Linear Programming Test Review. Day 6
|
|
- Vanessa Alisha Allen
- 5 years ago
- Views:
Transcription
1 Linear Programming Test Review Day 6
2 Warm-Up: Test Review Practice A machine can produce either nuts or bolts, but not both at the same time. The machine can be used at most 8 hours a day. Furthermore, at most 6 hours a day can be used for making nuts and at most 5 hours a day can be used for making bolts. There is a $2 profit for each hour the machine makes nuts and a $3 profit for each hour the machine makes bolts. How many hours per day should the machine make each item in order to maximize profit? What is the maximum profit? Decision Variables: x 1 = x 2 = Constraints: (sketch graph) Objective Function: BE PREPARED TO SHARE Optimal Solution: Done early? Start the Juan Clocks Handout
3 Sharing your Warm-up answers We will use the document camera to share what you wrote for your decision variables, constraints and objective functions. Then we will graph the constraints.
4 A B C D E F Today s Date Here Juan s Clocks Profit Maximization 5 Decision Variable Decision 6 Value Pine Clocks (x 1 ) Oak Clocks (x 2 ) 7 8 Objective Function ($) Total Profit = B8*B6 + C8*C6 10 Constraints Used Available 11 Maximum Varnish 12 Maximum Time 1 4 = B11*B6 + C11*C6 2 2 = B12*B6 + C12*C OR =sumproduct(b12:c12, B6:C6)
5 Homework Questions
6 Today s Agenda Formulating the SK8Man problem on paper You need to do this in your notes. Practice with EXCEL Review TEST FRIDAY!!
7 Title: SK8MAN, Inc. You will need to reference the following table, which is homework packet page 8. Please take that out. Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Ringer x 6 Chinese You also need notebook paper to write down the L.P. formulation.
8 Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Title: SK8MAN, Inc. Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Ringer x 6 Chinese Writing the constraints: (Use your answers to the reading guide packet p. 8.) A. Amount of available North American veneers is 840 and each skateboard requires 7 veneers. 7x 7x 840 Therefore the constraint would be B. Amount of available Chinese veneers is and each skateboard requires 7 veneers. 7( x x x x ) 1470 Therefore the constraint would be
9 Product Type of Maple Shaping Time (min) Screen Printing Time (min) Sporty x 1 Chinese Title: SK8MAN, Inc. Fancy x 2 North American Pool- Runner x 3 Chinese Pool- Beauty x 4 North American Recluse x 5 Chinese Writing the constraints: (Use your guided reading notes.) C. Amount of available time for shaping is Using the information in the table the constraint would be D. Amount of available time for screen printing is. Using the information in the table the constraint would be Ringer x 6 Chinese x 15x 4x 10x 7x 4x x 10x 8x 10x 5x 6x
10 Title: SK8MAN, Inc. Nonnegativity constraints: dummy constraints x 1 0 x2 0 x3 0 x 4 0 x5 0 x6 0 Number of trucks available per week is 700 and each skateboard requires two trucks. Therefore, the constraint would be. 2( x1 x2 x3 x4 x5 x6 ) 700
11 Solving Using Microsoft EXCEL There is no way to find the optimal solution by hand by visualizing a graphical solution so we need Excel SOLVER!
12 Solving and doing Sensitivity Analysis Using Microsoft EXCEL Use the Excel Sheet and Answer Report to answer questions ) What is the optimal solution? 2) What is the profit for this product mix? 3) Identify the constraints that are binding. 4) Identify the constraints that are non-binding. 5) Interpret the meaning of these binding and non-binding constraints in terms of the problem context. 6) Find the slack for each constraint. Interpret the meaning of slack in terms of the problem context.
13 Quiz Corrections & Practice 1) On a new sheet of notebook paper, work with the person beside you to make corrections to your Unit 3 LP quiz. 2) When you are both finished, staple your corrections on top of your quiz and bring it to me to check. 3) Done with Corrections and the Warm-Up worksheet? You may start working on Test Review homework.
14 CHECK YOUR CONSTRAINTS Test Review Warm-Up Handout Juan makes two types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz. of varnish. It takes 2 hours to assemble an oak clock, which requires 4 oz. of varnish. Juan has 16 oz. of varnish in stock and he can work 20 hours. If he makes $3 profit on each pine clock and $4 profit on each oak clock, how many of each type should he make to maximize his profits? Decision Variables: x 1 = # of pine clocks x 2 = # of oak clocks Constraints: (sketch graph) Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 Objective Function: $3x 1 + $4x 2
15 COLLECTED HANDOUT Test Review Warm-Up Juan makes two types of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz. of varnish. It takes 2 hours to assemble an oak clock, which requires 4 oz. of varnish. Juan has 16 oz. of varnish in stock and he can work 20 hours. If he makes $3 profit on each pine clock and $4 profit on each oak clock, how many of each type should he make to maximize his profits? Decision Variables: x 1 = # of pine clocks x 2 = # of oak clocks Constraints: Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 Objective Function: Remember to label your axes with units AND decision variables! $3x 1 + $4x 2
16 X 2 : number of oak clocks Your Graph should look like... Constraints: Varnish x 1 + 4x 2 16 Hours 2 x 1 + 2x 2 20 Non- x 1 0 Negativity x 2 0 You will have to use Matrices or Elimination Method to find this point. X 1 : number of pine clocks
17 X 2 : number of oak clocks Your Completed Table & Solution Corner Objective Value Points (x 1, x 2 ) Function 3x 1 + 4x 2 (0, 0) 3(0) + 4(0) 0 (10, 0) 3(10) + 4(0) 30 (0, 4) 3(0) + 4(4) 16 (8, 2) 3(8) + 4(2) 32 X 1 : number of pine clocks Juan should make 8 pine clocks and 2 oak clocks for a maximum profit of $32.
18 Extra Review on Next Slides
19 Arrival Problem Half Sheet The graph below is created using the following equations: A 3x 2x 48 9x 8x 24 C D E B Find the coordinates of point A, B, C, D and E Discuss with your partner the best approach to figuring this out.
20 Arrival Problem--Answers The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E C A D E B 3x 2x 48 9x 8x 24 A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) Next slide has detailed solutions
21 Arrival Problem--Solutions The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E C A D A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) E 3x 2x 48 B (0,24) and (16,0) are on this line 9x 8x (0,3) and,0 9 Next slide has detailed solutions for point E
22 Arrival Problem--Solutions The graph below is created using the two equations in blue. Find the coordinates of point A, B, C, D and E A Use Matrices or Elimination (see below) to Find the Coordinates of Point E C D E 3x 2x 48 4( 3x 2x 48) B 9x 8x 24 A (0,24) B (16, 0) C (-24/9, 0) D (0,3) E (8,12) 21x 168 x1 8 9(8) 8x 24 x
23 For some review, More about Computer Flips.
24 Continuing Computer Flips (packet p. 5) Adding an Additional Constraint TAKE NOTES The students at Computer Flips notice that they are getting a lot of returns. Every computer that was returned had a problem with one of the add-ons. They realize that they need to test their finished products before shipping them. The student who will do the testing will work 10 hours per week. It takes her 20 minutes to test a Simplex and 24 minutes to test an Omniplex. Write the additional constraint for testing time. 20x1 24x2 600 minutes OR x1 x2 10 hours 3 5 Which one would you consider to be the best to work with? Adapted with permission from NCSU MINDSET materials
25 Now, we are ready to continue At the top of a NEW piece of graph paper, write down the following constraints. We will graph using the intercepts. Simplex models Omniplex Models 60x 1 120x Install Minutes + Intercepts on the Next Slide 20x 4x Testing Minutes + x 0 1 X 0 2
26 Now... Determine the x 1 -intercept and x 2 intercept for each of the following constraints. 60x 120x 2400 x intercept: (, 40 ) 0 x intercept:(, ) 20x 24x 600 x intercept: (, 30 0 ) x intercept:(, 0 25 ) 0 20
27 X 2 : weekly production rate of Omniplex computers The graph would look like... Corner Points (x 1, x 2 ) Objective Function 200x x 2 Value X 1 : weekly production rate of Simplex computers You will have to use Matrices or Elimination Method to find this point. Let s try together! 60x 120x x 24x 600 For reference, the Elimination Method work is on the next slide
28 Elimination to Find Corner Point 60x 120x x 120x x 24x 600 3( 20x 24x 600) 60x 120x x 72x x 600 x x 120(12.5) x 900 x 15
29 X 2 : weekly production rate of Omniplex computers Your Completed Table Corner Points (x 1, x 2 ) Objective Function 200x x 2 Value (0,0) 200(0) + 300(0) 0 (0, 20) 200(0) + 300(20) 6000 (30, 0) 200(30) + 300(0) 6000 (15, 12.5) 200(15) + 300(12.5) 6750 X 1 : weekly production rate of Simplex computers
30 X 2 : weekly production rate of Omniplex computers Now, Interpret the Solution Corner Points (x 1, x 2 ) Objective Function 200x x 2 Value (0,0) 200(0) + 300(0) 0 (0, 20) 200(0) + 300(20) 6000 (30, 0) 200(30) + 300(0) 6000 (15, 12.5) 200(15) + 300(12.5) 6750 X 1 : weekly production rate of Simplex computers Computer Flips can make 15 Simplex and 12.5 Omniplex Computers per week (on average) for a profit of $6750 (on average).
31 Practice Write the constraints and the objective function for the following:
Linear Programming Test Review. Day 6
Linear Programming Test Review Day 6 Arrival Instructions Take out: Your homework, calculator, and the unit outline Pick up: 1. Warm-Up Test Review Day 2. Test Review HW Sheet A sponge Yes, it is different
More informationDay 4 ~ Increasing/Decreasing, and Extrema. A Graphical Approach
Day 4 ~ Increasing/Decreasing, and Extrema A Graphical Approach Warm Up ~ Day 4 1) Find the a) domain b) x & y intercepts c) range d) discontinuities e) end behavior using limit notation ) g( x) 3x 7x
More informationDeterministic Operations Research, ME 366Q and ORI 391 Chapter 2: Homework #2 Solutions
Deterministic Operations Research, ME 366Q and ORI 391 Chapter 2: Homework #2 Solutions 11. Consider the following linear program. Maximize z = 6x 1 + 3x 2 subject to x 1 + 2x 2 2x 1 + x 2 20 x 1 x 2 x
More informationEast Greenwich Mathematics Summer Review Material for Students Entering Algebra I, Part II Directions:
East Greenwich Mathematics Summer Review Material for Students Entering Algebra I, Part II 2016-2017 Directions: In order to earn full credit, show all work in the spaces provided. Do all work without
More informationSection 4.1 Solving Systems of Linear Inequalities
Section 4.1 Solving Systems of Linear Inequalities Question 1 How do you graph a linear inequality? Question 2 How do you graph a system of linear inequalities? Question 1 How do you graph a linear inequality?
More informationStudy Unit 3 : Linear algebra
1 Study Unit 3 : Linear algebra Chapter 3 : Sections 3.1, 3.2.1, 3.2.5, 3.3 Study guide C.2, C.3 and C.4 Chapter 9 : Section 9.1 1. Two equations in two unknowns Algebraically Method 1: Elimination Step
More informationDefine the word inequality
Warm Up: Define the word inequality Agenda: Objective- Students can solve linear inequalities in one variable, including equations with coefficients represented by letters. Define Inequalities One & Two
More informationOne box per group ( star group of 6)
4 markers 2 erasers One box per group ( star group of 6) 1 pencil (just in case) Some small post-it notes 1 glue stick One person from each group collect all items and place them back into the box. Concept
More informationUnit 9: Quadratics Intercept Form
For Teacher Use Packet Score: Name: Period: Algebra 1 Unit 9: Quadratics Intercept Form Note & Homework Packet Date Topic/Assignment HW Page 9-A Graphing Parabolas in Intercept Form 9-B Solve Quadratic
More information28 (Late Start) 7.2a Substitution. 7.1b Graphing with technology Feb 2. 4 (Late Start) Applications/ Choosing a method
Unit 7: Systems of Linear Equations NAME: The calendar and all assignments are subject to change. Students will be notified of any changes during class, so it is their responsibility to pay attention and
More informationUnit S Student Success Sheet (SSS) Trigonometric Identities Part 3 (section 5.5)
Unit S Student Success Sheet (SSS) Trigonometric Identities Part 3 (section 5.5) Standards: Trig 11.0 Segerstrom High School -- Math Analysis Honors Name: Period: Thinkbinder Study Group: www.bit.ly/chatunits
More informationGUIDED NOTES 5.6 RATIONAL FUNCTIONS
GUIDED NOTES 5.6 RATIONAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Use arrow notation. Solve applied problems involving rational functions. Find the domains of rational functions. Identify
More informationUnit 4 Day 4 & 5. Piecewise Functions
Unit 4 Day 4 & 5 Piecewise Functions Warm Up 1. Why does the inverse variation have a vertical asymptote? 2. Graph. Find the asymptotes. Write the domain and range using interval notation. a. b. f(x)=
More informationShenandoah University. (PowerPoint) LESSON PLAN *
Shenandoah University (PowerPoint) LESSON PLAN * NAME DATE 10/28/04 TIME REQUIRED 90 minutes SUBJECT Algebra I GRADE 6-9 OBJECTIVES AND PURPOSE (for each objective, show connection to SOL for your subject
More informationOptimization Methods in Management Science
Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 2 First Group of Students) Students with first letter of surnames A H Due: February 21, 2013 Problem Set Rules: 1. Each student
More informationMSA 640 Homework #2 Due September 17, points total / 20 points per question Show all work leading to your answers
Name MSA 640 Homework #2 Due September 17, 2010 100 points total / 20 points per question Show all work leading to your answers 1. The annual demand for a particular type of valve is 3,500 units. The cost
More informationGraphical and Computer Methods
Chapter 7 Linear Programming Models: Graphical and Computer Methods Quantitative Analysis for Management, Tenth Edition, by Render, Stair, and Hanna 2008 Prentice-Hall, Inc. Introduction Many management
More informationIntroduction to Spectroscopy: Analysis of Copper Ore
Introduction to Spectroscopy: Analysis of Copper Ore Using a Buret and Volumetric Flask: 2.06 ml of solution delivered 2.47 ml of solution delivered 50.00 ml Volumetric Flask Reading a buret: Burets are
More informationMath 3 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 3 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 informationApril 2, 2015 Unit 7 Systems of Linear Equations Standards: MCC8.EE.8
April 2, 2015 The equations 5x + 2y = 48 and 3x + 2y = 32 represent money collected from a school's basketball game for two separate days. If x represents the cost for each adult ticket and y represents
More informationLinear programming I João Carlos Lourenço
Decision Support Models Linear programming I João Carlos Lourenço joao.lourenco@ist.utl.pt Academic year 2012/2013 Readings: Hillier, F.S., Lieberman, G.J., 2010. Introduction to Operations Research, 9th
More informationIntroduction. Formulating LP Problems LEARNING OBJECTIVES. Requirements of a Linear Programming Problem. LP Properties and Assumptions
Valua%on and pricing (November 5, 2013) LEARNING OBJETIVES Lecture 10 Linear Programming (part 1) Olivier J. de Jong, LL.M., MM., MBA, FD, FFA, AA www.olivierdejong.com 1. Understand the basic assumptions
More informationSection 1.1 Solving Linear Equations and Inequalities a + 4b c
Section 1 Solving Linear Equations and Inequalities A. Evaluating Expressions Examples Evaluate the following if a = 7, b =, and c = ( a + c ) + b. a + b c B. The Distributive Property Try the Following
More informationLinear Functions, Equations, and Inequalities
CHAPTER Linear Functions, Equations, and Inequalities Inventory is the list of items that businesses stock in stores and warehouses to supply customers. Businesses in the United States keep about.5 trillion
More informationFebruary 22, Introduction to the Simplex Algorithm
15.53 February 22, 27 Introduction to the Simplex Algorithm 1 Quotes for today Give a man a fish and you feed him for a day. Teach him how to fish and you feed him for a lifetime. -- Lao Tzu Give a man
More informationChapter 3 Introduction to Linear Programming PART 1. Assoc. Prof. Dr. Arslan M. Örnek
Chapter 3 Introduction to Linear Programming PART 1 Assoc. Prof. Dr. Arslan M. Örnek http://homes.ieu.edu.tr/~aornek/ise203%20optimization%20i.htm 1 3.1 What Is a Linear Programming Problem? Linear Programming
More informationProblem #1. The following matrices are augmented matrices of linear systems. How many solutions has each system? Motivate your answer.
Exam #4 covers the material about systems of linear equations and matrices (CH. 4.1-4.4, PART II); systems of linear inequalities in two variables (geometric approach) and linear programming (CH.5.1-5.2,
More informationLesson 27 Linear Programming; The Simplex Method
Lesson Linear Programming; The Simplex Method Math 0 April 9, 006 Setup A standard linear programming problem is to maximize the quantity c x + c x +... c n x n = c T x subject to constraints a x + a x
More informationIntroduction to Spectroscopy: Analysis of Copper Ore
Introduction to Spectroscopy: Analysis of Copper Ore Using a Buret and Volumetric Flask: 2.06 ml of solution 2.47 ml of solution 50.00 ml delivered delivered Volumetric Flask Reading a buret: Burets are
More informationUnit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
More informationSUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 215
SUMMER MATH PACKET ADVANCED ALGEBRA A COURSE 5 Updated May 0 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of
More informationLimiting Reactants An analogy and learning cycle approach
Limiting Reactants An analogy and learning cycle approach Introduction This lab builds on the previous one on conservation of mass by looking at a chemical reaction in which there is a limiting reactant.
More informationAP Physics 1 LCHS Summer Work
AP Physics 1 LCHS Summer Work 2018-2019 The problems below are a review of the math skills that you need to succeed in AP Physics 1. This class is based on a knowledge of the math and concepts of Physics.
More informationConstructing Potential Energy Diagrams
potential ENERGY diagrams Visual Quantum Mechanics Teaching Guide ACTIVITY 2B Constructing Potential Energy Diagrams Goal In this activity, you will explore energy diagrams for magnets in repulsive configurations.
More informationUnit 5. Linear equations and inequalities OUTLINE. Topic 13: Solving linear equations. Topic 14: Problem solving with slope triangles
Unit 5 Linear equations and inequalities In this unit, you will build your understanding of the connection between linear functions and linear equations and inequalities that can be used to represent and
More informationLesson 6: Switching Between Forms of Quadratic Equations Unit 5 Quadratic Functions
(A) Lesson Context BIG PICTURE of this UNIT: CONTEXT of this LESSON: How do we analyze and then work with a data set that shows both increase and decrease What is a parabola and what key features do they
More informationMomentum Balances & Quadratic Equations
Momentum Balances & Quadratic Equations ABET Course Outcomes: C.1 formulate and solve engineering problems using linear and quadratic equations By the end of this class you should be able to: Solve problems
More informationChesapeake Campus Chemistry 111 Laboratory
Chesapeake Campus Chemistry 111 Laboratory Objectives Calculate the density of a sugar solution. Evaluate lab sources of error and their effect on an experiment. Introduction The density of an object is
More informationFall 2017 Math 108 Week Week 1 Task List
Fall 2017 Math 108 Week 1 29 Week 1 Task List This week we will cover Sections 1.1, 1.2, and 1.4 in your e-text. Work through each of the following tasks, carefully filling in the following pages in your
More information(b) For the change in c 1, use the row corresponding to x 1. The new Row 0 is therefore: 5 + 6
Chapter Review Solutions. Write the LP in normal form, and the optimal tableau is given in the text (to the right): x x x rhs y y 8 y 5 x x x s s s rhs / 5/ 7/ 9 / / 5/ / / / (a) For the dual, just go
More information56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker
56:171 Operations Research Midterm Exam - October 26, 1989 Instructor: D.L. Bricker Answer all of Part One and two (of the four) problems of Part Two Problem: 1 2 3 4 5 6 7 8 TOTAL Possible: 16 12 20 10
More informationhttps://ipass.lsrhs.net/school/ipass/assignmentsheet.html?m=8... Earth Sci (1) (Lab) Sanders-Fleming BLOCK: 4 12/03/ /21/2016
Earth Sci (1) (Lab) Sanders-Fleming BLOCK: 4 12/03/2015 01/21/2016 ES1 B4 Weather 14 Jan 2016 Date What it is Due Date Thr 12/03/2015 In class today: Foucault Pendulum Lab Concept: Proving earth's rotation
More informationPHYSICS! Unit 1 Study Plan and Review Packet
Name Do this! PHYSICS! Unit 1 Study Plan and Review Packet Not this! Test Review Checklist Review the unit objectives. - On the next page, read the unit objectives and check off your current standing on
More informationLesson: Slope. Warm Up. Unit #2: Linear Equations. 2) If f(x) = 7x 5, find the value of the following: f( 2) f(3) f(0)
Warm Up 1) 2) If f(x) = 7x 5, find the value of the following: f( 2) f(3) f(0) Oct 15 10:21 AM Unit #2: Linear Equations Lesson: Slope Oct 15 10:05 AM 1 Students will be able to find the slope Oct 16 12:19
More informationLinear Programming. H. R. Alvarez A., Ph. D. 1
Linear Programming H. R. Alvarez A., Ph. D. 1 Introduction It is a mathematical technique that allows the selection of the best course of action defining a program of feasible actions. The objective of
More informationAn Introduction to Linear Programming
An Introduction to Linear Programming Linear Programming Problem Problem Formulation A Maximization Problem Graphical Solution Procedure Extreme Points and the Optimal Solution Computer Solutions A Minimization
More informationWelcome! Please do the following: 1) Take out Colored Pen. 2) Take out U1H2. 3) Pick up one whiteboard per person.
Welcome! Please do the following: 1) Take out Colored Pen. 2) Take out U1H2. 3) Pick up one whiteboard per person. Tear Out Pages: Ø Pg. 14, 16 (Put away; this is homework) Welcome! U1H3: Pg. 14 #6-14
More informationMathematics GRADE 8 Teacher Packet
COMMON CORE Standards Plus Mathematics GRADE 8 Teacher Packet Copyright 01 Learning Plus Associates All Rights Reserved; International Copyright Secured. Permission is hereby granted to teachers to reprint
More informationMon Tue Wed Thu Fri 6 7. Limits Matching Lab: We Belong Together. Folders, Popsicle Sticks
August 201 Week Week 3 Week 2 Week 1 6 1 st Day of School 1.3A TSW find the limit of a function using the limit laws & algebraic methods. Intro to WebAssign 20 1. TSW will evaluate infinite limits. TSW
More informationChapter 4 The Simplex Algorithm Part I
Chapter 4 The Simplex Algorithm Part I Based on Introduction to Mathematical Programming: Operations Research, Volume 1 4th edition, by Wayne L. Winston and Munirpallam Venkataramanan Lewis Ntaimo 1 Modeling
More informationGUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS
GUIDED NOTES 6.1 EXPONENTIAL FUNCTIONS LEARNING OBJECTIVES In this section, you will: Evaluate exponential functions. Find the equation of an exponential function. Use compound interest formulas. Evaluate
More informationIntroduction to Uncertainty and Treatment of Data
Introduction to Uncertainty and Treatment of Data Introduction The purpose of this experiment is to familiarize the student with some of the instruments used in making measurements in the physics laboratory,
More informationWelcome Pre- Calc! Updates: U2Q1 is Friday ( , 2.6, basic quest. on 2.7 )
Welcome Pre- Calc! HW#2: Pg. 76 #9 (Elimination) 10 (Substitution) 11 (Graphing Calc.) 13 (method of your choice) Classify each system as consistent independent, dependent, or inconsistent. Updates: U2Q1
More informationAlgebra 2 C Midterm Exam Review Topics Exam Date: A2: Wednesday, January 21 st A4: Friday, January 23 rd
Algebra C Midterm Exam Review Topics Exam Date: A: Wednesday, January 1 st A4: Friday, January rd Chapter 1 Properties and Operations Number Sets Properties of Real Numbers Square Roots Simplifying Algebraic
More informationx 1 2x 2 +x 3 = 0 2x 2 8x 3 = 8 4x 1 +5x 2 +9x 3 = 9
Sec 2.1 Row Operations and Gaussian Elimination Consider a system of linear equations x 1 2x 2 +x 3 = 0 2x 2 8x 3 = 8 4x 1 +5x 2 +9x 3 = 9 The coefficient matrix of the system is The augmented matrix of
More informationMath 101: Final Exam Review Sheet
Math 101: Final Exam Review Sheet (Answers are at the end.) Exam Coverage: Everything we learned in the course. Exam Date: Friday, December 11, 2015 Exam Time: 10:30 am 12:30 pm (Arrive at least 10 minutes
More informationWARMUP. Draw the Lewis dot diagram and determine the VSEPR shape for the following compounds. This is practice for the quiz. OF2
WARMUP Draw the Lewis dot diagram and determine the VSEPR shape for the following compounds. This is practice for the quiz. OF2 HBr SiO2 We will take a quiz over VSEPR and assemble a foldable. I will complete
More informationDo Now 18 Balance Point. Directions: Use the data table to answer the questions. 2. Explain whether it is reasonable to fit a line to the data.
Do Now 18 Do Now 18 Balance Point Directions: Use the data table to answer the questions. 1. Calculate the balance point.. Explain whether it is reasonable to fit a line to the data.. The data is plotted
More informationUnit 2 Day 3 MATRICES. MATRIX Applications Quiz 1
Unit 2 Day 3 MATRICES MATRIX Applications Quiz 1 Remember: Phones OFF Warm-Up and in Blue Pockets! Check the list. Tracey, Danica, and Sherri bought snacks for a girls sleepover. They each bought the items
More informationEureka Math. Grade 8, Module 4. Student File_B. Contains Exit Ticket, and Assessment Materials
A Story of Ratios Eureka Math Grade 8, Module 4 Student File_B Contains, and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced,
More informationOptimization Methods in Management Science
Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 1 Second Group of Students (with first letter of surnames I Z) Problem Set Rules: Due: February 12, 2013 1. Each student should
More information1617 GSE Alg. I Reasoning w/linear Equalities & Inequalities Touchstone
High School HS Algebra 1 1617 GSE Alg. I Reasoning w/linear Equalities & Inequalities Touchstone 1. A number of apples were shared evenly among 4 students. Each student was also given 2 pears. Each student
More informationThe Simplex Method of Linear Programming
The Simplex Method of Linear Programming Online Tutorial 3 Tutorial Outline CONVERTING THE CONSTRAINTS TO EQUATIONS SETTING UP THE FIRST SIMPLEX TABLEAU SIMPLEX SOLUTION PROCEDURES SUMMARY OF SIMPLEX STEPS
More informationDemo 1: Solving LP problem graphically and with Excel Solver
MS-C2105 Introduction to Optimization Solutions 2 Ehtamo Demo 1: Solving LP problem graphically and with Excel Solver Solve the linear optimization problem graphically and with Excel Solver. a) max 8 +
More information56:171 Operations Research Fall 1998
56:171 Operations Research Fall 1998 Quiz Solutions D.L.Bricker Dept of Mechanical & Industrial Engineering University of Iowa 56:171 Operations Research Quiz
More informationSpring 2018 IE 102. Operations Research and Mathematical Programming Part 2
Spring 2018 IE 102 Operations Research and Mathematical Programming Part 2 Graphical Solution of 2-variable LP Problems Consider an example max x 1 + 3 x 2 s.t. x 1 + x 2 6 (1) - x 1 + 2x 2 8 (2) x 1,
More informationUnit 3 Day 4. Solving Equations with Rational Exponents and Radicals
Unit Day 4 Solving Equations with Rational Exponents and Radicals Day 4 Warm Up You know a lot about inverses in mathematics we use them every time we solve equations. Write down the inverse operation
More informationUnit X: Quadratic Functions and Equations Algebra 1 8 Class Meetings Revised July Essential Questions. Enduring Understandings and Unit Goals
Revised July 2016 Essential Questions What are the characteristics of quadratic functions? How can you solve a quadratic equation? How can you use functions to model real-world situations? Enduring Understandings
More information7.1 Solving Systems of Equations
Date: Precalculus Notes: Unit 7 Systems of Equations and Matrices 7.1 Solving Systems of Equations Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities.
More informationGrade 8 + DIGITAL. EL Strategies. DOK 1-4 RTI Tiers 1-3. Flexible Supplemental K-8 ELA & Math Online & Print
Standards PLUS Flexible Supplemental K-8 ELA & Math Online & Print Grade 8 SAMPLER Mathematics EL Strategies DOK 1-4 RTI Tiers 1-3 15-20 Minute Lessons Assessments Consistent with CA Testing Technology
More informationPart III: A Simplex pivot
MA 3280 Lecture 31 - More on The Simplex Method Friday, April 25, 2014. Objectives: Analyze Simplex examples. We were working on the Simplex tableau The matrix form of this system of equations is called
More informationFURTHER MATHEMATICS Units 3 & 4 - Written Examination 2
THIS BOX IS FOR ILLUSTRATIVE PURPOSES ONLY 2016 Examination Package - Trial Examination 4 of 5 Figures STUDENT NUMBER Letter Words FURTHER MATHEMATICS Units 3 & 4 - Written Examination 2 (TSSM s 2014 trial
More informationInstructor Notes for Module 5
Instructor Notes for Module 5 M5 I1 and I The Bottle Problem Modeling Co-Varying Relationships Learning outcomes: Student will be able to: Given bottles of various shapes, sketch a graph of the co-varying
More informationhave tried with your racer that are working well? you would like to make to your car?
1. What is energy? 2. What are some things you have tried with your racer that are working well? 3. What are some changes you would like to make to your car? Chapter 5 Section 1 Energy is the ability to
More informationObjective: SWBAT identify a compound as either ionic, covalent or an acid after reviewing terminology at 85% accuracy.
Mrs. Chausse s Chemistry Gifted Lesson Plan: Unit 11 Chemical Reactions January 4 18, 2018 Wednesday, January 3, 2018 Naming and writing formulas Objective: SWBAT identify a compound as either ionic, covalent
More informationDETERMINATION OF AN EQUILIBRIUM CONSTANT
DETERMINATION OF AN EQUILIBRIUM CONSTANT In this experiment the equilibrium properties of the reaction between the iron(iii) ion and the thiocyanate ion will be studied. The relevant chemical equation
More informationBachelor s Degree Programme Operations Research (Valid from 1st January, 2012 to 30th November, 2012.)
AOR-01 ASSIGNMENT BOOKLET Bachelor s Degree Programme Operations Research (Valid from 1st January, 2012 to 30th November, 2012.) It is compulsory to submit the assignment before filling in the exam form.
More informationUnit 7 Systems and Linear Programming
Unit 7 Systems and Linear Programming PREREQUISITE SKILLS: students should be able to solve linear equations students should be able to graph linear equations students should be able to create linear equations
More informationCONIC SECTIONS TEST FRIDAY, JANUARY 5 TH
CONIC SECTIONS TEST FRIDAY, JANUARY 5 TH DAY 1 - CLASSIFYING CONICS 4 Conics Parabola Circle Ellipse Hyperbola DAY 1 - CLASSIFYING CONICS GRAPHICALLY Parabola Ellipse Circle Hyperbola DAY 1 - CLASSIFYING
More informationKINETICS II - THE IODINATION OF ACETONE Determining the Activation Energy for a Chemical Reaction
KINETICS II - THE IODINATION OF ACETONE Determining the Activation Energy for a Chemical Reaction The rate of a chemical reaction depends on several factors: the nature of the reaction, the concentrations
More informationUnit Plan: Matrices (4 weeks or 20 instructional Days)
Unit Plan: Matrices (4 weeks or 20 instructional Days) Day 1: Review of Systems of Equations in 2 and 3 Variables Objectives: (M.4HS.CVM.18 )(CCSS.Math.Content.HSF-IF.B.6) Students will be able to solve
More informationInstructor Notes for Module 5
Instructor Notes for Module 5 M5_I3 Transformations of Polynomial Functions The Pre-class assignment for this section (PC3) on IMathAS consists of problem #1 on p. 195 in the workbook and a discussion
More informationAcceleration and Force: I
Lab Section (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Acceleration and Force: I Name Partners Pre-Lab You are required to finish this section before coming to the lab, which will be checked
More informationAlgebra Curriculum Map Overview
Algebra Curriculum Map Overview 2016-2017 Unit Approximate Number of Days 1 Expressions, Equations, and Functions 17 2 Linear Equations 18 3 Linear Functions 14 4 Linear Relations and Functions 7 5 Inequalities
More informationIntroduction to LP. Types of Linear Programming. There are five common types of decisions in which LP may play a role
Linear Programming RK Jana Lecture Outline Introduction to Linear Programming (LP) Historical Perspective Model Formulation Graphical Solution Method Simplex Method Introduction to LP Continued Today many
More informationDistributed Real-Time Control Systems. Lecture Distributed Control Linear Programming
Distributed Real-Time Control Systems Lecture 13-14 Distributed Control Linear Programming 1 Linear Programs Optimize a linear function subject to a set of linear (affine) constraints. Many problems can
More informationMATH 445/545 Homework 1: Due February 11th, 2016
MATH 445/545 Homework 1: Due February 11th, 2016 Answer the following questions Please type your solutions and include the questions and all graphics if needed with the solution 1 A business executive
More informationChemistry Day 15. Monday, October 1 st Tuesday, October 2 nd, 2018
Chemistry Day 15 Monday, October 1 st Tuesday, October 2 nd, 2018 Do-Now: VN: Nuclear Review 1. Write down today s FLT 2. What does the atomic number represent? 3. What does alpha decay do to the atomic
More informationAlgebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella
1 Algebra II (Common Core) Summer Assignment Due: September 11, 2017 (First full day of classes) Ms. Vella In this summer assignment, you will be reviewing important topics from Algebra I that are crucial
More informationAlgebra 1 Bassam Raychouni Grade 8 Math. Greenwood International School Course Description. Course Title: Head of Department:
Course Title: Head of Department: Bassam Raychouni (bassam@greenwood.sh.ae) Teacher(s) + e-mail: Femi Antony (femi.a@greenwood.sch.ae) Cycle/Division: High School Grade Level: 9 or 10 Credit Unit: 1 Duration:
More informationCottonwood Classical Preparatory School CCPS Pre-Calculus with Statistics Summer Packet
Cottonwood Classical Preparatory School CCPS Pre-Calculus with Statistics Summer Packet Greetings Pre-Calculus Student: Welcome to another successful year in math class at CCPS. The summer packet is one
More informationHerndon High School Geometry Honors Summer Assignment
Welcome to Geometry! This summer packet is for all students enrolled in Geometry Honors at Herndon High School for Fall 07. The packet contains prerequisite skills that you will need to be successful in
More informationTHANK YOU FOR YOUR PURCHASE!
THANK YOU FOR YOUR PURCHASE! The resources included in this purchase were designed and created by me. I hope that you find this resource helpful in your classroom. Please feel free to contact me with any
More informationWelcome to Chemistry 376
CHM 376 Spring 2014 Welcome to Chemistry 376 This course will give you a chance to explore the experimental world of physical chemistry. In each of six experiments you will study how simple physical models,
More informationST. JOSEPH S COLLEGE OF ARTS & SCIENCE (AUTONOMOUS) CUDDALORE-1
ST. JOSEPH S COLLEGE OF ARTS & SCIENCE (AUTONOMOUS) CUDDALORE-1 SUB:OPERATION RESEARCH CLASS: III B.SC SUB CODE:EMT617S SUB INCHARGE:S.JOHNSON SAVARIMUTHU 2 MARKS QUESTIONS 1. Write the general model of
More informationMath 141:512. Practice Exam 1 (extra credit) Due: February 6, 2019
Math 141:512 Due: February 6, 2019 Practice Exam 1 (extra credit) This is an open book, extra credit practice exam which covers the material that Exam 1 will cover (Sections 1.3, 1.4, 2.1, 2.2, 2.3, 2.4,
More informationThe steps in Raya s solution to 2.5 (6.25x + 0.5) = 11 are shown. Select the correct reason for line 4 of Raya s solution.
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. Unit 2: Reasoning with Linear Equations and Inequalities The perimeter
More informationMHF4U Unit 2 Polynomial Equation and Inequalities
MHF4U Unit 2 Polynomial Equation and Inequalities Section Pages Questions Prereq Skills 82-83 # 1ac, 2ace, 3adf, 4, 5, 6ace, 7ac, 8ace, 9ac 2.1 91 93 #1, 2, 3bdf, 4ac, 5, 6, 7ab, 8c, 9ad, 10, 12, 15a,
More informationOptimization Methods in Management Science
Problem Set Rules: Optimization Methods in Management Science MIT 15.053, Spring 2013 Problem Set 1 (Second Group of Students) Students with first letter of surnames G Z Due: February 12, 2013 1. Each
More informationWatertown Public Schools Algebra 2 Honors/CP Summer Packet
Name Date Teacher Watertown Public Schools Algebra 2 Honors/CP Summer Packet Summer 2018 This packet contains topics that you are expected to know prior to entering Algebra 2. You have learned these skills
More information