Instantaneous Rate of Change:

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Instantaneous Rate of Change:"

Transcription

1 Instantaneous Rate of Cange: Last section we discovered tat te average rate of cange in F(x) can also be interpreted as te slope of a scant line. Te average rate of cange involves te cange in F(x) over a designated interval [x 1, x ] or between te two points (x 1, F(x 1 )) and (x, F(x )). Te secant line passes troug tese two points. A logical extrapolation would indicate tat te instantaneous rate of cange in F(x) at a point x would be te same as te slope of a tangent line toucing te grap of F(x) at tat same point (x, F(x)). In te case of average rate of cange you can find te slope of te secant line easily because you ave two points to work wit. But in te case of te instantaneous rate of cange tere is only one point and terefore it is not possible to directly calculate te slope of te tangent line. Is it possible to find tis value tat will be bot te slope of te tangent line and te instantaneous rate of cange? We will now define a process tat will elp us do just tat. Finding Instantaneous Rate of Cange and te Slope of te Tangent Line. We will again consider te following profit function from te last section: P(q) = -q + 4q 5 were q represents quantity of items sold. Profit and slope of tangent line P Profit Tangent line x Figure 1 We would like te find te instantaneous rate of cange in profit for q = 4. We will find te instantaneous rate of cange numerically. Tis metod proceeds as follows: Since we can not find te slope of te tangent line toucing te grap at q = 4 by direct calculation we will try to guess its value as precisely as possible. In order to do tis we must find slopes of secant lines tat are very close to te tangent line at te point

2 (4, 78). First we will find te slope of secant line troug te points ( 4, 78) and (41, 83). Ten we will repeat te process for points ( 4, 78) and ( 4.5, ). We ten will repeat again for te points ( 4, 78), (4.5, ) and so on. In order to be very precise we will also need to consider points on te left side of q = 4 as well suc as ( 39.99, ) and ( 4, 78). Te process is better illustrated in te table below. using points from te rigt side of q = 4 x1 x P(x1) P(x) [P(x) - P(x1)] / [x - x1] using points from te left side of q = 4 x1 x P(x1) P(x) [P(x) - P(x1)] / [x - x1] Figure As we look at te slopes of te many secant lines in te cart above we notice tat as te secant lines pass troug points tat are closer and closer togeter teir slopes seem to be closer and closer to te same value bot on te rigt and left side of te point (4, 78). Terefore we will make an estimate (or a good guess) tat te slope of te tangent line is m = 4. Te closer te points involved in calculations te better te estimate. In te process above we found te slope of te tangent line numerically. We will also learn to find it algebraically. In order to develop te algebraic approac we need to go back to idea of a scant line. If we ave a function f(x) a secant line intersect te grap of tis function in two places. Terefore it sares two points wit f(x). We will call tose two points (x, f(x)) and (x +, f(x + )). In order to find te slope of tis second line we need to find te cange in te y-values divided by te cange in te x-values. Tis will give us te secant line f ( x+ ) f( x) f( x+ ) f( x) slope m = =. Tis is a generalization of te slope of ( x+ ) x any secant line. If we now tink back to te numerical process above, you will remember tat we found te slopes of several secant lines. Eac secant line was closer to te tangent line of interest tan te secant line before it and te slope values were getting closer to te slope value of te tangent line. For eac succeeding secant line te two points used were closer

3 togeter tan before. One way to express tis cange in points is to use te notation Limit. Tis notation means tat te distance between te x values is getting smaller and smaller. Te distance is approacing te value zero. Terefore te limit of te slope values of te secant lines will approac te slope value of te tangent. Te symbolic notation tat describes tis process is given as follows. dy f ( x + ) f ( x) y dx x x ( dy y is notation for te slope of te tangent line and is notation for te slope of te dx x secant line. Tis formula is also called te Derivative (to be discussed in a later section) Now we will look at an example ow to find te slope of a tangent line toucing te grap of f(x) at some value x = a. Example 1 Find te slope of te tangent line toucing te grap ( ) x 3 f x = + at x = 5. Solution: Before we start te problem we will review notation: Review te following evaluations: f(x) = x + 3 f(4) = (4) + 3 = 19 f(7) = (7) + 3 = 5 f(m) = m + 3 f(x + ) = (x + ) + 3 Now we will now proceed to solve te example problem. dy f ( x + ) f ( x) Te slope of te tangent line is given to be so we need to dx make te appropriate substitution as follows: dy [( x + ) + 3] [ x + 3] x + x x 3 ( x + ) limit limit Limit dx = = = = lim it( x + ) = x = (5) = 1. Terefore te slope of te tangent line toucing te grap at x = 5 is m = 1. See te grap below.

4 F(x) and tangent line x f(x) T Figure 3 In order to verify tat tis is te correct Tangent line we will find te equation of te tangent line tat touces te grap of f( x) = x + 3 at x = 5. Solution: In order to find te equation of any line you need te following: 1) a point ) a slope 3) te point slope form of te equation y y1 = m( x x1) Since we are interested in te tangent line tat touces te grap of f( x) = x + 3 at 4) x = 5, we need to find tat te y value f(5) = 8. So now we ave te point (5, 8). Next we find te slope to te tangent line at x = 5. Since te derivative of dy = x is te general slope formula for any tangent line toucing f (x) we found dx dy above ave tat te slope of te tangent line at x = 5 is 1 dx =. Finally we use te point and slope information to create te equation y y1 = m( x x1) to get y 8 = 1( x 5). Now simplify to get y = 1x wic is te tangent line equation pictured in Figure 3 above. Example Find te instantaneous rate of cange in Revenue R = te quantity of items sold. q + 4q at q = 3 were q is a) Using te table in Figure as a guide, find te instantaneous rate of cange numerically from te left side. Calculate te slope of secants lines troug te following points were 1) q 1 =3 and q = 9.5 3) q 1 =3 and q = 9.99 ) q 1 =3 and q = 9.9 4) q 1 =3 and q = 9.999

5 b) Now find te instantaneous rate of cange numerically from te rigt side. Calculate te slope of secants lines troug te following points were 1) q 1 =3 and q = 3.1 3) q 1 =3 and q = 3.1 ) q 1 =3 and q = 3.1 4) q 1 =3 and q = 3.1 c) Use your calculations from a) and b) to estimate te slope of te tangent line at q = 3 wic will also te instantaneous rate of cange in revenue at q = 3. Now we will do te problem again algebraically: To find te instantaneous rate of cange at q = 3, we need to find dr R( x + ) f ( x) ( ( q + ) + 4( q + )) ( q + 4 q) dq o q 4q 4q 4 q 4 q ( 4q 4) = -4q + 4 = -4(3) + 4 = Te Marginal Concept Revisited: In te section on average rate of cange we defined te term marginal as te average rate cange in te dependent variable over a one unit cange in te independent variable. Wen considering instantaneous rate of cange you are considering a very small cange in te independent variable. However wen you are considering a cange in quantity of items te smallest cange will be a unit of one. ( you would not make a fraction of an item to sell) Terefore te instantaneous rate of cange wen quantity is involved will be equivalent to te definition of marginal.

2.1: The Derivative and the Tangent Line Problem

2.1: The Derivative and the Tangent Line Problem .1.1.1: Te Derivative and te Tangent Line Problem Wat is te deinition o a tangent line to a curve? To answer te diiculty in writing a clear deinition o a tangent line, we can deine it as te iting position

More information

2 Limits and Derivatives

2 Limits and Derivatives 2 Limits and Derivatives 2.7 Tangent Lines, Velocity, and Derivatives A tangent line to a circle is a line tat intersects te circle at exactly one point. We would like to take tis idea of tangent line

More information

7.6 Complex Fractions

7.6 Complex Fractions Section 7.6 Comple Fractions 695 7.6 Comple Fractions In tis section we learn ow to simplify wat are called comple fractions, an eample of wic follows. 2 + 3 Note tat bot te numerator and denominator are

More information

Tangent Lines and Rates of Change

Tangent Lines and Rates of Change Tangent Lines and Rates of Cange 9-2-2005 Given a function y = f(x), ow do you find te slope of te tangent line to te grap at te point P(a, f(a))? (I m tinking of te tangent line as a line tat just skims

More information

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q =

f(x) f(a) x a Our intuition tells us that the slope of the tangent line to the curve at the point P is m P Q = Lecture 6 : Derivatives and Rates of Cange In tis section we return to te problem of finding te equation of a tangent line to a curve, y f(x) If P (a, f(a)) is a point on te curve y f(x) and Q(x, f(x))

More information

f(a + h) f(a) f (a) = lim

f(a + h) f(a) f (a) = lim Lecture 7 : Derivative AS a Function In te previous section we defined te derivative of a function f at a number a (wen te function f is defined in an open interval containing a) to be f (a) 0 f(a + )

More information

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function

Lecture 10: What is a Function, definition, piecewise defined functions, difference quotient, domain of a function Lecture 10: Wat is a Function, definition, piecewise defined functions, difference quotient, domain of a function A function arises wen one quantity depends on anoter. Many everyday relationsips between

More information

Math 113 HW #5 Solutions

Math 113 HW #5 Solutions Mat 3 HW #5 Solutions. Exercise.5.6. Suppose f is continuous on [, 5] and te only solutions of te equation f(x) = 6 are x = and x =. If f() = 8, explain wy f(3) > 6. Answer: Suppose we ad tat f(3) 6. Ten

More information

Similar interpretations can be made for total revenue and total profit functions.

Similar interpretations can be made for total revenue and total profit functions. EXERCISE 3-7 Tings to remember: 1. MARGINAL COST, REVENUE, AND PROFIT If is te number of units of a product produced in some time interval, ten: Total Cost C() Marginal Cost C'() Total Revenue R() Marginal

More information

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation

Sections 3.1/3.2: Introducing the Derivative/Rules of Differentiation Sections 3.1/3.2: Introucing te Derivative/Rules of Differentiation 1 Tangent Line Before looking at te erivative, refer back to Section 2.1, looking at average velocity an instantaneous velocity. Here

More information

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12)

Differential Calculus: Differentiation (First Principles, Rules) and Sketching Graphs (Grade 12) OpenStax-CNX moule: m39313 1 Differential Calculus: Differentiation (First Principles, Rules) an Sketcing Graps (Grae 12) Free Hig Scool Science Texts Project Tis work is prouce by OpenStax-CNX an license

More information

Derivatives Math 120 Calculus I D Joyce, Fall 2013

Derivatives Math 120 Calculus I D Joyce, Fall 2013 Derivatives Mat 20 Calculus I D Joyce, Fall 203 Since we ave a good understanding of its, we can develop derivatives very quickly. Recall tat we defined te derivative f x of a function f at x to be te

More information

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION

MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION MATHEMATICS FOR ENGINEERING DIFFERENTIATION TUTORIAL 1 - BASIC DIFFERENTIATION Tis tutorial is essential pre-requisite material for anyone stuing mecanical engineering. Tis tutorial uses te principle of

More information

SAT Subject Math Level 1 Facts & Formulas

SAT Subject Math Level 1 Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses

More information

The EOQ Inventory Formula

The EOQ Inventory Formula Te EOQ Inventory Formula James M. Cargal Matematics Department Troy University Montgomery Campus A basic problem for businesses and manufacturers is, wen ordering supplies, to determine wat quantity of

More information

Average and Instantaneous Rates of Change: The Derivative

Average and Instantaneous Rates of Change: The Derivative 9.3 verage and Instantaneous Rates of Cange: Te Derivative 609 OBJECTIVES 9.3 To define and find average rates of cange To define te derivative as a rate of cange To use te definition of derivative to

More information

The Derivative as a Function

The Derivative as a Function Section 2.2 Te Derivative as a Function 200 Kiryl Tsiscanka Te Derivative as a Function DEFINITION: Te derivative of a function f at a number a, denoted by f (a), is if tis limit exists. f (a) f(a+) f(a)

More information

ACT Math Facts & Formulas

ACT Math Facts & Formulas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationals: fractions, tat is, anyting expressable as a ratio of integers Reals: integers plus rationals plus special numbers suc as

More information

Compute the derivative by definition: The four step procedure

Compute the derivative by definition: The four step procedure Compute te derivative by definition: Te four step procedure Given a function f(x), te definition of f (x), te derivative of f(x), is lim 0 f(x + ) f(x), provided te limit exists Te derivative function

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

CHAPTER 8: DIFFERENTIAL CALCULUS

CHAPTER 8: DIFFERENTIAL CALCULUS CHAPTER 8: DIFFERENTIAL CALCULUS 1. Rules of Differentiation As we ave seen, calculating erivatives from first principles can be laborious an ifficult even for some relatively simple functions. It is clearly

More information

Proof of the Power Rule for Positive Integer Powers

Proof of the Power Rule for Positive Integer Powers Te Power Rule A function of te form f (x) = x r, were r is any real number, is a power function. From our previous work we know tat x x 2 x x x x 3 3 x x In te first two cases, te power r is a positive

More information

Finite Volume Discretization of the Heat Equation

Finite Volume Discretization of the Heat Equation Lecture Notes 3 Finite Volume Discretization of te Heat Equation We consider finite volume discretizations of te one-dimensional variable coefficient eat equation, wit Neumann boundary conditions u t x

More information

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry.

This supplement is meant to be read after Venema s Section 9.2. Throughout this section, we assume all nine axioms of Euclidean geometry. Mat 444/445 Geometry for Teacers Summer 2008 Supplement : Similar Triangles Tis supplement is meant to be read after Venema s Section 9.2. Trougout tis section, we assume all nine axioms of uclidean geometry.

More information

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line.

f(x + h) f(x) h as representing the slope of a secant line. As h goes to 0, the slope of the secant line approaches the slope of the tangent line. Derivative of f(z) Dr. E. Jacobs Te erivative of a function is efine as a limit: f (x) 0 f(x + ) f(x) We can visualize te expression f(x+) f(x) as representing te slope of a secant line. As goes to 0,

More information

ACTIVITY: Deriving the Area Formula of a Trapezoid

ACTIVITY: Deriving the Area Formula of a Trapezoid 4.3 Areas of Trapezoids a trapezoid? How can you derive a formula for te area of ACTIVITY: Deriving te Area Formula of a Trapezoid Work wit a partner. Use a piece of centimeter grid paper. a. Draw any

More information

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion

1 Density functions, cummulative density functions, measures of central tendency, and measures of dispersion Density functions, cummulative density functions, measures of central tendency, and measures of dispersion densityfunctions-intro.tex October, 9 Note tat tis section of notes is limitied to te consideration

More information

Verifying Numerical Convergence Rates

Verifying Numerical Convergence Rates 1 Order of accuracy Verifying Numerical Convergence Rates We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, suc as te grid size or time step, and

More information

CHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0

CHAPTER TWO. f(x) Slope = f (3) = Rate of change of f at 3. x 3. f(1.001) f(1) Average velocity = 1.1 1 1.01 1. s(0.8) s(0) 0.8 0 CHAPTER TWO 2.1 SOLUTIONS 99 Solutions for Section 2.1 1. (a) Te average rate of cange is te slope of te secant line in Figure 2.1, wic sows tat tis slope is positive. (b) Te instantaneous rate of cange

More information

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations

Section 3.3. Differentiation of Polynomials and Rational Functions. Difference Equations to Differential Equations Difference Equations to Differential Equations Section 3.3 Differentiation of Polynomials an Rational Functions In tis section we begin te task of iscovering rules for ifferentiating various classes of

More information

Chapter 7 Numerical Differentiation and Integration

Chapter 7 Numerical Differentiation and Integration 45 We ave a abit in writing articles publised in scientiþc journals to make te work as Þnised as possible, to cover up all te tracks, to not worry about te blind alleys or describe ow you ad te wrong idea

More information

Area of a Parallelogram

Area of a Parallelogram Area of a Parallelogram Focus on After tis lesson, you will be able to... φ develop te φ formula for te area of a parallelogram calculate te area of a parallelogram One of te sapes a marcing band can make

More information

SAT Math Facts & Formulas

SAT Math Facts & Formulas Numbers, Sequences, Factors SAT Mat Facts & Formuas Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reas: integers pus fractions, decimas, and irrationas ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences:

More information

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.)

Lecture 10. Limits (cont d) One-sided limits. (Relevant section from Stewart, Seventh Edition: Section 2.4, pp. 113.) Lecture 10 Limits (cont d) One-sided its (Relevant section from Stewart, Sevent Edition: Section 2.4, pp. 113.) As you may recall from your earlier course in Calculus, we may define one-sided its, were

More information

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu

Math 229 Lecture Notes: Product and Quotient Rules Professor Richard Blecksmith richard@math.niu.edu Mat 229 Lecture Notes: Prouct an Quotient Rules Professor Ricar Blecksmit ricar@mat.niu.eu 1. Time Out for Notation Upate It is awkwar to say te erivative of x n is nx n 1 Using te prime notation for erivatives,

More information

2.2 Derivative as a Function

2.2 Derivative as a Function 2.2 Derivative as a Function Recall that we defined the derivative as f (a) = lim h 0 f(a + h) f(a) h But since a is really just an arbitrary number that represents an x-value, why don t we just use x

More information

New Vocabulary volume

New Vocabulary volume -. Plan Objectives To find te volume of a prism To find te volume of a cylinder Examples Finding Volume of a Rectangular Prism Finding Volume of a Triangular Prism 3 Finding Volume of a Cylinder Finding

More information

1 Derivatives of Piecewise Defined Functions

1 Derivatives of Piecewise Defined Functions MATH 1010E University Matematics Lecture Notes (week 4) Martin Li 1 Derivatives of Piecewise Define Functions For piecewise efine functions, we often ave to be very careful in computing te erivatives.

More information

Math Test Sections. The College Board: Expanding College Opportunity

Math Test Sections. The College Board: Expanding College Opportunity Taking te SAT I: Reasoning Test Mat Test Sections Te materials in tese files are intended for individual use by students getting ready to take an SAT Program test; permission for any oter use must be sougt

More information

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution

1.6. Analyse Optimum Volume and Surface Area. Maximum Volume for a Given Surface Area. Example 1. Solution 1.6 Analyse Optimum Volume and Surface Area Estimation and oter informal metods of optimizing measures suc as surface area and volume often lead to reasonable solutions suc as te design of te tent in tis

More information

An inquiry into the multiplier process in IS-LM model

An inquiry into the multiplier process in IS-LM model An inquiry into te multiplier process in IS-LM model Autor: Li ziran Address: Li ziran, Room 409, Building 38#, Peing University, Beijing 00.87,PRC. Pone: (86) 00-62763074 Internet Address: jefferson@water.pu.edu.cn

More information

Chapter 11. Limits and an Introduction to Calculus. Selected Applications

Chapter 11. Limits and an Introduction to Calculus. Selected Applications Capter Limits and an Introduction to Calculus. Introduction to Limits. Tecniques for Evaluating Limits. Te Tangent Line Problem. Limits at Infinit and Limits of Sequences.5 Te Area Problem Selected Applications

More information

Geometric Stratification of Accounting Data

Geometric Stratification of Accounting Data Stratification of Accounting Data Patricia Gunning * Jane Mary Horgan ** William Yancey *** Abstract: We suggest a new procedure for defining te boundaries of te strata in igly skewed populations, usual

More information

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula

Determine the perimeter of a triangle using algebra Find the area of a triangle using the formula Student Name: Date: Contact Person Name: Pone Number: Lesson 0 Perimeter, Area, and Similarity of Triangles Objectives Determine te perimeter of a triangle using algebra Find te area of a triangle using

More information

An Interest Rate Model

An Interest Rate Model An Interest Rate Model Concepts and Buzzwords Building Price Tree from Rate Tree Lognormal Interest Rate Model Nonnegativity Volatility and te Level Effect Readings Tuckman, capters 11 and 12. Lognormal

More information

FINITE DIFFERENCE METHODS

FINITE DIFFERENCE METHODS FINITE DIFFERENCE METHODS LONG CHEN Te best known metods, finite difference, consists of replacing eac derivative by a difference quotient in te classic formulation. It is simple to code and economic to

More information

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution

EC201 Intermediate Macroeconomics. EC201 Intermediate Macroeconomics Problem set 8 Solution EC201 Intermediate Macroeconomics EC201 Intermediate Macroeconomics Prolem set 8 Solution 1) Suppose tat te stock of mone in a given econom is given te sum of currenc and demand for current accounts tat

More information

In other words the graph of the polynomial should pass through the points

In other words the graph of the polynomial should pass through the points Capter 3 Interpolation Interpolation is te problem of fitting a smoot curve troug a given set of points, generally as te grap of a function. It is useful at least in data analysis (interpolation is a form

More information

6. Differentiating the exponential and logarithm functions

6. Differentiating the exponential and logarithm functions 1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

More information

2.28 EDGE Program. Introduction

2.28 EDGE Program. Introduction Introduction Te Economic Diversification and Growt Enterprises Act became effective on 1 January 1995. Te creation of tis Act was to encourage new businesses to start or expand in Newfoundland and Labrador.

More information

Surface Areas of Prisms and Cylinders

Surface Areas of Prisms and Cylinders 12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

More information

4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a

4.4 The Derivative. 51. Disprove the claim: If lim f (x) = L, then either lim f (x) = L or. 52. If lim x a. f (x) = and lim x a. g(x) =, then lim x a Capter 4 Real Analysis 281 51. Disprove te claim: If lim f () = L, ten eiter lim f () = L or a a lim f () = L. a 52. If lim a f () = an lim a g() =, ten lim a f + g =. 53. If lim f () = an lim g() = L

More information

Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic

Areas and Centroids. Nothing. Straight Horizontal line. Straight Sloping Line. Parabola. Cubic Constructing Sear and Moment Diagrams Areas and Centroids Curve Equation Sape Centroid (From Fat End of Figure) Area Noting Noting a x 0 Straigt Horizontal line /2 Straigt Sloping Line /3 /2 Paraola /4

More information

Research on the Anti-perspective Correction Algorithm of QR Barcode

Research on the Anti-perspective Correction Algorithm of QR Barcode Researc on te Anti-perspective Correction Algoritm of QR Barcode Jianua Li, Yi-Wen Wang, YiJun Wang,Yi Cen, Guoceng Wang Key Laboratory of Electronic Tin Films and Integrated Devices University of Electronic

More information

Average rate of change

Average rate of change Average rate of change 1 1 Average rate of change A fundamental philosophical truth is that everything changes. 1 Average rate of change A fundamental philosophical truth is that everything changes. In

More information

SAT Math Must-Know Facts & Formulas

SAT Math Must-Know Facts & Formulas SAT Mat Must-Know Facts & Formuas Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Rationas: fractions, tat is, anyting expressabe as a ratio of integers Reas: integers pus rationas

More information

Theoretical calculation of the heat capacity

Theoretical calculation of the heat capacity eoretical calculation of te eat capacity Principle of equipartition of energy Heat capacity of ideal and real gases Heat capacity of solids: Dulong-Petit, Einstein, Debye models Heat capacity of metals

More information

The differential amplifier

The differential amplifier DiffAmp.doc 1 Te differential amplifier Te emitter coupled differential amplifier output is V o = A d V d + A c V C Were V d = V 1 V 2 and V C = (V 1 + V 2 ) / 2 In te ideal differential amplifier A c

More information

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE

ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE ON LOCAL LIKELIHOOD DENSITY ESTIMATION WHEN THE BANDWIDTH IS LARGE Byeong U. Park 1 and Young Kyung Lee 2 Department of Statistics, Seoul National University, Seoul, Korea Tae Yoon Kim 3 and Ceolyong Park

More information

The modelling of business rules for dashboard reporting using mutual information

The modelling of business rules for dashboard reporting using mutual information 8 t World IMACS / MODSIM Congress, Cairns, Australia 3-7 July 2009 ttp://mssanz.org.au/modsim09 Te modelling of business rules for dasboard reporting using mutual information Gregory Calbert Command, Control,

More information

Area Formulas with Applications

Area Formulas with Applications Formulas wit Applications Ojective To review and use formulas for perimeter, circumference, and area. www.everydaymatonline.com epresentations etoolkit Algoritms Practice EM Facts Worksop Game Family Letters

More information

Projective Geometry. Projective Geometry

Projective Geometry. Projective Geometry Euclidean versus Euclidean geometry describes sapes as tey are Properties of objects tat are uncanged by rigid motions» Lengts» Angles» Parallelism Projective geometry describes objects as tey appear Lengts,

More information

Writing Mathematics Papers

Writing Mathematics Papers Writing Matematics Papers Tis essay is intended to elp your senior conference paper. It is a somewat astily produced amalgam of advice I ave given to students in my PDCs (Mat 4 and Mat 9), so it s not

More information

Torchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK

Torchmark Corporation 2001 Third Avenue South Birmingham, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK News Release Torcmark Corporation 2001 Tird Avenue Sout Birmingam, Alabama 35233 Contact: Joyce Lane 972-569-3627 NYSE Symbol: TMK TORCHMARK CORPORATION REPORTS FOURTH QUARTER AND YEAR-END 2004 RESULTS

More information

Definition of derivative

Definition of derivative Definition of derivative Contents 1. Slope-The Concept 2. Slope of a curve 3. Derivative-The Concept 4. Illustration of Example 5. Definition of Derivative 6. Example 7. Extension of the idea 8. Example

More information

Introduction to Calculus

Introduction to Calculus Introduction to Calculus Contents 1 Introduction to Calculus 3 11 Introduction 3 111 Origin of Calculus 3 112 The Two Branches of Calculus 4 12 Secant and Tangent Lines 5 13 Limits 10 14 The Derivative

More information

CHAPTER 7. Di erentiation

CHAPTER 7. Di erentiation CHAPTER 7 Di erentiation 1. Te Derivative at a Point Definition 7.1. Let f be a function defined on a neigborood of x 0. f is di erentiable at x 0, if te following it exists: f 0 fx 0 + ) fx 0 ) x 0 )=.

More information

Volumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m

Volumes of Pyramids and Cones. Use the Pythagorean Theorem to find the value of the variable. h 2 m. 1.5 m 12 in. 8 in. 2.5 m -5 Wat You ll Learn To find te volume of a pramid To find te volume of a cone... And W To find te volume of a structure in te sape of a pramid, as in Eample Volumes of Pramids and Cones Ceck Skills You

More information

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1)

M(0) = 1 M(1) = 2 M(h) = M(h 1) + M(h 2) + 1 (h > 1) Insertion and Deletion in VL Trees Submitted in Partial Fulfillment of te Requirements for Dr. Eric Kaltofen s 66621: nalysis of lgoritms by Robert McCloskey December 14, 1984 1 ackground ccording to Knut

More information

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade? Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te

More information

Schedulability Analysis under Graph Routing in WirelessHART Networks

Schedulability Analysis under Graph Routing in WirelessHART Networks Scedulability Analysis under Grap Routing in WirelessHART Networks Abusayeed Saifulla, Dolvara Gunatilaka, Paras Tiwari, Mo Sa, Cenyang Lu, Bo Li Cengjie Wu, and Yixin Cen Department of Computer Science,

More information

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY

SAMPLE DESIGN FOR THE TERRORISM RISK INSURANCE PROGRAM SURVEY ASA Section on Survey Researc Metods SAMPLE DESIG FOR TE TERRORISM RISK ISURACE PROGRAM SURVEY G. ussain Coudry, Westat; Mats yfjäll, Statisticon; and Marianne Winglee, Westat G. ussain Coudry, Westat,

More information

Derivatives as Rates of Change

Derivatives as Rates of Change Derivatives as Rates of Change One-Dimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions

More information

MAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules

MAT1A01: Differentiation of Polynomials & Exponential Functions + the Product & Quotient Rules MAT1A01: Differentiation of Polynomials & Exponential Functions + te Prouct & Quotient Rules Dr Craig 17 April 2013 Reminer Mats Learning Centre: C-Ring 512 My office: C-Ring 533A (Stats Dept corrior)

More information

2.23 Gambling Rehabilitation Services. Introduction

2.23 Gambling Rehabilitation Services. Introduction 2.23 Gambling Reabilitation Services Introduction Figure 1 Since 1995 provincial revenues from gambling activities ave increased over 56% from $69.2 million in 1995 to $108 million in 2004. Te majority

More information

Operation go-live! Mastering the people side of operational readiness

Operation go-live! Mastering the people side of operational readiness ! I 2 London 2012 te ultimate Up to 30% of te value of a capital programme can be destroyed due to operational readiness failures. 1 In te complex interplay between tecnology, infrastructure and process,

More information

Marginal Cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for 2

Marginal Cost. Example 1: Suppose the total cost in dollars per week by ABC Corporation for 2 Math 114 Marginal Functions in Economics Marginal Cost Suppose a business owner is operating a plant that manufactures a certain product at a known level. Sometimes the business owner will want to know

More information

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz

- 1 - Handout #22 May 23, 2012 Huffman Encoding and Data Compression. CS106B Spring 2012. Handout by Julie Zelenski with minor edits by Keith Schwarz CS106B Spring 01 Handout # May 3, 01 Huffman Encoding and Data Compression Handout by Julie Zelenski wit minor edits by Keit Scwarz In te early 1980s, personal computers ad ard disks tat were no larger

More information

Working Capital 2013 UK plc s unproductive 69 billion

Working Capital 2013 UK plc s unproductive 69 billion 2013 Executive summary 2. Te level of excess working capital increased 3. UK sectors acieve a mixed performance 4. Size matters in te supply cain 6. Not all companies are overflowing wit cas 8. Excess

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1

Note nine: Linear programming CSE 101. 1 Linear constraints and objective functions. 1.1 Introductory example. Copyright c Sanjoy Dasgupta 1 Copyrigt c Sanjoy Dasgupta Figure. (a) Te feasible region for a linear program wit two variables (see tet for details). (b) Contour lines of te objective function: for different values of (profit). Te

More information

Techniques of Differentiation Selected Problems. Matthew Staley

Techniques of Differentiation Selected Problems. Matthew Staley Techniques of Differentiation Selected Problems Matthew Staley September 10, 011 Techniques of Differentiation: Selected Problems 1. Find /dx: (a) y =4x 7 dx = d dx (4x7 ) = (7)4x 6 = 8x 6 (b) y = 1 (x4

More information

Catalogue no. 12-001-XIE. Survey Methodology. December 2004

Catalogue no. 12-001-XIE. Survey Methodology. December 2004 Catalogue no. 1-001-XIE Survey Metodology December 004 How to obtain more information Specific inquiries about tis product and related statistics or services sould be directed to: Business Survey Metods

More information

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53

4.1 Right-angled Triangles 2. 4.2 Trigonometric Functions 19. 4.3 Trigonometric Identities 36. 4.4 Applications of Trigonometry to Triangles 53 ontents 4 Trigonometry 4.1 Rigt-angled Triangles 4. Trigonometric Functions 19 4.3 Trigonometric Identities 36 4.4 pplications of Trigonometry to Triangles 53 4.5 pplications of Trigonometry to Waves 65

More information

Continuous Functions, Smooth Functions and the Derivative

Continuous Functions, Smooth Functions and the Derivative UCSC AMS/ECON 11A Supplemental Notes # 4 Continuous Functions, Smooth Functions and the Derivative c 2004 Yonatan Katznelson 1. Continuous functions One of the things that economists like to do with mathematical

More information

The Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios

The Dynamics of Movie Purchase and Rental Decisions: Customer Relationship Implications to Movie Studios Te Dynamics of Movie Purcase and Rental Decisions: Customer Relationsip Implications to Movie Studios Eddie Ree Associate Professor Business Administration Stoneill College 320 Wasington St Easton, MA

More information

Grade 12 Assessment Exemplars

Grade 12 Assessment Exemplars Grade Assessment Eemplars Learning Outcomes and. Assignment : Functions - Memo. Investigation: Sequences and Series Memo/Rubric 5. Control Test: Number Patterns, Finance and Functions - Memo 7. Project:

More information

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.

Derivatives and Graphs. Review of basic rules: We have already discussed the Power Rule. Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,

More information

2.2. Instantaneous Velocity

2.2. Instantaneous Velocity 2.2. Instantaneous Velocity toc Assuming that your are not familiar with the technical aspects of this section, when you think about it, your knowledge of velocity is limited. In terms of your own mathematical

More information

Binary Search Trees. Adnan Aziz. Heaps can perform extract-max, insert efficiently O(log n) worst case

Binary Search Trees. Adnan Aziz. Heaps can perform extract-max, insert efficiently O(log n) worst case Binary Searc Trees Adnan Aziz 1 BST basics Based on CLRS, C 12. Motivation: Heaps can perform extract-max, insert efficiently O(log n) worst case Has tables can perform insert, delete, lookup efficiently

More information

2.13 Solid Waste Management. Introduction. Scope and Objectives. Conclusions

2.13 Solid Waste Management. Introduction. Scope and Objectives. Conclusions Introduction Te planning and delivery of waste management in Newfoundland and Labrador is te direct responsibility of municipalities and communities. Te Province olds overall responsibility for te development

More information

22.1 Finding the area of plane figures

22.1 Finding the area of plane figures . Finding te area of plane figures a cm a cm rea of a square = Lengt of a side Lengt of a side = (Lengt of a side) b cm a cm rea of a rectangle = Lengt readt b cm a cm rea of a triangle = a cm b cm = ab

More information

Blue Pelican Calculus First Semester

Blue Pelican Calculus First Semester Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 2011-2013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function

More information

Slope and Rate of Change

Slope and Rate of Change Chapter 1 Slope and Rate of Change Chapter Summary and Goal This chapter will start with a discussion of slopes and the tangent line. This will rapidly lead to heuristic developments of limits and the

More information

Learning Objectives for Math 165

Learning Objectives for Math 165 Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given

More information

INDEX OF PERSONS EMPLOYED IN RETAIL TRADE (2010=100.0)

INDEX OF PERSONS EMPLOYED IN RETAIL TRADE (2010=100.0) NDX OF PRSONS MPLOYD N RTAL TRAD (20000.0. ntroduction Te ndex of Persons mployed in Retail Trade is a quarterly index tat was compiled for te first time in 2004 wit 200000.0 as te base year and backdated

More information

RISK ASSESSMENT MATRIX

RISK ASSESSMENT MATRIX U.S.C.G. AUXILIARY STANDARD AV-04-4 Draft Standard Doc. AV- 04-4 18 August 2004 RISK ASSESSMENT MATRIX STANDARD FOR AUXILIARY AVIATION UNITED STATES COAST GUARD AUXILIARY NATIONAL OPERATIONS DEPARTMENT

More information

Advice for Undergraduates on Special Aspects of Writing Mathematics

Advice for Undergraduates on Special Aspects of Writing Mathematics Advice for Undergraduates on Special Aspects of Writing Matematics Abstract Tere are several guides to good matematical writing for professionals, but few for undergraduates. Yet undergraduates wo write

More information

THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present. Prepared for public release by:

THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present. Prepared for public release by: THE NEISS SAMPLE (DESIGN AND IMPLEMENTATION) 1997 to Present Prepared for public release by: Tom Scroeder Kimberly Ault Division of Hazard and Injury Data Systems U.S. Consumer Product Safety Commission

More information