarithmic Functions
More informationBayes Theorem (4A) Young Won Lim 3/5/18
Bayes Theorem (4A) Copyright (c) 2017-2018 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
More informationPropositional Logic Resolution (6A) Young W. Lim 12/12/16
Propositional Logic Resolution (6A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More informationSequential Circuit Timing. Young Won Lim 11/6/15
Copyright (c) 2011 2015 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
More informationDIFFERENTIATION RULES
3 DIFFERENTIATION RULES DIFFERENTIATION RULES 3. The Product and Quotient Rules In this section, we will learn about: Formulas that enable us to differentiate new functions formed from old functions by
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 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
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 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
More informationGroup Delay and Phase Delay (1A) Young Won Lim 7/19/12
Group Delay and Phase Delay (A) 7/9/2 Copyright (c) 2 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationGeneral Vector Space (3A) Young Won Lim 11/19/12
General (3A) /9/2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 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
More informationMath 104: l Hospital s rule, Differential Equations and Integration
Math 104: l Hospital s rule, and Integration Ryan Blair University of Pennsylvania Tuesday January 22, 2013 Math 104:l Hospital s rule, andtuesday Integration January 22, 2013 1 / 8 Outline 1 l Hospital
More informationPropositional Logic Logical Implication (4A) Young W. Lim 4/11/17
Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationChapter 2: Differentiation
Chapter 2: Differentiation Winter 2016 Department of Mathematics Hong Kong Baptist University 1 / 75 2.1 Tangent Lines and Their Slopes This section deals with the problem of finding a straight line L
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationReview for the Final Exam
Math 171 Review for the Final Exam 1 Find the limits (4 points each) (a) lim 4x 2 3; x x (b) lim ( x 2 x x 1 )x ; (c) lim( 1 1 ); x 1 ln x x 1 sin (x 2) (d) lim x 2 x 2 4 Solutions (a) The limit lim 4x
More informationPropositional Logic Resolution (6A) Young W. Lim 12/31/16
Propositional Logic Resolution (6A) Young W. Lim Copyright (c) 2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License,
More information= π + sin π = π + 0 = π, so the object is moving at a speed of π feet per second after π seconds. (c) How far does it go in π seconds?
Mathematics 115 Professor Alan H. Stein April 18, 005 SOLUTIONS 1. Define what is meant by an antiderivative or indefinite integral of a function f(x). Solution: An antiderivative or indefinite integral
More informationAudio Signal Generation. Young Won Lim 2/2/18
Generation Copyright (c) 2016-2018 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
More informationDifferential calculus. Background mathematics review
Differential calculus Background mathematics review David Miller Differential calculus First derivative Background mathematics review David Miller First derivative For some function y The (first) derivative
More informationCapacitors in an AC circuit
Capacitors in an AC circuit Copyright (c) 2011 2017 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
More informationGroup Velocity and Phase Velocity (1A) Young Won Lim 5/26/12
Group Velocity and Phase Velocity (1A) Copyright (c) 211 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationMath 3B: Lecture 11. Noah White. October 25, 2017
Math 3B: Lecture 11 Noah White October 25, 2017 Introduction Midterm 1 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Introduction Midterm 1 Average is 73%. This is
More informationHyperbolic Functions (1A)
Hyperbolic Functions (A) 08/3/04 Copyright (c) 0-04 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or
More informationCapacitor in an AC circuit
Capacitor in an AC circuit Copyright (c) 2011 2017 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
More informationDigital Signal Octave Codes (0A)
Digital Signal Periodic Conditions Copyright (c) 2009-207 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2
More informationBandpass Sampling (2B) Young Won Lim 3/27/12
andpass Sapling (2) Copyright (c) 2009-2012 Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later
More informationPropositional Logic Logical Implication (4A) Young W. Lim 4/21/17
Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More information