Question 2: How do you evaluate a limit from a graph?

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Question 2: How do you evaluate a limit from a graph?"

Transcription

1 Question 2: How do you evaluate a limit from a graph? In the question before this one, we used a table to observe the output values from a function as the input values approach some value from the left of right. With a little practice, we can evaluate limits using a graph to find the values of a function. Suppose we have the graph of a function like the one below. We can use this graph to evaluate the two-sided limit. x2 As with the limits we calculated from tables, we must evaluate the one-sided limits near x 2. To calculate the limit, x2 we must examine the graph at x values that are slightly smaller than x 2. 9

2 Figure 1 - As the x values get closer and closer to 2 from values slightly smaller than 2, the y values approach 4. In Figure 1, a red dashed vertical line is positioned slightly to the left of 2. The height of the line indicates the y value at that x value. A red dashed horizontal line locates the y value on the graph. As the vertical line moves closer and closer to 2, the horizontal line gets closer and closer to the y value 4. This means the limit as x approaches 2 from the left is 4 or 4. x2 The same strategy allows us to solve the one-sided limit x2 10

3 Figure 2 - As the x values get closer and closer to 2 from values slightly larger than 2, the y values approach 4. The red dashed vertical line in Figure 2 locates an x value slightly larger than 2. The red dashed horizontal line gives the corresponding value on the y axis. As the vertical line moves closer and closer to 2, the horizontal line moves closer and closer to 4. In other words, for x values closer and closer to 2, the y values are closer and closer to 4. The limit from the right is 4. x2 Since the limits from the left and right are both equal to 4, the two-sided limit is also equal to 4, 4. x2 11

4 Example 4 Find the Limit Graphically Suppose f( x) is given by the graph below. Evaluate each of the limits below. a. x1 Solution To evaluate this limit, we need to examine y values on the graph as x gets closer and closer to 1 from the left side of 1. This region of the graph is shown in the graph to the below. 12

5 Let us locate an x value and its corresponding y value in this region. y x Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 1. Notice that the y value at x 1, f (1) 2, is not the x1 same as the limit. 13

6 b. x1 Solution In this one sided limit, the x values are on the right side of 1. y x As the point moves to the left towards x 1, the point moves up vertically towards 1. This means that the closer the point gets to x 1, the closer the y value gets to 1 or 1 x1 c. x1 Solution For the two sided limit to exist, the one sided limits must be equal. In this case they are both equal to 1. Since they are both equal to 1, the two sided limit is also equal to 1, lim f ( x) 1 x1 Notice that none of these limits have anything to do with the fact that f (1) 2. This is because we are using x values approaching 1, not equal to 1. 14

7 Example 5 Find the Limit Graphically Suppose f( x) is given by the graph below. Evaluate each of the limits below. a. x1 Solution To left of x 1, the graph looks like the graph in Example 1. 15

8 y x Notice that as x moves horizontally closer and closer to 1, the corresponding y value moves vertically closer and closer to 1. This tells us that 1. x1 b. x1 Solution As the point moves to the left towards x 1, the point moves up vertically towards 2. 16

9 y x This means that the closer the point gets to x 1, the closer the y value gets to 2 or 2. x1 c. x1 Solution For the two sided limit to exist, the one sided limits must be equal. In this case, they are not equal. From the left side the limit is equal to 1 and from the right side the limit is equal to 2, so does not exist x1 The vertical gap in the graph at x 1is what leads to different values in the one sided limits. In Example 1 there was a horizontal gap at x 1, but not a vertical gap since the two pieces of the graph come together at x 1. In each of these examples, we evaluate the one-sided limits to find the two-sided limit. If the one-sided limits are equal to some value, the two-sided limit is equal to the same 17

10 value. If the one-sided limits do not match, the two-sided limit does not exist. In the next example, we examine a function for which the one-sided limit does not exist. Example 6 Find the Limit Graphically Suppose f( x ) is given by the graph below. Evaluate the limit. x5 Solution This function has a vertical asymptote at x 5. The vertical asymptote is shown on the graph as a blue dashed line. The one-sided limit is a left hand limit. Locate points on the left side of x 5 with red dashed lines. 18

11 As the vertical line gets closer and closer to 5, the horizontal line gets higher and higher. This indicates that the y values do not get closer to any value as x gets closer to 5 from the left. The one-sided limit does not exist. 19

WHICH TYPE OF GRAPH SHOULD YOU CHOOSE?

WHICH TYPE OF GRAPH SHOULD YOU CHOOSE? PRESENTING GRAPHS WHICH TYPE OF GRAPH SHOULD YOU CHOOSE? CHOOSING THE RIGHT TYPE OF GRAPH You will usually choose one of four very common graph types: Line graph Bar graph Pie chart Histograms LINE GRAPHS

More information

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if

Asymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.

More information

Kevin James. MTHSC 102 Section 1.5 Exponential Functions and Models

Kevin James. MTHSC 102 Section 1.5 Exponential Functions and Models MTHSC 102 Section 1.5 Exponential Functions and Models Exponential Functions and Models Definition Algebraically An exponential function has an equation of the form f (x) = ab x. The constant a is called

More information

PLOTTING DATA AND INTERPRETING GRAPHS

PLOTTING DATA AND INTERPRETING GRAPHS PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they

More information

Horizontal and Vertical Asymptotes of Graphs of Rational Functions

Horizontal and Vertical Asymptotes of Graphs of Rational Functions PRECALCULUS AND ADVANCED OPICS Horizontal and Vertical Asymptotes of Graphs of Rational Functions Student Outcomes Students identify vertical and horizontal asymptotes of rational functions. Lesson Notes

More information

How to Graph Trigonometric Functions

How to Graph Trigonometric Functions How to Graph Trigonometric Functions This handout includes instructions for graphing processes of basic, amplitude shifts, horizontal shifts, and vertical shifts of trigonometric functions. The Unit Circle

More information

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply?

Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Prompt Students are studying multiplying binomials (factoring and roots) ax + b and cx + d. A student asks What if we divide instead of multiply? Commentary In our foci, we are assuming that we have a

More information

3.4 Limits at Infinity - Asymptotes

3.4 Limits at Infinity - Asymptotes 3.4 Limits at Infinity - Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line

More information

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd

Power functions: f(x) = x n, n is a natural number The graphs of some power functions are given below. n- even n- odd 5.1 Polynomial Functions A polynomial unctions is a unction o the orm = a n n + a n-1 n-1 + + a 1 + a 0 Eample: = 3 3 + 5 - The domain o a polynomial unction is the set o all real numbers. The -intercepts

More information

Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts.

Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x- and y-intercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D.

More information

Curve Sketching. MATH 1310 Lecture 26 1 of 14 Ronald Brent 2016 All rights reserved.

Curve Sketching. MATH 1310 Lecture 26 1 of 14 Ronald Brent 2016 All rights reserved. Curve Sketching 1. Domain. Intercepts. Symmetry. Asymptotes 5. Intervals of Increase or Decrease 6. Local Maimum and Minimum Values 7. Concavity and Points of Inflection 8. Sketch the curve MATH 110 Lecture

More information

Math 1314 Lesson 13 Analyzing Other Types of Functions

Math 1314 Lesson 13 Analyzing Other Types of Functions Math 1314 Lesson 13 Analyzing Other Types of Functions Asymptotes We will need to identify any vertical or horizontal asymptotes of the graph of a function. A vertical asymptote is a vertical line x a

More information

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated

List the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible

More information

Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions

Functions. MATH 160, Precalculus. J. Robert Buchanan. Fall 2011. Department of Mathematics. J. Robert Buchanan Functions Functions MATH 160, Precalculus J. Robert Buchanan Department of Mathematics Fall 2011 Objectives In this lesson we will learn to: determine whether relations between variables are functions, use function

More information

Math 234 February 28. I.Find all vertical and horizontal asymptotes of the graph of the given function.

Math 234 February 28. I.Find all vertical and horizontal asymptotes of the graph of the given function. Math 234 February 28 I.Find all vertical and horizontal asymptotes of the graph of the given function.. f(x) = /(x 3) x 3 = 0 when x = 3 Vertical Asymptotes: x = 3 H.A.: /(x 3) = 0 /(x 3) = 0 Horizontal

More information

3.5 Summary of Curve Sketching

3.5 Summary of Curve Sketching 3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) -intercepts occur when f() = 0 (b) y-intercept occurs when = 0 3. Symmetry: Is it even or

More information

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6

The slope m of the line passes through the points (x 1,y 1 ) and (x 2,y 2 ) e) (1, 3) and (4, 6) = 1 2. f) (3, 6) and (1, 6) m= 6 6 Lines and Linear Equations Slopes Consider walking on a line from left to right. The slope of a line is a measure of its steepness. A positive slope rises and a negative slope falls. A slope of zero means

More information

1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some

1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number

More information

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as:

Limits. Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: Limits Limits: Graphical Solutions Graphical Limits Let be a function defined on the interval [-6,11] whose graph is given as: The limits are defined as the value that the function approaches as it goes

More information

Animated example of Mr Coscia s trading

Animated example of Mr Coscia s trading 1 Animated example of Mr Coscia s trading 4 An example of Mr Coscia's trading (::. to ::.69) 3 2 The chart explained: horizontal axis shows the timing of the trades in milliseconds right hand vertical

More information

Graphing Equations. with Color Activity

Graphing Equations. with Color Activity Graphing Equations with Color Activity Students must re-write equations into slope intercept form and then graph them on a coordinate plane. 2011 Lindsay Perro Name Date Between The Lines Re-write each

More information

2.5 Transformations of Functions

2.5 Transformations of Functions 2.5 Transformations of Functions Section 2.5 Notes Page 1 We will first look at the major graphs you should know how to sketch: Square Root Function Absolute Value Function Identity Function Domain: [

More information

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0 College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

More information

The Point-Slope Form

The Point-Slope Form 7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope

More information

Grade level: secondary Subject: mathematics Time required: 45 to 90 minutes

Grade level: secondary Subject: mathematics Time required: 45 to 90 minutes TI-Nspire Activity: Paint Can Dimensions By: Patsy Fagan and Angela Halsted Activity Overview Problem 1 explores the relationship between height and volume of a right cylinder, the height and surface area,

More information

Graphing Rational Functions

Graphing Rational Functions Graphing Rational Functions A rational function is defined here as a function that is equal to a ratio of two polynomials p(x)/q(x) such that the degree of q(x) is at least 1. Examples: is a rational function

More information

Unit 7: Radical Functions & Rational Exponents

Unit 7: Radical Functions & Rational Exponents Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

More information

containing Kendall correlations; and the OUTH = option will create a data set containing Hoeffding statistics.

containing Kendall correlations; and the OUTH = option will create a data set containing Hoeffding statistics. Getting Correlations Using PROC CORR Correlation analysis provides a method to measure the strength of a linear relationship between two numeric variables. PROC CORR can be used to compute Pearson product-moment

More information

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa

Apr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,

More information

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

More information

chapter >> Making Decisions Section 2: Making How Much Decisions: The Role of Marginal Analysis

chapter >> Making Decisions Section 2: Making How Much Decisions: The Role of Marginal Analysis chapter 7 >> Making Decisions Section : Making How Much Decisions: The Role of Marginal Analysis As the story of the two wars at the beginning of this chapter demonstrated, there are two types of decisions:

More information

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.

Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote

More information

LIMITS AND CONTINUITY

LIMITS AND CONTINUITY LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from

More information

Unit 10: Quadratic Relations

Unit 10: Quadratic Relations Date Period Unit 0: Quadratic Relations DAY TOPIC Distance and Midpoint Formulas; Completing the Square Parabolas Writing the Equation 3 Parabolas Graphs 4 Circles 5 Exploring Conic Sections video This

More information

Linear functions Increasing Linear Functions. Decreasing Linear Functions

Linear functions Increasing Linear Functions. Decreasing Linear Functions 3.5 Increasing, Decreasing, Max, and Min So far we have been describing graphs using quantitative information. That s just a fancy way to say that we ve been using numbers. Specifically, we have described

More information

8.2 Gantt Charts. Dr. Tarek A. Tutunji Philadelphia University, Jordan. Engineering Skills, Philadelphia University

8.2 Gantt Charts. Dr. Tarek A. Tutunji Philadelphia University, Jordan. Engineering Skills, Philadelphia University 8.2 Gantt Charts Philadelphia University, Jordan Overview In the previous section, an introduction to Project Management was provided. In this sequence, Gant Charts will be introduced. Graphical Project

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Exploring Asymptotes. Student Worksheet. Name. Class. Rational Functions

Exploring Asymptotes. Student Worksheet. Name. Class. Rational Functions Exploring Asymptotes Name Student Worksheet Class Rational Functions 1. On page 1.3 of the CollegeAlg_Asymptotes.tns fi le is a graph of the function 1 f(x) x 2 9. a. Is the function defi ned over all

More information

1 Lesson 3: Presenting Data Graphically

1 Lesson 3: Presenting Data Graphically 1 Lesson 3: Presenting Data Graphically 1.1 Types of graphs Once data is organized and arranged, it can be presented. Graphic representations of data are called graphs, plots or charts. There are an untold

More information

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) =

Math 2250 Exam #1 Practice Problem Solutions. g(x) = x., h(x) = Math 50 Eam # Practice Problem Solutions. Find the vertical asymptotes (if any) of the functions g() = + 4, h() = 4. Answer: The only number not in the domain of g is = 0, so the only place where g could

More information

Working with Spreadsheets

Working with Spreadsheets osborne books Working with Spreadsheets UPDATE SUPPLEMENT 2015 The AAT has recently updated its Study and Assessment Guide for the Spreadsheet Software Unit with some minor additions and clarifications.

More information

Unit 8: Normal Calculations

Unit 8: Normal Calculations Unit 8: Normal Calculations Summary of Video In this video, we continue the discussion of normal curves that was begun in Unit 7. Recall that a normal curve is bell-shaped and completely characterized

More information

This activity will show you how to draw graphs of algebraic functions in Excel.

This activity will show you how to draw graphs of algebraic functions in Excel. This activity will show you how to draw graphs of algebraic functions in Excel. Open a new Excel workbook. This is Excel in Office 2007. You may not have used this version before but it is very much the

More information

Costing and Break-Even Analysis

Costing and Break-Even Analysis W J E C B U S I N E S S S T U D I E S A L E V E L R E S O U R C E S. 28 Spec. Issue 2 Sept 212 Page 1 Costing and Break-Even Analysis Specification Requirements- Classify costs: fixed, variable and semi-variable.

More information

Lecture 3: Derivatives and extremes of functions

Lecture 3: Derivatives and extremes of functions Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16

More information

Acceleration Introduction: Objectives: Methods:

Acceleration Introduction: Objectives: Methods: Acceleration Introduction: Acceleration is defined as the rate of change of velocity with respect to time, thus the concepts of velocity also apply to acceleration. In the velocity-time graph, acceleration

More information

Math Rational Functions

Math Rational Functions Rational Functions Math 3 Rational Functions A rational function is the algebraic equivalent of a rational number. Recall that a rational number is one that can be epressed as a ratio of integers: p/q.

More information

Algebra II A Final Exam

Algebra II A Final Exam Algebra II A Final Exam Multiple Choice Identify the choice that best completes the statement or answers the question. Evaluate the expression for the given value of the variable(s). 1. ; x = 4 a. 34 b.

More information

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions

Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions Pre-Calculus Notes Vertical, Horizontal, and Slant Asymptotes of Rational Functions A) Vertical Asymptotes A rational function, in lowest terms, will have vertical asymptotes at the real zeros of the denominator

More information

Background: Electric resistance, R, is defined by:

Background: Electric resistance, R, is defined by: RESISTANCE & OHM S LAW (PART I and II) - 8 Objectives: To understand the relationship between applied voltage and current in a resistor and to verify Ohm s Law. To understand the relationship between applied

More information

Principles of Math 12 - Transformations Practice Exam 1

Principles of Math 12 - Transformations Practice Exam 1 Principles of Math 2 - Transformations Practice Exam www.math2.com Transformations Practice Exam Use this sheet to record your answers. NR. 2. 3. NR 2. 4. 5. 6. 7. 8. 9. 0.. 2. NR 3. 3. 4. 5. 6. 7. NR

More information

Manual for simulation of EB processing. Software ModeRTL

Manual for simulation of EB processing. Software ModeRTL 1 Manual for simulation of EB processing Software ModeRTL How to get results. Software ModeRTL. Software ModeRTL consists of five thematic modules and service blocks. (See Fig.1). Analytic module is intended

More information

EXPONENTIAL FUNCTIONS 8.1.1 8.1.6

EXPONENTIAL FUNCTIONS 8.1.1 8.1.6 EXPONENTIAL FUNCTIONS 8.1.1 8.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the

More information

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization

Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued

More information

Business Statistics & Presentation of Data BASIC MATHEMATHICS MATH0101

Business Statistics & Presentation of Data BASIC MATHEMATHICS MATH0101 Business Statistics & Presentation of Data BASIC MATHEMATHICS MATH0101 1 STATISTICS??? Numerical facts eg. the number of people living in a certain town, or the number of cars using a traffic route each

More information

Describing Relationships between Two Variables

Describing Relationships between Two Variables Describing Relationships between Two Variables Up until now, we have dealt, for the most part, with just one variable at a time. This variable, when measured on many different subjects or objects, took

More information

Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range:

Algebra 2 Notes AII.7 Functions: Review, Domain/Range. Function: Domain: Range: Name: Date: Block: Functions: Review What is a.? Relation: Function: Domain: Range: Draw a graph of a : a) relation that is a function b) relation that is NOT a function Function Notation f(x): Names the

More information

For additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3.

For additional information, see the Math Notes boxes in Lesson B.1.3 and B.2.3. EXPONENTIAL FUNCTIONS B.1.1 B.1.6 In these sections, students generalize what they have learned about geometric sequences to investigate exponential functions. Students study exponential functions of the

More information

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010

MATH 121 FINAL EXAM FALL 2010-2011. December 6, 2010 MATH 11 FINAL EXAM FALL 010-011 December 6, 010 NAME: SECTION: Instructions: Show all work and mark your answers clearly to receive full credit. This is a closed notes, closed book exam. No electronic

More information

Statistics Chapter 2

Statistics Chapter 2 Statistics Chapter 2 Frequency Tables A frequency table organizes quantitative data. partitions data into classes (intervals). shows how many data values are in each class. Test Score Number of Students

More information

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION 4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,

More information

Making Visio Diagrams Come Alive with Data

Making Visio Diagrams Come Alive with Data Making Visio Diagrams Come Alive with Data An Information Commons Workshop Making Visio Diagrams Come Alive with Data Page Workshop Why Add Data to A Diagram? Here are comparisons of a flow chart with

More information

MEASURES OF VARIATION

MEASURES OF VARIATION NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are

More information

LIFTING A LOAD NAME. Materials: Spring scale or force probes

LIFTING A LOAD NAME. Materials: Spring scale or force probes Lifting A Load 1 NAME LIFTING A LOAD Problems: Can you lift a load using only one finger? Does it always take the same amount of force to lift a load? Where should you press to lift a load with the least

More information

EXPERIMENT GRAPHING IN EXCEL

EXPERIMENT GRAPHING IN EXCEL EXPERIMENT GRAPHING IN EXCEL Introduction In this lab you will learn how to use Microsoft Excel to plot and analyze data that you obtain while doing experiments. In this lab you learn how to Enter data

More information

Smart Magnets for Precision Alignment

Smart Magnets for Precision Alignment Smart Magnets for Precision Alignment Smart Magnets for Precision Alignment in Product Design Correlated Magnetics Research creates specially engineered smart magnets for use in product design that are

More information

Graphing calculators Transparencies (optional)

Graphing calculators Transparencies (optional) What if it is in pieces? Piecewise Functions and an Intuitive Idea of Continuity Teacher Version Lesson Objective: Length of Activity: Students will: Recognize piecewise functions and the notation used

More information

THE COST OF PRODUCTION

THE COST OF PRODUCTION Chulalongkorn University: BBA International Program, Faculty of Commerce and Accountancy 2900 (Section ) Chairat Aemkulwat Economics I: Microeconomics Spring 205 Solution to Selected Questions: CHAPTER

More information

CONDUCT YOUR EXPERIMENT/COLLECT YOUR DATA AND RECORD YOUR RESULTS WRITE YOUR CONCLUSION

CONDUCT YOUR EXPERIMENT/COLLECT YOUR DATA AND RECORD YOUR RESULTS WRITE YOUR CONCLUSION CONDUCT YOUR EXPERIMENT/COLLECT YOUR DATA AND RECORD YOUR RESULTS WRITE YOUR CONCLUSION Due Date: February 9, 2010 Conducting Your Experiment Adapted with permission from www.sciencebuddies.org 1) Before

More information

Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

More information

Break-even analysis. On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart.

Break-even analysis. On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart. Break-even analysis On page 256 of It s the Business textbook, the authors refer to an alternative approach to drawing a break-even chart. In order to survive businesses must at least break even, which

More information

Rational Functions 5.2 & 5.3

Rational Functions 5.2 & 5.3 Math Precalculus Algebra Name Date Rational Function Rational Functions 5. & 5.3 g( ) A function is a rational function if f ( ), where g( ) and h( ) are polynomials. h( ) Vertical asymptotes occur at

More information

= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )).

= f x 1 + h. 3. Geometrically, the average rate of change is the slope of the secant line connecting the pts (x 1 )). Math 1205 Calculus/Sec. 3.3 The Derivative as a Rates of Change I. Review A. Average Rate of Change 1. The average rate of change of y=f(x) wrt x over the interval [x 1, x 2 ]is!y!x ( ) - f( x 1 ) = y

More information

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º.

Using the Quadrant. Protractor. Eye Piece. You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements >90º. Using the Quadrant Eye Piece Protractor Handle You can measure angles of incline from 0º ( horizontal ) to 90º (vertical ). Ignore measurements 90º. Plumb Bob ø

More information

DeVry University and Keller Graduate School of Management C O - B R A N D E D GUIDELIN ES

DeVry University and Keller Graduate School of Management C O - B R A N D E D GUIDELIN ES C O - B R A N D E D GUIDELIN ES 2009 Table of contents Introduction... 1-4 Look and feel... 5-7 Layout examples Vertical... 8-9 Horizontal... 10-11 Brochures and printed materials... 12-13 Web... 14 What

More information

Review Sheet for Third Midterm Mathematics 1300, Calculus 1

Review Sheet for Third Midterm Mathematics 1300, Calculus 1 Review Sheet for Third Midterm Mathematics 1300, Calculus 1 1. For f(x) = x 3 3x 2 on 1 x 3, find the critical points of f, the inflection points, the values of f at all these points and the endpoints,

More information

Long Run Economic Growth Agenda. Long-run Economic Growth. Long-run Growth Model. Long-run Economic Growth. Determinants of Long-run Growth

Long Run Economic Growth Agenda. Long-run Economic Growth. Long-run Growth Model. Long-run Economic Growth. Determinants of Long-run Growth Long Run Economic Growth Agenda Long-run economic growth. Determinants of long-run growth. Production functions. Long-run Economic Growth Output is measured by real GDP per capita. This measures our (material)

More information

SPC Response Variable

SPC Response Variable SPC Response Variable This procedure creates control charts for data in the form of continuous variables. Such charts are widely used to monitor manufacturing processes, where the data often represent

More information

Briefing document: How to create a Gantt chart using a spreadsheet

Briefing document: How to create a Gantt chart using a spreadsheet Briefing document: How to create a Gantt chart using a spreadsheet A Gantt chart is a popular way of using a bar-type chart to show the schedule for a project. It is named after Henry Gantt who created

More information

MINITAB ASSISTANT WHITE PAPER

MINITAB ASSISTANT WHITE PAPER MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

More information

Presentation of data

Presentation of data 2 Presentation of data Using various types of graph and chart to illustrate data visually In this chapter we are going to investigate some basic elements of data presentation. We shall look at ways in

More information

0 0 such that f x L whenever x a

0 0 such that f x L whenever x a Epsilon-Delta Definition of the Limit Few statements in elementary mathematics appear as cryptic as the one defining the limit of a function f() at the point = a, 0 0 such that f L whenever a Translation:

More information

Essay 5 Tutorial for a Three-Dimensional Heat Conduction Problem Using ANSYS Workbench

Essay 5 Tutorial for a Three-Dimensional Heat Conduction Problem Using ANSYS Workbench Essay 5 Tutorial for a Three-Dimensional Heat Conduction Problem Using ANSYS Workbench 5.1 Introduction The problem selected to illustrate the use of ANSYS software for a three-dimensional steadystate

More information

Breakeven, Leverage, and Elasticity

Breakeven, Leverage, and Elasticity Breakeven, Leverage, and Elasticity Dallas Brozik, Marshall University Breakeven Analysis Breakeven analysis is what management is all about. The idea is to compare where you are now to where you might

More information

2. More Use of the Mouse in Windows 7

2. More Use of the Mouse in Windows 7 65 2. More Use of the Mouse in Windows 7 The mouse has become an essential part of the computer. But it is actually a relatively new addition. The mouse did not become a standard part of the PC until Windows

More information

Appendix 2.1 Tabular and Graphical Methods Using Excel

Appendix 2.1 Tabular and Graphical Methods Using Excel Appendix 2.1 Tabular and Graphical Methods Using Excel 1 Appendix 2.1 Tabular and Graphical Methods Using Excel The instructions in this section begin by describing the entry of data into an Excel spreadsheet.

More information

AP Calculus AB Summer Packet DUE August 8, Welcome!

AP Calculus AB Summer Packet DUE August 8, Welcome! AP Calculus AB Summer Packet 06--DUE August 8, 06 Welcome! This packet includes a sampling of problems that students entering AP Calculus AB should be able to answer without hesitation. The questions are

More information

Week 7 - Game Theory and Industrial Organisation

Week 7 - Game Theory and Industrial Organisation Week 7 - Game Theory and Industrial Organisation The Cournot and Bertrand models are the two basic templates for models of oligopoly; industry structures with a small number of firms. There are a number

More information

Module 49 Consumer and Producer Surplus

Module 49 Consumer and Producer Surplus What you will learn in this Module: The meaning of consumer surplus and its relationship to the demand curve The meaning of producer surplus and its relationship to the supply curve Module 49 Consumer

More information

Numerical integration of a function known only through data points

Numerical integration of a function known only through data points Numerical integration of a function known only through data points Suppose you are working on a project to determine the total amount of some quantity based on measurements of a rate. For example, you

More information

Confidence Intervals for Cpk

Confidence Intervals for Cpk Chapter 297 Confidence Intervals for Cpk Introduction This routine calculates the sample size needed to obtain a specified width of a Cpk confidence interval at a stated confidence level. Cpk is a process

More information

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule.

If Σ is an oriented surface bounded by a curve C, then the orientation of Σ induces an orientation for C, based on the Right-Hand-Rule. Oriented Surfaces and Flux Integrals Let be a surface that has a tangent plane at each of its nonboundary points. At such a point on the surface two unit normal vectors exist, and they have opposite directions.

More information

SPSS Manual for Introductory Applied Statistics: A Variable Approach

SPSS Manual for Introductory Applied Statistics: A Variable Approach SPSS Manual for Introductory Applied Statistics: A Variable Approach John Gabrosek Department of Statistics Grand Valley State University Allendale, MI USA August 2013 2 Copyright 2013 John Gabrosek. All

More information

Chapter 04 Firm Production, Cost, and Revenue

Chapter 04 Firm Production, Cost, and Revenue Chapter 04 Firm Production, Cost, and Revenue Multiple Choice Questions 1. A key assumption about the way firms behave is that they a. Minimize costs B. Maximize profit c. Maximize market share d. Maximize

More information

A. a change in demand. B. a change in quantity demanded. C. a change in quantity supplied. D. unit elasticity. E. a change in average variable cost.

A. a change in demand. B. a change in quantity demanded. C. a change in quantity supplied. D. unit elasticity. E. a change in average variable cost. 1. The supply of gasoline changes, causing the price of gasoline to change. The resulting movement from one point to another along the demand curve for gasoline is called A. a change in demand. B. a change

More information

Creating Population Pyramids Using Microsoft Excel

Creating Population Pyramids Using Microsoft Excel Creating Population Pyramids Using Microsoft Excel Population pyramids are one of the most basic illustrative tools used in demography to show the age structure of a population. This document will show

More information

chapter >> Consumer and Producer Surplus Section 1: Consumer Surplus and the Demand Curve

chapter >> Consumer and Producer Surplus Section 1: Consumer Surplus and the Demand Curve chapter 6 A consumer s willingness to pay for a good is the maximum price at which he or she would buy that good. >> Consumer and Producer Surplus Section 1: Consumer Surplus and the Demand Curve The market

More information

Autonomous Equations / Stability of Equilibrium Solutions. y = f (y).

Autonomous Equations / Stability of Equilibrium Solutions. y = f (y). Autonomous Equations / Stability of Equilibrium Solutions First order autonomous equations, Equilibrium solutions, Stability, Longterm behavior of solutions, direction fields, Population dynamics and logistic

More information

Experiment P-17 Magnetic Field Strength

Experiment P-17 Magnetic Field Strength 1 Experiment P-17 Magnetic Field Strength Objectives To learn about basic properties of magnets. To study the relationship between magnetic field strength and the distance from the magnet. Modules and

More information

GRAPH OF A RATIONAL FUNCTION

GRAPH OF A RATIONAL FUNCTION GRAPH OF A RATIONAL FUNCTION Find vertical asmptotes and draw them. Look for common factors first. Vertical asmptotes occur where the denominator becomes zero as long as there are no common factors. Find

More information