Developing Transfer Functions from Heat & Material Balances
|
|
|
- Marilynn Hood
- 6 years ago
- Views:
Transcription
1 Colorado Sool of Mine CHEN43 Stirred ank Heater Develoing ranfer untion from Heat & Material Balane Examle ranfer untion Stirred ank Heater,,, A,,,,, We will develo te tranfer funtion for a tirred tank eater by etting u te eat & material balane equation for te ae were te volume migt ange (ine flow out of te tank i ontrolled by a valve) e overall ma balane on ti ytem will be: d V were te volume in te tank, V, migt be anging wit time (ine te liquid level migt be anging) If we imlify te yial roertie by auming a ontant liquid denity,, and if te valve a linear flow arateriti, Cv, ten: d A d A Cv Remember tat te teady tate verion of ti equation i: Cv e energy balane will be: d VHˆ Hˆ Hˆ Jon Jeura (jjeura@mineedu) - - Coyrigt 27 Aril 23, 27
2 Colorado Sool of Mine CHEN43 Stirred ank Heater We an lit aart te time derivative term: ˆ d H ˆ d V V H Hˆ Hˆ and ten inert te ma balane ODE for te eond time derivative: d Hˆ V Hˆ Hˆ Hˆ Doing a little bit of mat give: d Hˆ V Hˆ Hˆ Hˆ Hˆ d Hˆ V Hˆ Hˆ d Hˆ V Hˆ Hˆ dhˆ V H H ˆ ˆ We would till like to onvert te entaly term into term tat diretly relate to temerature Even wit a temerature deendent eat aaity te time derivative an be lit into: ˆ d VC ˆ ˆ H H If we aume tat te eat aaitie are ontant and tat tere i no reation, ten: and: ˆ Hˆ C H ˆ and Hˆ H ˆ ref ref ref, ref, ˆ d VC ˆ C Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
3 Colorado Sool of Mine CHEN43 Stirred ank Heater or final imliity let aume te denity i eentially ontant, o and: ˆ d VC ˆ C e teady tate verion of ti equation i: Cˆ Canging temerature, ontant inlet flow rate Let develo te tranfer funtion uing tee ODE a our tarting oint Let aume tat te only indeendent variable tat an ange are and We will old ontant, o V (and ene ) will be ontant, too So we only need to deal wit te energy balane equation Wen we linearize ti we need to ut it in term of te deviation variable and (for te indeendent variable) and (for te deendent variable): ˆ d VC ˆ C C ˆ C ˆ ˆ d VC ˆ ˆ C C ˆ d VC ˆ ˆ C C Note tat ti equation ow ow te tirred tank fluid temerature i affeted by ange in te oter temerature i ODE get onverted to tranfer funtion by taking te Lalae tranform : VCˆ ˆ ˆ C C ˆ ˆ ˆ VC C C Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
4 Colorado Sool of Mine CHEN43 Stirred ank Heater Cˆ VCˆ ˆ ˆ ˆ C VC C i ow tat we ave two tranfer funtion: were: Cˆ ˆ ˆ VC C and ˆ VC ˆ C Notie tat bot of tee tranfer funtion ome from te ame firt order ODE, o bot are referred to a firt order tranfer funtion We ll ee in a later etion tat we would like to ut te equation into a tandard form tat arie wen te multilier on te term i unity (ie, one) i an be done by dividing everyting by te exiting fator on te term, C ˆ Starting wit te deviation variable form of te ODE: VCˆ ˆ d C Cˆ ˆ ˆ C C We an imlify te form of te equation by defining new ymbol for te grouing: o: VCˆ Cˆ, Cˆ K Cˆ, and K C d K K K K K K and te tranfer funtion are now: ˆ Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
5 Colorado Sool of Mine CHEN43 Stirred ank Heater K and K A blok diagram for te tirred tank eater an be drawn a follow Allow Inlet low Rate to Cange e reviou et of tranfer funtion were derived from te energy balane ODE linearized only for ange in temerature If te inlet flow rate alo ange ten we mut alo take ti into aount in bot ODE Wen we linearize tee ODE we need to ut tem in term of te deviation variable,, and (for te indeendent variable) and & (for te deendent variable) e overall material balane beome: d A Cv Cv Cv Cv Cv d A Cv d A Cv Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
6 Colorado Sool of Mine CHEN43 d A C v and: Stirred ank Heater e energy balane beome (remember to firt divide by to ut all variable on te leftand ide of te ODE): ˆ ˆ d C A C Cˆ Cˆ Cˆ Cˆ ˆ ˆ ˆ d C C AC Cˆ 2 2 Cˆ ˆ ˆ ˆ d C C AC Cˆ 2 2 Cˆ Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
7 Colorado Sool of Mine CHEN43 Stirred ank Heater ˆ ˆ ˆ ˆ d C C C AC Cˆ 2 2 We an multily troug by te teady tate level, ˆ d A ˆ ˆ ˆ C C C C ˆ C Notie tat te term multilying i te teady tate energy balane and i zero, o: A ˆ ˆ ˆ ˆ C d C C C Now we an get te tranfer funtion by onverting to Lalae variable & algebraially maniulating irt from te ma balane: A C v A C v : A C v Next from te energy balane: ˆ ˆ ˆ ˆ A C C C C Cˆ A ˆ ˆ C C Cˆ ˆ ˆ ˆ ˆ A C C A C C Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
8 Colorado Sool of Mine CHEN43 Stirred ank Heater Now tere are tree tranfer funtion from te energy balane: were: Cˆ A ˆ ˆ C C Cˆ A ˆ ˆ C C A ˆ ˆ C C ere i only one tranfer funtion from te ma balane: were A C v e full blok diagram relating te indeendent & deendent variable are own in te following figure Jon Jeura (jjeura@mineedu) Coyrigt 27 Aril 23, 27
Analysis of Feedback Control Systems
Colorado Shool of Mine CHEN403 Feedbak Control Sytem Analyi of Feedbak Control Sytem ntrodution to Feedbak Control Sytem 1 Cloed oo Reone 3 Breaking Aart the Problem to Calulate the Overall Tranfer Funtion
Constrained Single Period Stochastic Uniform Inventory Model With Continuous Distributions of Demand and Varying Holding Cost
Journal of Matemati and Statiti (1): 334-338, 6 ISSN 1549-3644 6 Siene Publiation Contrained Single Period Stoati Uniform Inventory Model Wit Continuou Ditribution of Demand and Varying Holding Cot 1 Hala,
This appendix derives Equations (16) and (17) from Equations (12) and (13).
Capital growt pat of te neoclaical growt model Online Supporting Information Ti appendix derive Equation (6) and (7) from Equation () and (3). Equation () and (3) owed te evolution of pyical and uman capital
8-4 P 2. = 12 kw. AIR T = const. Therefore, Q &
8-4 8-4 Air i compreed teadily by a compreor. e air temperature i mataed contant by eat rejection to te urroundg. e rate o entropy cange o air i to be determed. Aumption i i a teady-low proce ce tere i
Chapter 10. Closed-Loop Control Systems
hapter 0 loed-loop ontrol Sytem ontrol Diagram of a Typical ontrol Loop Actuator Sytem F F 2 T T 2 ontroller T Senor Sytem T TT omponent and Signal of a Typical ontrol Loop F F 2 T Air 3-5 pig 4-20 ma
Section 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
Introduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
Math 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
Internal Model Control
Internal Model Control Part o a et o leture note on Introdution to Robut Control by Ming T. Tham 2002 The Internal Model Prinile The Internal Model Control hiloohy relie on the Internal Model Prinile,
. Compute the following limits.
Today: Tangent Lines and te Derivative at a Point Warmup:. Let f(x) =x. Compute te following limits. f( + ) f() (a) lim f( +) f( ) (b) lim. Let g(x) = x. Compute te following limits. g(3 + ) g(3) (a) lim
Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.
Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope
Chapter 6 Control Systems Design by Root-Locus Method. Lag-Lead Compensation. Lag lead Compensation Techniques Based on the Root-Locus Approach.
hapter 6 ontrol Sytem Deign by Root-Lou Method Lag-Lead ompenation Lag lead ompenation ehnique Baed on the Root-Lou Approah. γ β K, ( γ >, β > ) In deigning lag lead ompenator, we onider two ae where γ
Velocity or 60 km/h. a labelled vector arrow, v 1
11.7 Velocity en you are outide and notice a brik wind blowing, or you are riding in a car at 60 km/, you are imply conidering te peed of motion a calar quantity. ometime, owever, direction i alo important
S Mobile Communications Services and Systems
S-7.60 Mobile ommuniation Serie and Sytem Tutorial, Noember 9, 004. One a pyiit obert Wood did not top i ar beind te red trai ligt. He exue imel by uing oppler eet. Beaue o oppler it te red ligt ad turned
Math 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
We name Functions f (x) or g(x) etc.
Section 2 1B: Function Notation Bot of te equations y 2x +1 and y 3x 2 are functions. It is common to ave two or more functions in terms of x in te same problem. If I ask you wat is te value for y if x
MVT and Rolle s Theorem
AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state
T Turbine 8. Boiler fwh fwh I Condenser 4 3 P II P I P III. (a) From the steam tables (Tables A-4, A-5, and A-6), = = 10 MPa. = 0.
- - A team poer plant operate on an ideal regenerative anke cycle it to open feedater eater. e poer put of te poer plant and te termal efficiency of te cycle are to be determed. Aumption Steady operatg
(b) The heat transfer can be determined from an energy balance on the system
8-5 Heat is transferred to a iston-cylinder device wit a set of stos. e work done, te eat transfer, te exergy destroyed, and te second-law efficiency are to be deterined. Assutions e device is stationary
EGN 3353C Fluid Mechanics
eture 5 Bukingham PI Theorem Reall dynami imilarity beteen a model and a rototye require that all dimenionle variable mut math. Ho do e determine the '? Ue the method of reeating variable 6 te Ste : Parameter
Hilbert-Space Integration
Hilbert-Space Integration. Introduction. We often tink of a PDE, like te eat equation u t u xx =, a an evolution equation a itorically wa done for ODE. In te eat equation example two pace derivative are
Analysis: The speed of the proton is much less than light speed, so we can use the
Section 1.3: Wave Proerties of Classical Particles Tutorial 1 Practice, age 634 1. Given: 1.8! 10 "5 kg # m/s; 6.63! 10 "34 J #s Analysis: Use te de Broglie relation, λ. Solution:! 6.63 " 10#34 kg $ m
Math 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
Numerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
Taylor Series and the Mean Value Theorem of Derivatives
1 - Taylor Series and te Mean Value Teorem o Derivatives Te numerical solution o engineering and scientiic problems described by matematical models oten requires solving dierential equations. Dierential
Logarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
MAT 1339-S14 Class 2
MAT 1339-S14 Class 2 July 07, 2014 Contents 1 Rate of Cange 1 1.5 Introduction to Derivatives....................... 1 2 Derivatives 5 2.1 Derivative of Polynomial function.................... 5 2.2 Te
Analysis of Step Response, Impulse and Ramp Response in the Continuous Stirred Tank Reactor System
ISSN: 454-50 Volume 0 - Iue 05 May 07 PP. 7-78 Analyi of Step Repone, Impule and Ramp Repone in the ontinuou Stirred Tank Reactor Sytem * Zohreh Khohraftar, Pirouz Derakhhi, (Department of hemitry, Science
1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.
004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following
Average Rate of Change
Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope
Notes on the function gsw_enthalpy_first_derivatives_ct_exact(sa,ct,p)
Notes on gsw_entaly_first_derivatives_c_exact 1 Notes on te function gsw_entaly_first_derivatives_c_exact(c) is function gsw_entaly_first_derivatives_c_exact(c) evaluates two of te first order artial derivatives
Tangent Lines-1. Tangent Lines
Tangent Lines- Tangent Lines In geometry, te tangent line to a circle wit centre O at a point A on te circle is defined to be te perpendicular line at A to te line OA. Te tangent lines ave te special property
Math 312 Lecture Notes Modeling
Mat 3 Lecture Notes Modeling Warren Weckesser Department of Matematics Colgate University 5 7 January 006 Classifying Matematical Models An Example We consider te following scenario. During a storm, a
Correction for Simple System Example and Notes on Laplace Transforms / Deviation Variables ECHE 550 Fall 2002
Correction for Simple Sytem Example and Note on Laplace Tranform / Deviation Variable ECHE 55 Fall 22 Conider a tank draining from an initial height of h o at time t =. With no flow into the tank (F in
Lesson 6: The Derivative
Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange
Function Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
3.4 Algebraic Limits. Ex 1) lim. Ex 2)
Calculus Maimus.4 Algebraic Limits At tis point, you sould be very comfortable finding its bot grapically and numerically wit te elp of your graping calculator. Now it s time to practice finding its witout
2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
(4.2) -Richardson Extrapolation
(.) -Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Suppose tat lim G 0 and lim F L. Te function F is said to converge to L as
De-Coupler Design for an Interacting Tanks System
IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) e-issn: 2278-1676,p-ISSN: 2320-3331, Volume 7, Issue 3 (Sep. - Oct. 2013), PP 77-81 De-Coupler Design for an Interacting Tanks System
MAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
MATH CALCULUS I 2.1: Derivatives and Rates of Change
MATH 12002 - CALCULUS I 2.1: Derivatives and Rates of Cange Professor Donald L. Wite Department of Matematical Sciences Kent State University D.L. Wite (Kent State University) 1 / 1 Introduction Our main
1. State whether the function is an exponential growth or exponential decay, and describe its end behaviour using limits.
Questions 1. State weter te function is an exponential growt or exponential decay, and describe its end beaviour using its. (a) f(x) = 3 2x (b) f(x) = 0.5 x (c) f(x) = e (d) f(x) = ( ) x 1 4 2. Matc te
Preface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
ME 375 FINAL EXAM Wednesday, May 6, 2009
ME 375 FINAL EXAM Wedneday, May 6, 9 Diviion Meckl :3 / Adam :3 (circle one) Name_ Intruction () Thi i a cloed book examination, but you are allowed three ingle-ided 8.5 crib heet. A calculator i NOT allowed.
Polynomial Functions. Linear Functions. Precalculus: Linear and Quadratic Functions
Concepts: definition of polynomial functions, linear functions tree representations), transformation of y = x to get y = mx + b, quadratic functions axis of symmetry, vertex, x-intercepts), transformations
Continuity. Example 1
Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)
Solutions to Homework #05 MATH ln z 2 + x 2 1 = 2x2 z 2 + x 2 + ln z2 + x 2. = x. ln z 2 + x 2 2z z 2 + x 2 = 2xz
Solutions to Homeork #05 MATH Kaai Section. (I) Exercise #. g x and g z : Product Rule: g x = x @ ln z + x + ln z + x @ [x] = x x z + x + ln z + x = x z + x + ln z + x x is eld constant. g z = x @ ln z
y = 3 2 x 3. The slope of this line is 3 and its y-intercept is (0, 3). For every two units to the right, the line rises three units vertically.
Mat 2 - Calculus for Management and Social Science. Understanding te basics of lines in te -plane is crucial to te stud of calculus. Notes Recall tat te and -intercepts of a line are were te line meets
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION
MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of
pancakes. A typical pancake also appears in the sketch above. The pancake at height x (which is the fraction x of the total height of the cone) has
Volumes One can epress volumes of regions in tree dimensions as integrals using te same strateg as we used to epress areas of regions in two dimensions as integrals approimate te region b a union of small,
Time (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
Exam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
The Derivative The rate of change
Calculus Lia Vas Te Derivative Te rate of cange Knowing and understanding te concept of derivative will enable you to answer te following questions. Let us consider a quantity wose size is described by
Notes on Planetary Motion
(1) Te motion is planar Notes on Planetary Motion Use 3-dimensional coordinates wit te sun at te origin. Since F = ma and te gravitational pull is in towards te sun, te acceleration A is parallel to te
Physics 207 Lecture 23
ysics 07 Lecture ysics 07, Lecture 8, Dec. Agenda:. Finis, Start. Ideal gas at te molecular level, Internal Energy Molar Specific Heat ( = m c = n ) Ideal Molar Heat apacity (and U int = + W) onstant :
4.2 - Richardson Extrapolation
. - Ricardson Extrapolation. Small-O Notation: Recall tat te big-o notation used to define te rate of convergence in Section.: Definition Let x n n converge to a number x. Suppose tat n n is a sequence
1 = The rate at which the entropy of the high temperature reservoir changes, according to the definition of the entropy, is
8-7 8-9 A reverible eat um wit eciied reervoir temerature i conidered. e entroy cange o two reervoir i to be calculated and it i to be determed i ti eat um atiie te creae entroy rcile. Aumtion e eat um
Feedforward Control identifiable disturbance measured,
Feeforwar Control So far, mot of the focu of thi coure ha been on feeback control. In certain ituation, the erformance of control ytem can be enhance greatly by the alication of feeforwar control. What
Chapter A 9.0-V battery is connected to a lightbulb, as shown below. 9.0-V Battery. a. How much power is delivered to the lightbulb?
Capter continued carges on te plates were reversed, te droplet would accelerate downward since all forces ten act in te same direction as gravity. 5. A 0.5-F capacitor is able to store 7.0 0 C of carge
Simulation and verification of a plate heat exchanger with a built-in tap water accumulator
Simulation and verification of a plate eat excanger wit a built-in tap water accumulator Anders Eriksson Abstract In order to test and verify a compact brazed eat excanger (CBE wit a built-in accumulation
A: Derivatives of Circular Functions. ( x) The central angle measures one radian. Arc Length of r
4: Derivatives of Circular Functions an Relate Rates Before we begin, remember tat we will (almost) always work in raians. Raians on't ivie te circle into parts; tey measure te size of te central angle
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 6. Differential Calculus 6.. Differentiation from First Principles. In tis capter, we will introduce
Physics 107 Problem 2.5 O. A. Pringle h Physics 107 Problem 2.6 O. A. Pringle
Pysis 07 Problem 25 O A Pringle 3 663 0 34 700 = 284 0 9 Joules ote I ad to set te zero tolerane ere e 6 0 9 ev joules onversion ator ev e ev = 776 ev Pysis 07 Problem 26 O A Pringle 663 0 34 3 ev
1 Limits and Continuity
1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function
How to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
The Derivative as a Function
Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a + )
Notes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
Bob Brown Math 251 Calculus 1 Chapter 3, Section 1 Completed 1 CCBC Dundalk
Bob Brown Mat 251 Calculus 1 Capter 3, Section 1 Completed 1 Te Tangent Line Problem Te idea of a tangent line first arises in geometry in te context of a circle. But before we jump into a discussion of
h=1 cm 1.03 g/cm 1.43 g/cm
ES 43/614: Introduction to Oceanograpy Solution Homeork # 1) Tere i uge lake it contant ater dept 1 cm and an extenion of 5 km. Te ater in te lake i till and unperturbed (noting move). No e drop object,
1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -
Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.
Math 124. Section 2.6: Limits at infinity & Horizontal Asymptotes. 1 x. lim
Mat 4 Section.6: Limits at infinity & Horizontal Asymptotes Tolstoy, Count Lev Nikolgevic (88-90) A man is like a fraction wose numerator is wat e is and wose denominator is wat e tinks of imself. Te larger
v v = = Mixing chamber: = 30 or, = s6 Then, and = 52.4% Turbine Boiler process heater Condenser 7 MPa Q in 0.6 MPa Q proces 10 kpa Q out
0-0- A cogeneration plant i to generate poer and proce eat. art o te team extracted rom te turbe at a relatively ig preure i ued or proce eatg. e poer produced and te utilization actor o te plant are to
Lab 6 Derivatives and Mutant Bacteria
Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge
Figure 1 Siemens PSSE Web Site
Stability Analyi of Dynamic Sytem. In the lat few lecture we have een how mall ignal Lalace domain model may be contructed of the dynamic erformance of ower ytem. The tability of uch ytem i a matter of
FOCUS ON THEORY. We recall that a function g(x) is differentiable at the point a if the limit
FOCUS ON THEORY 653 DIFFERENTIABILITY Notes on Differentiabilit In Section 13.3 we gave an informal introduction to te concet of differentiabilit. We called a function f (; ) differentiable at a oint (a;
NUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
LEARNING FROM MISTAKES
AP Central Quetion of te Mont May 3 Quetion of te Mont By Lin McMullin LEARNING FROM MISTAKES Ti i te firt Quetion of te Mont tat ill appear on te Calculu ection of AP Central. Tee are not AP Exam quetion,
Excerpt from "Calculus" 2013 AoPS Inc.
Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities
Polynomials 3: Powers of x 0 + h
near small binomial Capter 17 Polynomials 3: Powers of + Wile it is easy to compute wit powers of a counting-numerator, it is a lot more difficult to compute wit powers of a decimal-numerator. EXAMPLE
EF 152 Exam #3, Fall, 2012 Page 1 of 6
EF 5 Exam #3, Fall, 0 Page of 6 Name: Setion: Guidelines: ssume 3 signifiant figures for all given numbers. Sow all of your work no work, no redit Write your final answer in te box provided - inlude units
does NOT exist. WHAT IF THE NUMBER X APPROACHES CANNOT BE PLUGGED INTO F(X)??????
MATH 000 Miterm Review.3 Te it of a function f ( ) L Tis means tat in a given function, f(), as APPROACHES c, a constant, it will equal te value L. Tis is c only true if f( ) f( ) L. Tat means if te verticle
Prediction of Coating Thickness
Prediction of Coating Tickness Jon D. Wind Surface Penomena CE 385M 4 May 1 Introduction Tis project involves te modeling of te coating of metal plates wit a viscous liquid by pulling te plate vertically
SIMPLE RANKINE CYCLE. 3 expander. boiler. pump. condenser 1 W Q. cycle cycle. net. shaft
SIMPLE RANKINE CYCLE um boiler exander condener Steady Flow, Oen Sytem - region ace Steady Flow Energy Equation for Procee m (u Pum Proce,, Boiler Proce,, V ρg) 0, 0, Exanion Proce,, 0, Condener Proce,,
Natural Convection Experiment Measurements from a Vertical Surface
OBJECTIVE Natural Convetion Experiment Measurements from a Vertial Surfae 1. To demonstrate te basi priniples of natural onvetion eat transfer inluding determination of te onvetive eat transfer oeffiient.
The Second Law implies:
e Send Law ilie: ) Heat Engine η W in H H L H L H, H H ) Ablute eerature H H L L Sale, L L W ) Fr a yle H H L L H 4) Fr an Ideal Ga Cyle H H L L L δ reerible ree d Claiu Inequality δ eerible Cyle fr a
Finite Difference Formulae for Unequal Sub- Intervals Using Lagrange s Interpolation Formula
Int. Journal o Mat. Analyi, Vol., 9, no. 7, 85-87 Finite Dierence Formulae or Unequal Sub- Interval Uing Lagrange Interpolation Formula Aok K. Sing a and B. S. Badauria b Department o Matematic, Faculty
MTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
The derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
f a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
Derivatives of Exponentials
mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function
Kelvin Planck Statement of the Second Law. Clausius Statement of the Second Law
Kelv Planck Statement of te Second aw It is imossible to construct an enge wic, oeratg a cycle, will roduce no oter effect tan te extraction of eat from a sgle reservoir and te erformance of an equivalent
Continuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
A Reconsideration of Matter Waves
A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,
Combining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
MATH1151 Calculus Test S1 v2a
MATH5 Calculus Test 8 S va January 8, 5 Tese solutions were written and typed up by Brendan Trin Please be etical wit tis resource It is for te use of MatSOC members, so do not repost it on oter forums
KNOWN: Air undergoes a polytropic process in a piston-cylinder assembly. The work is known.
PROBLEM.7 A hown in Fig. P.7, 0 ft of air at T = 00 o R, 00 lbf/in. undergoe a polytropic expanion to a final preure of 5.4 lbf/in. The proce follow pv. = contant. The work i W = 94.4 Btu. Auming ideal