Recitation 3: More Applications of the Derivative


 Tamsyn Ross
 5 years ago
 Views:
Transcription
1 Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech edge consists of distinct unordered pir of distinct vertices For exmple the pentgon 4 3 cn be thought of s the grph with V = { } E = { {1 2} {2 3} {3 4} {4 5} {5 1} }. Given grph G nd vertex v V we cn define the degree of v to be the number of edges tht hve v s one of their endpoints. For exmple every vertex in the pentgon bove hs degree 2 s ech vertex is the endpoint for precisely two edges. A tringle decomposition of grph G is wy to brek its edge set E prt into disjoint tringles. It s not too hrd to show tht if tringle decomposition exists then every vertex needs to hve even degree: this is becuse ech of tringle s vertices hs precisely two edges from the tringle hitting it so if we ve broken ll of our edges into tringles every vertex hs degree equl to 2 (the number of tringles tht use tht vertex) which is even. The number of edges in E hs to be multiple of 3 becuse ech tringle uses three edges nd they re ll disjoint. Suppose tht G is grph on n vertices 1. (Known not too hrd:) Show tht if the degree of every vertex is n 1 (i.e. every vertex is connected to every other vertex) these conditions re lso sufficient: i.e. if ll of the degrees re even nd the number of edges is multiple of 3 tringle decomposition exists. 2. (Known firly hrd:) Show tht if the degree of every vertex is ( ) n i.e. the vertices re connected to lmost every other vertex these conditions re still sufficient. 1
2 3. (Current reserch results I m working on) Show tht if the degree of every vertex is ( ) n then these conditions re still sufficient. 4. (Conjecture probbly wildly difficult) Show tht if the degree of every vertex is 3 4 n then these conditions re still sufficient. 2 Directionl Derivtives In our lst clss we spent lot of time working on the ide of derivtive in R n. Specificlly we introduced the ide of prtil derivtives used these to construct the totl derivtive ( liner pproximtion to our function t point which ws similr to the liner pproximtion ide tht we hd for the derivtive in R 1 ) nd discussed some of the geometric interprettions nd pplictions of these ides. However these techniques were not the only wys to tlk bout the derivtive! Another different wy is to generlize the ide of slope from R 1 vi the concept of the directionl derivtive: we discuss how to do this below. In R 1 the derivtive of function f : R R t some point is the slope of the grph f(x) = y t the point ( f()): essentilly we re mesuring the chnge in f(x) s we move long the xxis. However for functions R n R we re no longer restricted to just the xxis; insted we cn move long ny vector in R n! This leds us to define the directionl derivtive of function f : R n R t some point long some direction v s the slope of f t the point s mesured in the direction v. More formlly: Definition. The directionl derivtive of function f : R n R t some point long some direction v is the derivtive f (; v) := d dt (f( + t v). To illustrte wht s going on here consider the following exmple: Question 2 Consider the function f(x y) = x 2 + y 2. Wht is the directionl derivtive of this function t the point (0 1) in the direction (0 1)? Solution. First to get good ide of wht s going on in this problem we grph our function: z y x f (;v) v 2
3 Visully if we look t the point (0 1) nd its slope in the direction (0 1) we cn see tht it should be 1 just by exmintion. So let s clculte nd see if our visul intuition mtches our mthemticl definition: f ((0 1); (0 1)) = d dt (f((0 1) + t (0 1))) = d dt (f(0 t 1)) = d dt ( (t 1) 2 ) = d dt ( t 1 ) = d dt ( ( (t 1))) [ becuse ner 0 (t 1) is negtive] = d dt (t 1) = 1 = 1. t=0 This mtches our visul picture nd our intuition. The following result cn mke clculting the directionl derivtive esier in the cse tht we lredy know the grdient of function: Theorem 3 Suppose tht f is differentible t some point : one notble cse where this hppens is when ll of f s prtil derivtives re continuous t s we mentioned lst clss. Then the directionl derivtive of function f : R n R t some point long some direction v is given by the dot product of the grdient of f t with v/ v. In other words f v (; v) := f v. To illustrte the use of this theorem return to our cone problem from erlier. There we hd f(x y) = x 2 + y 2 ; thus if we hold y constnt we cn see tht Similrly by holding y constnt we hve f x = 2x 2 x 2 + y = x 2 x 2 + y. 2 f y = y x 2 + y 2. 3
4 Therefore we know tht the directionl derivtive of f t (0 1) in the direction (0 1) is given by ( ) ( ) f f (0) (0 1) (0 1) (0 1) = x y ( 1) ( 1) (0 1) ( 1) 2 = (0 1) (0 1) = 1 which mtches our erlier nswer. 3 HigherOrder Derivtives nd their Applictions Another thing we could wnt to do with the derivtive motivted by wht we were ble to do in R 1 is the concept of higherorder derivtives. These re reltively esy to define for prtil derivtives: Definition. Given function f : R n R we cn define its secondorder prtil derivtives s the following: = ( ) f. x i x j x i x j In other words the secondorder prtil derivtives re simply ll of the functions you cn get by tking two consecutive prtil derivtives of your function f. A useful theorem for clculting these prtil derivtives is the following: Theorem 4 A function f : R n R is clled C 2 t some point if ll of its secondorder prtil derivtives re continuous t tht point. If function is C 2 then the order in which secondorder prtil derivtives re clculted doesn t mtter: i.e. for ny i j. x i x j = x j x i It bers noting tht if the conditions of this theorem re not met then the order for computing secondorder prtil derivtives my ctully mtter! One such exmple is the function { x 3 y xy 3 (x y) (0 0) f(x y) = x 2 +y 2 0 (x y) = (0 0) At (0 0) you cn clculte tht xy = 1 1 = yx : result tht occurs becuse the secondorder prtils of this function re not continuous. 4
5 However the interesting spects of higherorder prtil derivtives re not relly in their clcultion; rther the pplictions of higherorder prtil derivtives re the things worth studying. In R for exmple we could turn the second derivtive of function into lot of informtion bout tht function: in prticulr we could use this second derivtive to determine whether given criticl point ws locl minim or mxim whether function is concve up or down t given point nd wht the secondorder Tylor pproximtion to tht function ws t point. Cn we do the sme for functions from R n to R? As it turns out the nswer is yes! The tool with which we do this is clled the Hessin which we define here: Definition. The Hessin of function f : R n R t some point H(f) (h) is the following function from R n to R: H(f) (h) = 1 x 2 (h 1 x 1 ()... h x 1 x n () 1 1 h 2... h n ).... h 2. x nx 1 ().... x nx n () h n The min useful property of the Hessin is the following: Theorem 5 Let f : R n R be function with welldefined secondorder prtils t some point nd H = H(f) be its Hessin. Pick ny two co ordintes x i x j in R n : then x i x j () = 2 H h i h j. In other words H s secondorder prtil derivtive re precisely the secondorder prtil derivtives of f t! So H is bsiclly function designed to hve the sme secondorder prtils s f t. One quick thing this theorem suggests is tht we could use H(f) to crete secondorder pproximtion to f t in similr fshion to how we used the derivtive to crete liner (i.e. firstorder) pproximtion to f. We define this below: Theorem 6 If f : R n R is function with continuous secondorder prtils we define the secondorder Tylor pproximtion to f t s the function T 2 (f) ( + h) = f() + ( f)() h + H(f) (h). You cn think of f() s the constnt or zeroth order prt ( f)() h s the liner prt nd H(f) (h) s the secondorder prt of this pproximtion. To illustrte how this process ctully cretes pretty decent pproximtion to f we clculte n exmple: 5
6 Exmple. Clculte the secondorder Tylor pproximtion to the function f(x y) = e xy t the origin. Answer. Clculting the second derivtives of f is pretty strightforwrd: f x = yexy f y = xexy x 2 = y2 e xy y 2 = x2 e xy xy = xyexy + e xy = yx. If we evlute these prtils t 0 nd plug them into the definition bove for T 2 (f) (00) we get T 2 (f) (00) ((0 0) + (h 1 h 2 )) = f(0 0) + ( f)(0 0) (h 1 h 2 ) + H(f) (00) (h 1 h 2 ) = 1 + (0 0) (h 1 h 2 ) + 1 ( ) ( ) (h h1 1 h 2 ) 1 0 h 2 = ( ) 2 (h h2 1 h 2 ) h 1 = (2h 1h 2 ) = 1 + h 1 h 2. So t the origin the secondorder Tylor pproximtion for f is just T 2 (f)(x y) = 1 + xy. The following grph with e xy in solid red nd T 2 in dshed blue shows tht it s ctully somewht decent pproximtion t (0 0): z y x 6
7 As well we cn use the second derivtives to serch for nd find locl minim nd mxim! We define these terms here: Definition. A point R n is clled locl mxim of function f : R n R iff there is some smll vlue r such tht for ny point x in B (r) not equl to we hve f(x) f(). A similr definition holds for locl minim. So: how cn we use the derivtive to find such locl mxim? Well it s cler tht (if our function is differentible in neighborhood round this point) tht no mtter how we move to leve this point our function must not increse in other words for ny direction v R n the directionl derivtive f ( v) must be 0. But this mens tht in fct ll of the directionl derivtives must be equl to 0! becuse if f ( v) ws < 0 then f ( v) would be > 0. This motivtes the following definitions nd bsiclly proves the following theorem: Definition. A point is clled sttionry point of some function f : R n R iff (f) = (0... 0). A point is clled criticl point if it is sttionry point or f is not differentible in ny neighborhood of. Theorem 7 A function f : R n R ttins its locl mxim nd minim only t criticl points. However it bers noting tht not every criticl or sttionry point is locl mxim or minim! A trivil exmple would be the function f(x y) = x 2 y 2 : the origin is sttionry point yet neither locl minim or mxim (s f(0 ɛ) < 0 < f(ɛ 0) nd thus there re positive nd negtive vlues of f ttined in ny bll round the origin where it is 0.) How cn we tell which sttionry points do wht? Well in onevrible clculus we used the ide of the second derivtive to determine wht ws going on! In specific we knew tht if the second derivtive of function f t some point ws negtive then tiny increses in our vrible t tht point would cuse the first derivtive to decrese nd tiny decreses in our vrible t tht point would cuse the negtive of the first derivtive to increse i.e. cuse the first derivtive to decrese nd therefore mke the function itself decrese! Therefore the second derivtive being negtive t sttionry point implied tht tht point ws locl mxim. In higher dimensions things re tricker t given point we no longer hve this ide of single second derivtive but insted hve mny different second derivtives like xy () nd (). Yet we cn still use the sme ides s before to figure out wht s going z 2 on! In prticulr in one dimension we sid tht we wnted tiny positive chnges of our vribles to mke the first functions decrese. In other words given ny of the prtils f x i we wnt ny positive chnges in the direction of this prtil to mke our function decrese i.e. we wnt the directionl derivtive of f x i to be negtive in ny direction v where ll of the coördintes of v re positive. (Positivity here stems from the sme reson tht in one dimension we hve tht the first derivtive is incresing for ll of the points to the left of mxim nd decresing for ll of the points to the right of mxim.) 7
8 So: this condition if we write it out is just sking tht for every i nd nonzero v tht ( 2 ) f () ()... () (v x 1 x i x 2 x i x n x 1 2 v vn) 2 i is negtive. If you choose to write this out s mtrix this ctully becomes the clim tht for ny v 0 we hve x v T 1 x 1 ()... x 1 x n ()..... v < 0. x nx 1 ()... x nx n () liner lgebr you my hopefully remember tht ny mtrix stisfying this condition is clled being negtivedefinite nd is equivlent to hving ll n of your eigenvlues existing nd being negtive. Similrly if we were looking for locl minim we would be sking tht the bove mtrix product is lwys positive: i.e. tht the mtrix is positivedefinite which is equivlent to ll of its eigenvlues being positive. But we ve seen this construction before! In prticulr this mtrixproduct thing bove is just the Hessin! Bsed on these observtions we mke the following definitions nd observtions: Definition. The Hessin H(f) of function f t some point is clled positivedefinite if for ll h 0 H(f) (h) > 0 nd similrly tht the Hessin is negtivedefinite if for ll h 0 H(f) (h) < 0. Theorem 8 The Hessin H(f) is positivedefinite if nd only if the mtrix x 1 x 1 ()... x 1 x n ()..... x nx 1 ()... x nx n () is positivedefinite. (The sme reltion holds for being negtivedefinite.) Theorem 9 A function f : R n R hs locl mxim t sttionry point if ll of its secondorder prtils exist nd re continuous in neighborhood of nd the Hesssin of f is negtivedefinite t. Similrly it hs locl minim if the Hessin is positivedefinite t. If the Hessin tkes on both positive nd negtive vlues there it s sddle point: there re directions you cn trvel where your function increse nd others where it will decrese. Finlly if the Hessin is identiclly 0 you hve no informtion s to wht your function my be up to: you could be in ny of the three bove cses. A quick exmple to illustrte how this gets used: 8
9 Exmple. For f(x y) = x 2 + y 2 g(x y) = x 2 y 2 nd h(x y) = x 2 y 2 find locl minim nd mxim. Solution. First by tking prtils it is cler tht the only point t which the grdient of these functions is (0 0) is the origin. There we hve tht H(f) = (00) [ ] H(g) = (00) [ ] H(h) = (00) [ nd thus tht f is positivedefinite t (00) g is negtivedefinite t (00) nd h is neither t (00) by exmining the eigenvlues. Thus f hs locl minim t (0 0) g hs locl mxim t (00) nd h hs sddle point t (00). ] 9
Recitation 3: Applications of the Derivative. 1 HigherOrder Derivatives and their Applications
Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 HigherOrder Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht
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 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 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 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 informationQuadratic Forms. Quadratic Forms
Qudrtic Forms Recll the Simon & Blume excerpt from n erlier lecture which sid tht the min tsk of clculus is to pproximte nonliner functions with liner functions. It s ctully more ccurte to sy tht we pproximte
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 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 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 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 informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
More informationHow do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?
XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk bout solving systems of liner equtions. These re problems tht give couple of equtions with couple of unknowns, like: 6 2 3 7 4
More informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
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, 3pm6pm, 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 informationAbstract inner product spaces
WEEK 4 Abstrct inner product spces Definition An inner product spce is vector spce V over the rel field R equipped with rule for multiplying vectors, such tht the product of two vectors is sclr, nd the
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 informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
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 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 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 informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
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 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 informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationRiemann is the Mann! (But Lebesgue may besgue to differ.)
Riemnn is the Mnn! (But Lebesgue my besgue to differ.) Leo Livshits My 2, 2008 1 For finite intervls in R We hve seen in clss tht every continuous function f : [, b] R hs the property tht for every ɛ >
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 informationLecture 2: Fields, Formally
Mth 08 Lecture 2: Fields, Formlly Professor: Pdric Brtlett Week UCSB 203 In our first lecture, we studied R, the rel numbers. In prticulr, we exmined how the rel numbers intercted with the opertions of
More informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
More informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 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 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 righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
More informationODE: Existence and Uniqueness of a Solution
Mth 22 Fll 213 Jerry Kzdn ODE: Existence nd Uniqueness of Solution The Fundmentl Theorem of Clculus tells us how to solve the ordinry differentil eqution (ODE) du = f(t) dt with initil condition u() =
More informationInfinite Geometric Series
Infinite Geometric Series Finite Geometric Series ( finite SUM) Let 0 < r < 1, nd let n be positive integer. Consider the finite sum It turns out there is simple lgebric expression tht is equivlent to
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
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 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 informationMAA 4212 Improper Integrals
Notes by Dvid Groisser, Copyright c 1995; revised 2002, 2009, 2014 MAA 4212 Improper Integrls The Riemnn integrl, while perfectly welldefined, is too restrictive for mny purposes; there re functions which
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 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 informationLecture 3: Equivalence Relations
Mthcmp Crsh Course Instructor: Pdric Brtlett Lecture 3: Equivlence Reltions Week 1 Mthcmp 2014 In our lst three tlks of this clss, we shift the focus of our tlks from proof techniques to proof concepts
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 informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
More informationRiemann Integrals and the Fundamental Theorem of Calculus
Riemnn Integrls nd the Fundmentl Theorem of Clculus Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University September 16, 2013 Outline Grphing Riemnn Sums
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More information1 Probability Density Functions
Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our
More informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
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 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 informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
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 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 informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
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 informationCS667 Lecture 6: Monte Carlo Integration 02/10/05
CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of
More informationLecture 1: Introduction to integration theory and bounded variation
Lecture 1: Introduction to integrtion theory nd bounded vrition Wht is this course bout? Integrtion theory. The first question you might hve is why there is nything you need to lern bout integrtion. You
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
More informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
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 informationSection 6.1 INTRO to LAPLACE TRANSFORMS
Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform
More informationAdvanced Calculus: MATH 410 Notes on Integrals and Integrability Professor David Levermore 17 October 2004
Advnced Clculus: MATH 410 Notes on Integrls nd Integrbility Professor Dvid Levermore 17 October 2004 1. Definite Integrls In this section we revisit the definite integrl tht you were introduced to when
More informationCS 188 Introduction to Artificial Intelligence Fall 2018 Note 7
CS 188 Introduction to Artificil Intelligence Fll 2018 Note 7 These lecture notes re hevily bsed on notes originlly written by Nikhil Shrm. Decision Networks In the third note, we lerned bout gme trees
More informationChapter 14. Matrix Representations of Linear Transformations
Chpter 4 Mtrix Representtions of Liner Trnsformtions When considering the Het Stte Evolution, we found tht we could describe this process using multipliction by mtrix. This ws nice becuse computers cn
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 informationSYDE 112, LECTURES 3 & 4: The Fundamental Theorem of Calculus
SYDE 112, LECTURES & 4: The Fundmentl Theorem of Clculus So fr we hve introduced two new concepts in this course: ntidifferentition nd Riemnn sums. It turns out tht these quntities re relted, but it is
More informationBest Approximation in the 2norm
Jim Lmbers MAT 77 Fll Semester 111 Lecture 1 Notes These notes correspond to Sections 9. nd 9.3 in the text. Best Approximtion in the norm Suppose tht we wish to obtin function f n (x) tht is liner combintion
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 informationInterpreting Integrals and the Fundamental Theorem
Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of
More information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationdifferent methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).
Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
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 informationNumerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
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 informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
More informationReview of Riemann Integral
1 Review of Riemnn Integrl In this chpter we review the definition of Riemnn integrl of bounded function f : [, b] R, nd point out its limittions so s to be convinced of the necessity of more generl integrl.
More information8 Laplace s Method and Local Limit Theorems
8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
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 informationMore on automata. Michael George. March 24 April 7, 2014
More on utomt Michel George Mrch 24 April 7, 2014 1 Automt constructions Now tht we hve forml model of mchine, it is useful to mke some generl constructions. 1.1 DFA Union / Product construction Suppose
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 informationIntroduction to the Calculus of Variations
Introduction to the Clculus of Vritions Jim Fischer Mrch 20, 1999 Abstrct This is selfcontined pper which introduces fundmentl problem in the clculus of vritions, the problem of finding extreme vlues
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
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 informationLecture 19: Continuous Least Squares Approximation
Lecture 19: Continuous Lest Squres Approximtion 33 Continuous lest squres pproximtion We begn 31 with the problem of pproximting some f C[, b] with polynomil p P n t the discrete points x, x 1,, x m for
More informationChapter 5 : Continuous Random Variables
STAT/MATH 395 A  PROBABILITY II UW Winter Qurter 216 Néhémy Lim Chpter 5 : Continuous Rndom Vribles Nottions. N {, 1, 2,...}, set of nturl numbers (i.e. ll nonnegtive integers); N {1, 2,...}, set of ll
More information1. GaussJacobi quadrature and Legendre polynomials. p(t)w(t)dt, p {p(x 0 ),...p(x n )} p(t)w(t)dt = w k p(x k ),
1. GussJcobi qudrture nd Legendre polynomils Simpson s rule for evluting n integrl f(t)dt gives the correct nswer with error of bout O(n 4 ) (with constnt tht depends on f, in prticulr, it depends on
More informationDuality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.
Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we
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 information