Section 4.3. By the Mean Value Theorem, for every i = 1, 2, 3,..., n, there exists a point c i in the interval [x i 1, x i ] such that


 Cory Kelley
 1 years ago
 Views:
Transcription
1 Difference Equtions to Differentil Equtions Section 4.3 The Fundmentl Theorem of Clculus We re now redy to mke the longpromised connection between differentition nd integrtion, between res nd tngent lines. We will look t two closely relted theorems, both of which re known s the Fundmentl Theorem of Clculus. We will cll the first of these the Fundmentl Theorem of Integrl Clculus. Suppose f is integrble on [, b] nd F is n ntiderivtive of f on (, b) which is continuous on [, b]. In prticulr, F () = f() for ll in (, b). Let P = {,,,..., n } be prtition of [, b] nd, s usul, let i = i i, i =,, 3,..., n. Now F (b) F () = F ( n ) F ( ) = F ( n ) + (F ( n ) F ( n )) + (F ( n ) F ( n )) + + (F ( ) F ( )) F ( ) = (F ( n ) F ( n )) + (F ( n ) F ( n )) + + (F ( ) F ( )) n = (F ( i ) F ( i )). i= (4.3.) By the Men Vlue Theorem, for every i =,, 3,..., n, there eists point c i in the intervl [ i, i ] such tht Since F (c i ) = f(c i ) nd i i = i, it follows tht Hence, putting (4.3.3) into (4.3.), F (c i ) = F ( i) F ( i ) i i. (4.3.) F ( i ) F ( i ) = f(c i ) i. (4.3.3) F (b) F () = n f(c i ) i. (4.3.4) i= Thus F (b) F () is equl to the vlue of Riemnn sum using the prtition P, nd so must lie between the upper nd lower sums for P. Tht is, we hve shown tht for ny prtition P, L(f, P ) F (b) F () U(f, P ). (4.3.5) Copyright c by Dn Sloughter
2 The Fundmentl Theorem of Clculus Section Figure 4.3. Region beneth the grph of f() = over the intervl [, ] But, since f is integrble, there is only one number tht hs this property, nmely, f()d. In other words, we hve shown tht f()d = F (b) F (). (4.3.6) Fundmentl Theorem of Integrl Clculus If f is integrble on [, b] nd F is n ntiderivtive of f on (, b) which is continuous on [, b], then f()d = F (b) F (). (4.3.7) This result revels sense in which integrtion is the inverse of differentition: The definite integrl of function f my be evluted esily, using (4.3.7), provided we cn find function F whose derivtive is f. It is common to write F () b for F (b) F (). With this nottion, (4.3.7) becomes f()d = F () b (4.3.8). Emple Since is n ntiderivtive of f() =, we hve F () = 3 3 d = 3 3 = 3 = 3. Thus the re under the prbol y = nd bove the intervl [, ] on the is is ectly 3. See Figure 4.3..
3 Section 4.3 The Fundmentl Theorem of Clculus Figure 4.3. Region beneth the grph of y = sin() over the intervl [, π] Note tht F in the previous emple is but one of n infinite number of ntiderivtives of f. We cn in fct use ny ntiderivtive of f we wnt in pplying (4.3.7), lthough we typiclly choose the simplest one we cn find. Emple Since G() = is n ntiderivtive of g() = + (you my check by differentiting G), we hve ( + )d = s we climed in Section 4.. ( ) = ( ) ( 3 ) = 6, Emple If A is the re under one rch of the curve y = sin(), then A = sin()d. Since F () = cos() is n ntiderivtive of f() = sin(), we hve A = sin()d = cos() π = cos(π) ( cos()) = + =. See Figure Emple Since F () =
4 4 The Fundmentl Theorem of Clculus Section 4.3 is n ntiderivtive of f() = 4 + (gin, you my check this by differentiting F ), we hve 3 ( 4 (4 + )d = 3 3 ) 3 + ( 8 = 3 9 ) + 6 ( 33 ) 4 = Emple Since is n ntiderivtive of f(t) = t, we hve 4 F (t) = 3 t 3 t dt = 3 t 3 4 = 6 3 = 6 3. As cn be seen from these emples, the Fundmentl Theorem of Integrl Clculus provides us with powerful tool for evluting definite integrls ectly. However, to utilize the theorem we must first find n ntiderivtive for the function we re integrting. This turns out to be difficult problem in generl, nd we will devote the net two sections, s well s prts of Chpter 6, to developing techniques to id in finding ntiderivtives. For emple, F () = 3 cos() sin() cos() 3 8 sin() is n ntiderivtive of f() = 3 sin(), s my be checked by differentition, but t this point it is not cler how to find such n ntiderivtive in the first plce. Moreover, there re integrble functions, even reltively simple ones such s f() = sin(), which do not hve ntiderivtives epressible in terms of the elementry functions studied in clculus. The Fundmentl Theorem of Integrl Clculus tells us tht if function f hs n ntiderivtive, then we my use tht ntiderivtive to evlute definite integrl of f, but it does not tell us which functions hve ntiderivtives. The Fundmentl Theorem of Differentil Clculus will tell us, in prt, tht every continuous function hs n ntiderivtive. Before beginning tht discussion, we need to etend the definition of the definite integrl slightly. The definition of f()d in Section 4. implicitly ssumes tht < b. For the work we re bout to do, we need to etend the definition to include b, s we did in Problem 9 of Section 4.. First of ll, if = b, it would seem resonble for the vlue of the definite
5 Section 4.3 The Fundmentl Theorem of Clculus 5 integrl to be since the region between the grph of the function nd the is hs been reduced to line segment. Hence we mke the following definition. Definition For ny function f defined t point, we define f()d =. (4.3.9) Note tht with this definition, the sttement f()d = f()d + c f()d, (4.3.) which we discussed in Section 4. in the cse < c < b, holds true even if = c, b = c, or = b = c. Now suppose we hve < b < c. Then from which it follows tht If we define f()d = f()d = c f()d + f()d b b f()d, (4.3.) f()d. (4.3.) f()d = f()d, (4.3.3) b then we my rewrite (4.3.) in the form of (4.3.). For this reson, we mke the following definition. Definition If b < nd f is integrble on [b, ], we define f()d = f()d. (4.3.4) b You my check tht with these two etensions to the definition of the definite integrl, we my now stte the following proposition. Proposition If f is integrble on closed intervl contining the points, b, nd c, then f()d = f()d + c f()d. (4.3.5)
6 6 The Fundmentl Theorem of Clculus Section 4.3 y = f( t) b Figure F () = f(t)dt is the re from to We my now return to our discussion of ntiderivtives nd the Fundmentl Theorem of Differentil Clculus. Suppose f is continuous on the intervl [, b]. We wnt to construct n ntiderivtive for f on (, b). From the Fundmentl Theorem of Integrl Clculus, we know tht if F is n ntiderivtive of f on (, b) which is continuous on [, b], then for ny in (, b) we would hve tht is, f(t)dt = F () F (), (4.3.6) F () = F () + f(t)dt. (4.3.7) Hence, if we re seeking n ntiderivtive for f, it mkes sense to define F () = f(t)dt (4.3.8) nd verify tht F () = f() for ll in (, b). Note tht F (), geometriclly, is the cumultive re between the grph of f nd the is from to, s shown in Figure We need to compute ( F F ( + h) F () +h ) () = lim = lim f(t)dt f(t)dt (4.3.9) h h h h for in (, b). Now so +h +h f(t)dt = f(t)dt f(t)dt + f(t)dt = +h +h f(t)dt, (4.3.) f(t)dt. (4.3.)
7 Section 4.3 The Fundmentl Theorem of Clculus 7 y = f( t) + h b Figure h f(t)dt f(t)dt = +h f(t)dt See Figure Thus F +h () = lim f(t)dt. (4.3.) h h Suppose h >. Since f is continuous, f hs minimum vlue m(h) nd mimum vlue M(h) on the intervl [, + h]. Hence m(h) f() M(h) for ll in [, + h], from which it follows tht +h m(h)dt +h f(t)dt Since m(h) nd M(h) re constnts, (4.3.3) implies Thus m(h)h m(h) h +h +h +h M(h)dt. (4.3.3) f(t)dt M(h)h. (4.3.4) f(t)dt M(h). (4.3.5) Now m(h) = f(c) for some c in [, + h]. Moreover, s h pproches, + h pproches, nd so c must lso pproch. Hence, since f is continuous, lim h h m(h) = lim f(c) = f(). (4.3.6) + + Similrly, lim M(h) = f(). (4.3.7) h +
8 8 The Fundmentl Theorem of Clculus Section 4.3 It now follows from (4.3.5) tht A similr rgument shows tht nd so we my conclude tht +h lim f(t)dt = f(). (4.3.8) h + h +h lim f(t)dt = f(), (4.3.9) h h F +h () = lim f(t)dt = f(). (4.3.3) h h Tht is, F () = is n ntiderivtive of f on (, b). f(t)dt Fundmentl Theorem of Differentil Clculus intervl [, b] nd F is the function on (, b) defined by If f is continuous on the closed F () = f(t)dt, (4.3.3) then F is differentible on (, b) with F () = f() for ll in (, b). In other words, for ll in (, b). d f(t)dt = f() (4.3.3) d It is worth noting tht (4.3.3) holds for < s well, s long s f is continuous on closed intervl which contins both nd. Emple Let sin(), if, f() =, if =. Then f is continuous on (, ), so F () = f(t)dt
9 Section 4.3 The Fundmentl Theorem of Clculus Figure Grphs of f() = sin() nd F () = sin(t) t dt is n ntiderivtive of f on (, ). In prticulr, F () = sin() for ll. The grphs of F nd f re shown in Figure Geometriclly, F () is the cumultive re between the grph of f nd the is from to nd F () is the rte t which re is ccumulting s increses. Since the rte t which re is ccumulting depends on the height of the curve, it is nturl to epect, nd the Fundmentl Theorem of Differentil Clculus confirms, tht F () = f(). The function F is known s the sine integrl function. It my be shown tht it is not representble in closed form in terms of the elementry functions of clculus. Emple Emple Using Leibniz nottion, Suppose d d G() = sin(t )dt = sin( ). 3 Then G() = F (h()), where h() = 3 nd sin(t )dt. Hence, using the chin rule, F () = sin(t )dt. G () = F (h())h () = sin((3) )(3) = 3 sin(9 ). Emple Suppose H() = + t 4 dt.
10 The Fundmentl Theorem of Clculus Section 4.3 Then, using (4.3.4), so H() = H () = d d + t 4 dt, + t 4 dt = + 4. Emple Suppose Then, using (4.3.5) nd (4.3.4), F () = + t4 dt + F () = + t4 dt. + t4 dt = + t4 dt + + t4 dt. Note tht there is nothing specil bout using in this decomposition, other thn the requirement tht the function f(t) = + t 4 be integrble on ll of the relevnt intervls. Now we hve F () = d + t4 dt + d + t4 dt d d = + () 4 () + + ( ) 4 () = To summrize this section, the Fundmentl Theorem of Integrl Clculus provides us with n elegnt method for evluting definite integrls, but is useful only when we cn find n ntiderivtive for the function being integrted. The Fundmentl Theorem of Differentil Clculus tells us tht every continuous function hs n ntiderivtive nd shows how to construct one using the definite integrl. Unfortuntely, this brings us in circle nd does not provide us with n effective mens for finding ntiderivtives to use in pplying the Fundmentl Theorem of Integrl Clculus. For emple, we know tht F () = sin(t) t is n ntiderivtive of f() = sin(), but this is of no help in evluting, sy, 4 sin() Hence in order to fully utilize the Fundmentl Theorem of Integrl Clculus in the evlution of definite integrls, we must develop some procedures to id in finding ntiderivtives. We will turn to this problem in the net section. d. dt
11 Section 4.3 The Fundmentl Theorem of Clculus Problems. Evlute the following definite integrls using the Fundmentl Theorem of Integrl Clculus. () (c) (e) (g) (i) 4 d 3 d (b) (d) ( )d (f) t dt sin()d (h) (j) π ( + )d 3 d d t + dt cos(z)dz. Evlute the following definite integrls using the Fundmentl Theorem of Integrl Clculus. () (c) (e) (g) (i) (k) 4 3 π π ( + ) d (b) + t dt sin()d (d) 4 (f) 4 sin(3)d (h) sin( )d sin( )d (j) π 3. For ech of the following functions, grph both f nd together over the given intervl. F () = (l) f(t)dt ( + ) d sec ()d 5 cos(3)d 8 cos(5θ)dθ ( + ) 5 d ( + ) 5 d () f() = sin() on [ π, π] (b) f() = sin( ) on [, ] (c) f() = on [ 3, 3] (d) f() = on [, ] + 4 +
12 The Fundmentl Theorem of Clculus Section Find the derivtives of ech of the following functions. () F () = (c) F () = (e) f() = sin (4t)dt (b) g() = 5. Evlute the following derivtives. () (c) d d d dt 5 cos 3 (t)dt (d) G(t) = ds (f) h(z) = + s + t dt (b) d d t sin (3)d (d) d d 3 3 t 3z z 3 t + dt 4 z dz sin(3t) t + t dt dt + t dt 6. Find the re of the region beneth one rch of the curve y = 3 sin(). 7. Let R be the region bounded by the curves y = nd y = ( ) nd the is. Find the re of R. 8. Eplin why the integrl ( )d is the re of the region bounded by the curves y = nd y =. Find this re. 9. Eplin why the integrl ( )d is the re of the region bounded by the curves y = nd y =. Find this re.
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
More information1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
More informationCurve Sketching. 96 Chapter 5 Curve Sketching
96 Chpter 5 Curve Sketching 5 Curve Sketching A B A B A Figure 51 Some locl mximum points (A) nd minimum points (B) If (x, f(x)) is point where f(x) reches locl mximum or minimum, nd if the derivtive of
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationIntroduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationDIFFERENTIATING UNDER THE INTEGRAL SIGN
DIFFEENTIATING UNDE THE INTEGAL SIGN KEITH CONAD I hd lerned to do integrls by vrious methods shown in book tht my high school physics techer Mr. Bder hd given me. [It] showed how to differentite prmeters
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationThe Riemann Integral. Chapter 1
Chpter The Riemnn Integrl now of some universities in Englnd where the Lebesgue integrl is tught in the first yer of mthemtics degree insted of the Riemnn integrl, but now of no universities in Englnd
More informationMath Review 1. , where α (alpha) is a constant between 0 and 1, is one specific functional form for the general production function.
Mth Review Vribles, Constnts nd Functions A vrible is mthemticl bbrevition for concept For emple in economics, the vrible Y usully represents the level of output of firm or the GDP of n economy, while
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationThe invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.
Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationPhysics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nmwide region t x
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationAll pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 12015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
More informationThe Fundamental Theorem of Calculus
Section 5.4 Te Funmentl Teorem of Clculus Kiryl Tsiscnk Te Funmentl Teorem of Clculus EXAMPLE: If f is function wose grp is sown below n g() = f(t)t, fin te vlues of g(), g(), g(), g(3), g(4), n g(5).
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationExponential and Logarithmic Functions
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
More informationOr more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
More informationReal Analysis and Multivariable Calculus: Graduate Level Problems and Solutions. Igor Yanovsky
Rel Anlysis nd Multivrible Clculus: Grdute Level Problems nd Solutions Igor Ynovsky 1 Rel Anlysis nd Multivrible Clculus Igor Ynovsky, 2005 2 Disclimer: This hndbook is intended to ssist grdute students
More informationSPECIAL PRODUCTS AND FACTORIZATION
MODULE  Specil Products nd Fctoriztion 4 SPECIAL PRODUCTS AND FACTORIZATION In n erlier lesson you hve lernt multipliction of lgebric epressions, prticulrly polynomils. In the study of lgebr, we come
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationAlgebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationThe Fundamental Theorem of Calculus for Lebesgue Integral
Divulgciones Mtemátics Vol. 8 No. 1 (2000), pp. 75 85 The Fundmentl Theorem of Clculus for Lebesgue Integrl El Teorem Fundmentl del Cálculo pr l Integrl de Lebesgue Diómedes Bárcens (brcens@ciens.ul.ve)
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More information2.4 Circular Waveguide
.4 Circulr Wveguide y x Figure.5: A circulr wveguide of rdius. For circulr wveguide of rdius (Fig..5, we cn perform the sme sequence of steps in cylindricl coordintes s we did in rectngulr coordintes to
More informationSection A4 Rational Expressions: Basic Operations
A Appendi A A BASIC ALGEBRA REVIEW 7. Construction. A rectngulr opentopped bo is to be constructed out of 9 by 6inch sheets of thin crdbord by cutting inch squres out of ech corner nd bending the
More informationNumerical Methods of Approximating Definite Integrals
6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,
More informationCHAPTER 11 Numerical Differentiation and Integration
CHAPTER 11 Numericl Differentition nd Integrtion Differentition nd integrtion re bsic mthemticl opertions with wide rnge of pplictions in mny res of science. It is therefore importnt to hve good methods
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationOperations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationHarvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More informationPure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationPROBLEMS 13  APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS  APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
More informationReal Analysis HW 10 Solutions
Rel Anlysis HW 10 Solutions Problem 47: Show tht funtion f is bsolutely ontinuous on [, b if nd only if for eh ɛ > 0, there is δ > 0 suh tht for every finite disjoint olletion {( k, b k )} n of open intervls
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More informationApplications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
More informationRIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
More informationNull Similar Curves with Variable Transformations in Minkowski 3space
Null Similr Curves with Vrile Trnsformtions in Minkowski spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. mil: mehmet.onder@yr.edu.tr
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationCURVES ANDRÉ NEVES. that is, the curve α has finite length. v = p q p q. a i.e., the curve of smallest length connecting p to q is a straight line.
CURVES ANDRÉ NEVES 1. Problems (1) (Ex 1 of 1.3 of Do Crmo) Show tht the tngent line to the curve α(t) (3t, 3t 2, 2t 3 ) mkes constnt ngle with the line z x, y. (2) (Ex 6 of 1.3 of Do Crmo) Let α(t) (e
More informationThe Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More information1. Area under a curve region bounded by the given function, vertical lines and the x axis.
Ares y Integrtion. Are uner urve region oune y the given funtion, vertil lines n the is.. Are uner urve region oune y the given funtion, horizontl lines n the y is.. Are etween urves efine y two given
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationExam 1 Study Guide. Differentiation and Antidifferentiation Rules from Calculus I
Exm Stuy Guie Mth 2020  Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the
More information3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
More informationVolumes of solids of revolution
Volumes of solids of revolution We sometimes need to clculte the volume of solid which cn be obtined by rotting curve bout the xxis. There is strightforwrd technique which enbles this to be done, using
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationMath 135 Circles and Completing the Square Examples
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
More informationPhysics 6010, Fall 2010 Symmetries and Conservation Laws: Energy, Momentum and Angular Momentum Relevant Sections in Text: 2.6, 2.
Physics 6010, Fll 2010 Symmetries nd Conservtion Lws: Energy, Momentum nd Angulr Momentum Relevnt Sections in Text: 2.6, 2.7 Symmetries nd Conservtion Lws By conservtion lw we men quntity constructed from
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationTHE RATIONAL NUMBERS CHAPTER
CHAPTER THE RATIONAL NUMBERS When divided by b is not n integer, the quotient is frction.the Bbylonins, who used number system bsed on 60, epressed the quotients: 0 8 s 0 60 insted of 8 s 7 60,600 0 insted
More informationRadius of the Earth  Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth  dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
More information6.5  Areas of Surfaces of Revolution and the Theorems of Pappus
Lecture_06_05.n 1 6.5  Ares of Surfces of Revolution n the Theorems of Pppus Introuction Suppose we rotte some curve out line to otin surfce, we cn use efinite integrl to clculte the re of the surfce.
More informationDIFFERENTIAL FORMS AND INTEGRATION
DIFFERENTIAL FORMS AND INTEGRATION TERENCE TAO The concept of integrtion is of course fundmentl in singlevrible clculus. Actully, there re three concepts of integrtion which pper in the subject: the indefinite
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationAA1H Calculus Notes Math1115, Honours 1 1998. John Hutchinson
AA1H Clculus Notes Mth1115, Honours 1 1998 John Hutchinson Author ddress: Deprtment of Mthemtics, School of Mthemticl Sciences, Austrlin Ntionl University Emil ddress: John.Hutchinson@nu.edu.u Contents
More informationCOMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
More informationINTERCHANGING TWO LIMITS. Zoran Kadelburg and Milosav M. Marjanović
THE TEACHING OF MATHEMATICS 2005, Vol. VIII, 1, pp. 15 29 INTERCHANGING TWO LIMITS Zorn Kdelburg nd Milosv M. Mrjnović This pper is dedicted to the memory of our illustrious professor of nlysis Slobodn
More informationMath 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
More information2 DIODE CLIPPING and CLAMPING CIRCUITS
2 DIODE CLIPPING nd CLAMPING CIRCUITS 2.1 Ojectives Understnding the operting principle of diode clipping circuit Understnding the operting principle of clmping circuit Understnding the wveform chnge of
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More information