2 Limits and Derivatives
|
|
- Maximillian Conley
- 8 years ago
- Views:
Transcription
1 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 and apply it to oter curves. y = f(x) P (a, f(a)) Te word tangent is derived from te Latin word tangens, wic means toucing. Tus a tangent to a curve is a line tat touces te curve. In oter words, a tangent line sould ave te same direction as te curve at te point of contact. Our goal will be to find an equation of te tangent line to a curve given by y = f(x) at te point P (a, f(a)). Note tat wen te x-coordinate is a, te y-coordinate is necessarily given by y = f(a). Let s begin by recalling te formula for te equation of a line. Point-slope Equation of a Line Te point slope formula for te equation 1
2 of a line is y y 1 = m(x x 1 ) were (x 1, y 1 ) is a point on te line, and m is te slope of te line. Tis is a good start. For our tangent line, we ave a point, P (a, f(a)). So we can let x 1 = a and y 1 = f(a). Next we need to find te slope of te tangent line, m tan. Slope of a Line Te formula for te slope of a line is m = y 2 y 1 x 2 x 1, were (x 1, y 1 ) and (x 2, y 2 ) are two points on te line. In order to use tis formula for slope, we need to know te coordinates of two points on a line. So far, we ave only one set of coordinates, x 1 = a, y 1 = f(a). We are going to get around tis problem by introducing a secant line. Tis will be a line troug te point P and anoter point Q(x, f(x)) on te curve. Te idea is tat te slope of te secant line will be almost te same as te slope of te tangent line provided tat te point Q is close to P. 2
3 Q(x, f(x)) Secant Line y = f(x) Tangent Line P (a, f(a)) Te secant line contains te point Q(x, f(x)) and te point P (a, f(a)). Terefore te slope of te secant line is given by m sec = rise run = Again, we are going to use te slope of te secant line as an approximation for te slope of te tangent line. To get a really good approximation, we want to make Q as close to P as possible. Tis is acieved by finding te limit of te slope of te secant line, m sec, as P approaces Q. Te slope of te tangent line, m tan, is ten defined as te limit of te slope of te secant line, m sec, as Q approaces P. Tat is, m tan Q P m sec To make Q approac P, we can let x approac a. Moreover, a formula for m sec was given above. We now give te following definition for te slope of te tangent line. 3
4 Definition Te slope of te tangent line to a curve y = f(x) at (a, f(a) is given by te formula m tan x a Tis was sort of te wole point of studying limits in te previous capter: we wanted to be able to make tis limit calculation. We can now do an example were we find te equation of te tangent line to a specific function. Example 1 Find te equation of te tangent line to te curve f(x) = x 2 at te point (1, 1). Solution Te point-slope formula for a line is y y 1 = m(x x 1 ). For tis example, x 1 = 1 and y 1 = 1. Note tat f(x) = x 2 and a = 1. We can now use te formula to find te slope of te tangent line, m tan. m tan x a x 1 f(x) f(1) x 1 x x 1 x 1 x 1 (x 1)(x + 1) (x 1) x 1 (x + 1) = = 2 We ave x 1 = 1, y 1 = 1 and m tan = 2. Terefore, wen substituting into te point-slope equation for te line we get y y 1 = m(x x 1 ) 4
5 y 1 = 2(x 1) y = 2x 1 Example 2 Find te slope of te tangent line at te given value of x = a using te formula given above for m tan. 1. f(x) = x 2 3x + 1, a = 2 2. f(x) = x, a = 4 3. f(x) = x 3, a = 3 Solution 1. f(x) = x 2 3x + 1, a = 2 2. f(x) = x, a = 4 m tan = x a lim = f(x) f(2) lim x 2 x 2 (x 2 3x + 1) ((2) 2 3(2) + 1) x 2 x 2 (x 2 3x + 1) ( 1) x 2 (x 2) x 2 3x + 2 x 2 (x 2) (x 1)(x 2) x 2 (x 2) (x 1) = 2 1 = 1 x 2 m tan = x a lim f(x) f(4) x 4 x 4 x 4 x 4 x 4 5
6 3. f(x) = x 3, a = 3 x 4 x 2 (x 4) x 4 x 2 (x 4) ( ) x + 2 x + 2 ( x) 2 (2) 2 x 4 (x 4)( x + 2) x 4 x 4 (x 4)( x + 2) 1 = x 4 x + 2 m tan x a x 3 f(x) f(2) x 3 x 3 x 3 (3) 3 x = 1 4 x 3 (x 3)(x 2 + 3x + (3) 3 ) (x 3) x 3 (x 2 + 3x + 9) = (3) 2 + 3(3) + 9 = 27 Example 3 Find te tangent line to te function f(x) = 1/x at (2, 1 2 ). Solution First we find te slope m tan were f(x) = 1/x and a = 2. m tan = lim x a = f(x) f(2) lim x 2 x 2 = 1 lim x 2 x 2 x 2 Multiply te numerator and denominator by 2x to clear te fraction. x 2 ( 1 x 1 2 )2x (x 2)2x 6
7 (2 x) x 2 (x 2)2x (x 2) x 2 (x 2)2x 1 x 2 2x 1 = 2 2 = 1 4 We can use te point-slope formula wit x 1 = 2, y 1 = f(2) = 1/2, and m tan = 1/4. Tis gives y y 1 = m tan (x x 1 ) y 1/2 = 1 (x 2) 4 y = x Velocity and Rate of Cange Suppose we drop a rock from a cliff in te Hig Sierras. Tere is an equation tat we can use to determine te position of a falling object after t seconds. If s(t) is te position in meters of a falling object after t seconds ten we ave te formula s(t) = 4.9t 2 m. For example, after 1 second, te rock as fallen s(1) = 4.9(1) 2 = 4.9 m. After 2 seconds, te rock as fallen s(2) = 4.9(2) 2 = 4.9(4) = 19.6 m. In fact, tis formula can be used to estimate te eigt of a cliff. We can make te approximation tat s(t) = 5t 2 m. Ten we can drop a rock off of a cliff and record ow long it takes for te rock to fall. Say te rock takes 5 seconds to fall. In tat case te distance tat te rock fell is approximately s(10) = 5 (5) 2 = 125 m. So te distance from te top of te cliff to te base is about 125 meters. Now let s talk about velocity. Using tis formula for position, or te distance tat a falling object as traveled, we can calculate te average velocity 7
8 of an object. Suppose we are interested in knowing ow fast a falling object is going after it as fallen a seconds. We ave te following formula for average velocity. v avg = s(t 2) s(t 1 ) t 2 t 1 So te units for average velocity will be meters per second in our example. To find ow fast te falling object is going after it as traveled a seconds, we can let t 1 = a and t 2 = t. Tis gives v avg = s(t) s(a) t a Common sense tells us tat to get te best approximation possible, we sould make te interval over wic we find te average velocity as small as possible. We would like to take te limit of te average velocity as t approaces a. Tis is ow we define instantaneous velocity. Te instantaneous velocity of of a particle at time t = a wit position function s(t) is given by s(t) s(a) v(a). t a t a Example 4 Suppose tat a ball is dropped from te upper deck of te CN Tower, 450 meters above te ground. Te position function of te ball is given by s(t) = 4.9t 2 meters after 2 seconds. Wat is te velocity of te ball after 5 seconds? Solution formula. We can find a formula for velocity at time 2 seconds using te v(2) = s(t) s(2) lim t 2 t 2 = 4.9t 2 4.9(2) 2 lim t 2 t 2 = 4.9(t 2 (2) 2 ) lim t 2 t 2 8
9 4.9(t 2)(t + 2) t 2 t 2 4.9(t + 2) = 4.9(2 + 2) = 4.9(4) = 19.6 m/s t 2 Example 5 Te position function of a particle is given by te equation s = t 3 were t is measured in seconds and s in meters. Find te velocity at time t = 2 seconds. Solution formula. We can find a formula for velocity at time 2 seconds using te v(2) = s(t) s(2) lim t 2 t 2 = (t 3 ) ((2) 3 ) lim t 2 t 2 t 2 (t 2)(t 2 + 2t + 4) t 2 t 2 (t 2 + 2t + 4) = (2) 2 + 2(2) + 4 = 12 m/s Derivatives In te previous subsection we gave a formula for te slope of te tangent line, m tan. Te slope of te tangent line is equal to wat is called te derivative of a function f at a number a. Te slope of te tangent line to a curve at a point (a, f(a)) equals te derivative. However, tere are many oter applications for te derivative of a function. Te bulk of tis capter will be spent studying te derivative as a pure mat entity. Definition Te derivative of a function f at a number a, denoted by f (a), is given by te following limit, provided te limit exists. f (a) x a 9
10 Example 6 Find te derivative of f(x) = x 2 + 2x + 5 at a = 2. Solution f (2) x 2 f(x) f(2) x 2 x 2 ((x) 2 + 2(x) + 5) ((2) 2 + 2(2) + 5) x 2 x 2 x 2 + 2x 8 x 2 x 2 (x + 4)(x 2) (x 2) x 2 (x + 4) = = 6 Te derivative can be different for different values of a. We would like to be able to find a general formula for te derivative f (a) and ten evaluate it for different values of a. Example 7 Let f(x) = x Compute a formula for f (a) using te definition of derivative. 2. Calculate f ( 1), f (0) and f (1). Solution 1. f (a) x a x 2 a 2 x a x a ()(x + a) x a (x + a) = a + a = 2a. 2. f ( 1) = 2( 1) = 1, f (0) = 2(0) = 0, and f (1) = 2(1) = 2. 10
11 f(x) = x 2 slope= 2 ( 1, 1) (0, 0) (1, 1) slope=2 slope=0 Derivative equals slope of te tangent line. Te slope of te tangent line can take different values for different points on te grap. We see tat for f(x) = x 2, we ave f ( 1) = 2, f (0) = 0, and f (1) = 2. Tere is an equivalent formula for derivative. If we make te substitution =, ten x = a +, and as x a, 0. So tat f (a) x a f(a + ) f(a) An Equivalent Definition for Derivative f at a number x = a is given by Te derivative of a function f (a) f(a + ) f(a) For certain functions, tis is an easier formula to work wit. Example 8 Find te derivative using te equivalent definition of derivative of f(x) = x 2 at a = 2. Solution 11
12 f (a) f(2 + ) f(2) (4 + ) (4 + ) = = 4 (2 + ) 2 (2) 2 Example 9 Find a formula for f (a) for f(x) = x 2 using te equivalent definition of derivative. Solution f (a) f(a + ) f(a) a 2 + 2a + 2 a 2 (a + ) 2 a 2 2a + 2 (2a + ) (2a + ) = 2a + 0 = 2a We see tat te derivative of a function depends on te number a and is terefore itself a function. Rater tat use te letter a, we can use te letter x. Definition Te derivative of a function f denoted f is given by f f(x + ) f(x) (x), provided tat te limit exists. Example 10 Find a formula for f (x) using te equivalent definition of derivative, were f(x) = x. Solution f f(x + ) f(x) (x) 1 ( ) x + x 12
13 1 ( ) x + + x x + x x + + x 1 1 = ( ) (x + ) x x + + x ( x + + x ) 1 x + + x 1 x x = 1 2 x Finding te derivative directly from te definition is long and difficult. We will introduce some rules for finding te derivative tat will make te job muc easier. Example 11 Te limit below represents te derivative of some function f at some number a. State suc an f and a. f (a) sec() 1 Solution Te formula used for f (a) appears to be te one involving. We compare. f (a) sec() 1 and f (a) f(a + ) f(a) Let s only look at te numerator of te two fractions. f(a + ) f(a) and sec() 1 Te terms f(a+) and sec() sould agree. Note tat sec() = sec(0+). We sould ave f(a + ) = sec(0 + ). Terefore, f(x) = sec(x) and a = 0. It sould follow tat f(0) = 1. We see tat f(a) = sec(0) = 1. Answer: f(x) = sec x, a = 0 Example 12 Te limit below represents te derivative of some function f at some number a. State suc an f and a. f cos(x) 1/2 (a) x π/3 x π/3 13
14 Solution We make a comparison. f (a) x a and f cos(x) 1/2 (a) x π/3 x π/3 We ave x a and x π/3. We conclude tat a = π/3. We can compare just te numerators of te fractions: and cos(x) 1/2. We conclude tat f(x) = cos x. Moreover, we sould ave f(a) = cos(a) = cos(π/3) = 1/2. Answer: f(x) = cos x and a = π/3. 14
15 Mat 180 Homework 2.7 Te Tangent Lines, Velocity, and Derivatives Sketc a plausible tangent line at te given point. 1. At x = π 2. At x = 0 3. At x = 0 4. At x = 1 15
16 5. Estimate te slope of te tangent line to te curve at (a) x = 0 (b) x = 0.5 (c) x = 1 (d) x = Estimate te slope of te tangent line to te curve at (a) x = 1 (b) x = 2 (c) x = 3 7. List te points A, B, C, and D in order of increasing slope of te tangent line. 16
17 Find te equation of te tangent line to te curve at te given point. 8. f(x) = x 2 2, (1, 1) 9. f(x) = x 2 2, (0, 2) 10. f(x) = 1/(x 2), (3, 1) 11. f(x) = x, (1, 1) 12. If a ball is trown into te air wit velocity of 40 ft/s, its eigt (in feet) after t seconds is given by y = 40 16t 2. find te velocity wen t = Te displacement (in meters) of a particle moving in a straigt line is given by te equation of motion s = 1/t 2, were t is measured in seconds. find te velocity of te particle at times t = a, t = 1, t = 2, and t = 3. For tese omework problems, use eiter definition of derivative given in tis section. or f (a) f(a + ) f(a) f (a) x a For te functions f given below, find a formula for f (a). Use te formula for f (a) to find f (0), f (5) and f (8). 14. f(x) = x f(x) = x 2 + x f(x) = 3x + 1. Eac limit represents te derivative of some function f at some number a. State suc an f and a in eac case. 17
18 17. f (1 + ) 10 1 (a) 18. f (a) f (a) x 5 2 x 32 x f tan x 1 (a) x π/4 x π/4 21. f (a) cos(π + ) f (a) t 1 t 4 + t 2 t 1 18
19 Mat 180 Homework Te Tangent Lines, Velocity, and Derivatives Solutions Tere is no tangent line at x = (a) 1 (b) 1/2 (c) 1/2 (d) 1 19
20 6. (a) 1 (b) 0 (c) 1 7. C, B, A, D 8. y = 2x 3 9. y = y = x y = x Te ball will be falling toward te ground at a velocity of 64 ft/s. 13. v(a) = 2/a 3 m/s, v(1) = 2 m/s, v(2) = 1/4 m/s, v(3) = 2/27 m/s. 14. f (a) = 3a 2. f (0) = 0, f (5) = 75, f (8) = f (a) = 2a + 1. f (0) = 1, f (5) = 11, f (8) = f(x) = 17. f(x) = x 10, a = x + 1. f (0) = 3/2, f (5) = 3/8, f (8) = 3/ f(x) = 4 x, a = f(x) = 2 x, a = f(x) = tan x, a = π/4 21. f(x) = cos x, a = π 22. f(t) = t 4 + t, a = 1 20
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 informationTangent 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 informationInstantaneous Rate of Change:
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
More informationSections 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 informationMath 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 informationf(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 informationAverage 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 information2.1: 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 informationLecture 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 informationVerifying 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 informationThe 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 informationCompute 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 informationMATHEMATICS 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 informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationSAT 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 informationAverage 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 informationSlope 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 informationCHAPTER 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 informationACT 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 informationThe 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 informationChapter 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= 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 informationAn 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 informationAverage 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 informationCHAPTER 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 informationCan 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 informationSection 2.3 Solving Right Triangle Trigonometry
Section.3 Solving Rigt Triangle Trigonometry Eample In te rigt triangle ABC, A = 40 and c = 1 cm. Find a, b, and B. sin 40 a a c 1 a 1sin 40 7.7cm cos 40 b c b 1 b 1cos40 9.cm A 40 1 b C B a B = 90 - A
More informationDerivatives 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 informationSection 3.7. Rolle s Theorem and the Mean Value Theorem. Difference Equations to Differential Equations
Difference Equations to Differential Equations Section.7 Rolle s Theorem and the Mean Value Theorem The two theorems which are at the heart of this section draw connections between the instantaneous rate
More informationCHAPTER 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 information1.6. 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 informationNew 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 informationcorrect-choice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More information2.2. 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 informationChapter 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 informationf(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 informationSection 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 informationSAT 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 informationHow To Ensure That An Eac Edge Program Is Successful
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 informationSection 1: Instantaneous Rate of Change and Tangent Lines Instantaneous Velocity
Chapter 2 The Derivative Business Calculus 74 Section 1: Instantaneous Rate of Change and Tangent Lines Instantaneous Velocity Suppose we drop a tomato from the top of a 100 foot building and time its
More information2.23 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 informationIn 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 informationSAMPLE 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 informationUnemployment insurance/severance payments and informality in developing countries
Unemployment insurance/severance payments and informality in developing countries David Bardey y and Fernando Jaramillo z First version: September 2011. Tis version: November 2011. Abstract We analyze
More informationResearch 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 informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More informationNote 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 informationTheoretical 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 informationSchedulability 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 informationNotes: 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 informationCollege Planning Using Cash Value Life Insurance
College Planning Using Cas Value Life Insurance CAUTION: Te advisor is urged to be extremely cautious of anoter college funding veicle wic provides a guaranteed return of premium immediately if funded
More information1.3.1 Position, Distance and Displacement
In the previous section, you have come across many examples of motion. You have learnt that to describe the motion of an object we must know its position at different points of time. The position of an
More informationDerivatives 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 informationSAT 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 information6. 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 informationA strong credit score can help you score a lower rate on a mortgage
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, JULY 3, 2007 A strong credit score can elp you score a lower rate on a mortgage By Sandra Block Sales of existing
More informationOptimized Data Indexing Algorithms for OLAP Systems
Database Systems Journal vol. I, no. 2/200 7 Optimized Data Indexing Algoritms for OLAP Systems Lucian BORNAZ Faculty of Cybernetics, Statistics and Economic Informatics Academy of Economic Studies, Bucarest
More informationMath 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 informationArea-Specific Recreation Use Estimation Using the National Visitor Use Monitoring Program Data
United States Department of Agriculture Forest Service Pacific Nortwest Researc Station Researc Note PNW-RN-557 July 2007 Area-Specific Recreation Use Estimation Using te National Visitor Use Monitoring
More informationDetermine 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 informationThe 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 informationSection 12.6: Directional Derivatives and the Gradient Vector
Section 26: Directional Derivatives and the Gradient Vector Recall that if f is a differentiable function of x and y and z = f(x, y), then the partial derivatives f x (x, y) and f y (x, y) give the rate
More informationM(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 informationDistances in random graphs with infinite mean degrees
Distances in random graps wit infinite mean degrees Henri van den Esker, Remco van der Hofstad, Gerard Hoogiemstra and Dmitri Znamenski April 26, 2005 Abstract We study random graps wit an i.i.d. degree
More informationON 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 information13 PERIMETER AND AREA OF 2D SHAPES
13 PERIMETER AND AREA OF D SHAPES 13.1 You can find te perimeter of sapes Key Points Te perimeter of a two-dimensional (D) sape is te total distance around te edge of te sape. l To work out te perimeter
More informationEC201 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 informationGeometric 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 informationStrategic trading and welfare in a dynamic market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt
More informationSection 13.5 Equations of Lines and Planes
Section 13.5 Equations of Lines and Planes Generalizing Linear Equations One of the main aspects of single variable calculus was approximating graphs of functions by lines - specifically, tangent lines.
More informationReadings this week. 1 Parametric Equations Supplement. 2 Section 10.1. 3 Sections 2.1-2.2. Professor Christopher Hoffman Math 124
Readings this week 1 Parametric Equations Supplement 2 Section 10.1 3 Sections 2.1-2.2 Precalculus Review Quiz session Thursday equations of lines and circles worksheet available at http://www.math.washington.edu/
More informationComputer Science and Engineering, UCSD October 7, 1999 Goldreic-Levin Teorem Autor: Bellare Te Goldreic-Levin Teorem 1 Te problem We æx a an integer n for te lengt of te strings involved. If a is an n-bit
More informationStrategic trading in a dynamic noisy market. Dimitri Vayanos
LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral
More informationPerimeter, Area and Volume of Regular Shapes
Perimeter, Area and Volume of Regular Sapes Perimeter of Regular Polygons Perimeter means te total lengt of all sides, or distance around te edge of a polygon. For a polygon wit straigt sides tis is te
More informationSection 2.5 Average Rate of Change
Section.5 Average Rate of Change Suppose that the revenue realized on the sale of a company s product can be modeled by the function R( x) 600x 0.3x, where x is the number of units sold and R( x ) is given
More informationComparison between two approaches to overload control in a Real Server: local or hybrid solutions?
Comparison between two approaces to overload control in a Real Server: local or ybrid solutions? S. Montagna and M. Pignolo Researc and Development Italtel S.p.A. Settimo Milanese, ITALY Abstract Tis wor
More informationWhat is Advanced Corporate Finance? What is finance? What is Corporate Finance? Deciding how to optimally manage a firm s assets and liabilities.
Wat is? Spring 2008 Note: Slides are on te web Wat is finance? Deciding ow to optimally manage a firm s assets and liabilities. Managing te costs and benefits associated wit te timing of cas in- and outflows
More informationVolumes 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 information1 The Collocation Method
CS410 Assignment 7 Due: 1/5/14 (Fri) at 6pm You must wor eiter on your own or wit one partner. You may discuss bacground issues and general solution strategies wit oters, but te solutions you submit must
More informationPre-trial Settlement with Imperfect Private Monitoring
Pre-trial Settlement wit Imperfect Private Monitoring Mostafa Beskar University of New Hampsire Jee-Hyeong Park y Seoul National University July 2011 Incomplete, Do Not Circulate Abstract We model pretrial
More informationShell and Tube Heat Exchanger
Sell and Tube Heat Excanger MECH595 Introduction to Heat Transfer Professor M. Zenouzi Prepared by: Andrew Demedeiros, Ryan Ferguson, Bradford Powers November 19, 2009 1 Abstract 2 Contents Discussion
More informationUNIFORM FLOW. Key words Uniform flow; most economical cross-section; discharge; velocity; erosion; sedimentation
Capter UNIFORM FLOW.. Introduction.. Basic equations in uniform open-cannel flow.3. Most economical cross-section.4. Cannel wit compound cross-section.5. Permissible velocity against erosion and sedimentation
More informationProjective 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 informationOn a Satellite Coverage
I. INTRODUCTION On a Satellite Coverage Problem DANNY T. CHI Kodak Berkeley Researc Yu T. su National Ciao Tbng University Te eart coverage area for a satellite in an Eart syncronous orbit wit a nonzero
More informationFor Sale By Owner Program. We can help with our for sale by owner kit that includes:
Dawn Coen Broker/Owner For Sale By Owner Program If you want to sell your ome By Owner wy not:: For Sale Dawn Coen Broker/Owner YOUR NAME YOUR PHONE # Look as professional as possible Be totally prepared
More informationCyber Epidemic Models with Dependences
Cyber Epidemic Models wit Dependences Maocao Xu 1, Gaofeng Da 2 and Souuai Xu 3 1 Department of Matematics, Illinois State University mxu2@ilstu.edu 2 Institute for Cyber Security, University of Texas
More informationFINITE 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 informationWorking 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 informationSection 1.1 Linear Equations: Slope and Equations of Lines
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
More informationChapter 10: Refrigeration Cycles
Capter 10: efrigeration Cycles Te vapor compression refrigeration cycle is a common metod for transferring eat from a low temperature to a ig temperature. Te above figure sows te objectives of refrigerators
More informationPressure. Pressure. Atmospheric pressure. Conceptual example 1: Blood pressure. Pressure is force per unit area:
Pressure Pressure is force per unit area: F P = A Pressure Te direction of te force exerted on an object by a fluid is toward te object and perpendicular to its surface. At a microscopic level, te force
More information4.4 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 informationTHE 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 informationSolutions by: KARATUĞ OZAN BiRCAN. PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set in
KOÇ UNIVERSITY, SPRING 2014 MATH 401, MIDTERM-1, MARCH 3 Instructor: BURAK OZBAGCI TIME: 75 Minutes Solutions by: KARATUĞ OZAN BiRCAN PROBLEM 1 (20 points): Let D be a region, i.e., an open connected set
More informationImproved dynamic programs for some batcing problems involving te maximum lateness criterion A P M Wagelmans Econometric Institute Erasmus University Rotterdam PO Box 1738, 3000 DR Rotterdam Te Neterlands
More informationWriting 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 informationHis solution? Federal law that requires government agencies and private industry to encrypt, or digitally scramble, sensitive data.
NET GAIN Scoring points for your financial future AS SEEN IN USA TODAY S MONEY SECTION, FEBRUARY 9, 2007 Tec experts plot to catc identity tieves Politicians to security gurus offer ideas to prevent data
More informationThe 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 informationOnce you have reviewed the bulletin, please
Akron Public Scools OFFICE OF CAREER EDUCATION BULLETIN #5 : Driver Responsibilities 1. Akron Board of Education employees assigned to drive Board-owned or leased veicles MUST BE FAMILIAR wit te Business
More informationDesign and Analysis of a Fault-Tolerant Mechanism for a Server-Less Video-On-Demand System
Design and Analysis of a Fault-olerant Mecanism for a Server-Less Video-On-Demand System Jack Y. B. Lee Department of Information Engineering e Cinese University of Hong Kong Satin, N.., Hong Kong Email:
More information