Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Motion. Acceleration. Part 2: Constant Acceleration. October Lab Phyiscs. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration."

Transcription

1 Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. = - o t = Δ Δt ccelertion = = - o t chnge of elocity elpsed time ccelertion is ector, lthough in one-dimensionl motion we only need the sign. Since only constnt ccelertion will be considered in this course, there is no need to differentite between erge nd instntneous ccelertion. Units for ccelertion ccelertion is the rte of chnge of. Units for ccelertion You cn derie the units by substituting the correct units into the right hnd side of these equtions. = Δ Δt m/s s = m/s C D displcement distnce speed elocity Ms. Leine

2 The unit for elocity is: 3 The metric unit for ccelertion is: m m m/s m/s C m/s D ft/s C m/s D ft/s 4 horse gllops with constnt ccelertion of 3m/s. Which sttement below is true? 5 Your elocity chnges from 60 m/s to the right to 00 m/s to the right in 0 s; wht is your erge ccelertion? C D The horse's elocity doesn't chnge. The horse moes 3m eery second. The horse's elocity increses 3m eery second. The horse's elocity increses 3m/s eery second. 6 Your elocity chnges from 60 m/s to the right to 0 m/s to the right in 0 s; wht is your erge ccelertion? 7 Your elocity chnges from 50 m/s to the left to 0 m/s to the right in 5 s; wht is your erge ccelertion? Ms. Leine

3 8 Your elocity chnges from 90 m/s to the right to 0 m/s to the right in 5.0 s; wht is your erge ccelertion? Kinemtics Eqution Return to Tble of Contents Motion t Constnt ccelertion = Δ Δt but since "Δ" mens chnge 9 Strting from rest, you ccelerte t 4.0 m/s for 6.0s. Wht is your finl elocity? Δ = - o nd = - o t t = - o - o = t Δt = t - to if we lwys let to = 0, Δt = t Soling for "" = o + t This eqution tells us how n object's elocity chnges s function of time. 0 Strting from rest, you ccelerte t 8.0 m/s for 9.0s. Wht is your finl elocity? You he n initil elocity of 5.0 m/s. You then eperience n ccelertion of -.5 m/s for 4.0s; wht is your finl elocity? Ms. Leine 3

4 You he n initil elocity of -3.0 m/s. You then eperience n ccelertion of.5 m/s for 9.0s; wht is your finl elocity? 3 How much time does it tke to ccelerte from n initil elocity of 0m/s to finl elocity of 00m/s if your ccelertion is.5 m/s? 4 How much time does it tke to come to rest if your initil elocity is 5.0 m/s nd your ccelertion is -.0 m/s? 5 n object ccelertes t rte of 3 m/s for 6 s until it reches elocity of 0 m/s. Wht ws its initil elocity? 6 n object ccelertes t rte of.5 m/s for 4 s until it reches elocity of 0 m/s. Wht ws its initil elocity? Grphing Motion t Constnt ccelertion In physics there is nother pproch in ddition to lgebric which is clled grphicl nlysis. The formul = 0 + t cn be interpreted by the grph. We just need to recll our memory from mth clsses where we lredy sw similr formul y = m + b. From these two formuls we cn some nlogies: y (dependent rible of ), 0 b (intersection with erticl is), t (independent rible), m ( slope of the grph- the rtio between rise nd run Δy/Δ). Ms. Leine 4

5 Motion t Constnt ccelertion elow we cn find the geometric eplntion to the ccelertion =Δ/Δt. If slope is equl to: m = Δy/Δ Motion t Constnt ccelertion The grph on the right hs slope of Δ/Δt, which is equl to ccelertion. Therefore, the slope of elocity s. time grph is equl to ccelertion. Then consider grph with elocity on the y-is nd time on the -is. Wht is the slope for the grph on the right? (slope) y =Δy/Δ (slope of elocity s. time) =Δ/Δt 7 The elocity s function of time is presented by the grph. Wht is the ccelertion? 8 The elocity s function of time is presented by the grph. Find the ccelertion. Motion t Constnt ccelertion The ccelertion grph s function of time cn be used to find the elocity of moing object. When the ccelertion is constnt the elocity is chnging by the sme mount ech second. This cn be shown on the grph s stright horizontl line. In order to find the chnge in elocity for certin limit of time we need to clculte the re under the ccelertion line tht is limited by the time interl. 9 The following grph shows ccelertion s function of time of moing object. Wht is the chnge in elocity during first 0 seconds? Ms. Leine 5

6 Free Fll Free Fll: ccelertion Due to Grity ll unsupported objects fll towrds Erth with the sme ccelertion. We cll this ccelertion the "ccelertion due to grity" nd it is denoted by g. g = 9.8 m/s Keep in mind, LL objects ccelerte towrds the erth t the sme rte. g is constnt! Return to Tble of Contents Click here to wtch Glileo's fmous eperiment performed on the moon It Wht stops hppens momentrily. t the top? = 0 It stops momentrily. = 0 It slows down. Wht hppens when it (negtie ccelertion) goes up? It Wht speeds hppens up when it (negtie goes down? ccelertion) It slows down. (negtie ccelertion) It speeds up. (negtie ccelertion) n object is thrown upwrd with initil elocity, o It returns Wht hppens with its when it originl lnds? elocity. n object is thrown upwrd with initil elocity, o It returns with its originl elocity. On the wy up: 0 t = 3 s t = s t = s t = 0 s On the wy down: 0 t = 0 s t = s t = s t = 3 s For ny object thrown stright up into the ir, this is wht the elocity s. time grph looks like. (m/s) n object is thrown upwrd with initil elocity, o It stops momentrily. = 0 It returns with its originl elocity but in the opposite direction. Ms. Leine 6

7 Lb Phyiscs elocity is constnt C elocity is decresing D ccelertion is decresing rock flls off cliff nd hits the ground 5 seconds lter. Wht elocity did it hit the ground with? 3 bll is thrown down off bridge with elocity of 5 m/s. Wht is its elocity seconds lter? 5 rocket is fired stright up from the ground. It returns to the ground 0 seconds lter. Wht ws its lunch speed? ccelertion is constnt n rrow is fired into the ir nd it reches its highest point 3 seconds lter. Wht ws its elocity when it ws fired? 4 Ms. Leine n corn flls from n ok tree. You note tht it tkes.5 seconds to hit the ground. How fst ws it going when it hit the ground? bll is dropped from rest nd flls (do not consider ir resistnce). Which is true bout its motion? 0 7

8 6 n object ccelertes from rest, with constnt ccelertion of 8.4 m/s, wht will its elocity be fter s? Grphing Return to Tble of Contents Velocity s. Time Grphs Similrly, the sme pproch cn be used to crete elocity s. time grph. elocity ersus time grph differs by hing the elocity on the erticl is. (m/s) Strting t the position, 0 = 4 m, you trel t constnt elocity of + m/s for 6s. b. Drw the Position ersus Time for your trel during this time. Drg nd drop the dt points on the grph in order to construct the s t pttern! elocity ersus time grph shows describes n objects elocity, it's displcement, nd ccelertion Drw line of best fit to obsere the pttern. Strting t the position, 0 = 4 m, you trel t constnt elocity of + m/s for 6s. c. Drw the Velocity ersus Time grph for your trip. Drg nd drop the dt points on the grph in order to construct the s t pttern! (m/s) Strting t the position, 0 = 0 m, you trel t constnt elocity of -m/s for 6s. b. Drw the Position ersus Time for your trel during this time. Drg nd drop the dt points on the grph in order to construct the s t pttern! Drw line of best fit to obsere the pttern. Drw line of best fit to obsere the pttern Ms. Leine 8

9 nlyzing Position s Time Grphs nlyzing Position s Time Grphs Recll erlier in this unit tht slope ws used to describe motion. positie slope is positie elocity, negtie slope is negtie elocity, nd slope of zero mens zero elocity. The slope in position s. time grph is Δ/Δt, which is equl to elocity. Δt Δ = Δ/Δt positie slope > 0 negtie slope < 0 zero slope = 0 Therefore, slope is equl to elocity on position s. time grph. positie elocity mens moing in the positie direction, negtie elocity mens moing in the negtie direction, nd zero elocity mens not moing t ll. The position ersus time grph, below, describes the motion of three different crs moing long the -is. b. Clculte the elocity of ech of the crs. Position The elocity s time grph, below, describes the motion of n object moing long the -is. 4 (m/s) Describe in words wht is hppening to the speed during the following interls. ) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s 73 The elocity s time grph, below, describes the motion of n object moing long the -is. 4 (m/s) Determine the erge speed during the following interls. ) 0s to s b) s to 3s c) 3s to 4 sec d) 4s to 5s e) 5s to 6s f) 4s to 6s Summry Kinemtics is the description of how objects moe with respect to defined reference frme. Displcement is the chnge in position of n object. erge elocity is the displcement diided by the time. erge ccelertion is the chnge in elocity diided by the time. Ms. Leine 9

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration.

Motion. Part 2: Constant Acceleration. Acceleration. October Lab Physics. Ms. Levine 1. Acceleration. Acceleration. Units for Acceleration. Moion Accelerion Pr : Consn Accelerion Accelerion Accelerion Accelerion is he re of chnge of velociy. = v - vo = Δv Δ ccelerion = = v - vo chnge of velociy elpsed ime Accelerion is vecor, lhough in one-dimensionl

More information

PHYSICS 211 MIDTERM I 21 April 2004

PHYSICS 211 MIDTERM I 21 April 2004 PHYSICS MIDERM I April 004 Exm is closed book, closed notes. Use only your formul sheet. Write ll work nd nswers in exm booklets. he bcks of pges will not be grded unless you so request on the front of

More information

Introduction to Mechanics Practice using the Kinematics Equations

Introduction to Mechanics Practice using the Kinematics Equations Introduction to Mechnics Prctice using the Kinemtics Equtions Ln Sheridn De Anz College Jn 24, 2018 Lst time finished deriing the kinemtics equtions some problem soling prctice Oeriew using kinemtics equtions

More information

Distance And Velocity

Distance And Velocity Unit #8 - The Integrl Some problems nd solutions selected or dpted from Hughes-Hllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl

More information

DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION

DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION DESCRIBING MOTION: KINEMATICS IN ONE DIMENSION Responses to Questions. A cr speedometer mesures only speed. It does not give ny informtion bout the direction, so it does not mesure velocity.. If the velocity

More information

20 MATHEMATICS POLYNOMIALS

20 MATHEMATICS POLYNOMIALS 0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of

More information

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.

MA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp. MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

More information

Section 14.3 Arc Length and Curvature

Section 14.3 Arc Length and Curvature Section 4.3 Arc Length nd Curvture Clculus on Curves in Spce In this section, we ly the foundtions for describing the movement of n object in spce.. Vector Function Bsics In Clc, formul for rc length in

More information

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40

Section 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40 Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since

More information

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1

Chapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1 Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the -coordinte of ech criticl vlue of g. Show

More information

Physics 207 Lecture 7

Physics 207 Lecture 7 Phsics 07 Lecture 7 Agend: Phsics 07, Lecture 7, Sept. 6 hpter 6: Motion in (nd 3) dimensions, Dnmics II Recll instntneous velocit nd ccelertion hpter 6 (Dnmics II) Motion in two (or three dimensions)

More information

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes

Test , 8.2, 8.4 (density only), 8.5 (work only), 9.1, 9.2 and 9.3 related test 1 material and material from prior classes Test 2 8., 8.2, 8.4 (density only), 8.5 (work only), 9., 9.2 nd 9.3 relted test mteril nd mteril from prior clsses Locl to Globl Perspectives Anlyze smll pieces to understnd the big picture. Exmples: numericl

More information

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

Chapter 4 Contravariance, Covariance, and Spacetime Diagrams Chpter 4 Contrvrince, Covrince, nd Spcetime Digrms 4. The Components of Vector in Skewed Coordintes We hve seen in Chpter 3; figure 3.9, tht in order to show inertil motion tht is consistent with the Lorentz

More information

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS

BIFURCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS BIFRCATIONS IN ONE-DIMENSIONAL DISCRETE SYSTEMS FRANCESCA AICARDI In this lesson we will study the simplest dynmicl systems. We will see, however, tht even in this cse the scenrio of different possible

More information

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81

FORM FIVE ADDITIONAL MATHEMATIC NOTE. ar 3 = (1) ar 5 = = (2) (2) (1) a = T 8 = 81 FORM FIVE ADDITIONAL MATHEMATIC NOTE CHAPTER : PROGRESSION Arithmetic Progression T n = + (n ) d S n = n [ + (n )d] = n [ + Tn ] S = T = T = S S Emple : The th term of n A.P. is 86 nd the sum of the first

More information

Big idea in Calculus: approximation

Big idea in Calculus: approximation Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:

More information

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:

Improper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows: Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 2014 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 5 NOVEMBER 014 Mrk Scheme: Ech prt of Question 1 is worth four mrks which re wrded solely for the correct nswer.

More information

Physics Honors. Final Exam Review Free Response Problems

Physics Honors. Final Exam Review Free Response Problems Physics Honors inl Exm Review ree Response Problems m t m h 1. A 40 kg mss is pulled cross frictionless tble by string which goes over the pulley nd is connected to 20 kg mss.. Drw free body digrm, indicting

More information

Calculus - Activity 1 Rate of change of a function at a point.

Calculus - Activity 1 Rate of change of a function at a point. Nme: Clss: p 77 Mths Helper Plus Resource Set. Copright 00 Bruce A. Vughn, Techers Choice Softwre Clculus - Activit Rte of chnge of function t point. ) Strt Mths Helper Plus, then lod the file: Clculus

More information

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs

Pre-Session Review. Part 1: Basic Algebra; Linear Functions and Graphs Pre-Session Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:

More information

Chapter 6 Continuous Random Variables and Distributions

Chapter 6 Continuous Random Variables and Distributions Chpter 6 Continuous Rndom Vriles nd Distriutions Mny economic nd usiness mesures such s sles investment consumption nd cost cn hve the continuous numericl vlues so tht they cn not e represented y discrete

More information

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs

MATHS NOTES. SUBJECT: Maths LEVEL: Higher TEACHER: Aidan Roantree. The Institute of Education Topics Covered: Powers and Logs MATHS NOTES The Institute of Eduction 06 SUBJECT: Mths LEVEL: Higher TEACHER: Aidn Rontree Topics Covered: Powers nd Logs About Aidn: Aidn is our senior Mths techer t the Institute, where he hs been teching

More information

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below

approaches as n becomes larger and larger. Since e > 1, the graph of the natural exponential function is as below . Eponentil nd rithmic functions.1 Eponentil Functions A function of the form f() =, > 0, 1 is clled n eponentil function. Its domin is the set of ll rel f ( 1) numbers. For n eponentil function f we hve.

More information

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.

PART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point. PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic

More information

MEE 214 (Dynamics) Tuesday Dr. Soratos Tantideeravit (สรทศ ต นต ธ รว ทย )

MEE 214 (Dynamics) Tuesday Dr. Soratos Tantideeravit (สรทศ ต นต ธ รว ทย ) MEE 14 (Dynmics) Tuesdy 8.30-11.0 Dr. Sortos Tntideerit (สรทศ ต นต ธ รว ทย ) sortos@oep.go.th Lecture Notes, Course updtes, Extr problems, etc No Homework Finl Exm (Dte & Time TBD) 1/03/58 MEE14 Dynmics

More information

5.1 Estimating with Finite Sums Calculus

5.1 Estimating with Finite Sums Calculus 5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during

More information

The Wave Equation I. MA 436 Kurt Bryan

The Wave Equation I. MA 436 Kurt Bryan 1 Introduction The Wve Eqution I MA 436 Kurt Bryn Consider string stretching long the x xis, of indeterminte (or even infinite!) length. We wnt to derive n eqution which models the motion of the string

More information

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60.

The final exam will take place on Friday May 11th from 8am 11am in Evans room 60. Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23

More information

pivot F 2 F 3 F 1 AP Physics 1 Practice Exam #3 (2/11/16)

pivot F 2 F 3 F 1 AP Physics 1 Practice Exam #3 (2/11/16) AP Physics 1 Prctice Exm #3 (/11/16) Directions: Ech questions or incomplete sttements below is followed by four suggested nswers or completions. Select one tht is best in ech cse nd n enter pproprite

More information

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1

Section 4.8. D v(t j 1 ) t. (4.8.1) j=1 Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions

More information

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0.

STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA. 0 if t < 0, 1 if t > 0. STEP FUNCTIONS, DELTA FUNCTIONS, AND THE VARIATION OF PARAMETERS FORMULA STEPHEN SCHECTER. The unit step function nd piecewise continuous functions The Heviside unit step function u(t) is given by if t

More information

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230

Polynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230 Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

More information

Problem set 2 The Ricardian Model

Problem set 2 The Ricardian Model Problem set 2 The Ricrdin Model Eercise 1 Consider world with two countries, U nd V, nd two goods, nd F. It is known tht the mount of work vilble in U is 150 nd in V is 84. The unit lbor requirements for

More information

5: The Definite Integral

5: The Definite Integral 5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce

More information

Math 8 Winter 2015 Applications of Integration

Math 8 Winter 2015 Applications of Integration Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl

More information

Linear Systems with Constant Coefficients

Linear Systems with Constant Coefficients Liner Systems with Constnt Coefficients 4-3-05 Here is system of n differentil equtions in n unknowns: x x + + n x n, x x + + n x n, x n n x + + nn x n This is constnt coefficient liner homogeneous system

More information

Partial Derivatives. Limits. For a single variable function f (x), the limit lim

Partial Derivatives. Limits. For a single variable function f (x), the limit lim Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles

More information

Topic 1 Notes Jeremy Orloff

Topic 1 Notes Jeremy Orloff Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble

More information

Precalculus Spring 2017

Precalculus Spring 2017 Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify

More information

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y

FUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x- x1)(x- x2) a y FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = (- 1)(- ) The hperbol: p q The eponentil function: b p q Importnt fetures: -intercept : Let = 0 -intercept : Let = 0 Turning points (Where pplicble)

More information

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

STRAND J: TRANSFORMATIONS, VECTORS and MATRICES Mthemtics SKE: STRN J STRN J: TRNSFORMTIONS, VETORS nd MTRIES J3 Vectors Text ontents Section J3.1 Vectors nd Sclrs * J3. Vectors nd Geometry Mthemtics SKE: STRN J J3 Vectors J3.1 Vectors nd Sclrs Vectors

More information

MATH 115: Review for Chapter 7

MATH 115: Review for Chapter 7 MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol

More information

Version 001 Review 1: Mechanics tubman (IBII ) During each of the three intervals correct

Version 001 Review 1: Mechanics tubman (IBII ) During each of the three intervals correct Version 001 Review 1: Mechnics tubmn (IBII20142015) 1 This print-out should hve 72 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Displcement

More information

4.1. Probability Density Functions

4.1. Probability Density Functions STT 1 4.1-4. 4.1. Proility Density Functions Ojectives. Continuous rndom vrile - vers - discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of

More information

Math 100 Review Sheet

Math 100 Review Sheet Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s

More information

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics SCHOOL OF ENGINEERING & BUIL ENVIRONMEN Mthemtics An Introduction to Mtrices Definition of Mtri Size of Mtri Rows nd Columns of Mtri Mtri Addition Sclr Multipliction of Mtri Mtri Multipliction 7 rnspose

More information

ESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability

ESCI 343 Atmospheric Dynamics II Lesson 14 Inertial/slantwise Instability ESCI 343 Atmospheric Dynmics II Lesson 14 Inertil/slntwise Instbility Reference: An Introduction to Dynmic Meteorology (3 rd edition), J.R. Holton Atmosphere-Ocen Dynmics, A.E. Gill Mesoscle Meteorology

More information

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω.

dx dt dy = G(t, x, y), dt where the functions are defined on I Ω, and are locally Lipschitz w.r.t. variable (x, y) Ω. Chpter 8 Stility theory We discuss properties of solutions of first order two dimensionl system, nd stility theory for specil clss of liner systems. We denote the independent vrile y t in plce of x, nd

More information

Version 001 HW#6 - Electromagnetism arts (00224) 1

Version 001 HW#6 - Electromagnetism arts (00224) 1 Version 001 HW#6 - Electromgnetism rts (00224) 1 This print-out should hve 11 questions. Multiple-choice questions my continue on the next column or pge find ll choices efore nswering. rightest Light ul

More information

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique?

How do we solve these things, especially when they get complicated? How do we know when a system has a solution, and when is it unique? XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

More information

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1

63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1 3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =

More information

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.

A. Limits - L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C. A. Limits - L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c

More information

Lecture 1. Functional series. Pointwise and uniform convergence.

Lecture 1. Functional series. Pointwise and uniform convergence. 1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is

More information

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t

More information

38 Riemann sums and existence of the definite integral.

38 Riemann sums and existence of the definite integral. 38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the x-xis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These

More information

PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3.

PROBLEM deceleration of the cable attached at B is 2.5 m/s, while that + ] ( )( ) = 2.5 2α. a = rad/s. a 3.25 m/s. = 3. PROLEM 15.105 A 5-m steel bem is lowered by mens of two cbles unwinding t the sme speed from overhed crnes. As the bem pproches the ground, the crne opertors pply brkes to slow the unwinding motion. At

More information

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings

16 Newton s Laws #3: Components, Friction, Ramps, Pulleys, and Strings Chpter 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings 16 Newton s Lws #3: Components, riction, Rmps, Pulleys, nd Strings When, in the cse of tilted coordinte system, you brek up the

More information

Phys 7221, Fall 2006: Homework # 6

Phys 7221, Fall 2006: Homework # 6 Phys 7221, Fll 2006: Homework # 6 Gbriel González October 29, 2006 Problem 3-7 In the lbortory system, the scttering ngle of the incident prticle is ϑ, nd tht of the initilly sttionry trget prticle, which

More information

As we have already discussed, all the objects have the same absolute value of

As we have already discussed, all the objects have the same absolute value of Lecture 3 Prjectile Mtin Lst time we were tlkin but tw-dimensinl mtin nd intrduced ll imprtnt chrcteristics f this mtin, such s psitin, displcement, elcit nd ccelertin Nw let us see hw ll these thins re

More information

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph.

u( t) + K 2 ( ) = 1 t > 0 Analyzing Damped Oscillations Problem (Meador, example 2-18, pp 44-48): Determine the equation of the following graph. nlyzing Dmped Oscilltions Prolem (Medor, exmple 2-18, pp 44-48): Determine the eqution of the following grph. The eqution is ssumed to e of the following form f ( t) = K 1 u( t) + K 2 e!"t sin (#t + $

More information

interatomic distance

interatomic distance Dissocition energy of Iodine molecule using constnt devition spectrometer Tbish Qureshi September 2003 Aim: To verify the Hrtmnn Dispersion Formul nd to determine the dissocition energy of I 2 molecule

More information

BRIEF NOTES ADDITIONAL MATHEMATICS FORM

BRIEF NOTES ADDITIONAL MATHEMATICS FORM BRIEF NOTES ADDITIONAL MATHEMATICS FORM CHAPTER : FUNCTION. : + is the object, + is the imge : + cn be written s () = +. To ind the imge or mens () = + = Imge or is. Find the object or 8 mens () = 8 wht

More information

5.2 Volumes: Disks and Washers

5.2 Volumes: Disks and Washers 4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of cross-section or slice. In this section, we restrict

More information

Lecture 1: Electrostatic Fields

Lecture 1: Electrostatic Fields Lecture 1: Electrosttic Fields Instructor: Dr. Vhid Nyyeri Contct: nyyeri@iust.c.ir Clss web site: http://webpges.iust.c. ir/nyyeri/courses/bee 1.1. Coulomb s Lw Something known from the ncient time (here

More information

Math 360: A primitive integral and elementary functions

Math 360: A primitive integral and elementary functions Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:

More information

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS 33 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS As simple ppliction of the results we hve obtined on lgebric extensions, nd in prticulr on the multiplictivity of extension degrees, we cn nswer (in

More information

ONLINE PAGE PROOFS. Anti-differentiation and introduction to integral calculus

ONLINE PAGE PROOFS. Anti-differentiation and introduction to integral calculus Anti-differentition nd introduction to integrl clculus. Kick off with CAS. Anti-derivtives. Anti-derivtive functions nd grphs. Applictions of nti-differentition.5 The definite integrl.6 Review . Kick off

More information

UCSD Phys 4A Intro Mechanics Winter 2016 Ch 4 Solutions

UCSD Phys 4A Intro Mechanics Winter 2016 Ch 4 Solutions USD Phys 4 Intro Mechnics Winter 06 h 4 Solutions 0. () he 0.0 k box restin on the tble hs the free-body dir shown. Its weiht 0.0 k 9.80 s 96 N. Since the box is t rest, the net force on is the box ust

More information

Numerical integration

Numerical integration 2 Numericl integrtion This is pge i Printer: Opque this 2. Introduction Numericl integrtion is problem tht is prt of mny problems in the economics nd econometrics literture. The orgniztion of this chpter

More information

Line Integrals. Partitioning the Curve. Estimating the Mass

Line Integrals. Partitioning the Curve. Estimating the Mass Line Integrls Suppose we hve curve in the xy plne nd ssocite density δ(p ) = δ(x, y) t ech point on the curve. urves, of course, do not hve density or mss, but it my sometimes be convenient or useful to

More information

Math 113 Exam 1-Review

Math 113 Exam 1-Review Mth 113 Exm 1-Review September 26, 2016 Exm 1 covers 6.1-7.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between

More information

Math 4200: Homework Problems

Math 4200: Homework Problems Mth 4200: Homework Problems Gregor Kovčič 1. Prove the following properties of the binomil coefficients ( n ) ( n ) (i) 1 + + + + 1 2 ( n ) (ii) 1 ( n ) ( n ) + 2 + 3 + + n 2 3 ( ) n ( n + = 2 n 1 n) n,

More information

Arithmetic & Algebra. NCTM National Conference, 2017

Arithmetic & Algebra. NCTM National Conference, 2017 NCTM Ntionl Conference, 2017 Arithmetic & Algebr Hether Dlls, UCLA Mthemtics & The Curtis Center Roger Howe, Yle Mthemtics & Texs A & M School of Eduction Relted Common Core Stndrds First instnce of vrible

More information

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011

Physics 9 Fall 2011 Homework 2 - Solutions Friday September 2, 2011 Physics 9 Fll 0 Homework - s Fridy September, 0 Mke sure your nme is on your homework, nd plese box your finl nswer. Becuse we will be giving prtil credit, be sure to ttempt ll the problems, even if you

More information

x ) dx dx x sec x over the interval (, ).

x ) dx dx x sec x over the interval (, ). Curve on 6 For -, () Evlute the integrl, n (b) check your nswer by ifferentiting. ( ). ( ). ( ).. 6. sin cos 7. sec csccot 8. sec (sec tn ) 9. sin csc. Evlute the integrl sin by multiplying the numertor

More information

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus

Unit #10 De+inite Integration & The Fundamental Theorem Of Calculus Unit # De+inite Integrtion & The Fundmentl Theorem Of Clculus. Find the re of the shded region ove nd explin the mening of your nswer. (squres re y units) ) The grph to the right is f(x) = -x + 8x )Use

More information

3.1 Review of Sine, Cosine and Tangent for Right Angles

3.1 Review of Sine, Cosine and Tangent for Right Angles Foundtions of Mth 11 Section 3.1 Review of Sine, osine nd Tngent for Right Tringles 125 3.1 Review of Sine, osine nd Tngent for Right ngles The word trigonometry is derived from the Greek words trigon,

More information

MATH STUDENT BOOK. 10th Grade Unit 5

MATH STUDENT BOOK. 10th Grade Unit 5 MATH STUDENT BOOK 10th Grde Unit 5 Unit 5 Similr Polygons MATH 1005 Similr Polygons INTRODUCTION 3 1. PRINCIPLES OF ALGEBRA 5 RATIOS AND PROPORTIONS 5 PROPERTIES OF PROPORTIONS 11 SELF TEST 1 16 2. SIMILARITY

More information

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:

Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are: (x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

More information

1 Error Analysis of Simple Rules for Numerical Integration

1 Error Analysis of Simple Rules for Numerical Integration cs41: introduction to numericl nlysis 11/16/10 Lecture 19: Numericl Integrtion II Instructor: Professor Amos Ron Scries: Mrk Cowlishw, Nthnel Fillmore 1 Error Anlysis of Simple Rules for Numericl Integrtion

More information

What else can you do?

What else can you do? Wht else cn you do? ngle sums The size of specil ngle types lernt erlier cn e used to find unknown ngles. tht form stright line dd to 180c. lculte the size of + M, if L is stright line M + L = 180c( stright

More information

The Definite Integral

The Definite Integral CHAPTER 3 The Definite Integrl Key Words nd Concepts: Definite Integrl Questions to Consider: How do we use slicing to turn problem sttement into definite integrl? How re definite nd indefinite integrls

More information

APPM 1360 Exam 2 Spring 2016

APPM 1360 Exam 2 Spring 2016 APPM 6 Em Spring 6. 8 pts, 7 pts ech For ech of the following prts, let f + nd g 4. For prts, b, nd c, set up, but do not evlute, the integrl needed to find the requested informtion. The volume of the

More information

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes

200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write

More information

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom

Continuous Random Variables Class 5, Jeremy Orloff and Jonathan Bloom Lerning Gols Continuous Rndom Vriles Clss 5, 8.05 Jeremy Orloff nd Jonthn Bloom. Know the definition of continuous rndom vrile. 2. Know the definition of the proility density function (pdf) nd cumultive

More information

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35

(0.0)(0.1)+(0.3)(0.1)+(0.6)(0.1)+ +(2.7)(0.1) = 1.35 7 Integrtion º½ ÌÛÓ Ü ÑÔÐ Up to now we hve been concerned with extrcting informtion bout how function chnges from the function itself. Given knowledge bout n object s position, for exmple, we wnt to know

More information

Instructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015

Instructor(s): Acosta/Woodard PHYSICS DEPARTMENT PHY 2049, Fall 2015 Midterm 1 September 29, 2015 Instructor(s): Acost/Woodrd PHYSICS DEPATMENT PHY 049, Fll 015 Midterm 1 September 9, 015 Nme (print): Signture: On m honor, I hve neither given nor received unuthorized id on this emintion. YOU TEST NUMBE

More information

MA Handout 2: Notation and Background Concepts from Analysis

MA Handout 2: Notation and Background Concepts from Analysis MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,

More information

The Trapezoidal Rule

The Trapezoidal Rule _.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion

More information

CS667 Lecture 6: Monte Carlo Integration 02/10/05

CS667 Lecture 6: Monte Carlo Integration 02/10/05 CS667 Lecture 6: Monte Crlo Integrtion 02/10/05 Venkt Krishnrj Lecturer: Steve Mrschner 1 Ide The min ide of Monte Crlo Integrtion is tht we cn estimte the vlue of n integrl by looking t lrge number of

More information

Lecture 3. Limits of Functions and Continuity

Lecture 3. Limits of Functions and Continuity Lecture 3 Limits of Functions nd Continuity Audrey Terrs April 26, 21 1 Limits of Functions Notes I m skipping the lst section of Chpter 6 of Lng; the section bout open nd closed sets We cn probbly live

More information

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties

What Is Calculus? 42 CHAPTER 1 Limits and Their Properties 60_00.qd //0 : PM Pge CHAPTER Limits nd Their Properties The Mistress Fellows, Girton College, Cmridge Section. STUDY TIP As ou progress through this course, rememer tht lerning clculus is just one of

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.

NUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by. NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with

More information

GRADE 4. Division WORKSHEETS

GRADE 4. Division WORKSHEETS GRADE Division WORKSHEETS Division division is shring nd grouping Division cn men shring or grouping. There re cndies shred mong kids. How mny re in ech shre? = 3 There re 6 pples nd go into ech bsket.

More information

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s).

different methods (left endpoint, right endpoint, midpoint, trapezoid, Simpson s). Mth 1A with Professor Stnkov Worksheet, Discussion #41; Wednesdy, 12/6/217 GSI nme: Roy Zho Problems 1. Write the integrl 3 dx s limit of Riemnn sums. Write it using 2 intervls using the 1 x different

More information

7.6 The Use of Definite Integrals in Physics and Engineering

7.6 The Use of Definite Integrals in Physics and Engineering Arknss Tech University MATH 94: Clculus II Dr. Mrcel B. Finn 7.6 The Use of Definite Integrls in Physics nd Engineering It hs been shown how clculus cn be pplied to find solutions to geometric problems

More information

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet

Alg. Sheet (1) Department : Math Form : 3 rd prep. Sheet Ciro Governorte Nozh Directorte of Eduction Nozh Lnguge Schools Ismili Rod Deprtment : Mth Form : rd prep. Sheet Alg. Sheet () [] Find the vlues of nd in ech of the following if : ) (, ) ( -5, 9 ) ) (,

More information

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions )

15 - TRIGONOMETRY Page 1 ( Answers at the end of all questions ) - TRIGONOMETRY Pge P ( ) In tringle PQR, R =. If tn b c = 0, 0, then Q nd tn re the roots of the eqution = b c c = b b = c b = c [ AIEEE 00 ] ( ) In tringle ABC, let C =. If r is the inrdius nd R is the

More information