Partial Derivatives: Suppose that z = f(x, y) is a function of two variables.
|
|
- Lynne Willis
- 5 years ago
- Views:
Transcription
1 Chaptr Functions o Two Variabls Applid Calculus 61 Sction : Calculus o Functions o Two Variabls Now that ou hav som amiliarit with unctions o two variabls it s tim to start appling calculus to hlp us solv problms with thm. In Chaptr w larnd about th drivativ or unctions o two variabls. Drivativs told us about th shap o th unction and lt us ind local ma and min w want to b abl to do th sam thing with a unction o two variabls. First lt s think. Imagin a surac th graph o a unction o two variabls. Imagin that th surac is smooth and has som hills and som valls. Concntrat on on point on our surac. What do w want th drivativ to tll us? It ought to tll us how quickl th hight o th surac changs as w mov. Wait which dirction do w want to mov? This is th rason that drivativs ar mor complicatd or unctions o svral variabls thr ar so man dirctions w could mov rom an point. It turns out that our ida o iing on variabl and watching what happns to th unction as th othr changs is th k to tnding th ida o drivativs to mor than on variabl. Partial Drivativs Partial Drivativs: Suppos that z = ( ) is a unction o two variabls. Th partial drivativ o with rspct to is th drivativ o th unction () whr w think o as th onl variabl and act as i is a constant. Th partial drivativ o with rspct to is th drivativ o th unction () whr w think o as th onl variabl and act as i is a constant. Th with rspct to or with rspct to part is rall important ou hav to know and tll which variabl ou ar thinking o as THE variabl. Gomtricall th partial drivativ with rspct to givs th slop o th curv as ou travl along a cross-sction a curv on th surac paralll to th -ais. Th partial drivativ with rspct to givs th slop o th cross-sction paralll to th -ais. Notation or th Partial Drivativ: Th partial drivativ o = () with rspct to is writtn as or z simpl z Th Libniz notation is or d d z W us an adaptation o th notation to man ind th partial drivativ o () with d rspct to : This chaptr is (c) 01. It was rmid b David Lippman rom Shana Calawa's rmi o Contmporar Calculus b Dal Homan. It is licnsd undr th Crativ Commons Attribution licns.
2 Chaptr Functions o Two Variabls Applid Calculus 6 To stimat a partial drivativ rom a tabl or contour diagram: Th partial drivativ with rspct to can b approimatd b looking at an avrag rat o chang or th slop o a scant lin ovr a vr tin intrval in th -dirction (holding constant). Th tinir th intrval th closr this is to th tru partial drivativ. To comput a partial drivativ rom a ormula: I () is givn as a ormula ou can ind th partial drivativ with rspct to algbraicall b taking th ordinar drivativ thinking o as th onl variabl (holding id). O cours vrthing hr works th sam wa i w r tring to ind th partial drivativ with rspct to just think o as our onl variabl and act as i is constant. Th ida o a partial drivativ works prctl wll or a unction o svral variabls ou ocus on on variabl to b THE variabl and act as i all th othr variabls ar constants. Eampl 1 Hr is a contour diagram or a unction g(). Us th diagram to answr th ollowing qustions: g a. Estimat g and b. Whr on this diagram is g gratst? Whr is g gratst? g mans w'r thinking o as th onl variabl so w ll hold id at =. That mans w ll b looking along th horizontal lin =. To stimat g w nd two unction valus. ( ) lis on th contour lin so w know that g( ) = 0.6. Th nt point as w mov to th right is g(.) = 0.7. a. Now w can ind th avrag rat o chang: g Avrag rat o chang = (chang in output) / (chang in input) W can do th sam thing b going to th nt point w can rad to th lt which is g(.) = g Thn th avrag rat o chang is g givn th inormation w hav or ou could Eithr o ths would b a in stimat o tak thir avrag. W can stimat that. 1 g.
3 Chaptr Functions o Two Variabls Applid Calculus 6 Estimat g th sam wa but moving on th vrtical lin. Using th nt point up w g gt th avrag rat o chang. 1. Using th nt point down w gt.8 g Taking thir avrag w stimat g b. g mans is m onl variabl and w'r thinking o as a constant. So w'r thinking about moving across th diagram on horizontal lins. g will b gratst whn th contour lins g ar closst togthr whn th surac is stpst thn th dnominator in will b small so g will b big. Scanning th graph w can s that th contour lins ar closst togthr whn w had to th lt or to th right rom about (0. 8) and (9 8). So 8) and (9 8). For g I want to look at vrtical lins. 1). g is gratst at about (0. g is gratst at about (.8) and ( Eampl Cold tmpraturs l coldr whn th wind is blowing. Windchill is th prcivd tmpratur and it dpnds on both th actual tmpratur and th wind spd a unction o two variabls! You can rad mor about windchill at Blow is a tabl (courts o th National Wathr Srvic) that shows th prcivd tmpratur or various tmpraturs and windspds. Not that th also includ th ormula but or this ampl w'll us th inormation in th tabl. a. What is th prcivd tmpratur whn th actual tmpratur is F and th wind is blowing at 1 mils pr hour? b. Suppos th actual tmpratur is F. Us inormation rom th tabl to dscrib how th prcivd tmpratur would chang i th wind spd incrasd rom 1 mils pr hour?
4 Chaptr Functions o Two Variabls Applid Calculus 6 a. Rading th tabl w s that th prcivd tmpratur is 1 F. b. This is a qustion about a partial drivativ. W r holding th tmpratur (T) id at F and asking what happns as wind spd (V) incrass rom 1 mils pr hour. W r thinking o V as th onl variabl so w want WindChill V = W V whn T = and V = 1. W ll ind th avrag rat o chang b looking in th column whr T = and ltting V incras and us that to approimat th partial drivativ. W 11 1 W V 0. V 0 1 What ar th units? W is masurd in F and V is masurd in mph so th units hr ar F/mph. And that lts us dscrib what happns: Th prcivd tmpratur would dcras b about. F or ach mph incras in wind spd. Eampl Find and at th points (0 0) and (1 1) i To ind tak th ordinar drivativ o with rspct to acting as i is constant: Not that th drivativ o th trm with rspct to is zro it s a constant. Similarl 8. Now w can valuat ths at th points:
5 Chaptr Functions o Two Variabls Applid Calculus and 00 0 both lat at (00). 1 1 and 1 1 ; this tlls us that th cross sctions paralll to th - and - as ar ; this tlls us that abov th point (1 1) th surac dcrass i ou mov to mor positiv valus and incrass i ou mov to mor positiv valus. Eampl Find and i ln mans is our onl variabl w r thinking o as a constant. Thn w ll just ind th ordinar drivativ. From s point o viw this is an ponntial unction dividd b a constant with a constant addd. Th constant pulls out in ront th drivativ o th ponntial unction is th sam thing and w nd to us th chain rul so w multipl b th drivativ o that ponnt (which is just 1): 1 mans that w r thinking o as th variabl acting as i is constant. From s point o viw is a quotint plus a product w ll nd th quotint rul and th product rul: ln Eampl Find i z z w w z z 1 z mans w act as i z is our onl variabl so w ll act as i all th othr variabls ( and w) ar constants and tak th ordinar drivativ. 1 z z w z z
6 Chaptr Functions o Two Variabls Applid Calculus 66 Using Partial Drivativs to Estimat Function Valus W can us th partial drivativs to stimat valus o a unction. Th gomtr is similar to th tangnt lin approimation in on variabl. Rcall th on-variabl cas: i is clos nough to a known point a thn a ' a a. In two variabls w do th sam thing in both dirctions at onc: Approimating Function Valus with Partial Drivativs To approimat th valu o ( ) ind som point (a b) whr 1. ( ) and (a b) ar clos that is and a ar clos and and b ar clos.. You know th act valus o (a b) and both partial drivativs thr. Thn a b a b a a b b Notic that th total chang in is bing approimatd b adding th approimat changs coming rom th and dirctions. Anothr wa to look at th sam ormula: How clos is clos? It dpnds on th shap o th graph o. In gnral th closr th bttr. Eampl 6 Us partial drivativs to stimat th valu o at ( ) Not that th point ( ) is clos to an as point (1 1). In act w alrad workd out th partial drivativs at (1 1): ; ; 11. W also know that So Not that it would hav bn possibl in this cas to simpl comput th act answr; Our stimat is not prct but it s prtt clos.
7 Chaptr Functions o Two Variabls Applid Calculus 67 Eampl 7 Hr is a contour diagram or a unction g(). Us partial drivativs to stimat th valu o g(..7). This is th sam diagram rom bor so w alrad stimatd th valu o th unction and th partial drivativs at th narb point (). g( ) is 0.6 our stimat o g. 1 and our stimat o g. 16. So g Not that in this cas w hav no wa to know how clos our stimat is to th actual valu.
8 Chaptr Functions o Two Variabls Applid Calculus 68. Erciss For problms 1 through 16 ind and or th unction givn ln
9 Chaptr Functions o Two Variabls Applid Calculus Hr is a tabl showing th unction A t r t r a. Estimat A t.0. b. Estimat A r.0 c. Us our answrs to parts a and b to stimat th valu o A..0 which shows th intrst arnd i 1000 dollars is dpositd in an account arning r annual intrst compoundd continuousl and lt thr or t ars. How clos ar our stimats rom parts a b and c? rt d. Th valus in th tabl cam rom A t r Hr is a tabl showing valus or th unction t h H t h H a. Estimat th valu o dt at ( 10). H b. Estimat th valu o dh at ( 10). c. Us our answrs to parts a and b to stimat th valu o H.616. d. Th valus in th tabl cam rom Ht h h 1t.9t which givs th hight in mtrs abov th ground atr t sconds o an objct that is thrown upward rom an initial hight o h mtrs with an initial vlocit o 1 mtrs pr scond. How clos ar our stimats rom parts a b and c? 19. Givn th unction a. Calculat and b. Us our answrs rom part a to stimat Givn th unction ln 10 and b. Us our answrs rom part a to stimat a. Calculat
10 Chaptr Functions o Two Variabls Applid Calculus 70 In problms 1-6 us th contour plot shown to stimat th dsird valu
Math 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationSection 11.6: Directional Derivatives and the Gradient Vector
Sction.6: Dirctional Drivativs and th Gradint Vctor Practic HW rom Stwart Ttbook not to hand in p. 778 # -4 p. 799 # 4-5 7 9 9 35 37 odd Th Dirctional Drivativ Rcall that a b Slop o th tangnt lin to th
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More information2008 AP Calculus BC Multiple Choice Exam
008 AP Multipl Choic Eam Nam 008 AP Calculus BC Multipl Choic Eam Sction No Calculator Activ AP Calculus 008 BC Multipl Choic. At tim t 0, a particl moving in th -plan is th acclration vctor of th particl
More information1973 AP Calculus AB: Section I
97 AP Calculus AB: Sction I 9 Minuts No Calculator Not: In this amination, ln dnots th natural logarithm of (that is, logarithm to th bas ).. ( ) d= + C 6 + C + C + C + C. If f ( ) = + + + and ( ), g=
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs for which f () is a ral numbr.. (4 6 ) d= 4 6 6
More informationare given in the table below. t (hours)
CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th
More informationA. Limits and Horizontal Asymptotes ( ) f x f x. f x. x "±# ( ).
A. Limits and Horizontal Asymptots What you ar finding: You can b askd to find lim x "a H.A.) problm is asking you find lim x "# and lim x "$#. or lim x "±#. Typically, a horizontal asymptot algbraically,
More informationy = 2xe x + x 2 e x at (0, 3). solution: Since y is implicitly related to x we have to use implicit differentiation: 3 6y = 0 y = 1 2 x ln(b) ln(b)
4. y = y = + 5. Find th quation of th tangnt lin for th function y = ( + ) 3 whn = 0. solution: First not that whn = 0, y = (1 + 1) 3 = 8, so th lin gos through (0, 8) and thrfor its y-intrcpt is 8. y
More informationas a derivative. 7. [3.3] On Earth, you can easily shoot a paper clip straight up into the air with a rubber band. In t sec
MATH6 Fall 8 MIDTERM II PRACTICE QUESTIONS PART I. + if
More informationMor Tutorial at www.dumblittldoctor.com Work th problms without a calculator, but us a calculator to chck rsults. And try diffrntiating your answrs in part III as a usful chck. I. Applications of Intgration
More informationCalculus concepts derivatives
All rasonabl fforts hav bn mad to mak sur th nots ar accurat. Th author cannot b hld rsponsibl for any damags arising from th us of ths nots in any fashion. Calculus concpts drivativs Concpts involving
More information6.1 Integration by Parts and Present Value. Copyright Cengage Learning. All rights reserved.
6.1 Intgration by Parts and Prsnt Valu Copyright Cngag Larning. All rights rsrvd. Warm-Up: Find f () 1. F() = ln(+1). F() = 3 3. F() =. F() = ln ( 1) 5. F() = 6. F() = - Objctivs, Day #1 Studnts will b
More informationHigher order derivatives
Robrto s Nots on Diffrntial Calculus Chaptr 4: Basic diffrntiation ruls Sction 7 Highr ordr drivativs What you nd to know alrady: Basic diffrntiation ruls. What you can larn hr: How to rpat th procss of
More information1997 AP Calculus AB: Section I, Part A
997 AP Calculus AB: Sction I, Part A 50 Minuts No Calculator Not: Unlss othrwis spcifid, th domain of a function f is assumd to b th st of all ral numbrs x for which f (x) is a ral numbr.. (4x 6 x) dx=
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Unit C Algbra Trigonomtr Gomtr Calculus Vctor gomtr Pag Algbra Molus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv
More informationMSLC Math 151 WI09 Exam 2 Review Solutions
Eam Rviw Solutions. Comput th following rivativs using th iffrntiation ruls: a.) cot cot cot csc cot cos 5 cos 5 cos 5 cos 5 sin 5 5 b.) c.) sin( ) sin( ) y sin( ) ln( y) ln( ) ln( y) sin( ) ln( ) y y
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationu r du = ur+1 r + 1 du = ln u + C u sin u du = cos u + C cos u du = sin u + C sec u tan u du = sec u + C e u du = e u + C
Tchniqus of Intgration c Donald Kridr and Dwight Lahr In this sction w ar going to introduc th first approachs to valuating an indfinit intgral whos intgrand dos not hav an immdiat antidrivativ. W bgin
More informationText: WMM, Chapter 5. Sections , ,
Lcturs 6 - Continuous Probabilit Distributions Tt: WMM, Chaptr 5. Sctions 6.-6.4, 6.6-6.8, 7.-7. In th prvious sction, w introduc som of th common probabilit distribution functions (PDFs) for discrt sampl
More informationCalculus II (MAC )
Calculus II (MAC232-2) Tst 2 (25/6/25) Nam (PRINT): Plas show your work. An answr with no work rcivs no crdit. You may us th back of a pag if you nd mor spac for a problm. You may not us any calculators.
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationThomas Whitham Sixth Form
Thomas Whitham Sith Form Pur Mathmatics Cor rvision gui Pag Algbra Moulus functions graphs, quations an inqualitis Graph of f () Draw f () an rflct an part of th curv blow th ais in th ais. f () f () f
More informationJOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH 241) Final Review Fall 2016
JOHNSON COUNTY COMMUNITY COLLEGE Calculus I (MATH ) Final Rviw Fall 06 Th Final Rviw is a starting point as you study for th final am. You should also study your ams and homwork. All topics listd in th
More informationContinuous probability distributions
Continuous probability distributions Many continuous probability distributions, including: Uniform Normal Gamma Eponntial Chi-Squard Lognormal Wibull EGR 5 Ch. 6 Uniform distribution Simplst charactrizd
More informationDifferential Equations
Prfac Hr ar m onlin nots for m diffrntial quations cours that I tach hr at Lamar Univrsit. Dspit th fact that ths ar m class nots, th should b accssibl to anon wanting to larn how to solv diffrntial quations
More informationChapter 1. Chapter 10. Chapter 2. Chapter 11. Chapter 3. Chapter 12. Chapter 4. Chapter 13. Chapter 5. Chapter 14. Chapter 6. Chapter 7.
Chaptr Binomial Epansion Chaptr 0 Furthr Probability Chaptr Limits and Drivativs Chaptr Discrt Random Variabls Chaptr Diffrntiation Chaptr Discrt Probability Distributions Chaptr Applications of Diffrntiation
More informationDifferentiation of Exponential Functions
Calculus Modul C Diffrntiation of Eponntial Functions Copyright This publication Th Northrn Albrta Institut of Tchnology 007. All Rights Rsrvd. LAST REVISED March, 009 Introduction to Diffrntiation of
More informationFourier Transforms and the Wave Equation. Key Mathematics: More Fourier transform theory, especially as applied to solving the wave equation.
Lur 7 Fourir Transforms and th Wav Euation Ovrviw and Motivation: W first discuss a fw faturs of th Fourir transform (FT), and thn w solv th initial-valu problm for th wav uation using th Fourir transform
More informationThings I Should Know Before I Get to Calculus Class
Things I Should Know Bfor I Gt to Calculus Class Quadratic Formula = b± b 4ac a sin + cos = + tan = sc + cot = csc sin( ± y ) = sin cos y ± cos sin y cos( + y ) = cos cos y sin sin y cos( y ) = cos cos
More informationChapter 3 Exponential and Logarithmic Functions. Section a. In the exponential decay model A. Check Point Exercises
Chaptr Eponntial and Logarithmic Functions Sction. Chck Point Erciss. a. A 87. Sinc is yars aftr, whn t, A. b. A A 87 k() k 87 k 87 k 87 87 k.4 Thus, th growth function is A 87 87.4t.4t.4t A 87..4t 87.4t
More informationMath-3. Lesson 5-6 Euler s Number e Logarithmic and Exponential Modeling (Newton s Law of Cooling)
Math-3 Lsson 5-6 Eulr s Numbr Logarithmic and Eponntial Modling (Nwton s Law of Cooling) f ( ) What is th numbr? is th horizontal asymptot of th function: 1 1 ~ 2.718... On my 3rd submarin (USS Springfild,
More informationAP Calculus BC AP Exam Problems Chapters 1 3
AP Eam Problms Captrs Prcalculus Rviw. If f is a continuous function dfind for all ral numbrs and if t maimum valu of f() is 5 and t minimum valu of f() is 7, tn wic of t following must b tru? I. T maimum
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationSystems of Equations
CHAPTER 4 Sstms of Equations 4. Solving Sstms of Linar Equations in Two Variabls 4. Solving Sstms of Linar Equations in Thr Variabls 4. Sstms of Linar Equations and Problm Solving Intgratd Rviw Sstms of
More informationGrade 12 (MCV4UE) AP Calculus Page 1 of 5 Derivative of a Function & Differentiability
Gra (MCV4UE) AP Calculus Pag of 5 Drivativ of a Function & Diffrntiabilit Th Drivativ at a Point f ( a h) f ( a) Rcall, lim provis th slop of h0 h th tangnt to th graph f ( at th point a, f ( a), an th
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationINTEGRATION BY PARTS
Mathmatics Rvision Guids Intgration by Parts Pag of 7 MK HOME TUITION Mathmatics Rvision Guids Lvl: AS / A Lvl AQA : C Edcl: C OCR: C OCR MEI: C INTEGRATION BY PARTS Vrsion : Dat: --5 Eampls - 6 ar copyrightd
More informationThe graph of y = x (or y = ) consists of two branches, As x 0, y + ; as x 0, y +. x = 0 is the
Copyright itutcom 005 Fr download & print from wwwitutcom Do not rproduc by othr mans Functions and graphs Powr functions Th graph of n y, for n Q (st of rational numbrs) y is a straight lin through th
More informationCh. 24 Molecular Reaction Dynamics 1. Collision Theory
Ch. 4 Molcular Raction Dynamics 1. Collision Thory Lctur 16. Diffusion-Controlld Raction 3. Th Matrial Balanc Equation 4. Transition Stat Thory: Th Eyring Equation 5. Transition Stat Thory: Thrmodynamic
More informationIntegration by Parts
Intgration by Parts Intgration by parts is a tchniqu primarily for valuating intgrals whos intgrand is th product of two functions whr substitution dosn t work. For ampl, sin d or d. Th rul is: u ( ) v'(
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More information4 x 4, and. where x is Town Square
Accumulation and Population Dnsity E. A city locatd along a straight highway has a population whos dnsity can b approimatd by th function p 5 4 th distanc from th town squar, masurd in mils, whr 4 4, and
More informationMultiple Short Term Infusion Homework # 5 PHA 5127
Multipl Short rm Infusion Homwork # 5 PHA 527 A rug is aministr as a short trm infusion. h avrag pharmacokintic paramtrs for this rug ar: k 0.40 hr - V 28 L his rug follows a on-compartmnt boy mol. A 300
More informationCOMPUTER GENERATED HOLOGRAMS Optical Sciences 627 W.J. Dallas (Monday, April 04, 2005, 8:35 AM) PART I: CHAPTER TWO COMB MATH.
C:\Dallas\0_Courss\03A_OpSci_67\0 Cgh_Book\0_athmaticalPrliminaris\0_0 Combath.doc of 8 COPUTER GENERATED HOLOGRAS Optical Scincs 67 W.J. Dallas (onday, April 04, 005, 8:35 A) PART I: CHAPTER TWO COB ATH
More informationUnit 6: Solving Exponential Equations and More
Habrman MTH 111 Sction II: Eonntial and Logarithmic Functions Unit 6: Solving Eonntial Equations and Mor EXAMPLE: Solv th quation 10 100 for. Obtain an act solution. This quation is so asy to solv that
More information1 1 1 p q p q. 2ln x x. in simplest form. in simplest form in terms of x and h.
NAME SUMMER ASSIGNMENT DUE SEPTEMBER 5 (FIRST DAY OF SCHOOL) AP CALC AB Dirctions: Answr all of th following qustions on a sparat sht of papr. All work must b shown. You will b tstd on this matrial somtim
More informationAnswer Homework 5 PHA5127 Fall 1999 Jeff Stark
Answr omwork 5 PA527 Fall 999 Jff Stark A patint is bing tratd with Drug X in a clinical stting. Upon admiion, an IV bolus dos of 000mg was givn which yildd an initial concntration of 5.56 µg/ml. A fw
More informationReview of Exponentials and Logarithms - Classwork
Rviw of Eponntials and Logarithms - Classwork In our stud of calculus, w hav amind drivativs and intgrals of polnomial prssions, rational prssions, and trignomtric prssions. What w hav not amind ar ponntial
More informationdx equation it is called a second order differential equation.
TOPI Diffrntial quations Mthods of thir intgration oncption of diffrntial quations An quation which spcifis a rlationship btwn a function, its argumnt and its drivativs of th first, scond, tc ordr is calld
More informationNote If the candidate believes that e x = 0 solves to x = 0 or gives an extra solution of x = 0, then withhold the final accuracy mark.
. (a) Eithr y = or ( 0, ) (b) Whn =, y = ( 0 + ) = 0 = 0 ( + ) = 0 ( )( ) = 0 Eithr = (for possibly abov) or = A 3. Not If th candidat blivs that = 0 solvs to = 0 or givs an tra solution of = 0, thn withhold
More informationLenses & Prism Consider light entering a prism At the plane surface perpendicular light is unrefracted Moving from the glass to the slope side light
Lnss & Prism Considr light ntring a prism At th plan surac prpndicular light is unrractd Moving rom th glass to th slop sid light is bnt away rom th normal o th slop Using Snll's law n sin( ϕ ) = n sin(
More informationExponential Functions
Eponntial Functions Dinition: An Eponntial Function is an unction tat as t orm a, wr a > 0. T numbr a is calld t bas. Eampl: Lt i.. at intgrs. It is clar wat t unction mans or som valus o. 0 0,,, 8,,.,.
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationSolution: APPM 1360 Final (150 pts) Spring (60 pts total) The following parts are not related, justify your answers:
APPM 6 Final 5 pts) Spring 4. 6 pts total) Th following parts ar not rlatd, justify your answrs: a) Considr th curv rprsntd by th paramtric quations, t and y t + for t. i) 6 pts) Writ down th corrsponding
More informationExam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.
Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r
More informationGradebook & Midterm & Office Hours
Your commnts So what do w do whn on of th r's is 0 in th quation GmM(1/r-1/r)? Do w nd to driv all of ths potntial nrgy formulas? I don't undrstand springs This was th first lctur I actually larnd somthing
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationy cos x = cos xdx = sin x + c y = tan x + c sec x But, y = 1 when x = 0 giving c = 1. y = tan x + sec x (A1) (C4) OR y cos x = sin x + 1 [8]
DIFF EQ - OPTION. Sol th iffrntial quation tan +, 0
More informationdy 1. If fx ( ) is continuous at x = 3, then 13. If y x ) for x 0, then f (g(x)) = g (f (x)) when x = a. ½ b. ½ c. 1 b. 4x a. 3 b. 3 c.
AP CALCULUS BC SUMMER ASSIGNMENT DO NOT SHOW YOUR WORK ON THIS! Complt ts problms during t last two wks of August. SHOW ALL WORK. Know ow to do ALL of ts problms, so do tm wll. Itms markd wit a * dnot
More informationProblem Set 6 Solutions
6.04/18.06J Mathmatics for Computr Scinc March 15, 005 Srini Dvadas and Eric Lhman Problm St 6 Solutions Du: Monday, March 8 at 9 PM in Room 3-044 Problm 1. Sammy th Shark is a financial srvic providr
More informationu x v x dx u x v x v x u x dx d u x v x u x v x dx u x v x dx Integration by Parts Formula
7. Intgration by Parts Each drivativ formula givs ris to a corrsponding intgral formula, as w v sn many tims. Th drivativ product rul yilds a vry usful intgration tchniqu calld intgration by parts. Starting
More informationENGR 323 BHW 15 Van Bonn 1/7
ENGR 33 BHW 5 Van Bonn /7 4.4 In Eriss and 3 as wll as man othr situations on has th PDF o X and wishs th PDF o Yh. Assum that h is an invrtibl untion so that h an b solvd or to ild. Thn it an b shown
More informationThe function y loge. Vertical Asymptote x 0.
Grad 1 (MCV4UE) AP Calculus Pa 1 of 6 Drivativs of Eponntial & Loarithmic Functions Dat: Dfinition of (Natural Eponntial Numr) 0 1 lim( 1 ). 7188188459... 5 1 5 5 Proprtis of and ln Rcall th loarithmic
More informationWhere k is either given or determined from the data and c is an arbitrary constant.
Exponntial growth and dcay applications W wish to solv an quation that has a drivativ. dy ky k > dx This quation says that th rat of chang of th function is proportional to th function. Th solution is
More informationMassachusetts Institute of Technology Department of Mechanical Engineering
Massachustts Institut of Tchnolog Dpartmnt of Mchanical Enginring. Introduction to Robotics Mid-Trm Eamination Novmbr, 005 :0 pm 4:0 pm Clos-Book. Two shts of nots ar allowd. Show how ou arrivd at our
More informationProblem Statement. Definitions, Equations and Helpful Hints BEAUTIFUL HOMEWORK 6 ENGR 323 PROBLEM 3-79 WOOLSEY
Problm Statmnt Suppos small arriv at a crtain airport according to Poisson procss with rat α pr hour, so that th numbr of arrivals during a tim priod of t hours is a Poisson rv with paramtr t (a) What
More informationAP Calculus Multiple-Choice Question Collection connect to college success
AP Calculus Multipl-Choic Qustion Collction 969 998 connct to collg succss www.collgboard.com Th Collg Board: Conncting Studnts to Collg Succss Th Collg Board is a not-for-profit mmbrship association whos
More information10. Limits involving infinity
. Limits involving infinity It is known from th it ruls for fundamntal arithmtic oprations (+,-,, ) that if two functions hav finit its at a (finit or infinit) point, that is, thy ar convrgnt, th it of
More information4. Money cannot be neutral in the short-run the neutrality of money is exclusively a medium run phenomenon.
PART I TRUE/FALSE/UNCERTAIN (5 points ach) 1. Lik xpansionary montary policy, xpansionary fiscal policy rturns output in th mdium run to its natural lvl, and incrass prics. Thrfor, fiscal policy is also
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationGeneral Notes About 2007 AP Physics Scoring Guidelines
AP PHYSICS C: ELECTRICITY AND MAGNETISM 2007 SCORING GUIDELINES Gnral Nots About 2007 AP Physics Scoring Guidlins 1. Th solutions contain th most common mthod of solving th fr-rspons qustions and th allocation
More informationFirst order differential equation Linear equation; Method of integrating factors
First orr iffrntial quation Linar quation; Mtho of intgrating factors Exampl 1: Rwrit th lft han si as th rivativ of th prouct of y an som function by prouct rul irctly. Solving th first orr iffrntial
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &D) AND CALCULUS. TIME : hrs Ma. Marks.75 Not: This qustion papr consists of thr sctions A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.If th portion
More informationWhat are those βs anyway? Understanding Design Matrix & Odds ratios
Ral paramtr stimat WILD 750 - Wildlif Population Analysis of 6 What ar thos βs anyway? Undrsting Dsign Matrix & Odds ratios Rfrncs Hosmr D.W.. Lmshow. 000. Applid logistic rgrssion. John Wily & ons Inc.
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More information3) Use the average steady-state equation to determine the dose. Note that only 100 mg tablets of aminophylline are available here.
PHA 5127 Dsigning A Dosing Rgimn Answrs provi by Jry Stark Mr. JM is to b start on aminophyllin or th tratmnt o asthma. H is a non-smokr an wighs 60 kg. Dsign an oral osing rgimn or this patint such that
More informationPipe flow friction, small vs. big pipes
Friction actor (t/0 t o pip) Friction small vs larg pips J. Chaurtt May 016 It is an intrsting act that riction is highr in small pips than largr pips or th sam vlocity o low and th sam lngth. Friction
More informationMAXIMA-MINIMA EXERCISE - 01 CHECK YOUR GRASP
EXERCISE - MAXIMA-MINIMA CHECK YOUR GRASP. f() 5 () 75 f'() 5. () 75 75.() 7. 5 + 5. () 7 {} 5 () 7 ( ) 5. f() 9a + a +, a > f'() 6 8a + a 6( a + a ) 6( a) ( a) p a, q a a a + + a a a (rjctd) or a a 6.
More informationAddition of angular momentum
Addition of angular momntum April, 0 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat th
More informationAP Calculus Multiple-Choice Question Collection
AP Calculus Multipl-Coic Qustion Collction 985 998 . f is a continuous function dfind for all ral numbrs and if t maimum valu of f () is 5 and t minimum valu of f () is 7, tn wic of t following must b
More informationAddition of angular momentum
Addition of angular momntum April, 07 Oftn w nd to combin diffrnt sourcs of angular momntum to charactriz th total angular momntum of a systm, or to divid th total angular momntum into parts to valuat
More information4.2 Design of Sections for Flexure
4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt
More informationChapter 13 GMM for Linear Factor Models in Discount Factor form. GMM on the pricing errors gives a crosssectional
Chaptr 13 GMM for Linar Factor Modls in Discount Factor form GMM on th pricing rrors givs a crosssctional rgrssion h cas of xcss rturns Hors rac sting for charactristic sting for pricd factors: lambdas
More informationSec 2.3 Modeling with First Order Equations
Sc.3 Modling with First Ordr Equations Mathmatical modls charactriz physical systms, oftn using diffrntial quations. Modl Construction: Translating physical situation into mathmatical trms. Clarly stat
More informationMAT 270 Test 3 Review (Spring 2012) Test on April 11 in PSA 21 Section 3.7 Implicit Derivative
MAT 7 Tst Rviw (Spring ) Tst on April in PSA Sction.7 Implicit Drivativ Rmmbr: Equation of t tangnt lin troug t point ( ab, ) aving slop m is y b m( a ). dy Find t drivativ y d. y y. y y y. y 4. y sin(
More informationProbability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers Roy D. Yates and David J.
Probability and Stochastic Procsss: A Frindly Introduction for Elctrical and Computr Enginrs Roy D. Yats and David J. Goodman Problm Solutions : Yats and Goodman,4.3. 4.3.4 4.3. 4.4. 4.4.4 4.4.6 4.. 4..7
More informationEXST Regression Techniques Page 1
EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy
More informationLecture 37 (Schrödinger Equation) Physics Spring 2018 Douglas Fields
Lctur 37 (Schrödingr Equation) Physics 6-01 Spring 018 Douglas Filds Rducd Mass OK, so th Bohr modl of th atom givs nrgy lvls: E n 1 k m n 4 But, this has on problm it was dvlopd assuming th acclration
More informationEngineering 323 Beautiful HW #13 Page 1 of 6 Brown Problem 5-12
Enginring Bautiful HW #1 Pag 1 of 6 5.1 Two componnts of a minicomputr hav th following joint pdf for thir usful liftims X and Y: = x(1+ x and y othrwis a. What is th probability that th liftim X of th
More informationMathematics 1110H Calculus I: Limits, derivatives, and Integrals Trent University, Summer 2018 Solutions to the Actual Final Examination
Mathmatics H Calculus I: Limits, rivativs, an Intgrals Trnt Univrsity, Summr 8 Solutions to th Actual Final Eamination Tim-spac: 9:-: in FPHL 7. Brought to you by Stfan B lan k. Instructions: Do parts
More informationProd.C [A] t. rate = = =
Concntration Concntration Practic Problms: Kintics KEY CHEM 1B 1. Basd on th data and graph blow: Ract. A Prod. B Prod.C..25.. 5..149.11.5 1..16.144.72 15..83.167.84 2..68.182.91 25..57.193.96 3..5.2.1
More informationIndeterminate Forms and L Hôpital s Rule. Indeterminate Forms
SECTION 87 Intrminat Forms an L Hôpital s Rul 567 Sction 87 Intrminat Forms an L Hôpital s Rul Rcogniz its that prouc intrminat forms Apply L Hôpital s Rul to valuat a it Intrminat Forms Rcall from Chaptrs
More informationChapter 4: Functions of Two Variables
Chapter 4 Functions o Two Variables Applied Calculus 39 Chapter 4: Functions o Two Variables PreCalculus Idea -- Topographical Maps I ou ve ever hiked, ou have probabl seen a topographical map. Here is
More informationSECTION where P (cos θ, sin θ) and Q(cos θ, sin θ) are polynomials in cos θ and sin θ, provided Q is never equal to zero.
SETION 6. 57 6. Evaluation of Dfinit Intgrals Exampl 6.6 W hav usd dfinit intgrals to valuat contour intgrals. It may com as a surpris to larn that contour intgrals and rsidus can b usd to valuat crtain
More informationMathematics. Complex Number rectangular form. Quadratic equation. Quadratic equation. Complex number Functions: sinusoids. Differentiation Integration
Mathmatics Compl numbr Functions: sinusoids Sin function, cosin function Diffrntiation Intgration Quadratic quation Quadratic quations: a b c 0 Solution: b b 4ac a Eampl: 1 0 a= b=- c=1 4 1 1or 1 1 Quadratic
More informationSetting up the problem. Constructing a Model. Stability Analysis of ODE. Cheyne-Stokes. A Specific Case of the ODE 7/20/2011
7/0/0 Rspiration: Using Dla Diffrntial Equations to Modl Rgulation of Vntilation SPWM 0 Trika Harris, Caitlin Hult, Linds Scopptta, and Amanda Sndgar Stting up th problm Brathing is th procss that movs
More informationMCE503: Modeling and Simulation of Mechatronic Systems Discussion on Bond Graph Sign Conventions for Electrical Systems
MCE503: Modling and Simulation o Mchatronic Systms Discussion on Bond Graph Sign Convntions or Elctrical Systms Hanz ichtr, PhD Clvland Stat Univrsity, Dpt o Mchanical Enginring 1 Basic Assumption In a
More information