MATH 2410 Review of Mixing Problems
|
|
- Florence Russell
- 5 years ago
- Views:
Transcription
1 MATH 2410 Review of Mixing Problems David Nichols The following examples explore two different kinds of mixing problems. The word problems are very similar, but the differential equations that result are solved using different methods. After the examples, there is an exercise for you to try on your own. xample 1 Suppose we start with a 100-gallon vat of pure water and then start pouring in ethanol at a rate of 2 gallons/minute. Some rotors in the vat mix the ethanol and water thoroughly. At the same time that we start pouring in the ethanol, we also open a drain at the bottom of the vat, so that the mixture of water & ethanol drains out at a rate of 2 gallons/minute. Set up a differential equation for the quantity (in gallons) (t) of ethanol in the vat at time t (in minutes), and then solve the differential equation. The differential equation we set up will look like this: = in out. Our job is to express the rate of ethanol moving in and out of the vat in mathematical terms. The in part is easy, because the problem told us the ethanol is poured in at a rate of 2 gallons/minute. So now we have this: = 2 out. What s the rate of ethanol leaving the vat? Well, we know that the mixture leaves the vat at 2 gallons/minute. But what proportion of that is ethanol? We have a name for the amount of ethanol in the whole vat:. Since the ethanol is mixed throughout the whole vat, and there are 100 gallons of mixture in the vat (notice this doesn t change because every minute 2 gallons enter the tank and 2 gallons leave the tank), the concentration of ethanol is 100. So the differential equation we re after is ( ) = 2 2 =
2 How do we solve this? The first thing we check is whether the equation is separable, since separable equations are (usually) the easiest to solve. Sure enough, we can separate the s and the t s and then integrate: 2 /50 = 1 2 /50 = 1 50 ln 2 50 = t + c 1 ln 2 50 = t 50 + c = c 3e t/ = c 4e t/50 50 = 2 c 4e t/50 (t) = 100 c 5 e t/50. Now we should figure out what the unknown constant c 5 is. To figure out this sort of thing, we normally plug in an initial condition. What s the initial condition here? We started with a tank of pure water, so at time t = 0 there was no ethanol. In other words, (0) = 0. We can plug this in: So c 5 = 100. Thus our final answer is 0 = 100 c 5 e 0 = 100 c 5. (t) = e t/50.
3 xample 2 Suppose we start with a 200-gallon vat of pure water and then start pouring in acetic acid (AcOH) at a rate of 4 gallons/minute. Some rotors in the vat mix the acetic acid and water thoroughly. At the same time that we start pouring in the acetic acid, we also open a drain at the bottom of the vat, so that the mixture of water & acetic acid drains out at a rate of 5 gallons/minute. Set up a differential equation for the quantity (in gallons) A(t) of acetic acid in the vat at time t (in minutes), and then solve the differential equation. Once again, we want to set up an equation of the form = in out, and once again we can start filling in numbers from the word problem: = 4 gal } min {{} rate of AcOH in 5 gal (concentration of acetic acid). } min {{} rate of mix out What is the concentration of acetic acid in the vat? It s the amount of acetic acid, which we are calling A, divided by the total volume of the mixture. But be careful here! The volume isn t 200. The volume starts at 200 gallons, but every minute thereafter we add 4 gallons and subtract 5 gallons. In other words, the vat loses 1 gallon every minute, so the actual volume is 200 t. Now we can complete the differential equation: ( ) A = t This time the equation is not separable. In fact, it s first order linear, so our first step to solve it is to write all the terms with A in them on the left and everything else on the right: + 5 A = t }{{} p(t) Next, we find an integrating factor using the formula I = e p(t), where p(t) is shorthand for the coefficient of A in the equation. I = e t 5 ln 200 t = e = 200 t 5.
4 It s worth pausing here to discuss the absolute value. Usually, these are very important. But in this case, the absolute value doesn t matter. Why? Because in the physical system we re modeling, 200 t represents the volume of mixture left in the mixing vat. We can never have a negative number of gallons left. So 200 t will never be negative, and we can ignore the absolute value and just write I = (200 t) 5. The next step in our solution is to multiply the whole equation by the integrating factor: 5 (200 t) + 5(200 t) 6 A = 4(200 t) 5. Once we ve done that, the left hand side of the equation always works out to be a case of the product rule, and always follows the same pattern: 5 (200 t) + 5(200 t) 6 A = d ( (200 t) 5 A ). this is the product rule So we can rewrite the left hand side of the equation and just integrate: d ( (200 t) 5 A ) = 4(200 t) 5 d ( (200 t) 5 A ) = 4(200 t) 5 (200 t) 5 A = (200 t) 4 + C A(t) = 200 t + C(200 t) 5. We solve for C by plugging in the initial condition A(0) = 0 as before: 0 = C(200) 5, so C = Now we can write out the complete answer: A(t) = 200 t (200 t) 5. It is helpful to graph the solution, in particular so that we can visualize whether our solution makes physical sense. A graph appears atop the next page.
5 y x xercise A drain is opened in a 150-liter tank of pure water so that the contents of the tank pour out at a rate of 3 liters/minute. At the same time, a chute is opened so that a solution of 50% hydrazine and 50% water starts pouring into the tank at a rate of 3 liters/minute. During this process, the contents of the tank are thoroughly mixed. Write out a differential equation describing the amount N(t) of hydrazine (in liters) over time (in minutes). Then solve the differential equation for N(t).
Lesson 3-2: Solving Linear Systems Algebraically
Yesterday we took our first look at solving a linear system. We learned that a linear system is two or more linear equations taken at the same time. Their solution is the point that all the lines have
More informationThree major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation
Section 2.3 Modeling : Key Terms: Three major steps in modeling: Construction of the Model Analysis of the Model Comparison with Experiment or Observation Mixing Problems Population Example Continuous
More informationMAT 311 Midterm #1 Show your work! 1. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE. y = (1 x 2 y 2 ) 1/3
MAT 3 Midterm # Show your work!. The existence and uniqueness theorem says that, given a point (x 0, y 0 ) the ODE y = ( x 2 y 2 ) /3 has a unique (local) solution with initial condition y(x 0 ) = y 0
More informationSolving Equations with Addition and Subtraction
OBJECTIVE: You need to be able to solve equations by using addition and subtraction. In math, when you say two things are equal to each other, you mean they represent the same value. We use the = sign
More informationHomework 2 Solutions Math 307 Summer 17
Homework 2 Solutions Math 307 Summer 17 July 8, 2017 Section 2.3 Problem 4. A tank with capacity of 500 gallons originally contains 200 gallons of water with 100 pounds of salt in solution. Water containing
More informationP (t) = rp (t) 22, 000, 000 = 20, 000, 000 e 10r = e 10r. ln( ) = 10r 10 ) 10. = r. 10 t. P (30) = 20, 000, 000 e
APPM 360 Week Recitation Solutions September 18 01 1. The population of a country is growing at a rate that is proportional to the population of the country. The population in 1990 was 0 million and in
More informationMath 392 Exam 1 Solutions Fall (10 pts) Find the general solution to the differential equation dy dt = 1
Math 392 Exam 1 Solutions Fall 20104 1. (10 pts) Find the general solution to the differential equation = 1 y 2 t + 4ty = 1 t(y 2 + 4y). Hence (y 2 + 4y) = t y3 3 + 2y2 = ln t + c. 2. (8 pts) Perform Euler
More information(a) x cos 3x dx We apply integration by parts. Take u = x, so that dv = cos 3x dx, v = 1 sin 3x, du = dx. Thus
Math 128 Midterm Examination 2 October 21, 28 Name 6 problems, 112 (oops) points. Instructions: Show all work partial credit will be given, and Answers without work are worth credit without points. You
More informationLesson 10 MA Nick Egbert
Overview There is no new material for this lesson, we just apply our knowledge from the previous lesson to some (admittedly complicated) word problems. Recall that given a first-order linear differential
More information8.a: Integrating Factors in Differential Equations. y = 5y + t (2)
8.a: Integrating Factors in Differential Equations 0.0.1 Basics of Integrating Factors Until now we have dealt with separable differential equations. Net we will focus on a more specific type of differential
More informationProblem Set. Assignment #1. Math 3350, Spring Feb. 6, 2004 ANSWERS
Problem Set Assignment #1 Math 3350, Spring 2004 Feb. 6, 2004 ANSWERS i Problem 1. [Section 1.4, Problem 4] A rocket is shot straight up. During the initial stages of flight is has acceleration 7t m /s
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationMath 120. x x 4 x. . In this problem, we are combining fractions. To do this, we must have
Math 10 Final Eam Review 1. 4 5 6 5 4 4 4 7 5 Worked out solutions. In this problem, we are subtracting one polynomial from another. When adding or subtracting polynomials, we combine like terms. Remember
More informationof 8 28/11/ :25
Paul's Online Math Notes Home Content Chapter/Section Downloads Misc Links Site Help Contact Me Differential Equations (Notes) / First Order DE`s / Modeling with First Order DE's [Notes] Differential Equations
More informationSections 6.1 and 6.2: Systems of Linear Equations
What is a linear equation? Sections 6.1 and 6.2: Systems of Linear Equations We are now going to discuss solving systems of two or more linear equations with two variables. Recall that solving an equation
More informationSolving Equations with Addition and Subtraction. Solving Equations with Addition and Subtraction. Solving Equations with Addition and Subtraction
OBJECTIVE: You need to be able to solve equations by using addition and subtraction. In math, when you say two things are equal to each other, you mean they represent the same value. We use the sign to
More informationUnit 4 Systems of Equations Systems of Two Linear Equations in Two Variables
Unit 4 Systems of Equations Systems of Two Linear Equations in Two Variables Solve Systems of Linear Equations by Graphing Solve Systems of Linear Equations by the Substitution Method Solve Systems of
More informationExam 2 Solutions, Math March 17, ) = 1 2
Eam Solutions, Math 56 March 7, 6. Use the trapezoidal rule with n = 3 to approimate (Note: The eact value of the integral is ln 5 +. (you do not need to verify this or use it in any way to complete this
More informationPre Algebra Section 4.2
Unit 4 - Equations Section 2 Solving One-Step Equations In this section we will be looking for solutions to equations. A solution is a number that can be plugged into an equation that keeps the equation
More informationCollege Algebra. Chapter 5 Review Created by: Lauren Atkinson. Math Coordinator, Mary Stangler Center for Academic Success
College Algebra Chapter 5 Review Created by: Lauren Atkinson Math Coordinator, Mary Stangler Center for Academic Success Note: This review is composed of questions from the chapter review at the end of
More informationSolution Guide for Chapter 10
Solution Guide for Chapter 10 Here are the solutions for the Doing the Math exercises in Kiss My Math! DTM from p.133-4 2. 8 7 + 3 =? So, let s distribute the to each term inside the parentheses. In order
More informationLesson 3-1: Solving Linear Systems by Graphing
For the past several weeks we ve been working with linear equations. We ve learned how to graph them and the three main forms they can take. Today we re going to begin considering what happens when we
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationLecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations
Lecture Notes for Math 251: ODE and PDE. Lecture 6: 2.3 Modeling With First Order Equations Shawn D. Ryan Spring 2012 1 Modeling With First Order Equations Last Time: We solved separable ODEs and now we
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationMath Problem Set #3 Solution 19 February 2001
Math 203-04 Problem Set #3 Solution 19 February 2001 Exercises: 1. B & D, Section 2.3, problem #3. In your answer, give both exact values and decimal approximations for the amount of salt in the tank at
More informationMATH 408N PRACTICE MIDTERM 1
02/0/202 Bormashenko MATH 408N PRACTICE MIDTERM Show your work for all the problems. Good luck! () (a) [5 pts] Solve for x if 2 x+ = 4 x Name: TA session: Writing everything as a power of 2, 2 x+ = (2
More informationConceptual Explanations: Radicals
Conceptual Eplanations: Radicals The concept of a radical (or root) is a familiar one, and was reviewed in the conceptual eplanation of logarithms in the previous chapter. In this chapter, we are going
More informationSection 4.6 Negative Exponents
Section 4.6 Negative Exponents INTRODUCTION In order to understand negative exponents the main topic of this section we need to make sure we understand the meaning of the reciprocal of a number. Reciprocals
More informationCOLLEGE ALGEBRA. Solving Equations and Inequalities. Paul Dawkins
COLLEGE ALGEBRA Solving Equations and Inequalities Paul Dawkins Table of Contents Preface... ii Solving Equations and Inequalities... 3 Introduction... 3 s and Sets... 4 Linear Equations... 8 Application
More informationSolving Systems of Equations
Solving Systems of Equations Solving Systems of Equations What are systems of equations? Two or more equations that have the same variable(s) Solving Systems of Equations There are three ways to solve
More information( ) ( ). ( ) " d#. ( ) " cos (%) " d%
Math 22 Fall 2008 Solutions to Homework #6 Problems from Pages 404-407 (Section 76) 6 We will use the technique of Separation of Variables to solve the differential equation: dy d" = ey # sin 2 (") y #
More informationCh. 3 Equations and Inequalities
Ch. 3 Equations and Inequalities 3.1 Solving Linear Equations Graphically There are 2 methods presented in this section for solving linear equations graphically. Normally I would not cover solving linear
More informationFinding Limits Graphically and Numerically
Finding Limits Graphically and Numerically 1. Welcome to finding limits graphically and numerically. My name is Tuesday Johnson and I m a lecturer at the University of Texas El Paso. 2. With each lecture
More informationMATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November
MATH 307 Introduction to Differential Equations Autumn 2017 Midterm Exam Monday November 6 2017 Name: Student ID Number: I understand it is against the rules to cheat or engage in other academic misconduct
More informationDifferential equations
Differential equations Math 27 Spring 2008 In-term exam February 5th. Solutions This exam contains fourteen problems numbered through 4. Problems 3 are multiple choice problems, which each count 6% of
More informationMath 308 Exam I Practice Problems
Math 308 Exam I Practice Problems This review should not be used as your sole source of preparation for the exam. You should also re-work all examples given in lecture and all suggested homework problems..
More informationWRITING EQUATIONS through 6.1.3
WRITING EQUATIONS 6.1.1 through 6.1.3 An equation is a mathematical sentence that conveys information to the reader. It uses variables and operation symbols (like +, -, /, =) to represent relationships
More information5.2 Infinite Series Brian E. Veitch
5. Infinite Series Since many quantities show up that cannot be computed exactly, we need some way of representing it (or approximating it). One way is to sum an infinite series. Recall that a n is the
More informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More informationf(r) = (r 1/2 r 1/2 ) 3 u = (ln t) ln t ln u = (ln t)(ln (ln t)) t(ln t) g (t) = t
Math 4, Autumn 006 Final Exam Solutions Page of 9. [ points total] Calculate the derivatives of the following functions. You need not simplfy your answers. (a) [4 points] y = 5x 7 sin(3x) + e + ln x. y
More informationMath 2250 Final Exam Practice Problem Solutions. f(x) = ln x x. 1 x. lim. lim. x x = lim. = lim 2
Math 5 Final Eam Practice Problem Solutions. What are the domain and range of the function f() = ln? Answer: is only defined for, and ln is only defined for >. Hence, the domain of the function is >. Notice
More informationMath Applied Differential Equations
Math 256 - Applied Differential Equations Notes Basic Definitions and Concepts A differential equation is an equation that involves one or more of the derivatives (first derivative, second derivative,
More informationAlex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1
Alex s Guide to Word Problems and Linear Equations Following Glencoe Algebra 1 What is a linear equation? It sounds fancy, but linear equation means the same thing as a line. In other words, it s an equation
More information1 Limits and continuity
1 Limits and continuity Question 1. Which of the following its can be evaluated by continuity ( plugging in )? sin(x) (a) x + 1 (d) x 3 x 2 + x 6 (b) e x sin(x) (e) x 2 + x 6 (c) x 2 x 2 + x 6 (f) n (
More informationMath Lecture 18 Notes
Math 1010 - Lecture 18 Notes Dylan Zwick Fall 2009 In our last lecture we talked about how we can add, subtract, and multiply polynomials, and we figured out that, basically, if you can add, subtract,
More information1.5. Applications. Theorem The solution of the exponential decay equation with N(0) = N 0 is N(t) = N 0 e kt.
6 Section Objective(s): The Radioactive Decay Equation Newton s Cooling Law Salt in a Water Tanks 151 Exponential Decay 15 Applications Definition 151 The exponential decay equation for N is N = k N, k
More informationNewton s Cooling Model in Matlab and the Cooling Project!
Newton s Cooling Model in Matlab and the Cooling Project! James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University March 10, 2014 Outline Your Newton
More informationExam 3, Math Fall 2016 October 19, 2016
Exam 3, Math 500- Fall 06 October 9, 06 This is a 50-minute exam. You may use your textbook, as well as a calculator, but your work must be completely yours. The exam is made of 5 questions in 5 pages,
More informationMath 138: Introduction to solving systems of equations with matrices. The Concept of Balance for Systems of Equations
Math 138: Introduction to solving systems of equations with matrices. Pedagogy focus: Concept of equation balance, integer arithmetic, quadratic equations. The Concept of Balance for Systems of Equations
More informationSection 2.6 Solving Linear Inequalities
Section 2.6 Solving Linear Inequalities INTRODUCTION Solving an inequality is much like solving an equation; there are, though, some special circumstances of which you need to be aware. In solving an inequality
More informationDIFFERENTIATION AND INTEGRATION PART 1. Mr C s IB Standard Notes
DIFFERENTIATION AND INTEGRATION PART 1 Mr C s IB Standard Notes In this PDF you can find the following: 1. Notation 2. Keywords Make sure you read through everything and the try examples for yourself before
More informationPOLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
More informationAlgebra 2/Pre-Calculus
Algebra /Pre-Calculus Name Introduction to Eponential Functions (Day 1, Eponential Functions) In this handout, we will introduce eponential functions. Definition We say f () is an eponential function if
More informationChapter 6: The Definite Integral
Name: Date: Period: AP Calc AB Mr. Mellina Chapter 6: The Definite Integral v v Sections: v 6.1 Estimating with Finite Sums v 6.5 Trapezoidal Rule v 6.2 Definite Integrals 6.3 Definite Integrals and Antiderivatives
More informationWord Problems. Mathematics Division, IMSP, UPLB
Word Problems Objectives Upon completion, you should be able to: Translate English statements into mathematical statements Use the techniques learned in solving linear, quadratic and systems of equations
More information6.5 Metric U.S. Customary Measurement Conversions
6. Metric U.S. Customary Measurement Conversions Since most of the world uses the metric system of measurement, we often need to know how to convert back and forth between U.S. Customary measurements and
More informationChapter Usual types of questions Tips What can go ugly. and, common denominator will be
C3 Cheat Sheet Chapter Usual types of questions Tips What can go ugly 1 Algebraic Almost always adding or subtracting Factorise everything in each fraction first. e.g. If denominators Blindly multiplying
More informationUnit 6 Study Guide: Equations. Section 6-1: One-Step Equations with Adding & Subtracting
Unit 6 Study Guide: Equations DUE DATE: A Day: Dec 18 th B Day: Dec 19 th Name Period Score / Section 6-1: One-Step Equations with Adding & Subtracting Textbook Reference: Page 437 Vocabulary: Equation
More informationMath 121, Chapter 1 and Complex Numbers Practice Questions Hints and Answers. 2. Multiply numerator and denominator by complex conjugate and simplify:
Math 2, Chapter and Complex Numbers Practice Questions Hints and Answers (a) (3 2i)( + i) = 2 + 3i 8i 2i 2 = 2 5i 2( ) = 5i (b) i 223 = (i ) 55 (i 3 ) = i 2 Multiply numerator and denominator by complex
More informationIntroduction to Differential Equations Math 286 X1 Fall 2009 Homework 2 Solutions
Introuction to Differential Equations Math 286 X1 Fall 2009 Homewk 2 Solutions 1. Solve each of the following ifferential equations: (a) y + 3xy = 0 (b) y + 3y = 3x (c) y t = cos(t)y () x 2 y x y = 3 Solution:
More informationPRINTABLE VERSION. Quiz 3. Question 1 Give the general solution to. f) None of the above. Question 2 Give the general solution to. 2/1/2016 Print Test
PRINTABLE VERSION Question 1 Give the general solution to Quiz 3 Question 2 Give the general solution to https://assessment.casa.uh.edu/assessment/printtest.htm 1/12 Question 3 + xy = 4 cos(3x) 3 Give
More informationMath 2214 Solution Test 1D Spring 2015
Math 2214 Solution Test 1D Spring 2015 Problem 1: A 600 gallon open top tank initially holds 300 gallons of fresh water. At t = 0, a brine solution containing 3 lbs of salt per gallon is poured into the
More information= (1 3 )= =4 3 +2=4. Now that we have it down to a simple two-step equation, we can solve like normal and get the following: =4 2
6.3 Solving Systems with Substitution While graphing is useful for an estimate, the main way that we can solve a system to get an exact answer is algebraically. There are a few useful methods to do this,
More informationAnswers to Sample Exam Problems
Math Answers to Sample Exam Problems () Find the absolute value, reciprocal, opposite of a if a = 9; a = ; Absolute value: 9 = 9; = ; Reciprocal: 9 ; ; Opposite: 9; () Commutative law; Associative law;
More informationChapter 5: Integrals
Chapter 5: Integrals Section 5.5 The Substitution Rule (u-substitution) Sec. 5.5: The Substitution Rule We know how to find the derivative of any combination of functions Sum rule Difference rule Constant
More informationLimits: How to approach them?
Limits: How to approach them? The purpose of this guide is to show you the many ways to solve it problems. These depend on many factors. The best way to do this is by working out a few eamples. In particular,
More informationSystems of Linear Equations and Inequalities
Systems of Linear Equations and Inequalities Alex Moore February 4, 017 1 What is a system? Now that we have studied linear equations and linear inequalities, it is time to consider the question, What
More information5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality
5.6 Solving Equations Using Both the Addition and Multiplication Properties of Equality Now that we have studied the Addition Property of Equality and the Multiplication Property of Equality, we can solve
More informationCHAPTER 7: TECHNIQUES OF INTEGRATION
CHAPTER 7: TECHNIQUES OF INTEGRATION DAVID GLICKENSTEIN. Introduction This semester we will be looking deep into the recesses of calculus. Some of the main topics will be: Integration: we will learn how
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More informationFind the orthogonal trajectories for the family of curves. 9. The family of parabolas symmetric with respect to the x-axis and vertex at the origin.
Exercises 2.4.1 Find the orthogonal trajectories for the family of curves. 1. y = Cx 3. 2. x = Cy 4. 3. y = Cx 2 + 2. 4. y 2 = 2(C x). 5. y = C cos x 6. y = Ce x 7. y = ln(cx) 8. (x + y) 2 = Cx 2 Find
More informationDerivatives and Continuity
Derivatives and Continuity As with limits, derivatives do not exist unless the right and left-hand derivatives both exist and are equal. There are three main instances when this happens: One, if the curve
More informationAlgebra 8.6 Simple Equations
Algebra 8.6 Simple Equations 1. Introduction Let s talk about the truth: 2 = 2 This is a true statement What else can we say about 2 that is true? Eample 1 2 = 2 1+ 1= 2 2 1= 2 4 1 = 2 2 4 2 = 2 4 = 4
More informationLesson 14: Solving Inequalities
Student Outcomes Students learn if-then moves using the addition and multiplication properties of inequality to solve inequalities and graph the solution sets on the number line. Classwork Exercise 1 (5
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Quadratic Equations
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR ANTHONY BROWN 31 Graphs of Quadratic Functions 3 Quadratic Equations In Chapter we looked at straight lines,
More informationMath 2602 Finite and Linear Math Fall 14. Homework 8: Core solutions
Math 2602 Finite and Linear Math Fall 14 Homework 8: Core solutions Review exercises for Chapter 5 page 183 problems 25, 26a-26b, 29. Section 8.1 on page 252 problems 8, 9, 10, 13. Section 8.2 on page
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationMath 119 Main Points of Discussion
Math 119 Main Points of Discussion 1. Solving equations: When you have an equation like y = 3 + 5, you should see a relationship between two variables, and y. The graph of y = 3 + 5 is the picture of this
More information1 a) Remember, the negative in the front and the negative in the exponent have nothing to do w/ 1 each other. Answer: 3/ 2 3/ 4. 8x y.
AP Calculus Summer Packer Key a) Remember, the negative in the front and the negative in the eponent have nothing to do w/ each other. Answer: b) Answer: c) Answer: ( ) 4 5 = 5 or 0 /. 9 8 d) The 6,, and
More informationMath 121, Chapter 1 and Complex Numbers Practice Questions Hints and Answers. 2. Multiply numerator and denominator by complex conjugate and simplify:
Math 121, Chapter 1 and Complex Numbers Practice Questions Hints and Answers 1. (a) (3 2i)( + 1i) = 12 + 3i 8i 2i 2 = 12 5i 2( 1) = 1 5i. (b) i 223 = (i ) 55 (i 3 ) = i. 2. Multiply numerator and denominator
More informationOPEN QUESTIONS FOR MIDDLE SCHOOL MATH. Marian Small NOVEMBER 2018
OPEN QUESTIONS FOR MIDDLE SCHOOL MATH Marian Small NOVEMBER 2018 1 LET S DO A LITTLE MATH The answer is 30% What might the question be? 2 maybe What is a percent less than half? What is 3/10? What is a
More informationAlgebra Year 9. Language
Algebra Year 9 Introduction In Algebra we do Maths with numbers, but some of those numbers are not known. They are represented with letters, and called unknowns, variables or, most formally, literals.
More informationQ 2.0.2: If it s 5:30pm now, what time will it be in 4753 hours? Q 2.0.3: Today is Wednesday. What day of the week will it be in one year from today?
2 Mod math Modular arithmetic is the math you do when you talk about time on a clock. For example, if it s 9 o clock right now, then it ll be 1 o clock in 4 hours. Clearly, 9 + 4 1 in general. But on a
More informationWhy? 2.2. What Do You Already Know? 2.2. Goals 2.2. Building Mathematical Language 2.2. Key Concepts 2.2
Section. Solving Basic Equations Why. You can solve some equations that arise in the real world by isolating a variable. You can use this method to solve the equation 1 400 + 1 (10) x = 460 to determine
More informationLesson 3: Using Linear Combinations to Solve a System of Equations
Lesson 3: Using Linear Combinations to Solve a System of Equations Steps for Using Linear Combinations to Solve a System of Equations 1. 2. 3. 4. 5. Example 1 Solve the following system using the linear
More informationMath 31 Lesson Plan. Day 2: Sets; Binary Operations. Elizabeth Gillaspy. September 23, 2011
Math 31 Lesson Plan Day 2: Sets; Binary Operations Elizabeth Gillaspy September 23, 2011 Supplies needed: 30 worksheets. Scratch paper? Sign in sheet Goals for myself: Tell them what you re going to tell
More informationUnit 1 Science Models & Graphing
Name: Date: 9/18 Period: Unit 1 Science Models & Graphing Essential Questions: What do scientists mean when they talk about models? How can we get equations from graphs? Objectives Explain why models are
More informationChapter 5: Integrals
Chapter 5: Integrals Section 5.3 The Fundamental Theorem of Calculus Sec. 5.3: The Fundamental Theorem of Calculus Fundamental Theorem of Calculus: Sec. 5.3: The Fundamental Theorem of Calculus Fundamental
More informationCore 1 Module Revision Sheet J MS. 1. Basic Algebra
Core 1 Module Revision Sheet The C1 exam is 1 hour 0 minutes long and is in two sections Section A (6 marks) 8 10 short questions worth no more than 5 marks each Section B (6 marks) questions worth 12
More informationDaily WeBWorK. 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8).
Daily WeBWorK 1. Below is the graph of the derivative f (x) of a function defined on the interval (0, 8). (a) On what intervals is f (x) concave down? f (x) is concave down where f (x) is decreasing, so
More informationAlgebra Review. Finding Zeros (Roots) of Quadratics, Cubics, and Quartics. Kasten, Algebra 2. Algebra Review
Kasten, Algebra 2 Finding Zeros (Roots) of Quadratics, Cubics, and Quartics A zero of a polynomial equation is the value of the independent variable (typically x) that, when plugged-in to the equation,
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More informationwe first add 7 and then either divide by x - 7 = 1 Adding 7 to both sides 3 x = x = x = 3 # 8 1 # x = 3 # 4 # 2 x = 6 1 =?
. Using the Principles Together Applying Both Principles a Combining Like Terms a Clearing Fractions and Decimals a Contradictions and Identities EXAMPLE Solve: An important strategy for solving new problems
More informationMATH 251 Examination I July 1, 2013 FORM A. Name: Student Number: Section:
MATH 251 Examination I July 1, 2013 FORM A Name: Student Number: Section: This exam has 12 questions for a total of 100 points. Show all your work! In order to obtain full credit for partial credit problems,
More informationChapter 2 Notes, Kohler & Johnson 2e
Contents 2 First Order Differential Equations 2 2.1 First Order Equations - Existence and Uniqueness Theorems......... 2 2.2 Linear First Order Differential Equations.................... 5 2.2.1 First
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationHot X: Algebra Exposed
Hot X: Algebra Exposed Solution Guide for Chapter 14 Here are the solutions for the Doing the Math exercises in Hot X: Algebra Exposed! DTM from p.204-206 2. Okay, our units are consistent: bracelets and
More informationDEplot(D(y)(x)=2*sin(x*y(x)),y(x),x=-2..2,[[y(1)=1]],y=-5..5)
Project #1 Math 181 Name: Email your project to ftran@mtsac.edu with your full name and class on the subject line of the email. Do not turn in a hardcopy of your project. Step 1: Initialize the program:
More informationApril 30, Name: Amy s Solutions. Discussion Section: N/A. Discussion TA: N/A
Math 1151, April 30, 010 Exam 3 (in-class) Name: Amy s Solutions Discussion Section: N/A Discussion TA: N/A This exam has 8 multiple-choice problems, each worth 5 points. When you have decided on a correct
More information