# Math 211A Homework. Edward Burkard. = tan (2x + z)

Size: px
Start display at page:

Transcription

1 Mth A Homework Ewr Burkr Eercises 5-C Eercise 8 Show tht the utonomous system: 5 Plne Autonomous Systems = e sin 3y + sin cos + e z, y = sin ( + 3y, z = tn ( + z hs n unstble criticl point t = y = z = 0 Solution First let s verify tht (0, 0, 0 is inee criticl point of our system: (0, 0, 0 = e0 sin 0 + sin 0 cos 0 + e 0 = = 0, y (0, 0, 0 = sin (0 + 0 = 0, z (0, 0, 0 = tn (0 + 0 = 0 so (0, 0, 0 is inee criticl point of the system To voi pin n heches, write our system s = F ( To check the if this is n unstble criticl point, we merely nee to look t the liner pproimtion of our system ner (0, 0, 0 Ner (0, 0, 0 the liner pproimtion is: First we compute DF : DF = n t (0, 0, 0 the mtri is: so our liner pproimtion is = F (0, 0, 0 + DF (0,0,0 ( (0, 0, 0 = DF (0,0,0 e sin 3y + cos 3 e cos 3y e z cos ( + 3y 3 cos ( + 3y 0 sec ( + z 0 sec ( + z 3 DF (0,0,0 = = 3 0 = A 0 To check if (0, 0, 0 is n unstble criticl point, we merely nee to know if t lest one of the eigenvlues of A hs positive rel prt So let s fin them (I will spre you the gruesome clcultions: First, the Chrcteristic Polynomil of A: p A (λ = λ 3 6λ + 6λ + 3

2 We merely wnt to know if there is n eigenvlue with positive rel prt In fct, there is, n though I will not ctully compute it, I will prove its eistence Notice tht the chrcteristic polynomil is continuous function from R to R Notice tht p A ( = = 4 > 0 n p A (3 = = 6 < 0, so by the intermeite vlue theorem, p A must hve zero in the intervl (, 3 This is sufficient to sy tht A hs positive rel eigenvlue (in fct, ll three re rel: one is negtive n the other two positive, n hence tht (0, 0, 0 is n unstble criticl point of our system 6 Eercises 6-A 6 Eistence n Uniqueness Theorems Eercise 8 Show tht the curves efine prmetriclly s solutions of the system: y = F = F y (6 z = F z re orthogonl to the surfces F (, y, z = c, c R Wht ifferentibility conition on F must be ssume to mke this system stisfy Lipschitz conition? Solution Let g(t be solution to (6 Then it s irection vector t ny time t is g (t Notice tht g (t = F for ll t Recll tht the grient of F is lwys perpeniculr to ny of its level sets, hence g (t is perpeniculr to F (, y, z = c for ny c R Thus ny solution to (6 is perpeniculr to the level sets of F In orer to stisfy Lipschitz conition in compct, conve region R, we nee F C (R, since then the system (6 is in C (R n hence we cn pply the Lemm in Section 6 6 Aitionl Eercises Eercise 8 Show tht, if X(t = ij (t is mtri whose columns re solutions of the homogeneous liner system X = A(tX, then ( t et X(t = et X(] ep kk (s Proof Assume tht both X n A re n n mtrices Borrowing the nottion bove, let A(t = ij (t Viewing X (t = A(tX(t, then the entries of X (t hve the form n ij(t = ik (t kj (t (6 Spring the pin of the computtion, let s tke the erivtive of the et X(t: n n n n n n et X(t = n n nn n n nn n n nn Focusing on the first eterminnt for moment, by eqution (6, we hve: k k k k k kn n n n nn k=

3 3 Notice tht if we subtrct times the secon row from the first row we lose the secon term in ech of the sums in the first row (recll tht ing multiples of one row of eterminnt to nother row oes not chnge the vlue of the eterminnt Similrly if we subtrct 3 times the thir row from the first row, we lose the thir term in ech of the sums in the first row, oing this for ech row leves us with only the k = term in ech sum, ie n n n n nn Now this is precisely et X(t Doing similr thing for ech of the other eterminnts in the sum for et X(t, we see tht: n et X(t = kk (t et X(t Let et X(t = y(t, n k= n kk (t = η(t; then the bove eqution reuces to: k= This is seprble eqution with solution: y (t = η(ty(t ( t y(t = y( ep η(s for some ( woul be etermine by n initil vlue Replcing y n η with their originl menings, we get the esire result: ( t n et X(t = et X(] ep kk (s k= Remrk My resoning for this problem ws bsiclly try to pt proof of the similr formul for seprble eqution (which I use for the eqution in y The proof of the formul use bove for the erivtive of the eterminnt of X(t is s follows: We cn write et X(t = δ(j,, j n j (t j (t njn (t, (j,,j n S n where S n is the permuttion group on n letters (the specific letters re {,, n} n δ(j,, j n is the prity of the permuttion Now tking the erivtive et X(t = δ(j,, j n j (t njn (t (j,,j n S n n = δ(j,, j n j (t k,jk (t kj k (t k+,jk+ (t njn (t (j,,j n S n k= n = δ(j,, j n j (t k,jk (t kj k (t k+,jk+ (t njn (t (j,,j n S n k= n = δ(j,, j n j (t k,jk (t kj k (t k+,jk+ (t njn (t k= (j,,j n S n which is precisely the foruml use bove

4 4 0 Eercises 0-D 0 Sturm-Liouville Systems Eercise 7 Derive Theorem 3 from the Sturm Comprison Theorem of Ch by introucing the new epenent vribles t = P (s n t = P (s Theorem 3: Theorem Let P ( P ( > 0 n Q ( Q( in the DEs: ( P ( u + Q(u = 0, ( P ( u + Q (u = 0 Then, between ny two zeros of nontrivil solution u( of the first DE, there lies t lest one zero of every rel solution of the secon DE, ecept when u( cu ( This implies P P n Q Q, ecept possibly in intervls where Q Q 0 The Sturm Comprison Theorem of Ch : Theorem Let f( n g( be nontrivil solutions of the DEs u + p( = 0 n v + q(v = 0, respectively, where p( q( Then f( vnishes t lest once between ny two zeros of g(, unless p( q( n f is constnt multiple of g Proof Recll Given functions g n h where g is invertible, the eqution f g = h cn be solve for f Let P, P, Q, n Q be efine s in Theorem 3 Since t = P (s n t = re both incresing P (s functions of (becuse ( P n P re positive functions they re invertible, n hence we my fin functions q(t ( n q (t such tht q = P (Q( n q = P (Q ( Notice tht, by efinition, since P (s P (s P P n Q Q, it follows tht q q Consier the ifferentil equtions given by: v + q(tv = 0 (0 n v + q (tv = 0 (0 Let v(t n v (t be nontrivil solutions to (0 n (0 respectively, n efine the new functions u( n u ( by ( u( = v P (s n u ( = v ( P (s

5 Let s see wht hppens when we mke the substitution t = cttywompus (ie chin rule glore : u = u = = P ( = P ( = u = u P ( P ( u ] = P u + P ( P (s ] u = P u + P u ( = P P (u + P (] u ] P u = P ( + P u ( = P ( P ( u ] Thus substituting t s bove into (0, we en up with the eqution: ( ] ( ( v + q v = u P (s P (s P (s which simplifies to since P ( > 0 Thus we similrly hve = P ( in (0 Now let s compute bunch of ( + P (Q(u( P ( u ] + P (Q(u( = 0 P ( u ] + Q(u( = 0 (03 P ( u ] + Q (u ( = 0 (04 when plugging t = into (0 P (s Recll tht q (t q(t Using the Sturm Comprison Theorem of Chpter on (0 n (0 we hve the result tht between ny two zeros of v(t there is t lest one zero of v (t, unless q(t q (t, in which cse v(t cv (t for some c R (c 0 Assuming tht v(t cv (t, then since to ny nontrivil solutions v n v of (0 n (0 respectively, there is unique corresponing u n u s bove, it follows tht between ny two zeros of u( there is t lest one zero of u ( Now if v(t cv (t, then q(t q (t n it follows tht u( cu ( Assume we re on n intervl such tht Q Q 0 Then since q(t q (t we hve tht P (Q( = P (Q ( = P (Q(, n since we re on n intervl where Q 0 we cn ivie both sies by Q to ttin: P ( P (, 5 on tht intervl On the other hn, if Q Q 0 on some intervl, then it is not necessry tht P P Eercise 8 For ny solution of u +q(u = 0, q( < 0, show tht the prouct u(u ( is n incresing function Infer tht nontrivil solution cn hve t most one zero

6 6 Proof Let u( be nontrivil solution of the DE bove Since u is solution of secon orer ifferentil eqution it is t lest twice ifferentible This implies tht the prouct u(u ( is ifferentible (n hence continuous Now let s look t its erivtive: u(u (] = u (u ( + u(u ( = u (] u(q(u( = u (] q(u(] 0 Hence u(u ( is n incresing function Now ssume tht u is nontrivil solution n tht it hs two zeros, sy n b ( < b, thus the function u(u ( hs zeros t n b This mens tht, since u(u ( is continuous, u(u ( is ecresing on some subintervl of, b] or u(u ( 0 But if u(u ( is ecresing, this implies tht u(u (] is negtive somewhere, which contricts the bove clcultion Hence it must be tht u(u ( 0 But if this is true, then since u( 0, it must be tht u ( 0 But then u( constnt, however, the only constnt solution to this ifferentil eqution is the trivil one Therefore it is not possible tht u(u ( 0, n hence it must be tht u( oes not hve more thn one zero

### Chapter Five - Eigenvalues, Eigenfunctions, and All That

Chpter Five - Eigenvlues, Eigenfunctions, n All Tht The prtil ifferentil eqution methos escrie in the previous chpter is specil cse of more generl setting in which we hve n eqution of the form L 1 xux,tl

### 5.4, 6.1, 6.2 Handout. As we ve discussed, the integral is in some way the opposite of taking a derivative. The exact relationship

5.4, 6.1, 6.2 Hnout As we ve iscusse, the integrl is in some wy the opposite of tking erivtive. The exct reltionship is given by the Funmentl Theorem of Clculus: The Funmentl Theorem of Clculus: If f is

### Basic Derivative Properties

Bsic Derivtive Properties Let s strt this section by remining ourselves tht the erivtive is the slope of function Wht is the slope of constnt function? c FACT 2 Let f () =c, where c is constnt Then f 0

### Notes on the Eigenfunction Method for solving differential equations

Notes on the Eigenfunction Metho for solving ifferentil equtions Reminer: Wereconsieringtheinfinite-imensionlHilbertspceL 2 ([, b] of ll squre-integrble functions over the intervl [, b] (ie, b f(x 2

### M 106 Integral Calculus and Applications

M 6 Integrl Clculus n Applictions Contents The Inefinite Integrls.................................................... Antierivtives n Inefinite Integrls.. Antierivtives.............................................................

### Sturm-Liouville Theory

LECTURE 1 Sturm-Liouville Theory In the two preceing lectures I emonstrte the utility of Fourier series in solving PDE/BVPs. As we ll now see, Fourier series re just the tip of the iceerg of the theory

### LECTURE 3. Orthogonal Functions. n X. It should be noted, however, that the vectors f i need not be orthogonal nor need they have unit length for

ECTURE 3 Orthogonl Functions 1. Orthogonl Bses The pproprite setting for our iscussion of orthogonl functions is tht of liner lgebr. So let me recll some relevnt fcts bout nite imensionl vector spces.

### Section 6.3 The Fundamental Theorem, Part I

Section 6.3 The Funmentl Theorem, Prt I (3//8) Overview: The Funmentl Theorem of Clculus shows tht ifferentition n integrtion re, in sense, inverse opertions. It is presente in two prts. We previewe Prt

### APPENDIX. Precalculus Review D.1. Real Numbers and the Real Number Line

APPENDIX D Preclculus Review APPENDIX D.1 Rel Numers n the Rel Numer Line Rel Numers n the Rel Numer Line Orer n Inequlities Asolute Vlue n Distnce Rel Numers n the Rel Numer Line Rel numers cn e represente

### VII. The Integral. 50. Area under a Graph. y = f(x)

VII. The Integrl In this chpter we efine the integrl of function on some intervl [, b]. The most common interprettion of the integrl is in terms of the re uner the grph of the given function, so tht is

### Overview of Calculus

Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L

### Conservation Law. Chapter Goal. 6.2 Theory

Chpter 6 Conservtion Lw 6.1 Gol Our long term gol is to unerstn how mthemticl moels re erive. Here, we will stuy how certin quntity chnges with time in given region (sptil omin). We then first erive the

### If we have a function f(x) which is well-defined for some a x b, its integral over those two values is defined as

Y. D. Chong (26) MH28: Complex Methos for the Sciences 2. Integrls If we hve function f(x) which is well-efine for some x, its integrl over those two vlues is efine s N ( ) f(x) = lim x f(x n ) where x

### ODE: Existence and Uniqueness of a Solution

Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =

### x ) dx dx x sec x over the interval (, ).

Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

### Instantaneous Rate of Change of at a :

AP Clculus AB Formuls & Justiictions Averge Rte o Chnge o on [, ]:.r.c. = ( ) ( ) (lger slope o Deinition o the Derivtive: y ) (slope o secnt line) ( h) ( ) ( ) ( ) '( ) lim lim h0 h 0 3 ( ) ( ) '( ) lim

### Physics 116C Solution of inhomogeneous ordinary differential equations using Green s functions

Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner

### Homework Problem Set 1 Solutions

Chemistry 460 Dr. Jen M. Stnr Homework Problem Set 1 Solutions 1. Determine the outcomes of operting the following opertors on the functions liste. In these functions, is constnt..) opertor: / ; function:

### f a L Most reasonable functions are continuous, as seen in the following theorem:

Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f

### How can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?

Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those

### 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS

4.5 The Funmentl Theorem of Clculus Contemporry Clculus 4.5 THE FUNDAMENTAL THEOREM OF CALCULUS This section contins the most importnt n most use theorem of clculus, THE Funmentl Theorem of Clculus. Discovere

### lim P(t a,b) = Differentiate (1) and use the definition of the probability current, j = i (

PHYS851 Quntum Mechnics I, Fll 2009 HOMEWORK ASSIGNMENT 7 1. The continuity eqution: The probbility tht prticle of mss m lies on the intervl [,b] t time t is Pt,b b x ψx,t 2 1 Differentite 1 n use the

### Best Approximation in the 2-norm

Jim Lmbers MAT 77 Fll Semester 1-11 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the -norm Suppose tht we wish to obtin function f n (x) tht is liner combintion

### sec x over the interval (, ). x ) dx dx x 14. Use a graphing utility to generate some representative integral curves of the function Curve on 5

Curve on Clcultor eperience Fin n ownlo (or type in) progrm on your clcultor tht will fin the re uner curve using given number of rectngles. Mke sure tht the progrm fins LRAM, RRAM, n MRAM. (You nee to

### Introduction to Complex Variables Class Notes Instructor: Louis Block

Introuction to omplex Vribles lss Notes Instructor: Louis Block Definition 1. (n remrk) We consier the complex plne consisting of ll z = (x, y) = x + iy, where x n y re rel. We write x = Rez (the rel prt

### Matrix & Vector Basic Linear Algebra & Calculus

Mtrix & Vector Bsic Liner lgebr & lculus Wht is mtrix? rectngulr rry of numbers (we will concentrte on rel numbers). nxm mtrix hs n rows n m columns M x4 M M M M M M M M M M M M 4 4 4 First row Secon row

### Recitation 3: More Applications of the Derivative

Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

### How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4

### dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

### Main topics for the First Midterm

Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 2-3, Sections 4.1-4.8, nd Sections 5.1-5.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the

### Improper Integrals. Type I Improper Integrals How do we evaluate an integral such as

Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph

### x dx does exist, what does the answer look like? What does the answer to

Review Guie or MAT Finl Em Prt II. Mony Decemer th 8:.m. 9:5.m. (or the 8:3.m. clss) :.m. :5.m. (or the :3.m. clss) Prt is worth 5% o your Finl Em gre. NO CALCULATORS re llowe on this portion o the Finl

### In Section 5.3 we considered initial value problems for the linear second order equation. y.a/ C ˇy 0.a/ D k 1 (13.1.4)

678 Chpter 13 Boundry Vlue Problems for Second Order Ordinry Differentil Equtions 13.1 TWO-POINT BOUNDARY VALUE PROBLEMS In Section 5.3 we considered initil vlue problems for the liner second order eqution

### Antiderivatives Introduction

Antierivtives 0. Introuction So fr much of the term hs been sent fining erivtives or rtes of chnge. But in some circumstnces we lrey know the rte of chnge n we wish to etermine the originl function. For

### Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte

### Chapter 8: Methods of Integration

Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln

### Lecture 3. Limits of Functions and Continuity

Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

### Math Lecture 23

Mth 8 - Lecture 3 Dyln Zwick Fll 3 In our lst lecture we delt with solutions to the system: x = Ax where A is n n n mtrix with n distinct eigenvlues. As promised, tody we will del with the question of

### Lecture 1. Functional series. Pointwise and uniform convergence.

1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

### Linear Algebra 1A - solutions of ex.4

Liner Algebr A - solutions of ex.4 For ech of the following, nd the inverse mtrix (mtritz hofkhit if it exists - ( 6 6 A, B (, C 3, D, 4 4 ( E i, F (inverse over C for F. i Also, pick n invertible mtrix

### The Fundamental Theorem of Calculus Part 2, The Evaluation Part

AP Clculus AB 6.4 Funmentl Theorem of Clculus The Funmentl Theorem of Clculus hs two prts. These two prts tie together the concept of integrtion n ifferentition n is regre by some to by the most importnt

### ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

### ( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

### 5.3 The Fundamental Theorem of Calculus

CHAPTER 5. THE DEFINITE INTEGRAL 35 5.3 The Funmentl Theorem of Clculus Emple. Let f(t) t +. () Fin the re of the region below f(t), bove the t-is, n between t n t. (You my wnt to look up the re formul

### W. We shall do so one by one, starting with I 1, and we shall do it greedily, trying

Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)

### 20 MATHEMATICS POLYNOMIALS

0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

### Exam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH-1105 Instructor: Attila Máté 1

Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH-115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixed-point itertion to converge when solving the eqution

### Introduction. Calculus I. Calculus II: The Area Problem

Introuction Clculus I Clculus I h s its theme the slope problem How o we mke sense of the notion of slope for curves when we only know wht the slope of line mens? The nswer, of course, ws the to efine

### p(t) dt + i 1 re it ireit dt =

Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)

### Review of basic calculus

Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below

### MATH 174A: PROBLEM SET 5. Suggested Solution

MATH 174A: PROBLEM SET 5 Suggested Solution Problem 1. Suppose tht I [, b] is n intervl. Let f 1 b f() d for f C(I; R) (i.e. f is continuous rel-vlued function on I), nd let L 1 (I) denote the completion

### Introduction. Calculus I. Calculus II: The Area Problem

Introuction Clculus I Clculus I h s its theme the slope problem How o we mke sense of the notion of slope for curves when we only know wht the slope of line mens? The nswer, of course, ws the to efine

### Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

### CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS

CHAPTER 9 BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS BASIC CONCEPTS OF DIFFERENTIAL AND INTEGRAL CALCULUS LEARNING OBJECTIVES After stuying this chpter, you will be ble to: Unerstn the bsics

### Math 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008

Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 2-5pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of

### Theoretical foundations of Gaussian quadrature

Theoreticl foundtions of Gussin qudrture 1 Inner product vector spce Definition 1. A vector spce (or liner spce) is set V = {u, v, w,...} in which the following two opertions re defined: (A) Addition of

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### Homework Assignment 5 Solution Set

Homework Assignment 5 Solution Set PHYCS 44 3 Februry, 4 Problem Griffiths 3.8 The first imge chrge gurntees potentil of zero on the surfce. The secon imge chrge won t chnge the contribution to the potentil

### Math 61CM - Solutions to homework 9

Mth 61CM - Solutions to homework 9 Cédric De Groote November 30 th, 2018 Problem 1: Recll tht the left limit of function f t point c is defined s follows: lim f(x) = l x c if for ny > 0 there exists δ

### Review of Calculus, cont d

Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some

### and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

### 1. Gauss-Jacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),

1. Guss-Jcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on

### (4.1) D r v(t) ω(t, v(t))

1.4. Differentil inequlities. Let D r denote the right hnd derivtive of function. If ω(t, u) is sclr function of the sclrs t, u in some open connected set Ω, we sy tht function v(t), t < b, is solution

### Math 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED

Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type

### A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

### Properties of the Riemann Integral

Properties of the Riemnn Integrl Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University Februry 15, 2018 Outline 1 Some Infimum nd Supremum Properties 2

### ax bx c (2) x a x a x a 1! 2!! gives a useful way of approximating a function near to some specific point x a, giving a power-series expansion in x

Elementr mthemticl epressions Qurtic equtions b b b The solutions to the generl qurtic eqution re (1) b c () b b 4c (3) Tlor n Mclurin series (power-series epnsion) The Tlor series n n f f f n 1!! n! f

### HW3, Math 307. CSUF. Spring 2007.

HW, Mth 7. CSUF. Spring 7. Nsser M. Abbsi Spring 7 Compiled on November 5, 8 t 8:8m public Contents Section.6, problem Section.6, problem Section.6, problem 5 Section.6, problem 7 6 5 Section.6, problem

### approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

. Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

### 3.4 THE DERIVATIVE AS A RATE OF CHANGE

3 CHAPTER 3 THE DERIVATIVE; THE PROCESS OF DIFFERENTIATION y y y y y y y = m( ) Figure 3.4. 3.4 THE DERIVATIVE AS A RATE OF CHANGE In the cse of liner function y = m + b, the grph is stright line n the

### MA Handout 2: Notation and Background Concepts from Analysis

MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

### Example Sheet 2 Solutions

Exmple Sheet Solutions. i L f, g f, L g efinition of joint L g, f property of inner prouct g, Lf efinition of joint Lf, g property of inner prouct ii L L f, g Lf, g L f, g liner opertor property f, L g

### EULER-LAGRANGE EQUATIONS. Contents. 2. Variational formulation 2 3. Constrained systems and d Alembert principle Legendre transform 6

EULER-LAGRANGE EQUATIONS EUGENE LERMAN Contents 1. Clssicl system of N prticles in R 3 1 2. Vritionl formultion 2 3. Constrine systems n Alembert principle. 4 4. Legenre trnsform 6 1. Clssicl system of

### Lecture Solution of a System of Linear Equation

ChE Lecture Notes, Dept. of Chemicl Engineering, Univ. of TN, Knoville - D. Keffer, 5/9/98 (updted /) Lecture 8- - Solution of System of Liner Eqution 8. Why is it importnt to e le to solve system of liner

### Riemann Sums and Riemann Integrals

Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct

### n f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1

The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the

### 21.6 Green Functions for First Order Equations

21.6 Green Functions for First Order Equtions Consider the first order inhomogeneous eqution subject to homogeneous initil condition, B[y] y() = 0. The Green function G( ξ) is defined s the solution to

### Chapter 2. Determinants

Chpter Determinnts The Determinnt Function Recll tht the X mtrix A c b d is invertible if d-bc0. The expression d-bc occurs so frequently tht it hs nme; it is clled the determinnt of the mtrix A nd is

### MORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)

MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give

### 5.2 Exponent Properties Involving Quotients

5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use

Mth 452 - Advnced Clculus II Line Integrls nd Green s Theorem The min gol of this chpter is to prove Stoke s theorem, which is the multivrible version of the fundmentl theorem of clculus. We will be focused

### SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION

Physics 8.06 Apr, 2008 SUPPLEMENTARY NOTES ON THE CONNECTION FORMULAE FOR THE SEMICLASSICAL APPROXIMATION c R. L. Jffe 2002 The WKB connection formuls llow one to continue semiclssicl solutions from n

### Math 231E, Lecture 33. Parametric Calculus

Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider

### Recitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications

Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 Higher-Order Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht

### Advanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004

Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when

### 2.4 Linear Inequalities and Interval Notation

.4 Liner Inequlities nd Intervl Nottion We wnt to solve equtions tht hve n inequlity symol insted of n equl sign. There re four inequlity symols tht we will look t: Less thn , Less thn or

### Matrices and Determinants

Nme Chpter 8 Mtrices nd Determinnts Section 8.1 Mtrices nd Systems of Equtions Objective: In this lesson you lerned how to use mtrices, Gussin elimintion, nd Guss-Jordn elimintion to solve systems of liner

### NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

### State space systems analysis (continued) Stability. A. Definitions A system is said to be Asymptotically Stable (AS) when it satisfies

Stte spce systems nlysis (continued) Stbility A. Definitions A system is sid to be Asymptoticlly Stble (AS) when it stisfies ut () = 0, t > 0 lim xt () 0. t A system is AS if nd only if the impulse response

### 1 nonlinear.mcd Find solution root to nonlinear algebraic equation f(x)=0. Instructor: Nam Sun Wang

nonlinermc Fin solution root to nonliner lgebric eqution ()= Instructor: Nm Sun Wng Bckgroun In science n engineering, we oten encounter lgebric equtions where we wnt to in root(s) tht stisies given eqution

### 1 Linear Least Squares

Lest Squres Pge 1 1 Liner Lest Squres I will try to be consistent in nottion, with n being the number of dt points, nd m < n being the number of prmeters in model function. We re interested in solving

### 18.06 Problem Set 4 Due Wednesday, Oct. 11, 2006 at 4:00 p.m. in 2-106

8. Problem Set Due Wenesy, Ot., t : p.m. in - Problem Mony / Consier the eight vetors 5, 5, 5,..., () List ll of the one-element, linerly epenent sets forme from these. (b) Wht re the two-element, linerly

### AP Calculus AB First Semester Final Review

P Clculus B This review is esigne to give the stuent BSIC outline of wht nees to e reviewe for the P Clculus B First Semester Finl m. It is up to the iniviul stuent to etermine how much etr work is require

### Math 113 Exam 2 Practice

Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

### When e = 0 we obtain the case of a circle.

3.4 Conic sections Circles belong to specil clss of cures clle conic sections. Other such cures re the ellipse, prbol, n hyperbol. We will briefly escribe the stnr conics. These re chosen to he simple

### Solution to HW 4, Ma 1c Prac 2016

Solution to HW 4 M c Prc 6 Remrk: every function ppering in this homework set is sufficiently nice t lest C following the jrgon from the textbook we cn pply ll kinds of theorems from the textbook without

### MUST-KNOW MATERIAL FOR CALCULUS

MUST-KNOW MATERIAL FOR CALCULUS MISCELLANEOUS: intervl nottion: (, b), [, b], (, b], (, ), etc. Rewrite ricls s frctionl exponents: 3 x = x 1/3, x3 = x 3/2 etc. An impliction If A then B is equivlent to