Exponential and Logarithmic Functions
|
|
|
- Ami Shields
- 9 years ago
- Views:
Transcription
1 Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define ech term or concept. Algebric functions Trnscendentl functions Nturl bse e Continuous compounding I. Eponentil Functions (Pge 28) The eponentil function f with bse is denoted b, where > 0,, nd is n rel number. How to recognize nd evlute eponentil functions with bse Emple : Use clcultor to evlute the epression / 5 5. II. Grphs of Eponentil Functions (Pges 29 22) For >, is the grph of its domin? = For >, is the grph of = incresing or decresing over incresing or decresing over How to grph eponentil functions nd use the One-to-One Propert its domin? For the grph of = or =, >, the domin is, the rnge is, nd the intercept is. Also, both grphs hve s horizontl smptote. Copright Houghton Mifflin Compn. All rights reserved. 57
2 58 Chpter Eponentil nd Logrithmic Functions Emple 2: Sketch the grph of the function f ( ) = The grph of the eponentil function psses the Test, nd therefore, the function is one-to-one function (nd, thus, hs n inverse function). Stte the One-to-One Propert for eponentil functions nd eplin how it m be used to solve simple eponentil equtions. III. The Nturl Bse e (Pge 222) The nturl eponentil function is given b the function. In this function, is the constnt nd is the vrible. How to recognize, evlute, nd grph eponentil functions with bse e / 5 Emple : Use clcultor to evlute the epression e. Copright Houghton Mifflin Compn. All rights reserved.
3 Section. Eponentil Functions nd Their Grphs 59 IV. Applictions of Eponentil Functions (Pges ) After t ers, the blnce A in n ccount with principl P nd nnul interest rte r (in deciml form) is given b the formuls: For n compoundings per er: How to use eponentil functions to model nd solve rel-life pplictions For continuous compounding: Emple 4: Find the mount in n ccount fter 0 ers if $6000 is invested t n interest rte of 7%, () compounded monthl. (b) compounded continuousl. Additionl notes Copright Houghton Mifflin Compn. All rights reserved.
4 60 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.
5 Section.2 Logrithmic Functions nd Their Grphs 6 Nme Section.2 Logrithmic Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph logrithmic functions. Importnt Vocbulr Define ech term or concept. Common logrithmic function Nturl logrithmic function I. Logrithmic Functions (Pges 229 2) For > 0, > 0, nd, where =, the logrithmic function with bse is defined s., which is red s How to recognize nd evlute logrithmic functions with bse The logrithmic function with bse is the of the eponentil function f ( ) =. The eqution = in eponentil form is equivlent to the eqution in logrithmic form. When evluting logrithms, remember tht logrithm is (n). This mens tht log is the to which must be rised to obtin. Emple : Use the definition of logrithmic function to evlute log Emple 2: Use clcultor to evlute log Copright Houghton Mifflin Compn. All rights reserved.
6 62 Chpter Eponentil nd Logrithmic Functions Complete the following properties of logrithms: ) log = 2) log = ) log = nd log = 4) If log = log, then. Emple : Solve the eqution log 7 = for. II. Grphs of Logrithmic Functions (Pges 2 22) To sketch the grph of = log, ou cn use the fct tht... How to grph logrithmic functions For >, is the grph of over its domin? = log incresing or decresing For the grph of = log, >, the domin is, the rnge is, nd the -intercept is. Also, the grph hs s verticl smptote. Emple 4: Sketch the grph of the function ( ) = log. f Copright Houghton Mifflin Compn. All rights reserved.
7 Section.2 Logrithmic Functions nd Their Grphs 6 Nme III. The Nturl Logrithmic Function (Pges 2 24) The nturl logrithm is written without bse; the bse is understood to be. How to recognize, evlute, nd grph nturl logrithmic function Complete the following properties of nturl logrithms: ) ln = 2) ln e = ) ln e = nd e ln = 4) If ln = ln, then. Emple 5: Use clcultor to evlute ln 0. Emple 6: Find the domin of the function f ( ) = ln( + ). IV. Applictions of Logrithmic Functions (Pge 25) Describe rel-life sitution in which logrithms re used. How to use logrithmic functions to model nd solve rel-life pplictions Emple 7: A principl P, invested t 6% interest nd compounded continuousl, increses to n mount K times the originl principl fter t ers, where t ln K is given b t =. How long will it tke the 0.06 originl investment to double in vlue? To triple in vlue? Copright Houghton Mifflin Compn. All rights reserved.
8 64 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.
9 Section. Properties of Logrithms 65 Nme Section. Properties of Logrithms Objective: In this lesson ou lerned how to use the chnge-of-bse formul to rewrite nd evlute logrithmic epressions nd how to use properties of logrithms to evlute, rewrite, epnd, or condense logrithmic epressions. I. Chnge of Bse (Pge 29) Let, b, nd be positive rel numbers such tht nd b. Use the Chnge-of-Bse Formul to rewrite log using bse b: log = How to use the chngeof-bse formul to rewrite nd evlute logrithmic epressions Eplin how to use clcultor to evlute log II. Properties of Logrithms (Pge 240) Let be positive number such tht ; let n be rel number; nd let u nd v be positive rel numbers. Complete the following properties of logrithms:. log ( uv) = How to use properties of logrithms to evlute or rewrite logrithmic epressions 2. u log = v n u. log = III. Rewriting Logrithmic Epressions (Pge 24) To epnd logrithmic epression mens to.... How to use properties of logrithms to epnd or condense logrithmic epressions Emple : Epnd the logrithmic epression 4 ln. 2 Copright Houghton Mifflin Compn. All rights reserved.
10 66 Chpter Eponentil nd Logrithmic Functions To condense logrithmic epression mens to.... Emple 2: Condense the logrithmic epression log + 4 log( ). IV. Applictions of Properties of Logrithms (Pge 242) One w of finding model for set of nonliner dt is to tke the nturl logrithm of ech of the -vlues nd -vlues of the dt set. If the points re grphed nd fll on stright line, then the -vlues nd the -vlues re relted b the eqution: How to use logrithmic functions to model nd solve rel-life pplictions stright line., where m is the slope of the Emple : Find nturl logrithmic eqution for the following dt tht epresses s function of Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.
11 Section.4 Eponentil nd Logrithmic Equtions 67 Nme Section.4 Eponentil nd Logrithmic Equtions Objective: In this lesson ou lerned how to solve eponentil nd logrithmic equtions. I. Introduction (Pge 246) Stte the One-to-One Propert for eponentil equtions. How to solve simple eponentil nd logrithmic equtions Stte the One-to-One Propert for logrithmic equtions. Stte the Inverse Properties for eponentil equtions nd for logrithmic equtions. Describe how the One-to-One Properties nd the Inverse Properties cn be used to solve eponentil nd logrithmic equtions. Emple : () Solve log 8 = for. (b) Solve 5 = 0.04 for. II. Solving Eponentil Equtions (Pges ) Describe how to solve the eponentil eqution 0 = 90. How to solve more complicted eponentil equtions Copright Houghton Mifflin Compn. All rights reserved.
12 68 Chpter Eponentil nd Logrithmic Functions 2 Emple 2: Solve e plces. 7 = 59 for. Round to three deciml III. Solving Logrithmic Equtions (Pges ) Describe how to solve the logrithmic eqution log6 (4 7) = log6 (8 ). How to solve more complicted logrithmic equtions Emple : Solve 4 ln 5 = 28 plces. for. Round to three deciml IV. Applictions of Eponentil nd Logrithmic Equtions (Pges ) Emple 4: Use the formul for continuous compounding, rt A = Pe, to find how long it will tke $500 to triple in vlue if it is invested t 2% interest, compounded continuousl. How to use eponentil nd logrithmic equtions to model nd solve rellife pplictions Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.
13 Section.5 Eponentil nd Logrithmic Models 69 Nme Section.5 Eponentil nd Logrithmic Models Objective: In this lesson ou lerned how to use eponentil growth models, eponentil dec models, Gussin models, logistic growth models, nd logrithmic models to solve rel-life problems. Importnt Vocbulr Define ech term or concept. Bell-shped curve Logistic curve Sigmoidl curve I. Introduction (Pge 257) The eponentil growth model is The eponentil dec model is. How to recognize the five most common tpes of models involving eponentil nd logrithmic functions The Gussin model is. The logistic growth model is Logrithmic models re nd II. Eponentil Growth nd Dec (Pges ) Emple : Suppose popultion is growing ccording to the 0.05t model P = 800e, where t is given in ers. () Wht is the initil size of the popultion? (b) How long will it tke this popultion to double? How to use eponentil growth nd dec functions to model nd solve rel-life problems Copright Houghton Mifflin Compn. All rights reserved.
14 70 Chpter Eponentil nd Logrithmic Functions To estimte the ge of ded orgnic mtter, scientists use the crbon dting model, which denotes the rtio R of crbon 4 to crbon 2 present t n time t (in ers). Emple 2: The rtio of crbon 4 to crbon 2 in fossil is R = 0 6. Find the ge of the fossil. III. Gussin Models (Pge 26) The Gussin model is commonl used in probbilit nd sttistics to represent popultions tht re. How to use Gussin functions to model nd solve rel-life problems For Gussin model, the verge vlue for popultion cn be found... Emple : Drw the bsic form of the grph of Gussin model. Copright Houghton Mifflin Compn. All rights reserved.
15 Section.5 Eponentil nd Logrithmic Models 7 Nme IV. Logistic Growth Models (Pge 262) Give n emple of rel-life sitution tht is modeled b logistic growth model. How to use logistic growth functions to model nd solve rel-life problems Emple 4: Drw the bsic form of the grph of logistic growth model. V. Logrithmic Models (Pge 26) Emple 5: The number of kitchen widgets (in millions) demnded ech er is given b the model = 2 + ln( + ), where = 0 represents the er 2000 nd 0. Find the er in which the number of kitchen widgets demnded will be 8.6 million. How to use logrithmic functions to model nd solve rel-life problems Copright Houghton Mifflin Compn. All rights reserved.
16 72 Chpter Eponentil nd Logrithmic Functions Additionl notes Homework Assignment Pge(s) Eercises Copright Houghton Mifflin Compn. All rights reserved.
Graphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
Operations with Polynomials
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
AP STATISTICS SUMMER MATH PACKET
AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.
www.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
Warm-up for Differential Calculus
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
Module Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
P.3 Polynomials and Factoring. P.3 an 1. Polynomial STUDY TIP. Example 1 Writing Polynomials in Standard Form. What you should learn
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
Unit 6: Exponents and Radicals
Eponents nd Rdicls -: The Rel Numer Sstem Unit : Eponents nd Rdicls Pure Mth 0 Notes Nturl Numers (N): - counting numers. {,,,,, } Whole Numers (W): - counting numers with 0. {0,,,,,, } Integers (I): -
AREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
Section 7-4 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
Use Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
MA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
Integration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
Binary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
Basic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
15.6. The mean value and the root-mean-square value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the root-men-squre vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
Polynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator
AP Clculus Finl Review Sheet When you see the words. This is wht you think of doing. Find the zeros Find roots. Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor.
Multiplication and Division - Left to Right. Addition and Subtraction - Left to Right.
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
Basically, logarithmic transformations ask, a number, to what power equals another number?
Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In
LECTURE #05. Learning Objective. To describe the geometry in and around a unit cell in terms of directions and planes.
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.
PROBLEMS 13 - APPLICATIONS OF DERIVATIVES Page 1
PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
Section 5-4 Trigonometric Functions
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 1470 - COLLEGE ALGEBRA (4 SEMESTER HOURS)
SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO DEPARTMENT SYLLABUS FOR COURSE IN MAT 470 - COLLEGE ALGEBRA (4 SEMESTER HOURS). COURSE DESCRIPTION: Polynomil, rdicl, rtionl, exponentil, nd logrithmic functions
Example A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
Answers (Anticipation Guide and Lesson 7-1)
Answers (Anticiption Guide nd Lesson 7-) NAME DATE PERID 7 Anticiption Guide Rdicl Equtions STEP Chpter 7 Glencoe Algebr Answers Chpter Resources Before ou begin Chpter 7 Red ech sttement. Decide whether
Reasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
Example 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
Distributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
Algebra Review. How well do you remember your algebra?
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
Mathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
Factoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
Bayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
Review Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
10.6 Applications of Quadratic Equations
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
Experiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
Numerical Methods of Approximating Definite Integrals
6 C H A P T E R Numericl Methods o Approimting Deinite Integrls 6. APPROXIMATING SUMS: L n, R n, T n, AND M n Introduction Not only cn we dierentite ll the bsic unctions we ve encountered, polynomils,
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
Or more simply put, when adding or subtracting quantities, their uncertainties add.
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
Introduction to Integration Part 2: The Definite Integral
Mthemtics Lerning Centre Introduction to Integrtion Prt : The Definite Integrl Mr Brnes c 999 Universit of Sdne Contents Introduction. Objectives...... Finding Ares 3 Ares Under Curves 4 3. Wht is the
COMPLEX FRACTIONS. section. Simplifying Complex Fractions
58 (6-6) Chpter 6 Rtionl Epressions undles tht they cn ttch while working together for 0 hours. 00 600 6 FIGURE FOR EXERCISE 9 95. Selling. George sells one gzine suscription every 0 inutes, wheres Theres
5.6 POSITIVE INTEGRAL EXPONENTS
54 (5 ) Chpter 5 Polynoils nd Eponents 5.6 POSITIVE INTEGRAL EXPONENTS In this section The product rule for positive integrl eponents ws presented in Section 5., nd the quotient rule ws presented in Section
Review guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS
RIGHT TRIANGLES AND THE PYTHAGOREAN TRIPLETS Known for over 500 yers is the fct tht the sum of the squres of the legs of right tringle equls the squre of the hypotenuse. Tht is +b c. A simple proof is
Introductory Mathematical Analysis
Introductory Mthemticl Anlysis For Business, Economics, nd the Life nd Socil Sciences Arb World Edition Ernest F. Heussler, Jr. The Pennsylvni Stte University Richrd S. Pul The Pennsylvni Stte University
4. DC MOTORS. Understand the basic principles of operation of a DC motor. Understand the operation and basic characteristics of simple DC motors.
4. DC MOTORS Almost every mechnicl movement tht we see round us is ccomplished by n electric motor. Electric mchines re mens o converting energy. Motors tke electricl energy nd produce mechnicl energy.
Lecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
Helicopter Theme and Variations
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
Unit 29: Inference for Two-Way Tables
Unit 29: Inference for Two-Wy Tbles Prerequisites Unit 13, Two-Wy Tbles is prerequisite for this unit. In ddition, students need some bckground in significnce tests, which ws introduced in Unit 25. Additionl
PHY 222 Lab 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
Econ 4721 Money and Banking Problem Set 2 Answer Key
Econ 472 Money nd Bnking Problem Set 2 Answer Key Problem (35 points) Consider n overlpping genertions model in which consumers live for two periods. The number of people born in ech genertion grows in
Applications to Physics and Engineering
Section 7.5 Applictions to Physics nd Engineering Applictions to Physics nd Engineering Work The term work is used in everydy lnguge to men the totl mount of effort required to perform tsk. In physics
NQF Level: 2 US No: 7480
NQF Level: 2 US No: 7480 Assessment Guide Primry Agriculture Rtionl nd irrtionl numers nd numer systems Assessor:.......................................... Workplce / Compny:.................................
Vectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
B Conic Sections. B.1 Conic Sections. Introduction to Conic Sections. Appendix B.1 Conic Sections B1
Appendi B. Conic Sections B B Conic Sections B. Conic Sections Recognize the four bsic conics: circles, prbols, ellipses, nd hperbols. Recognize, grph, nd write equtions of prbols (verte t origin). Recognize,
Physics 43 Homework Set 9 Chapter 40 Key
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
3 The Utility Maximization Problem
3 The Utility Mxiiztion Proble We hve now discussed how to describe preferences in ters of utility functions nd how to forulte siple budget sets. The rtionl choice ssuption, tht consuers pick the best
Derivatives 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
Integration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
Cypress Creek High School IB Physics SL/AP Physics B 2012 2013 MP2 Test 1 Newton s Laws. Name: SOLUTIONS Date: Period:
Nme: SOLUTIONS Dte: Period: Directions: Solve ny 5 problems. You my ttempt dditionl problems for extr credit. 1. Two blocks re sliding to the right cross horizontl surfce, s the drwing shows. In Cse A
The Velocity Factor of an Insulated Two-Wire Transmission Line
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS
CHAPTER ALGEBRAIC FRACTIONS,AND EQUATIONS AND INEQUALITIES INVOLVING FRACTIONS Although people tody re mking greter use of deciml frctions s they work with clcultors, computers, nd the metric system, common
Math 314, Homework Assignment 1. 1. Prove that two nonvertical lines are perpendicular if and only if the product of their slopes is 1.
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
and thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 249-25 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
Pure C4. Revision Notes
Pure C4 Revision Notes Mrch 0 Contents Core 4 Alger Prtil frctions Coordinte Geometry 5 Prmetric equtions 5 Conversion from prmetric to Crtesin form 6 Are under curve given prmetriclly 7 Sequences nd
LESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
Answer, Key Homework 10 David McIntyre 1
Answer, Key Homework 10 Dvid McIntyre 1 This print-out should hve 22 questions, check tht it is complete. Multiple-choice questions my continue on the next column or pge: find ll choices efore mking your
Firm Objectives. The Theory of the Firm II. Cost Minimization Mathematical Approach. First order conditions. Cost Minimization Graphical Approach
Pro. Jy Bhttchry Spring 200 The Theory o the Firm II st lecture we covered: production unctions Tody: Cost minimiztion Firm s supply under cost minimiztion Short vs. long run cost curves Firm Ojectives
v T R x m Version PREVIEW Practice 7 carroll (11108) 1
Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
Thinking out of the Box... Problem It s a richer problem than we ever imagined
From the Mthemtics Techer, Vol. 95, No. 8, pges 568-574 Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd
Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.
The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only
Radius of the Earth - Radii Used in Geodesy James R. Clynch February 2006
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
Higher. Exponentials and Logarithms 160
hsn uknt Highr Mthmtics UNIT UTCME Eponntils nd Logrithms Contnts Eponntils nd Logrithms 6 Eponntils 6 Logrithms 6 Lws of Logrithms 6 Eponntils nd Logrithms to th Bs 65 5 Eponntil nd Logrithmic Equtions
9.3. The Scalar Product. Introduction. Prerequisites. Learning Outcomes
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
COMPONENTS: COMBINED LOADING
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
All pay auctions with certain and uncertain prizes a comment
CENTER FOR RESEARC IN ECONOMICS AND MANAGEMENT CREAM Publiction No. 1-2015 All py uctions with certin nd uncertin prizes comment Christin Riis All py uctions with certin nd uncertin prizes comment Christin
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL - INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
Exam 1 Study Guide. Differentiation and Anti-differentiation Rules from Calculus I
Exm Stuy Guie Mth 2020 - Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the
CS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
The Definite Integral
Chpter 4 The Definite Integrl 4. Determining distnce trveled from velocity Motivting Questions In this section, we strive to understnd the ides generted by the following importnt questions: If we know
Version 001 Summer Review #03 tubman (IBII20142015) 1
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
trademark and symbol guidelines FOR CORPORATE STATIONARY APPLICATIONS reviewed 01.02.2007
trdemrk nd symbol guidelines trdemrk guidelines The trdemrk Cn be plced in either of the two usul configurtions but horizontl usge is preferble. Wherever possible the trdemrk should be plced on blck bckground.
2005-06 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 25-6 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
National greenhouse gas inventory data for the period 1990 2010
United Ntions FCCC/SBI/2012/31 Distr.: Generl 16 November 2012 Originl: English Subsidiry Body for Implementtion Thirty-seventh session Doh, 26 November to 1 December 2012 Item 3(b) of the provisionl gend
PHY 140A: Solid State Physics. Solution to Homework #2
PHY 140A: Solid Stte Physics Solution to Homework # TA: Xun Ji 1 October 14, 006 1 Emil: [email protected] Problem #1 Prove tht the reciprocl lttice for the reciprocl lttice is the originl lttice.
Vendor Rating for Service Desk Selection
Vendor Presented By DATE Using the scores of 0, 1, 2, or 3, plese rte the vendor's presenttion on how well they demonstrted the functionl requirements in the res below. Also consider how efficient nd functionl
Health insurance exchanges What to expect in 2014
Helth insurnce exchnges Wht to expect in 2014 33096CAEENABC 02/13 The bsics of exchnges As prt of the Affordble Cre Act (ACA or helth cre reform lw), strting in 2014 ALL Americns must hve minimum mount
Homework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
Scalar and Vector Quantities. A scalar is a quantity having only magnitude (and possibly phase). LECTURE 2a: VECTOR ANALYSIS Vector Algebra
Sclr nd Vector Quntities : VECTO NLYSIS Vector lgebr sclr is quntit hving onl mgnitude (nd possibl phse). Emples: voltge, current, chrge, energ, temperture vector is quntit hving direction in ddition to