# Test , 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

Save this PDF as:

Size: px
Start display at page:

Download "Test , 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"

## Transcription

1 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 integrtion vi rectngles re between two curves vi rectngles volume by cylindricl disk or rectngulr box slices totl work vi the work for ech slice = force for ech slice displcement of tht slice series diverges when sequence terms do not get smller. (when they do get smller nything cn hppen)

2 8. Are nd Volume (Slice nd Conquer) Are by slicing into rectngles with known length Volume by slicing into regions we know the re of Riemnn sums with x or y b dx or b dy π( 2 5 y i) 2 y 5( 2 5 y)2 dy 0 Wht I wnt you to show me... picture, slice, Riemnn sum, integrl

3 8. Are Steps Sketch grph of the functions to find the enclosed region 2 Sketch picture of Riemnn slice on your grph. 3 Bse of the rectngle? Circle: x or y 4 Which function is lrger in tht vrible (top for x, right for y)? 5 Wht is the height of the rectngle (top-bottom or right-left)? 6 Wht is the Riemnn sum pproximtion? height bse = 7 Wht is nd b (lgebr finds the intersection points)? 8 Write the integrl?

4 8.2 Volume Steps Sketch grph of the object you wnt to find the volume of 2 Sketch picture of Riemnn slice on your grph 3 Wht shpe is it? Circle: box (length width height) or cylinder/disk (π rdius 2 height) 4 Infinitesiml prt of the slice? Circle: x or y or h 5 Sketch digrm nd show work to solve for ny lengths you need 6 Circle ny we used: Pythgoren theorem or similr tringles 7 Wht is the Riemnn sum pproximtion? 8 Wht is nd b? 9 Write the integrl?

5 8.2 Volume (Revolutions) nd Arc Length Volume by revolving region bout n xis. Slice. Riemnn sums with x or y b dx or b dy π( 2 5 y i) 2 y 0 5( 2 5 y)2 dy Common forms: b b πr 2 dx nd π(r 2 outer r 2 inner )dx = b πr 2 outer dx b Key is to figure out the rdius (or rdii) vi pics πr 2 inner dx Wht I wnt you to show me... resoning for rdius, integrl

6 8.2 Volume (Revolutions) nd Arc Length Pythgoren looks good until we see BOTH x, y y x slope = f (x), so y f (x) x rc length x 2 + (f (x) x) 2 = + (f (x)) 2 x b rc length = + (f (x)) 2 dx Wht I wnt you to show me... f, integrl

7 8.4 Vrying Density Density over length, re or volume, chnging only in dimension (Clc III for others) Substnce mss with density vrying over volume g/cm 3 Popultion quntity like people/squre mile, bcteri/cc. Slice so density is pproximtely constnt: If δ = f (x), then slice x, for constnt density slices mss= density length x, re, or volume δ(x) dx If δ = f (r), then slice from center outwrd, rings popultion= density re b 2πrδ(r) dr Figure out slicing vrible, then δ length or n δ 8./8.2 re or volume in tht vrible

8 8.5 Work: Vrying Force Work is force distnce displced only pplies if the force is constnt while it is exerted over the distnce Integrls pply when we vry the force. The ide is to slice so tht the force is pproximtely constnt on slice for its displcement. Then like Hook s Lw to stretch (nd hold) spring, where F(x) = kx constnt for displcement x nd W = F(x)dx Sometimes need to clculte the force, like when it is column of wter: mss = density volume, F = mss g Often we won t need to multiply by g like when we hve density tht lredy hs force component: weight (force in lbs) = volume of slice 62.4 lbs/ft 3 work on slice = volume 62.4 lbs/ft 3 slice displcement

9 Slicing for Volume, Density nd Work Prctice Sheet cylindricl disk volume = π rdius of slice 2 y 2 totl cone volume: 5 0 π( 2y 5 )2 dy Similr : rdius of slice y = 2 5 so r = 2y 5 3 density δ(y) of the cone vries with it s height y: mss = 5 0 δ(y) volume = 5 0 δ(y)π( 2y 5 )2 dy 4 Work to pump the wter out if cone filled to height of 4ft. = F d = ( 62.5lb/ft 3 volume ) d ech slice displced = 4 0 (62.5 (π( 2y 5 )2 dy) (5 y)

10 9. Sequences list of terms s, s 2,...s n,... often rrnged in fixed pttern lgebric, numeric nd grphicl representtions new vocb: monotone, lternting, recursive, bounded lim s n? converges or diverges? n

11 9.2 Series: Geometric rtio between ny two consecutive terms is constnt. sum of the first n terms: ( x n ). Creful of # terms nd x ( x n ) strting index. lim = if x < n x x i = i Exmple: 2 2 i=0 i= 2

12 9.3 Series: Prtil Sums n nd convergence? [9.3, 9.4, 9.5, chpter 0] n= n th prtil sum: n i where i my not be geometric i= sequence of prtil sums S n converges series does so exmine lim n th prtil sums n Exmple: n(n+) S n = n n+ lim S n = n n=

13 9.3: Limits nd Linerity for Convergence or Divergence terms not getting smller: lim n 0 or DNE, then prtil n sums diverge nd so does the series. Exmple: Linerity: n converges to S nd n= nd k is ny constnt, then n= 5+n 2n+ b n converges to T, n= k n + b n converges to n= ks + T. Appliction : dd two geometric series (converge) Appliction 2: dd divergent & convergent series (diverge) Exmple: convergent n= 2 n= n + ( ) n. If convergent, then subtrct n 2 nd the result should converge.

14 9.3: Integrl Test Bounds If series hs terms tht re decresing nd positive, the integrl test not only tells us bout convergence, but lso bounds the series: =f() f (x)dx n + f (x)dx

15 n= n= 9.2 Geometric Series versus 9.3 p-series rtio between ny two consecutive terms is constnt. sum of the first n terms: ( x n ). Creful of # terms nd x ( x n ) strting index. lim if x < n = x x converges if p > nd diverges if p. np n= () n = geo series, x =.5 < conv to.5.5 n 2 = p series: p = 2 > conv by integrl test: terms dec +: x 2 dx = lim b b x 2 dx = lim b x n= n 2 + first term = + b = 0

16 Is this geometric series? yes no Geometric Series: x i where x is the common rtio nd is constnt. x i = i=0 x i=0 n i=0 x i = ( x n+ ). x provided x <. 2 Cn we pply the Terms not Getting Smller? yes no Terms not Getting Smller: For n, if the lim n 0, n > then the infinite series does not converge. 3 Are the terms decresing nd positive eventully, nd if so is this n integrl we cn do? yes no Integrl Test: For n, if the terms re decresing nd n > 0, then the series behves the sme wy s n dn, & f (x)dx n st term + f (x)dx.

17

18 Internlize Mteril Mke it Your Own

### 7.6 The Use of Definite Integrals in Physics and Engineering

Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

### 63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

### AP Calculus BC Review Applications of Integration (Chapter 6) noting that one common instance of a force is weight

AP Clculus BC Review Applictions of Integrtion (Chpter Things to Know n Be Able to Do Fin the re between two curves by integrting with respect to x or y Fin volumes by pproximtions with cross sections:

### FINALTERM EXAMINATION 2011 Calculus &. Analytical Geometry-I

FINALTERM EXAMINATION 011 Clculus &. Anlyticl Geometry-I Question No: 1 { Mrks: 1 ) - Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...

### The problems that follow illustrate the methods covered in class. They are typical of the types of problems that will be on the tests.

ADVANCED CALCULUS PRACTICE PROBLEMS JAMES KEESLING The problems tht follow illustrte the methods covered in clss. They re typicl of the types of problems tht will be on the tests. 1. Riemnn Integrtion

### The 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

### 5: 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

### Math 113 Exam 2 Practice

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

### Not for reproduction

AREA OF A SURFACE OF REVOLUTION cut h FIGURE FIGURE πr r r l h FIGURE A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundry of solid of revolution of the type

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

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

### Conducting 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,

### x 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx

. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

### MA123, 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. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### SULIT /2 3472/2 Matematik Tambahan Kertas 2 2 ½ jam 2009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING

SULIT 1 347/ 347/ Mtemtik Tmbhn Kerts ½ jm 009 SEKOLAH-SEKOLAH MENENGAH ZON A KUCHING PEPERIKSAAN PERCUBAAN SIJIL PELAJARAN MALAYSIA 009 MATEMATIK TAMBAHAN Kerts Du jm tig puluh minit JANGAN BUKA KERTAS

### 7 - Continuous random variables

7-1 Continuous rndom vribles S. Lll, Stnford 2011.01.25.01 7 - Continuous rndom vribles Continuous rndom vribles The cumultive distribution function The uniform rndom vrible Gussin rndom vribles The Gussin

### PDE Notes. Paul Carnig. January ODE s vs PDE s 1

PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................

### We 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

### Mapping the delta function and other Radon measures

Mpping the delt function nd other Rdon mesures Notes for Mth583A, Fll 2008 November 25, 2008 Rdon mesures Consider continuous function f on the rel line with sclr vlues. It is sid to hve bounded support

### Calculus MATH 172-Fall 2017 Lecture Notes

Clculus MATH 172-Fll 2017 Lecture Notes These notes re concise summry of wht hs been covered so fr during the lectures. All the definitions must be memorized nd understood. Sttements of importnt theorems

### 4.1. Probability Density Functions

STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

### Partial Differential Equations

Prtil Differentil Equtions Notes by Robert Piché, Tmpere University of Technology reen s Functions. reen s Function for One-Dimensionl Eqution The reen s function provides complete solution to boundry

### Discrete Least-squares Approximations

Discrete Lest-squres Approximtions Given set of dt points (x, y ), (x, y ),, (x m, y m ), norml nd useful prctice in mny pplictions in sttistics, engineering nd other pplied sciences is to construct curve

### Orthogonal Polynomials and Least-Squares Approximations to Functions

Chpter Orthogonl Polynomils nd Lest-Squres Approximtions to Functions **4/5/3 ET. Discrete Lest-Squres Approximtions Given set of dt points (x,y ), (x,y ),..., (x m,y m ), norml nd useful prctice in mny

### III. AB Review. The material in this section is a review of AB concepts Illegal to post on Internet

III. AB Review The mteril in this section is review of AB concepts. www.mstermthmentor.com - 181 - Illegl to post on Internet R1: Bsic Differentition The derivtive of function is formul for the slope of

### Lecture 8. Band theory con.nued

Lecture 8 Bnd theory con.nued Recp: Solved Schrodinger qu.on for free electrons, for electrons bound in poten.l box, nd bound by proton. Discrete energy levels rouse. The Schrodinger qu.on pplied to periodic

### Arithmetic & Algebra. NCTM National Conference, 2017

NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

### MAC-solutions of the nonexistent solutions of mathematical physics

Proceedings of the 4th WSEAS Interntionl Conference on Finite Differences - Finite Elements - Finite Volumes - Boundry Elements MAC-solutions of the nonexistent solutions of mthemticl physics IGO NEYGEBAUE

### EVALUATING DEFINITE INTEGRALS

Chpter 4 EVALUATING DEFINITE INTEGRALS If the defiite itegrl represets re betwee curve d the x-xis, d if you c fid the re by recogizig the shpe of the regio, the you c evlute the defiite itegrl. Those

### Matrix Eigenvalues and Eigenvectors September 13, 2017

Mtri Eigenvlues nd Eigenvectors September, 7 Mtri Eigenvlues nd Eigenvectors Lrry Cretto Mechnicl Engineering 5A Seminr in Engineering Anlysis September, 7 Outline Review lst lecture Definition of eigenvlues

### C1M14. Integrals as Area Accumulators

CM Integrls s Are Accumultors Most tetbooks do good job of developing the integrl nd this is not the plce to provide tht development. We will show how Mple presents Riemnn Sums nd the ccompnying digrms

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

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

### Calculus 2: Integration. Differentiation. Integration

Clculus 2: Integrtion The reverse process to differentition is known s integrtion. Differentition f() f () Integrtion As it is the opposite of finding the derivtive, the function obtined b integrtion is

### F is n ntiderivtive èor èindeæniteè integrlè off if F 0 èxè =fèxè. Nottion: F èxè = ; it mens F 0 èxè=fèxè ëthe integrl of f of x dee x" Bsic list: xn

Mth 70 Topics for third exm Chpter 3: Applictions of Derivtives x7: Liner pproximtion nd diæerentils Ide: The tngent line to grph of function mkes good pproximtion to the function, ner the point of tngency.

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Convex Sets and Functions

B Convex Sets nd Functions Definition B1 Let L, +, ) be rel liner spce nd let C be subset of L The set C is convex if, for ll x,y C nd ll [, 1], we hve 1 )x+y C In other words, every point on the line

### Physics Honors. Final Exam Review Free Response Problems

Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting

### Week 7 Riemann Stieltjes Integration: Lectures 19-21

Week 7 Riemnn Stieltjes Integrtion: Lectures 19-21 Lecture 19 Throughout this section α will denote monotoniclly incresing function on n intervl [, b]. Let f be bounded function on [, b]. Let P = { = 0

### E1: CALCULUS - lecture notes

E1: CALCULUS - lecture notes Ştefn Blint Ev Kslik, Simon Epure, Simin Mriş, Aureli Tomoiogă Contents I Introduction 9 1 The notions set, element of set, membership of n element in set re bsic notions of

### ( ) dx ; f ( x ) is height and Δx is

Mth : 6.3 Defiite Itegrls from Riem Sums We just sw tht the exct re ouded y cotiuous fuctio f d the x xis o the itervl x, ws give s A = lim A exct RAM, where is the umer of rectgles i the Rectgulr Approximtio

### Answers to Exercises. c 2 2ab b 2 2ab a 2 c 2 a 2 b 2

Answers to Eercises CHAPTER 9 CHAPTER LESSON 9. CHAPTER 9 CHAPTER. c 9. cm. cm. b 5. cm. d 0 cm 5. s cm. c 8.5 cm 7. b cm 8.. cm 9. 0 cm 0. s.5 cm. r cm. 7 ft. 5 m.. cm 5.,, 5. 8 m 7. The re of the lrge

### Non-Linear & Logistic Regression

Non-Liner & Logistic Regression If the sttistics re boring, then you've got the wrong numbers. Edwrd R. Tufte (Sttistics Professor, Yle University) Regression Anlyses When do we use these? PART 1: find

### DIFFRACTION OF LIGHT

DIFFRACTION OF LIGHT The phenomenon of bending of light round the edges of obstcles or nrrow slits nd hence its encrochment into the region of geometricl shdow is known s diffrction. P Frunhofer versus

### Continuous Random Variables

STAT/MATH 395 A - PROBABILITY II UW Winter Qurter 217 Néhémy Lim Continuous Rndom Vribles Nottion. The indictor function of set S is rel-vlued function defined by : { 1 if x S 1 S (x) if x S Suppose tht

### I. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=

11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel

### Waveguide Guide: A and V. Ross L. Spencer

Wveguide Guide: A nd V Ross L. Spencer I relly think tht wveguide fields re esier to understnd using the potentils A nd V thn they re using the electric nd mgnetic fields. Since Griffiths doesn t do it

### 11.1 Exponential Functions

. Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

### Flow in porous media

Red: Ch 2. nd 2.2 PART 4 Flow in porous medi Drcy s lw Imgine point (A) in column of wter (figure below); the point hs following chrcteristics: () elevtion z (2) pressure p (3) velocity v (4) density ρ

### Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report

Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block

### MATH 409 Advanced Calculus I Lecture 19: Riemann sums. Properties of integrals.

MATH 409 Advnced Clculus I Lecture 19: Riemnn sums. Properties of integrls. Drboux sums Let P = {x 0,x 1,...,x n } be prtition of n intervl [,b], where x 0 = < x 1 < < x n = b. Let f : [,b] R be bounded

### Chapter 4. Additional Variational Concepts

Chpter 4 Additionl Vritionl Concepts 137 In the previous chpter we considered clculus o vrition problems which hd ixed boundry conditions. Tht is, in one dimension the end point conditions were speciied.

### ECON 331 Lecture Notes: Ch 4 and Ch 5

Mtrix Algebr ECON 33 Lecture Notes: Ch 4 nd Ch 5. Gives us shorthnd wy of writing lrge system of equtions.. Allows us to test for the existnce of solutions to simultneous systems. 3. Allows us to solve

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

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

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

### Measuring Electron Work Function in Metal

n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

### HT Module 2 Paper solution. Module 2. Q6.Discuss Electrical analogy of combined heat conduction and convection in a composite wall.

HT Module 2 Pper solution Qulity Solutions wwwqulitytutorilin Module 2 Q6Discuss Electricl nlogy of combined het conduction nd convection in composite wll M-16-Q1(c)-5m Ans: It is frequently convient to

### Math 115 ( ) Yum-Tong Siu 1. Lagrange Multipliers and Variational Problems with Constraints. F (x,y,y )dx

Mth 5 2006-2007) Yum-Tong Siu Lgrnge Multipliers nd Vritionl Problems with Constrints Integrl Constrints. Consider the vritionl problem of finding the extremls for the functionl J[y] = F x,y,y )dx with

### Diophantine Steiner Triples and Pythagorean-Type Triangles

Forum Geometricorum Volume 10 (2010) 93 97. FORUM GEOM ISSN 1534-1178 Diophntine Steiner Triples nd Pythgoren-Type Tringles ojn Hvl bstrct. We present connection between Diophntine Steiner triples (integer

### Anonymous 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

### f (x) dx = f(b) f(a) f (x i ) x i i=1

1 Cse Study: Flood Wtch c 00 Donld Kreider nd Dwight L Animtion: Flood Wtch To get you going on the Cse Study! In this section, we hve lerned tht if we re given the continuous derivtive f of function on

### LECTURE NOTE #12 PROF. ALAN YUILLE

LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, K-mens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of

### Lecture 9: LTL and Büchi Automata

Lecture 9: LTL nd Büchi Automt 1 LTL Property Ptterns Quite often the requirements of system follow some simple ptterns. Sometimes we wnt to specify tht property should only hold in certin context, clled

### Physics 102. Final Examination. Spring Semester ( ) P M. Fundamental constants. n = 10P

ε µ0 N mp M G T Kuwit University hysics Deprtment hysics 0 Finl Exmintin Spring Semester (0-0) My, 0 Time: 5:00 M :00 M Nme.Student N Sectin N nstructrs: Drs. bdelkrim, frsheh, Dvis, Kkj, Ljk, Mrfi, ichler,

### University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2010 Homework Assignment 4; Due at 5p.m. on 2/01/10

University of Wshington Deprtment of Chemistry Chemistry 45 Winter Qurter Homework Assignment 4; Due t 5p.m. on // We lerned tht the Hmiltonin for the quntized hrmonic oscilltor is ˆ d κ H. You cn obtin

### Plate Theory. Section 11: PLATE BENDING ELEMENTS

Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A

### Using integration tables

Using integrtion tbles Integrtion tbles re inclue in most mth tetbooks, n vilble on the Internet. Using them is nother wy to evlute integrls. Sometimes the use is strightforwr; sometimes it tkes severl

### Chapter 6: Transcendental functions: Table of Contents: 6.3 The natural exponential function. 6.2 Inverse functions and their derivatives

Chpter 6: Trnscendentl functions: In this chpter we will lern differentition nd integrtion formuls for few new functions, which include the nturl nd generl eponentil nd the nturl nd generl logrithmic function

### 7. Numerical evaluation of definite integrals

7. Numericl evlution of definite integrls Tento učení text yl podpořen z Operčního progrmu Prh - Adptilit Hn Hldíková Numericl pproximtion of definite integrl is clled numericl qudrture, the formuls re

### Basic Angle Rules 5. A Short Hand Geometric Reasons. B Two Reasons. 1 Write in full the meaning of these short hand geometric reasons.

si ngle Rules 5 6 Short Hnd Geometri Resons 1 Write in full the mening of these short hnd geometri resons. Short Hnd Reson Full Mening ) se s isos Δ re =. ) orr s // lines re =. ) sum s t pt = 360. d)

### Role of Missing Carotenoid in Reducing the Fluorescence of Single Monomeric Photosystem II Core Complexes

Electronic Supplementry Mteril (ESI for Physicl Chemistry Chemicl Physics. This journl is the Owner Societies 017 Supporting Informtion Role of Missing Crotenoid in Reducing the Fluorescence of Single

### Using QM for Windows. Using QM for Windows. Using QM for Windows LEARNING OBJECTIVES. Solving Flair Furniture s LP Problem

LEARNING OBJECTIVES Vlu%on nd pricing (November 5, 2013) Lecture 11 Liner Progrmming (prt 2) 10/8/16, 2:46 AM Olivier J. de Jong, LL.M., MM., MBA, CFD, CFFA, AA www.olivierdejong.com Solving Flir Furniture

### UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

### The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2

The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published

### (a) A partition P of [a, b] is a finite subset of [a, b] containing a and b. If Q is another partition and P Q, then Q is a refinement of P.

Chpter 7: The Riemnn Integrl When the derivtive is introdued, it is not hrd to see tht the it of the differene quotient should be equl to the slope of the tngent line, or when the horizontl xis is time

### From the Numerical. to the Theoretical in. Calculus

From the Numericl to the Theoreticl in Clculus Teching Contemporry Mthemtics NCSSM Ferury 6-7, 003 Doug Kuhlmnn Phillips Acdemy Andover, MA 01810 dkuhlmnn@ndover.edu How nd Why Numericl Integrtion Should

### 2

1 Notes for Numericl Anlysis Mth 5466 by S. Adjerid Virgini Polytechnic Institute nd Stte University (A Rough Drft) 2 Contents 1 Differentition nd Integrtion 5 1.1 Introduction............................

### Series Solutions of ODEs. Special Functions

CHAPTER 5 Series Solutions of ODEs. Specil Functions In the previous chpters, we hve seen tht liner ODEs with constnt coefficients cn be solved by lgebric methods, nd tht their solutions re elementry functions

### Polytechnic Institute of NYU MA 2122 Worksheet 4

Polytechnic Institute of NYU MA 222 Worksheet 4 Print Nme: ignture: ID #: Instructor/ection: / Directions: how ll your work for every problem. Problem Possible Points 20 2 5 3 5 4 0 5 0 6 5 7 5 Totl 00

### Riemann Stieltjes Integration - Definition and Existence of Integral

- Definition nd Existence of Integrl Dr. Adity Kushik Directorte of Distnce Eduction Kurukshetr University, Kurukshetr Hryn 136119 Indi. Prtition Riemnn Stieltjes Sums Refinement Definition Given closed

### 1 computation of E {K(r, n)}

1 Anlysis of the sptil orgniztion of molecules with robust sttistics: Supplementry Mteril Thibult Lgche 1,,, Gbriel Lng 3, Nthlie Suvonnet 4,, Jen-Christophe Olivo-Mrin 1,, 1 Unité d Anlyse d Imges Quntittive,

### Lecture 11: Potential Gradient and Capacitor Review:

Lectue 11: Potentil Gdient nd Cpcito Review: Two wys to find t ny point in spce: Sum o Integte ove chges: q 1 1 q 2 2 3 P i 1 q i i dq q 3 P 1 dq xmple of integting ove distiution: line of chge ing of

### NON-DETERMINISTIC FSA

Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

### Data Assimilation. Alan O Neill Data Assimilation Research Centre University of Reading

Dt Assimiltion Aln O Neill Dt Assimiltion Reserch Centre University of Reding Contents Motivtion Univrite sclr dt ssimiltion Multivrite vector dt ssimiltion Optiml Interpoltion BLUE 3d-Vritionl Method

### Functions of bounded variation

Division for Mthemtics Mrtin Lind Functions of bounded vrition Mthemtics C-level thesis Dte: 2006-01-30 Supervisor: Viktor Kold Exminer: Thoms Mrtinsson Krlstds universitet 651 88 Krlstd Tfn 054-700 10

### f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

### Exponents and Polynomials

C H A P T E R 5 Eponents nd Polynomils ne sttistic tht cn be used to mesure the generl helth of ntion or group within ntion is life epectncy. This dt is considered more ccurte thn mny other sttistics becuse

### a a a a a a a a a a a a a a a a a a a a a a a a In this section, we introduce a general formula for computing determinants.

Section 9 The Lplce Expnsion In the lst section, we defined the determinnt of (3 3) mtrix A 12 to be 22 12 21 22 2231 22 12 21. In this section, we introduce generl formul for computing determinnts. Rewriting

### METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS

Journl of Young Scientist Volume III 5 ISSN 44-8; ISSN CD-ROM 44-9; ISSN Online 44-5; ISSN-L 44 8 METHODS OF APPROXIMATING THE RIEMANN INTEGRALS AND APPLICATIONS An ALEXANDRU Scientific Coordintor: Assist

### Definite Integral. The Left and Right Sums

Clculus Li Vs Defiite Itegrl. The Left d Right Sums The defiite itegrl rises from the questio of fidig the re betwee give curve d x-xis o itervl. The re uder curve c be esily clculted if the curve is give

### A short introduction to local fractional complex analysis

A short introduction to locl rctionl complex nlysis Yng Xio-Jun Deprtment o Mthemtics Mechnics, hin University o Mining Technology, Xuhou mpus, Xuhou, Jingsu, 228, P R dyngxiojun@63com This pper presents

### Lecture 8 Wrap-up Part1, Matlab

Lecture 8 Wrp-up Prt1, Mtlb Homework Polic Plese stple our homework (ou will lose 10% credit if not stpled or secured) Submit ll problems in order. This mens to plce ever item relting to problem 3 (our

### a n+2 a n+1 M n a 2 a 1. (2)

Rel Anlysis Fll 004 Tke Home Finl Key 1. Suppose tht f is uniformly continuous on set S R nd {x n } is Cuchy sequence in S. Prove tht {f(x n )} is Cuchy sequence. (f is not ssumed to be continuous outside

### Linear static analysis of perforated plates with round and staggered holes under their self-weights

Liner sttic nlysis of perforted pltes with round nd stggered holes under their self-weights Mustf Hlûk Srçoğlu*, Uğur Albyrk Online Publiction Dte: 8 Nov 2015 URL: http://www.jresm.org/rchive/resm2015.25me0910.html

### Parallel Projection Theorem (Midpoint Connector Theorem):

rllel rojection Theorem (Midpoint onnector Theorem): The segment joining the midpoints of two sides of tringle is prllel to the third side nd hs length one-hlf the third side. onversely, If line isects

### 4. CHEMICAL KINETICS

4. CHEMICAL KINETICS Synopsis: The study of rtes of chemicl rections mechnisms nd fctors ffecting rtes of rections is clled chemicl kinetics. Spontneous chemicl rection mens, the rection which occurs on

### The contact stress problem for a piecewisely defined punch indenting an elastic half space. Jacques Woirgard

The contct stress prolem for pieceisely defined punch indenting n elstic hlf spce Jcques Woirgrd 5 Rue du Châteu de l Arceu 8633 Notre Dme d Or FRANC Astrct A solution for the contct stress prolem of n