MATH 13 FINAL STUDY GUIDE, WINTER 2012
|
|
- Emil Neal
- 5 years ago
- Views:
Transcription
1 MATH 13 FINAL TUY GUI, WINTR 2012 This is ment to be quick reference guide for the topics you might wnt to know for the finl. It probbly isn t comprehensive, but should cover most of wht we studied in this clss. There re no exmples, so be sure to consult your old homeworks, notes, or textbook for those. 1. ontent from erlier clsses (vectors, differentition, etc.) Know bout bsic opertions on vectors, such s ddition, sclr multipliction, dot product, cross product, nd their pplictions to geometric problems Know bout properties of lines nd plnes in R 3 Know how to prmeterize curves, compute rc length, nd find tngent lines to curves Know bout prtil derivtives nd their pplictions to computing tngent plnes to surfces Know bout the grdient nd directionl derivtives 2. ouble integrtion Integrtion over rectngles Integrtion over more generl domins in the plne How to interchnge order of integrtion (Fubini s Theorem) Polr coordintes Applictions of double integrtion to finding re, mss of lmin, center of mss, etc. 3. Triple integrtion Integrtion over rectngulr prisms Integrtion over more generl three-dimensionl solids How to interchnge order of integrtion (Fubini s Theorem) ylindricl coordintes phericl coordintes Jcobin, chnge of vribles formul (this covers both double nd triple integrls) Applictions of triple integrtion to finding volume, mss of solid, center of mss, etc. 4. Line integrtion, vector fields Prmeterizing common curves (line segments, grphs of functions y = f(x), circles, ellipses, etc) lculting line integrls of sclr functions over curves lculting line integrls of vector fields over curves The Fundmentl Theorem of lculus for line integrls The vrious properties of conservtive vector fields 1
2 2 MATH 13 FINAL TUY GUI, WINTR 2012 How to pply the di erentil criterion P y = Q x for being conservtive vector field, simply connected domins Applictions of line integrls to clculting rc length, center of mss of wire, work, etc. Green s Theorem 5. urfce integrls Prmeterizing common surfces (plnes, cylinders, spheres, grphs of functions z = f(x, y), etc) lculting tngent plne nd norml vectors from prmeteriztion for surfce (in prticulr, using the fundmentl vector product) lculting surfce integrls of sclr functions over surfce Understnding wht orienttion on surfces mens lculting surfce integrls of vector fields over surfce lculting divergence nd curl of vector fields in R 3 How to pply the di erentil criterion r F = 0 for being conservtive vector field, simply connected domins The ivergence Theorem tokes Theorem Applictions of surfce integrls to finding surfce re, etc 6. Formuls A non-comprehensive collection of formuls from this clss: Polr coordintes: x = r cos,y = r sin, if is described by pple pple,g 1 ( ) pple r pple g 2 ( ), then f(x, y) da = g2 ( ) g 1 ( ) f(r cos,rsin )rdrd. ylindricl coordintes: x = r cos,y = r sin,z = z, if is described by pple pple,r 1 pple r pple r 2,z 1 pple z pple z 2 (z i my be function of r,, while r i might be function of ), then f(x, y, z) dv = r2 z2 r 1 z 1 f(r cos,rsin,z)rdzdrd. phericl coordintes: x = sin cos,y = sin sin,z = cos,if is described by 1 pple pple 2,..., then f(x, y, z) dv = f( sin cos, sin sin, cos ) 2 sin d d d. Jcobin: If T is function from the uv plne to the xy plne, with x = x(u, v),y = y(u, v), then the Jcobin of T is defined to be
3 MATH 13 FINAL TUY GUI, WINTR J(u, v) v) = x u x v y u y v = x u y v x v y u. The bsolute vlue of the Jcobin represents locl re mgnifiction fctor. There is corresponding formul for trnsformtion from uvw spce to xyz spce, involving 3 3determinnt. hnge of vribles formul: if is some region in the uv plne, nd R = T () (for exmple, if T is the polr chnge of coordintes, nd is the rectngle 0 pple r pple 1, 0 pple pple 2, then T () is the unit disc centered t the origin), then T () f(x, y) da = f(x(u, v),y(u, v)) J(u, v) du dv. There is corresponding formul for chnge of vribles from uvw spce to xyz spce. Line integrls of sclr functions: If is curve prmeterized by r(t) = hx(t),y(t)i,pple t pple b, ndf(x, y) somefunctionon, then f(x, y) ds = b f(r(t)) r 0 (t) dt = b f(x(t),y(t)) p x 0 (t) 2 + y 0 (t) 2 dt. When f(x, y) = 1,this specilizes to the formul for rc length of. There is corresponding formul for curves which lie in R n. Line integrls of vector fields: If is curve prmeterized s bove, nd F(x, y) somevectorfieldon, then F dr = b F(r(t)) r 0 (t) dt = b F(x(t),y(t)) hx 0 (t),y 0 (t)i dt. If F represents force, this line integrl equls the work this force does on prticle which moves long the pth. The Fundmentl Theorem of lculus for Line Integrls: if F = rf on curve prmeterized s bove, then F dr = f(r(b)) f(r()). onservtive vector fields: their properties. A vector field F is defined to be conservtive on domin if there exists function f on such tht F = rf. This is equivlent to the following properties: For ny closed curve in, F dr =0. For ny two curves 1, 2 in which strt nd end t the sme point, F dr = F dr. 1 2 If vector field FhP, Qi is conservtive, then P y = Q x in ll of. However, in generl the converse is true only if is simply connected. For F in R 3,the condition P y = Q x is replced by r F = 0.
4 4 MATH 13 FINAL TUY GUI, WINTR 2012 Green s Theorem: If is simple closed curve in the plne, its interior, F = hp, Qi some 1 vector field on, nd is given the positive (counterclockwise) orienttion, then F dr = P y da. Q x Fundmentl vector product: if surfce is prmeterized by r(u, v), the fundmentl vector product is the cross product r u r v, nd is in generl function of u, v. Its vlue t (u, v) isnormlvectortothesurfce t the point r(u, v), nd its length represents locl re mgnifiction fctor. urfce integrl of sclr function: if surfce is prmeterized by r(u, v) = hx(u, v), y(u, v), z(u, v)i, (u, v) 2, where is some domin in the uv plne, then f(x, y, z) d = f(r(u, v)) r u r v da = f(x(u, v),y(u, v),z(u, v)) r u r v da. When f(x, y, z) =1,thisspecilizestotheformulforthesurfcereof. urfce integrl of vector field: if is prmeterized s bove, given the orienttion which points in the sme direction s r u r v,ndf(x, y, z) is vector field on, then F d = F n d = F(r(u, v)) (r u r v ) da. The ivergence Theorem: if is some solid, the boundry of with outwrd orienttion, nd F 1 vector field over ll of, then F n d = r F dv. tokes Theorem: If is some oriented surfce, its boundry with induced orienttion from, ndf some 1 vector field on, then F dr = r F d. 7. trtegies for solving problems A list of strtegies we hve lerned for solving vrious types of problems. If you run into n iterted integrl which you cn t seem to evlute, try switching the order of integrtion or chnging coordinte systems. Polr coordintes re probbly useful when deling with circles, sectors of circles, nnuli, or other geometric figures relted to circles. ylindricl coordintes re good when deling with cylinders, prboloids, or cones. phericl coordintes re good when deling with spheres or pieces of spheres.
5 MATH 13 FINAL TUY GUI, WINTR When clculting the line integrl of vector field, you cn sometimes pply the Fundmentl Theorem of lculus for line integrls (if you cn find potentil function for F, which is not lwys possible or fesible) to skip prmeteriztion of the curve. You cn show tht F is conservtive on in vriety of wys. You cn either explicitly construct potentil function vi prtil integrtion, or check tht P y = Q x if is simply connected. You cn show tht F is not conservtive on in vriety of wys. You cn either check tht P y 6= Q x, or find closed curve in for which the integrl of F long is not equl to 0, or show tht no potentil function exists by trying to do prtil integrtion nd rriving t contrdiction. When clculting the line integrl of vector field over simple closed curve, sometimes Green s Theorem cn simplify the clcultion. This is especilly true if is polygonl, like rectngle, or some other curve which hs somewht complicted prmeteriztion. When clculting the surfce integrl of vector field, you cn sometimes skip lot of steps if you find out tht F n is constnt on nd is some geometric shpe which you cn esily clculte the re of. The ivergence Theorem is useful when you find surfce integrl of vector field which is complicted but whose divergence is simple, nd/or if the surfce encloses polyhedrl solid. tokes Theorem is sometimes useful for clculting line integrls if r F hs simpleform(likeconstntvectorfield),ndyoucnsometimesusetokes Theorem to interchnge the clcultion of surfce integrl F d of vector field F which equls r G for some G to the clcultion of F d, if, 0 hve the sme boundry curve. 0
MATH Summary of Chapter 13
MATH 21-259 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 informationdf dt f () b f () a dt
Vector lculus 16.7 tokes Theorem Nme: toke's Theorem is higher dimensionl nlogue to Green's Theorem nd the Fundmentl Theorem of clculus. Why, you sk? Well, let us revisit these theorems. Fundmentl Theorem
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
More informationReview for final
eview for 263.02 finl 14.1 Functions of severl vribles. Find domin nd rnge. Evlute. ketch grph. rw nd interpret level curves. (Functions of three vribles hve level surfces.) Mtch surfces with level curves.
More informationJim Lambers MAT 280 Spring Semester Lecture 26 and 27 Notes
Jim Lmbers MAT 280 pring emester 2009-10 Lecture 26 nd 27 Notes These notes correspond to ection 8.6 in Mrsden nd Tromb. ifferentil Forms To dte, we hve lerned the following theorems concerning the evlution
More informationMath 6A Notes. Written by Victoria Kala SH 6432u Office Hours: R 12:30 1:30pm Last updated 6/1/2016
Prmetric Equtions Mth 6A Notes Written by Victori Kl vtkl@mth.ucsb.edu H 6432u Office Hours: R 12:30 1:30pm Lst updted 6/1/2016 If x = f(t), y = g(t), we sy tht x nd y re prmetric equtions of the prmeter
More informationFinal Exam Solutions, MAC 3474 Calculus 3 Honors, Fall 2018
Finl xm olutions, MA 3474 lculus 3 Honors, Fll 28. Find the re of the prt of the sddle surfce z xy/ tht lies inside the cylinder x 2 + y 2 2 in the first positive) octnt; is positive constnt. olution:
More informationMath 223, Fall 2010 Review Information for Final Exam
1. Generl Informtion Mth 223, Fll 2010 Review Informtion for Finl Exm Time, dte nd plce of finl exm: Mondy, ecember 13, 10:30 AM 1:00 PM, Wescoe 4051 (the usul clssroom). Pln to rrive 15 minutes erly so
More informationFINAL REVIEW. 1. Vector Fields, Work, and Flux Suggested Problems:
FINAL EVIEW 1. Vector Fields, Work, nd Flux uggested Problems: { 14.1 7, 13, 16 14.2 17, 25, 27, 29, 36, 45 We dene vector eld F (x, y) to be vector vlued function tht mps ech point in the plne to two
More informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255 - Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationSection 14.3 Arc Length and Curvature
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
More informationSection 17.2 Line Integrals
Section 7. Line Integrls Integrting Vector Fields nd Functions long urve In this section we consider the problem of integrting functions, both sclr nd vector (vector fields) long curve in the plne. We
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationMath 32B Discussion Session Session 7 Notes August 28, 2018
Mth 32B iscussion ession ession 7 Notes August 28, 28 In tody s discussion we ll tlk bout surfce integrls both of sclr functions nd of vector fields nd we ll try to relte these to the mny other integrls
More informationStudent Handbook for MATH 3300
Student Hndbook for MATH 3300 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0.5 0 0.5 0.5 0 0.5 If people do not believe tht mthemtics is simple, it is only becuse they do not relize how complicted life is. John Louis
More information1 Line Integrals in Plane.
MA213 thye Brief Notes on hpter 16. 1 Line Integrls in Plne. 1.1 Introduction. 1.1.1 urves. A piece of smooth curve is ssumed to be given by vector vlued position function P (t) (or r(t) ) s the prmeter
More informationMAS 4156 Lecture Notes Differential Forms
MAS 4156 Lecture Notes Differentil Forms Definitions Differentil forms re objects tht re defined on mnifolds. For this clss, the only mnifold we will put forms on is R 3. The full definition is: Definition:
More informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationNote 16. Stokes theorem Differential Geometry, 2005
Note 16. Stokes theorem ifferentil Geometry, 2005 Stokes theorem is the centrl result in the theory of integrtion on mnifolds. It gives the reltion between exterior differentition (see Note 14) nd integrtion
More informationSpace Curves. Recall the parametric equations of a curve in xy-plane and compare them with parametric equations of a curve in space.
Clculus 3 Li Vs Spce Curves Recll the prmetric equtions of curve in xy-plne nd compre them with prmetric equtions of curve in spce. Prmetric curve in plne x = x(t) y = y(t) Prmetric curve in spce x = x(t)
More information4. Calculus of Variations
4. Clculus of Vritions Introduction - Typicl Problems The clculus of vritions generlises the theory of mxim nd minim. Exmple (): Shortest distnce between two points. On given surfce (e.g. plne), nd the
More informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
More informationContents. 4.1 Line Integrals. Calculus III (part 4): Vector Calculus (by Evan Dummit, 2018, v. 3.00) 4 Vector Calculus
lculus III prt 4): Vector lculus by Evn Dummit, 8, v. 3.) ontents 4 Vector lculus 4. Line Integrls................................................. 4. Surfces nd Surfce Integrls........................................
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 informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: olumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm-6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationI. INTEGRAL THEOREMS. A. Introduction
1 U Deprtment of Physics 301A Mechnics - I. INTEGRAL THEOREM A. Introduction The integrl theorems of mthemticl physics ll hve their origin in the ordinry fundmentl theorem of clculus, i.e. xb x df dx dx
More informationSo the `chnge of vribles formul' for sphericl coordintes reds: W f(x; y; z) dv = R f(ρ cos sin ffi; ρ sin sin ffi; ρ cos ffi) ρ 2 sin ffi dρ d dffi So
Mth 28 Topics for third exm Techniclly, everything covered on the first two exms, plus hpter 15: Multiple Integrls x4: Double integrls with polr coordintes Polr coordintes describe point in the plne by
More informationUNIFORM CONVERGENCE. Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3
UNIFORM CONVERGENCE Contents 1. Uniform Convergence 1 2. Properties of uniform convergence 3 Suppose f n : Ω R or f n : Ω C is sequence of rel or complex functions, nd f n f s n in some sense. Furthermore,
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 informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
More informationUniversity of. d Class. 3 st Lecture. 2 nd
University of Technology Electromechnicl Deprtment Energy Brnch Advnced Mthemtics Line Integrl nd d lss st Lecture nd Advnce Mthemtic Line Integrl lss Electromechnicl Engineer y Dr.Eng.Muhmmd.A.R.Yss Dr.Eng
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 two-dimensionl region, ρ, φ β, is bounded by conducting surfces t φ =, ρ =, nd φ = β held t zero
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
More informationReview 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
More informationSurface Integrals of Vector Fields
Mth 32B iscussion ession Week 7 Notes Februry 21 nd 23, 2017 In lst week s notes we introduced surfce integrls, integrting sclr-vlued functions over prmetrized surfces. As with our previous integrls, we
More informationPHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS
PHYS 4390: GENERAL RELATIVITY LECTURE 6: TENSOR CALCULUS To strt on tensor clculus, we need to define differentition on mnifold.a good question to sk is if the prtil derivtive of tensor tensor on mnifold?
More information(b) Let S 1 : f(x, y, z) = (x a) 2 + (y b) 2 + (z c) 2 = 1, this is a level set in 3D, hence
Problem ( points) Find the vector eqution of the line tht joins points on the two lines L : r ( + t) i t j ( + t) k L : r t i + (t ) j ( + t) k nd is perpendiculr to both those lines. Find the set of ll
More information( 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
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO - Ares Under Functions............................................ 3.2 VIDEO - Applictions
More information50. Use symmetry to evaluate xx D is the region bounded by the square with vertices 5, Prove Property 11. y y CAS
68 CHAPTE MULTIPLE INTEGALS 46. e da, 49. Evlute tn 3 4 da, where,. [Hint: Eploit the fct tht is the disk with center the origin nd rdius is smmetric with respect to both es.] 5. Use smmetr to evlute 3
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 informationCollection of formulas Matematikk 3 (IRF30017)
ollection of formuls Mtemtikk 3 (IF30017) onic sections onic sections on stndrd form with foci on the x-xis: Ellipse: Hyperbol: Prbol: x 2 2 + y2 b 2 = 1, > b, foci: (±c, 0), c = 2 b 2. x 2 2 y2 b 2 =
More informationLine 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 informationChapter One: Calculus Revisited
Chpter One: Clculus Revisited 1 Clculus of Single Vrible Question in your mind: How do you understnd the essentil concepts nd theorems in Clculus? Two bsic concepts in Clculus re differentition nd integrtion
More informationJim Lambers MAT 280 Spring Semester Lecture 17 Notes. These notes correspond to Section 13.2 in Stewart and Section 7.2 in Marsden and Tromba.
Jim Lmbers MAT 28 Spring Semester 29- Lecture 7 Notes These notes correspond to Section 3.2 in Stewrt nd Section 7.2 in Mrsden nd Tromb. Line Integrls Recll from single-vrible clclus tht if constnt force
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
More informationRecitation 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
More information10 Vector Integral Calculus
Vector Integrl lculus Vector integrl clculus extends integrls s known from clculus to integrls over curves ("line integrls"), surfces ("surfce integrls") nd solids ("volume integrls"). These integrls hve
More informationMultiple Integrals. Review of Single Integrals. Planar Area. Volume of Solid of Revolution
Multiple Integrls eview of Single Integrls eding Trim 7.1 eview Appliction of Integrls: Are 7. eview Appliction of Integrls: Volumes 7.3 eview Appliction of Integrls: Lengths of Curves Assignment web pge
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationTable of Contents. 1. Limits The Formal Definition of a Limit The Squeeze Theorem Area of a Circle
Tble of Contents INTRODUCTION 5 CROSS REFERENCE TABLE 13 1. Limits 1 1.1 The Forml Definition of Limit 1. The Squeeze Theorem 34 1.3 Are of Circle 43. Derivtives 53.1 Eploring Tngent Lines 54. Men Vlue
More informationn 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
More information(6.5) Length and area in polar coordinates
86 Chpter 6 SLICING TECHNIQUES FURTHER APPLICATIONS Totl mss 6 x ρ(x)dx + x 6 x dx + 9 kg dx + 6 x dx oment bout origin 6 xρ(x)dx x x dx + x + x + ln x ( ) + ln 6 kg m x dx + 6 6 x x dx Centre of mss +
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re open-ended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnk-out-n-nswer problems! (There re plenty of those in the
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
More informationCalculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
More informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
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 informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationLecture 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
More informationHeat flux and total heat
Het flux nd totl het John McCun Mrch 14, 2017 1 Introduction Yesterdy (if I remember correctly) Ms. Prsd sked me question bout the condition of insulted boundry for the 1D het eqution, nd (bsed on glnce
More informationLecture Summaries for Multivariable Integral Calculus M52B
These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.
More informationAnti-derivatives/Indefinite Integrals of Basic Functions
Anti-derivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s one-minute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationA 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
More informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn
More informationFinal Exam - Review MATH Spring 2017
Finl Exm - Review MATH 5 - Spring 7 Chpter, 3, nd Sections 5.-5.5, 5.7 Finl Exm: Tuesdy 5/9, :3-7:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More informationMain 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
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 informationMATH 253 WORKSHEET 24 MORE INTEGRATION IN POLAR COORDINATES. r dr = = 4 = Here we used: (1) The half-angle formula cos 2 θ = 1 2
MATH 53 WORKSHEET MORE INTEGRATION IN POLAR COORDINATES ) Find the volume of the solid lying bove the xy-plne, below the prboloid x + y nd inside the cylinder x ) + y. ) We found lst time the set of points
More informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 4-6 in CASA Mteril - Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 14-17 in CASA You Might Be Interested
More informationr = cos θ + 1. dt ) dt. (1)
MTHE 7 Proble Set 5 Solutions (A Crdioid). Let C be the closed curve in R whose polr coordintes (r, θ) stisfy () Sketch the curve C. r = cos θ +. (b) Find pretriztion t (r(t), θ(t)), t [, b], of C in polr
More informationLine Integrals. Chapter Definition
hpter 2 Line Integrls 2.1 Definition When we re integrting function of one vrible, we integrte long n intervl on one of the xes. We now generlize this ide by integrting long ny curve in the xy-plne. It
More informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationRecitation 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
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 informationReview 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
More informationMA 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,
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationdf dx There is an infinite number of different paths from
Integrl clculus line integrls Feb 7, 18 From clculus, in the cse of single vrible x1 F F x F x f x dx, where f x 1 x df dx Now, consider the cse tht two vribles re t ply. Suppose,, df M x y dx N x y dy
More informationMath Advanced Calculus II
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
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More information1. Review. t 2. t 1. v w = vw cos( ) where is the angle between v and w. The above leads to the Schwarz inequality: v w vw.
1. Review 1.1. The Geometry of Curves. AprmetriccurveinR 3 is mp R R 3 t (t) = (x(t),y(t),z(t)). We sy tht is di erentile if x, y, z re di erentile. We sy tht it is C 1 if, in ddition, the derivtives re
More informationMathematics. Area under Curve.
Mthemtics Are under Curve www.testprepkrt.com Tle of Content 1. Introduction.. Procedure of Curve Sketching. 3. Sketching of Some common Curves. 4. Are of Bounded Regions. 5. Sign convention for finding
More informationdt. However, we might also be curious about dy
Section 0. The Clculus of Prmetric Curves Even though curve defined prmetricly my not be function, we cn still consider concepts such s rtes of chnge. However, the concepts will need specil tretment. For
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 information6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
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 informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
More informationCHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx
CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t
More information13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
More information