Section 14.3 Arc Length and Curvature


 Augustine Boyd
 1 years ago
 Views:
Transcription
1 Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in terms of prmetric equtions (in spce) ws determined. A similr formul holds for 3spce. Result.. If r(t) = f(t) i + g(t) j + h(t) k defines smooth curve C nd f (t), g (t) nd h (t) re ll continuous, then the rc length long the portion of the curve with t b (provided it is trversed only is) is given by the formul or L = b [f (t)] + [g (t)] + [h (t)] dt L = b r (t) dt. The formul is stright forwrd to work with: Exmple.. Find the length of for t. r(t) = i + t j + t 3 k This is stright forwrd clcultions: L = + 4t + 9t 4 dt = t 4 + 9t dt. Substituting u = 4 + 9t, we hve du/dt = 8t, dt = du/8t, nd when t =, u = 4, nd t = gives u = 3, t t dt = t u 4 8t du = 3 [ ] 3 udu = u 3 4 = 7 [ Often we re not interested in specific distnce prticle hs trveled, but rther formul to determine how fr prticle hs trveled in terms of me vrible (this is useful in things like irline flight or spce trvel), we need to determine wy to do this. Suppose r(u) is vector function for curve C which trverses C only once for u t ].
2 (notice we hve chnged the prmeter to u nd we re using t s time vrible ). We define the rclength function s for C by r (u) du = [f (u)] + [g (u)] + [h (u)] du. Notice tht the integrl defines function in the vrible t. Al note tht by the fundmentl Theorem of Clculus, this implies ds dt = r (t). The function s(t) mesures the distnce from the point r() to the point r(t), s on s we plug in vlue for t, it will provide us with this numericl distnce. We illustrte. Exmple.3. Find the distnce formul for r(u) = u i + ( 3u) j + (5 + 4u) k from u =. Use the formul to determine the distnce trveled fter seconds, nd how long it tkes for the prticle to trvel 8 units. The formul is stright forwrd: + ( 3) + 4 du = 9t This mens fter seconds, the prticle will hve trveled 9 units. In order to trvel 8 units, we need 9t = 8, or t = 9/8.67 seconds. Notice tht in the lst exmple, we obtined formul for s in terms of t. In prticulr, we could lve this formul for t in terms of s, t = s/ 9, nd then subsitute into the originl prmeteriztion for r nd thus we would hve prmeteriztion for r in terms of s insted of t. We cll such prmetriztion prmeteriztion with respect to rclength. This is often preferred method of prmeteriztion since it depends only upon the curve itself (or the length of the curve), nd not prticulr coordinte system. In order to determine prmeteriztion with respect to rclength of curve with vector eqution r(t), we do the following: (i) Solve the distnce formul r (u) du for the prmeteriztion you re given. You should hve function of s in terms of t. (ii) Cn you lve the function s(t) for t? If not, you cnnot reprmeterize this wy. Otherwise, lve for t ( t will be function of s, t(s)).
3 (iii) Substitute the function t(s) in for u (nd renme s s u) in the originl prmetriztion  this is now prmeteriztion with respect to rclength. Exmple.4. Find prmeteriztion with respect to rclength for r(u) = u i + ( 3u) j + (5 + 4u) k from u =. Recll, we hve t = s/ 9. Therefore, + ( 3) + 4 du = 9t r(t) = u ( 3u) (5 + 4u) i + j + k is prmeteriztion with respect to rclength.. Curvture Recll tht if C is smooth curve defined by the vector function r(t), nd r (t), then the unit tngent vector is given by T(t) = r(t)/ r (t) which indictes the direction of the curve. Since T(t) provides the direction of r(t), the rte of chnge of T with respect to s, the distnce function, mesures how quickly the direction of C is chnging (see figure below)  when d T/ds is lrge, it mens the direction of C is chnging quickly over distnce, nd when it is smll, it mens it is not chnging drmticlly. 3 This motivtes the following definition: Definition.. The curvture of curve is κ = d T ds where T is the unit tngent vector with prmeteriztion with respect to rclength s. By the wy we hve defined κ, it seems difficult to clculte  first we need to determine the prmeteriztion with respect to rclength of T (which s we sw previously, is not esy). However, by using the chin rule, we cn void doing this.
4 4 Result.. κ(t) cn be clculted using the following formul: κ(t) = T (t) r (t). Observe tht this completely voids hving to find s. We illustrte. Exmple.3. Find the curvture of r(t) = 3t i+4 sin (t) j +4 cos (t) k. First we find the unit norml vector, T(t) = r (t) r (t) = 3 i + 4 cos(t) j 4 sin (t) k (3 + 4 ) Then we hve Then we hve T (t) = 4 5 sin (t) j 4 5 cos (t) k. = 3 5 i+ 4 5 cos (t) j 4 5 sin (t) k. κ(t) = 4 sin (t) j 4 cos (t) 4 k i + 4 cos (t) j 4 sin(t) k = 5 5 = 4 5. Notice tht this is constnt (it does not depend upon t) which mens tht the curvture is constnt. This is l pprent from the grph below where we cn see the tngent vectors re chnging t constnt rte: There re other wys to clculte curvture which do not rely upon finding the tngent vector nd insted use crossproduct. Result.4. The curvture of the curve C given by r(t) is κ(t) = r (t) r (t) r (t) 3. Exmple.5. Find the curvture of r(t) = t i + e t j + e t k. We shll pply the recent formul: we hve r (t) = i + e t j e t j, nd r (t) = e t j + e t k, r (t) r (t) = i e t j + e t k r (t) r (t) = 4 + e t + e t = (e t + e t ) = (e t + e t )
5 We l hve r (t) 3 = (4 + e t + e t ) 3/ = ( (e t + e t )) 3 κ(t) = (e t + e t ). For the specil cse of plne curve y = f(x), ( dimensionl curve), the zcoordinte is lwys zero, we cn tke the prmeter to be x vector eqution will be Then we shll hve r(x) = x i + f(x) j. r (x) r (x) = ( i + f (x) j) (f (x) j) = f (x) k = f (x), nd r (x) 3 = ( i + f (x) j) 3 = + (f (x)) 3 = + (f (x)) 3/. Thus we hve Result.6. If y = f(x), then the curvture t ny point x is given by the formul f (x) κ(x) = + (f (x)) 3 Exmple.7. Find the curvture of y = x 3 t (, ). We just pply the formul, κ() = 6 3/ 3. Norml nd Binorml Vectors We hve lredy seen tht t ny point on curve, there is vector clled the unit tngent vector which tells us the direction the curve is going. There re two other vectors closely relted to this vector which l provide informtion bout the curve C. Specificlly, provided the unit tngent vector is nonzero, we cn find two other vectors which re perpendiculr to it nd re mutully perpendiculr to ech other (giving mething like coordinte xis t the point). We define them s follows: Definition 3.. Suppose C is curve with vector eqution r(t) nd let T(t) be its unit tngent vector defined s Then we define: T(t) = r (t) r (t). 5
6 6 (i) The principl unit norml vector N(t) defined s N(t) = T (t) T (t). (ii) The binorml vector B(t) defined s B(t) = T(t) B(t). All three vectors re mutully perpendiculr. Clcultion of these vectors though cumberme, is firly stright forwrd. We give n exmple. Exmple 3.. Find the vectors T, N nd B for r(t) = t i+t 3 /3 j+t k t the point (, /3, ). We hve T(t) = t i + t j + k (4t4 + 4t + = t i + t j + k (t + ) To clculte T (t), we use the generlized product rule: = t i + t j + k. (t + ) T 4t (t) = (t + ) (t i + t j + k) + (t + ) ( i + 4t j) = i + 8t 3 j + 4t (t + ) k) + + ) i + (8t 3 + 4t) j) (8t (t + ) ((4t = (t + ) (( 4t ) i+4t j 4t k) = Then we l hve N(t) = T (t) = (t + ) (( t ) i+t j t k). (t + ) ( 4t + 4t 4 + 4t + 4t ) = (t +) (( t ) i + t j t k) (t +) At (, /3, ) we hve t =, T() = i + j + k 9 = (t + ) t + (( t ) i+t j t k). = 3 i + 3 j + 3 k nd N() = 3 i + 3 j 3 k, B() = ( 3 i + 3 j + 3 k) ( 3 i + 3 j 3 k) = 3 i + 3 j + 3 k.
7 Recll tht given pir of (nonprllel) vectors nd point, there is plne which contins both vectors nd the point. The plnes scited to the vectors we hve introduced bove provide useful informtion bout curve C with vector r(t), we introduce me terminology for these plnes: Definition 3.3. Suppose tht r(t) is vector eqution for curve C. For fixed vlue, t =, we define the following: (i) The plne contining the point r() nd vectors N nd B is clled the norml plne to C t P = r(). It contins ll vectors orthogonl to C t t =. (ii) The plne contining the point r() nd vectors N nd T is clled the osculting plne to C t P = r(). It is the plne which C most closely lies in t P. Exmple 3.4. Find the equtions for the norml plne nd the osculting plne to r(t) = t i + t 3 /3 j + t k t the point (, /3, ). We hve lredy determined the three required vectors: T() = 3 i + 3 j + 3 k 7 nd nd N() = 3 i + 3 j 3 k, B() = 3 i + 3 j + 3 k. The norml plne contins B nd N, norml vector to the plne will be T (which is perpendiculr to both B nd N). This n eqution will be 3 (x ) + 3 (y 3 ) + (z ) =. 3 The osculting plne will hve B s norml vector, will hve eqution 3 (x ) + 3 (y 3 ) + (z ) =. 3
Line and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
More informationPolynomial 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
More informationMATH Summary of Chapter 13
MATH 21259 ummry of hpter 13 1. Vector Fields re vector functions of two or three vribles. Typiclly, vector field is denoted by F(x, y, z) = P (x, y, z)i+q(x, y, z)j+r(x, y, z)k where P, Q, R re clled
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationChapter 4 Contravariance, Covariance, and Spacetime Diagrams
Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz
More informationLine Integrals. Partitioning the Curve. Estimating the Mass
Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More informationBest Approximation. Chapter The General Case
Chpter 4 Best Approximtion 4.1 The Generl Cse In the previous chpter, we hve seen how n interpolting polynomil cn be used s n pproximtion to given function. We now wnt to find the best pproximtion to given
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationConducting Ellipsoid and Circular Disk
1 Problem Conducting Ellipsoid nd Circulr Disk Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 (September 1, 00) Show tht the surfce chrge density σ on conducting ellipsoid,
More information20 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
More informationJackson 2.26 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jckson 2.26 Homework Problem Solution Dr. Christopher S. Bird University of Msschusetts Lowell PROBLEM: The twodimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationNumerical integration
2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter
More informationAnonymous Math 361: Homework 5. x i = 1 (1 u i )
Anonymous Mth 36: Homewor 5 Rudin. Let I be the set of ll u (u,..., u ) R with u i for ll i; let Q be the set of ll x (x,..., x ) R with x i, x i. (I is the unit cube; Q is the stndrd simplex in R ). Define
More informationPractice final exam solutions
University of Pennsylvni Deprtment of Mthemtics Mth 26 Honors Clculus II Spring Semester 29 Prof. Grssi, T.A. Asher Auel Prctice finl exm solutions 1. Let F : 2 2 be defined by F (x, y (x + y, x y. If
More informationIntegrals along Curves.
Integrls long Curves. 1. Pth integrls. Let : [, b] R n be continuous function nd let be the imge ([, b]) of. We refer to both nd s curve. If we need to distinguish between the two we cll the function the
More informationNUMERICAL 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
More informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More informationWeek 10: Riemann integral and its properties
Clculus nd Liner Algebr for Biomedicl Engineering Week 10: Riemnn integrl nd its properties H. Führ, Lehrstuhl A für Mthemtik, RWTH Achen, WS 07 Motivtion: Computing flow from flow rtes 1 We observe the
More information2. VECTORS AND MATRICES IN 3 DIMENSIONS
2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2dimensionl Vectors x A point in 3dimensionl spce cn e represented y column vector of the form y z zxis yxis z x y xxis Most of the
More information7.6 The Use of Definite Integrals in Physics and Engineering
Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems
More information5.5 The Substitution Rule
5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n ntiderivtive is not esily recognizble, then we re in
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationLecture 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
More informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More information4 VECTORS. 4.0 Introduction. Objectives. Activity 1
4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply
More informationThe problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.
ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion
More informationUnit #10 De+inite Integration & The Fundamental Theorem Of Calculus
Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = x + 8x )Use
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More information13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS
33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in
More informationl 2 p2 n 4n 2, the total surface area of the
Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n nsided regulr polygon of perimeter p n with vertices on C. Form cone
More informationSTEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.
STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t
More information(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35
7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know
More informationdx 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
More informationSome Methods in the Calculus of Variations
CHAPTER 6 Some Methods in the Clculus of Vritions 6. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting
More informationMotion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.
Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. =  o t = Δ Δt ccelertion = =  o t chnge of elocity elpsed time ccelertion is ector, lthough
More informationThe Definite Integral
CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls
More informationSynoptic Meteorology I: Finite Differences September Partial Derivatives (or, Why Do We Care About Finite Differences?
Synoptic Meteorology I: Finite Differences 1618 September 2014 Prtil Derivtives (or, Why Do We Cre About Finite Differences?) With the exception of the idel gs lw, the equtions tht govern the evolution
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationContinuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is relvlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht
More information1 Error Analysis of Simple Rules for Numerical Integration
cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion
More informationDEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS
3 DEFINITION OF ASSOCIATIVE OR DIRECT PRODUCT AND ROTATION OF VECTORS This chpter summrizes few properties of Cli ord Algebr nd describe its usefulness in e ecting vector rottions. 3.1 De nition of Associtive
More informationExperiment 2: Equipotentials and Electric Fields
Chpter 4 Experiment 2: Equipotentils nd Electric Fields 4.1 Introduction One wy to look t the force between chrges is to sy tht the chrge lters the spce round it by generting n electric field E. Any other
More informationu( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 218, pp 4448): Determine the equation of the following graph.
nlyzing Dmped Oscilltions Prolem (Medor, exmple 218, pp 4448): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $
More informationCalculus 2: Integration. Differentiation. Integration
Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is
More informationChapter 2. Constraints, Lagrange s equations
Chpter Constrints, Lgrnge s equtions Section Constrints The position of the prticle or system follows certin rules due to constrints: Holonomic constrint: f (r. r,... r n, t) = 0 Constrints tht re not
More informationBRIEF NOTES ADDITIONAL MATHEMATICS FORM
BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationMTH 122 Fall 2008 Essex County College Division of Mathematics Handout Version 10 1 October 14, 2008
MTH 22 Fll 28 Essex County College Division of Mthemtics Hndout Version October 4, 28 Arc Length Everyone should be fmilir with the distnce formul tht ws introduced in elementry lgebr. It is bsic formul
More informationChapter 4. Additional Variational Concepts
Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.
More informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationUniversitaireWiskundeCompetitie. Problem 2005/4A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that
Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de
More information3.4 Numerical integration
3.4. Numericl integrtion 63 3.4 Numericl integrtion In mny economic pplictions it is necessry to compute the definite integrl of relvlued function f with respect to "weight" function w over n intervl [,
More information3.1 Review of Sine, Cosine and Tangent for Right Angles
Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,
More informationMapping the delta function and other Radon measures
Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support
More informationEnergy Bands Energy Bands and Band Gap. Phys463.nb Phenomenon
Phys463.nb 49 7 Energy Bnds Ref: textbook, Chpter 7 Q: Why re there insultors nd conductors? Q: Wht will hppen when n electron moves in crystl? In the previous chpter, we discussed free electron gses,
More information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More information15  TRIGONOMETRY Page 1 ( Answers at the end of all questions )
 TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the
More information200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes
PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write
More informationThe Wave Equation I. MA 436 Kurt Bryan
1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string
More informationUNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE
UNIFORM CONVERGENCE MA 403: REAL ANALYSIS, INSTRUCTOR: B. V. LIMAYE 1. Pointwise Convergence of Sequence Let E be set nd Y be metric spce. Consider functions f n : E Y for n = 1, 2,.... We sy tht the sequence
More informationCalculus MATH 172Fall 2017 Lecture Notes
Clculus MATH 172Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems
More informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
More informationFundamentals of Analytical Chemistry
Homework Fundmentls of nlyticl hemistry hpter 9 0, 1, 5, 7, 9 cids, Bses, nd hpter 9(b) Definitions cid Releses H ions in wter (rrhenius) Proton donor (Bronsted( Lowry) Electronpir cceptor (Lewis) hrcteristic
More informationBIFURCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS
BIFRCATIONS IN ONEDIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible
More informationCBSEXII2015 EXAMINATION. Section A. 1. Find the sum of the order and the degree of the following differential equation : = 0
CBSEXII EXMINTION MTHEMTICS Pper & Solution Time : Hrs. M. Mrks : Generl Instruction : (i) ll questions re compulsory. There re questions in ll. (ii) This question pper hs three sections : Section, Section
More informationA Convergence Theorem for the Improper Riemann Integral of Banach Spacevalued Functions
Interntionl Journl of Mthemticl Anlysis Vol. 8, 2014, no. 50, 24512460 HIKARI Ltd, www.mhikri.com http://dx.doi.org/10.12988/ijm.2014.49294 A Convergence Theorem for the Improper Riemnn Integrl of Bnch
More informationTest , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes
Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl
More informationV. DEMENKO MECHANICS OF MATERIALS LECTURE 6 Plane Bending Deformation. Diagrams of Internal Forces (Continued)
V. DEMENKO MECHNCS OF MTERLS 015 1 LECTURE 6 Plne ending Deformtion. Digrms of nternl Forces (Continued) 1 Construction of ending Moment nd Shering Force Digrms for Two Supported ems n this mode of loding,
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationHomework Assignment 3 Solution Set
Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.
More informationContour and surface integrals
Contour nd surfce integrls Contour integrls of the sclr type These re integrls of the type I = f@rd l C where C is contour nd l is n infinitesiml element of the contour length. If the contour cn be described
More informationLecture 6: Singular Integrals, Open Quadrature rules, and Gauss Quadrature
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics  A Peirce UBC Lecture 6: Singulr Integrls, Open Qudrture rules, nd Guss Qudrture (Compiled 6 August 7) In this lecture we discuss the
More informationPDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
More informationImproper 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:
More informationPhys 7221, Fall 2006: Homework # 6
Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 37 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which
More informationDiscrete Leastsquares Approximations
Discrete Lestsqures Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve
More informationProblem 1. Brachistochrone timeofflight
PHY 320 Homework 3 Solution Spring 207 Problem. Brchistochrone timeofflight From clss discussion nd the textbook, we know tht the functionl to be minimized is the time: 2 t = d s = 2 +(y') v 2 v d x
More informationMath 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
More informationMACsolutions of the nonexistent solutions of mathematical physics
Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences  Finite Elements  Finite Volumes  Boundry Elements MACsolutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE
More informationWaveguide Guide: A and V. Ross L. Spencer
Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it
More informationSULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAHSEKOLAH MENENGAH ZON A KUCHING
SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAHSEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
More information4.1. Probability Density Functions
STT 1 4.14. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of
More informationPHYSICS 211 MIDTERM I 21 April 2004
PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of
More information[ ( ) ( )] Section 6.1 Area of Regions between two Curves. Goals: 1. To find the area between two curves
Gols: 1. To find the re etween two curves Section 6.1 Are of Regions etween two Curves I. Are of Region Between Two Curves A. Grphicl Represention = _ B. Integrl Represention [ ( ) ( )] f x g x dx = C.
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
More informationChapter 1  Functions and Variables
Business Clculus 1 Chpter 1  Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:
More informationLesson Notes: Week 40Vectors
Lesson Notes: Week 40Vectors Vectors nd Sclrs vector is quntity tht hs size (mgnitude) nd direction. Exmples of vectors re displcement nd velocity. sclr is quntity tht hs size but no direction. Exmples
More informationF is n ntiderivtive èor èindeæniteè integrlè off if F 0 èxè =fèxè. Nottion: F èxè = ; it mens F 0 èxè=fèxè ëthe integrl of f of x dee x" Bsic list: xn
Mth 70 Topics for third exm Chpter 3: Applictions of Derivtives x7: Liner pproximtion nd diæerentils Ide: The tngent line to grph of function mkes good pproximtion to the function, ner the point of tngency.
More informationUSA Mathematical Talent Search Round 1 Solutions Year 21 Academic Year
1/1/21. Fill in the circles in the picture t right with the digits 18, one digit in ech circle with no digit repeted, so tht no two circles tht re connected by line segment contin consecutive digits.
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
More informationThe Basic Functional 2 1
2 The Bsic Functionl 2 1 Chpter 2: THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 2.1 Introduction..................... 2 3 2.2 The First Vrition.................. 2 3 2.3 The Euler Eqution..................
More informationPythagoras Theorem. Pythagoras
11 Pythgors Theorem Pythgors Theorem. onverse of Pythgors theorem. Pythgoren triplets Proof of Pythgors theorem nd its converse Problems nd riders bsed on Pythgors theorem. This unit fcilittes you in,
More informationChapter 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
More informationf (x)dx = f(b) f(a). a b f (x)dx is the limit of sums
Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx
More information