Differentiation Applications 1: Related Rates

Save this PDF as:

Size: px
Start display at page:

Download "Differentiation Applications 1: Related Rates"

Transcription

1 Differentiatin Applicatins 1: Related Rates 151 Differentiatin Applicatins 1: Related Rates Mdel 1: Sliding Ladder 10 ladder y 10 ladder 10 ladder A 10 ft ladder is leaning against a wall when the bttm begins t slip ut. Assume that the bttm f the ladder slides away frm the wall at a cnstant rate f 2 ft/s, as represented by the arrw n each drawing. At any mment, the bttm f the ladder is x feet frm the wall, and the tp f the ladder is y feet frm the grund, as shwn in the center diagram. Fcus n the questin: What is the velcity f the tp f the ladder when it is a given distance abve the grund? as yu wrk thrugh the first few pages f this activity. Cnstruct Yur Understanding Questins (t d in class) 1. Label the distances x and y in the first and last drawings in Mdel When a. y = 8 ft, what is x? b. y = ft, what is x? Write dwn any equatins yu used t find yur answers. x 2 ft/s 2 ft/s 2 ft/s. Suppse we want t knw the velcity f the tp f the ladder when y = 8 feet. A gd way t begin answering this questin is t take an inventry f what is knwn and unknwn, assigning variables t each. D this by cmpleting the table belw. Descriptin at the mment f interest: (when y = 8 ft) Symbl r Variable Value If unknwn, write? Distance f the tp f the ladder frm the grund y 8 feet Distance f the bttm f the ladder frm the wall Velcity f the bttm f the ladder, which is the rate f change in x r y [circle ne] with respect t time. Units Velcity f the tp f the ladder, which is the rate f change in x r y [circle ne] with respect t time.

2 152 Differentiatin Applicatins 1: Related Rates 4. (Check yur wrk) There shuld be nly ne unknwn in the table in Mdel 1, and this shuld crrespnd t the questin (in bld) in Mdel 1. If yu listed a value f? fr bth velcities, read Mdel 1 mre carefully and change ne f these t an actual value. 5. Nte that x and y in Mdel 1 are changing with time ( t ), s each is a functin f t. a. Cnstruct a descriptin f what these symbls mean in the cntext f Mdel 1. i. dx ii. dy b. (Check yur wrk) If yu have nt already dne s, circle x r y in the last tw rws n the table in Questin. Write dx and dy in the apprpriate bxes f the table, and check that yur descriptins abve match the descriptins in the table. c. Is dy psitive r negative? Circle ne, and explain yur reasning. d. (Check yur wrk) Is yur answer t part c cnsistent with the fact that dx ppsite signs in this prblem? and dy have 6. (Check yur wrk) Did yu cite the equatin x y 100 in Questin 2? 2 2 a. If nt, g back and check yur wrk. b. Starting frm this equatin, use implicit differentiatin t generate an expressin fr dy, in terms f the variables and rates f change n the table in Mdel 1. (If yu are stuck, mve nt parts c-e n the next page.)

3 Differentiatin Applicatins 1: Related Rates 15 c. (Check yur wrk) Which is the derivative f the term dy dx 2x 2x 2x dx dy dx dx 2 x with respect t t? 2 dy x dx Circle ne and state what differentiatin rule is being applied here. 2 dx x d. Using this same differentiatin rule, find the derivative with respect t t f the term 2 y. e. Use yur answers t parts c and d, abve, t cnstruct an answer t part b. Check the resulting expressin fr dy with anther grup. 7. Use what yu have learned s far in this activity t find the velcity f the tp f the ladder in Mdel 1 a. When it is 8 feet abve the grund? Draw a picture and shw yur wrk. b. When it is feet abve the grund? Draw a picture and shw yur wrk.

4 154 Differentiatin Applicatins 1: Related Rates 8. Helium is leaking ut f a spherical weather balln at a cnstant rate f cubic feet per secnd. What is the rate f change in the radius f the balln at the mment when the vlume f the balln is.5 ft? It is helpful t draw a picture and yu may summarize the infrmatin in a table like the ne in Mdel 1. If yu are stuck, try the next questin, then cme back t cmplete 4 this questin. Nte that the vlume f a sphere is given by V r Descriptin at the mment f interest: when Symbl r Variable Value If unknwn, write? Units 9. (Check yur wrk) The previus questin can be slved by answering each f the fllwing questins. If yu skipped any f these, answer the questin, then g back cnsider hw it applies t the previus questin. a. What defines the mment f interest fr this prblem? (e.g., in Mdel 1, the mment f interest was defined by the time at which the tp f the ladder was 8 ft. frm the grund, r y = 8 ft.) b. What variables and rates f change are invlved in this prblem? Write a descriptin and assign a symbl and units t each ne. (Enter these n the table.)

5 Differentiatin Applicatins 1: Related Rates 155 c. Write dwn an equatin relating the variables in the prblem. d. Cnsider the equatin, y k x, where k is a cnstant and the variables x and y are each a functin f time, t. i. Use implicit differentiatin t generate a new equatin cntaining the rate f change f x with respect t t (and ther quantities), then slve fr this rate. Shw yur wrk. ii. Explain hw this relates t the weather balln prblem. e. What is the radius when the balln vlume is.5 ft? f. What is the sign f dv? g. What is the rate f change in the radius f the balln at the mment when the vlume f the balln is.5 ft? 10. (Check yur wrk) The fllwing is a checklist f steps that culd be useful in slving a related rates prblem. Which steps did yu use in slving Questin 8? Which steps did yu skip? Put a check mark next t the nes yu used. Draw a picture and assign symbls t relevant quantities. Summarize the infrmatin prvided in the prblem (e.g., make a table): Nte which quantities are changing ver time, and use crrect ntatin t express their rate f change with respect t time. Write a descriptin f what each variable and rate f change means, being mindful f units f measure. Identify which quantities are knwn and which are unknwn. Write dwn an equatin relating the variables yu identified. If pssible, use gemetry t eliminate ne r mre variables. Use implicit differentiatin t generate a new equatin relating the variables and their rates f change, then slve fr the unknwn. Answer the questin asked in the prblem remembering t state the units f measure.

6 156 Differentiatin Applicatins 1: Related Rates Extend Yur Understanding Questins (t d in r ut f class) 11. Tw straight rads intersect in the middle f twn. One runs nrth-suth, the ther east-west. Nah is driving nrth frm the intersectin at 20 miles per hur. Ellie is driving east frm the intersectin at 0 miles per hur. When Nah is 0. miles frm the intersectin and Ellie is 0.4 miles frm the intersectin, hw fast are they mving apart? (Try nt t lk back at yur checklist unless yu are stuck.) 12. The cnical water tank shwn at right is filling at a rate f 1 m (1000 L) per hur. 2meters At what rate is the water level rising when the depth f the water is 2 m? 1 2 Recall that, fr a cne, V r h. (Try nt t lk back at yur checklist unless yu are stuck.) r h 5meters

7 Differentiatin Applicatins 1: Related Rates (Check yur wrk) Jeff generates the fllwing slutin t the previus questin: r 2 h 5 (by similar triangles) s h r 25 At the mment f interest: h 2, s Substituting r 0.8 int 1 (0.8) V h h V 1 2(2) r r h gives dv dh dh dv Differentiating 0.67 and slving gives 1.5 dv Plugging in 1 m dh /hr gives the answer as 1.5 m/hr a. (Check yur wrk) Jeff did a gd jb setting up the variables; and his equatin is crrect. Are these chices cnsistent with yur entries in the table n the previus page? If nt, g back and check yur wrk. r 2h 5 b. The crrect answer t this questin is dh m/hr Jeff made a very cmmn errr fr this type f prblem. He eliminated a variable by substituting in an actual value t sn. Find Jeff s errr, crrect it, and slve the prblem t get the right answer. Shw yur wrk. 2h c. (Check yur wrk) T slve this prblem Jeff shuld have substituted r in the 5 vlume equatin befre using implicit differentiatin (nt the value 0.8). Is this cnsistent with yur answer t part b? If nt, g back and revise yur wrk.

8 158 Differentiatin Applicatins 1: Related Rates Ntes

SPH3U1 Lesson 06 Kinematics

SPH3U1 Lesson 06 Kinematics PROJECTILE MOTION LEARNING GOALS Students will: Describe the mtin f an bject thrwn at arbitrary angles thrugh the air. Describe the hrizntal and vertical mtins f a prjectile. Slve prjectile mtin prblems.

More information

Trigonometric Ratios Unit 5 Tentative TEST date

Trigonometric Ratios Unit 5 Tentative TEST date 1 U n i t 5 11U Date: Name: Trignmetric Ratis Unit 5 Tentative TEST date Big idea/learning Gals In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin

More information

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b

MODULE 1. e x + c. [You can t separate a demominator, but you can divide a single denominator into each numerator term] a + b a(a + b)+1 = a + b . REVIEW OF SOME BASIC ALGEBRA MODULE () Slving Equatins Yu shuld be able t slve fr x: a + b = c a d + e x + c and get x = e(ba +) b(c a) d(ba +) c Cmmn mistakes and strategies:. a b + c a b + a c, but

More information

Section 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law

Section 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law Sectin 5.8 Ntes Page 1 5.8 Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential

More information

Fall 2013 Physics 172 Recitation 3 Momentum and Springs

Fall 2013 Physics 172 Recitation 3 Momentum and Springs Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3.

More information

Physics 2010 Motion with Constant Acceleration Experiment 1

Physics 2010 Motion with Constant Acceleration Experiment 1 . Physics 00 Mtin with Cnstant Acceleratin Experiment In this lab, we will study the mtin f a glider as it accelerates dwnhill n a tilted air track. The glider is supprted ver the air track by a cushin

More information

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y=

making triangle (ie same reference angle) ). This is a standard form that will allow us all to have the X= y= Intrductin t Vectrs I 21 Intrductin t Vectrs I 22 I. Determine the hrizntal and vertical cmpnents f the resultant vectr by cunting n the grid. X= y= J. Draw a mangle with hrizntal and vertical cmpnents

More information

Physics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1

Physics 212. Lecture 12. Today's Concept: Magnetic Force on moving charges. Physics 212 Lecture 12, Slide 1 Physics 1 Lecture 1 Tday's Cncept: Magnetic Frce n mving charges F qv Physics 1 Lecture 1, Slide 1 Music Wh is the Artist? A) The Meters ) The Neville rthers C) Trmbne Shrty D) Michael Franti E) Radiatrs

More information

M thematics. National 5 Practice Paper E. Paper 1. Duration 1 hour. Total marks 40

M thematics. National 5 Practice Paper E. Paper 1. Duration 1 hour. Total marks 40 N5 M thematics Natinal 5 Practice Paper E Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain

More information

Pipetting 101 Developed by BSU CityLab

Pipetting 101 Developed by BSU CityLab Discver the Micrbes Within: The Wlbachia Prject Pipetting 101 Develped by BSU CityLab Clr Cmparisns Pipetting Exercise #1 STUDENT OBJECTIVES Students will be able t: Chse the crrect size micrpipette fr

More information

PHYS 314 HOMEWORK #3

PHYS 314 HOMEWORK #3 PHYS 34 HOMEWORK #3 Due : 8 Feb. 07. A unifrm chain f mass M, lenth L and density λ (measured in k/m) hans s that its bttm link is just tuchin a scale. The chain is drpped frm rest nt the scale. What des

More information

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) =

AP Physics. Summer Assignment 2012 Date. Name. F m = = + What is due the first day of school? a. T. b. = ( )( ) = P Physics Name Summer ssignment 0 Date I. The P curriculum is extensive!! This means we have t wrk at a fast pace. This summer hmewrk will allw us t start n new Physics subject matter immediately when

More information

Physics 101 Math Review. Solutions

Physics 101 Math Review. Solutions Physics 0 Math eview Slutins . The fllwing are rdinary physics prblems. Place the answer in scientific ntatin when apprpriate and simplify the units (Scientific ntatin is used when it takes less time t

More information

AP Physics Kinematic Wrap Up

AP Physics Kinematic Wrap Up AP Physics Kinematic Wrap Up S what d yu need t knw abut this mtin in tw-dimensin stuff t get a gd scre n the ld AP Physics Test? First ff, here are the equatins that yu ll have t wrk with: v v at x x

More information

Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education Cambridge Assessment Internatinal Educatin Cambridge Internatinal General Certificate f Secndary Educatin ADDITIONAL MATHEMATICS 0606/1 Paper 1 MARK SCHEME Maximum Mark: 80 Published This mark scheme is

More information

Cambridge Assessment International Education Cambridge Ordinary Level. Published

Cambridge Assessment International Education Cambridge Ordinary Level. Published Cambridge Assessment Internatinal Educatin Cambridge Ordinary Level ADDITIONAL MATHEMATICS 4037/1 Paper 1 Octber/Nvember 017 MARK SCHEME Maximum Mark: 80 Published This mark scheme is published as an aid

More information

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving.

This section is primarily focused on tools to aid us in finding roots/zeros/ -intercepts of polynomials. Essentially, our focus turns to solving. Sectin 3.2: Many f yu WILL need t watch the crrespnding vides fr this sectin n MyOpenMath! This sectin is primarily fcused n tls t aid us in finding rts/zers/ -intercepts f plynmials. Essentially, ur fcus

More information

Lim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)?

Lim f (x) e. Find the largest possible domain and its discontinuity points. Why is it discontinuous at those points (if any)? THESE ARE SAMPLE QUESTIONS FOR EACH OF THE STUDENT LEARNING OUTCOMES (SLO) SET FOR THIS COURSE. SLO 1: Understand and use the cncept f the limit f a functin i. Use prperties f limits and ther techniques,

More information

M thematics. National 5 Practice Paper D. Paper 1. Duration 1 hour. Total marks 40

M thematics. National 5 Practice Paper D. Paper 1. Duration 1 hour. Total marks 40 N5 M thematics Natinal 5 Practice Paper D Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain

More information

NUMBERS, MATHEMATICS AND EQUATIONS

NUMBERS, MATHEMATICS AND EQUATIONS AUSTRALIAN CURRICULUM PHYSICS GETTING STARTED WITH PHYSICS NUMBERS, MATHEMATICS AND EQUATIONS An integral part t the understanding f ur physical wrld is the use f mathematical mdels which can be used t

More information

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations

Plan o o. I(t) Divide problem into sub-problems Modify schematic and coordinate system (if needed) Write general equations STAPLE Physics 201 Name Final Exam May 14, 2013 This is a clsed bk examinatin but during the exam yu may refer t a 5 x7 nte card with wrds f wisdm yu have written n it. There is extra scratch paper available.

More information

2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS

2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 2004 AP CHEMISTRY FREE-RESPONSE QUESTIONS 6. An electrchemical cell is cnstructed with an pen switch, as shwn in the diagram abve. A strip f Sn and a strip f an unknwn metal, X, are used as electrdes.

More information

Figure 1a. A planar mechanism.

Figure 1a. A planar mechanism. ME 5 - Machine Design I Fall Semester 0 Name f Student Lab Sectin Number EXAM. OPEN BOOK AND CLOSED NOTES. Mnday, September rd, 0 Write n ne side nly f the paper prvided fr yur slutins. Where necessary,

More information

Math 105: Review for Exam I - Solutions

Math 105: Review for Exam I - Solutions 1. Let f(x) = 3 + x + 5. Math 105: Review fr Exam I - Slutins (a) What is the natural dmain f f? [ 5, ), which means all reals greater than r equal t 5 (b) What is the range f f? [3, ), which means all

More information

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec

Corrections for the textbook answers: Sec 6.1 #8h)covert angle to a positive by adding period #9b) # rad/sec U n i t 6 AdvF Date: Name: Trignmetric Functins Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will study trignmetric functins frm grade, hwever everything will be dne in radian measure.

More information

Five Whys How To Do It Better

Five Whys How To Do It Better Five Whys Definitin. As explained in the previus article, we define rt cause as simply the uncvering f hw the current prblem came int being. Fr a simple causal chain, it is the entire chain. Fr a cmplex

More information

1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south?

1.2.1 Vectors. 1 P age. Examples What is the reference vector angle for a vector that points 50 degrees east of south? 1.2.1 Vectrs Definitins Vectrs are represented n paper by arrws directin = magnitude = Examples f vectrs: Examples What is the reference vectr angle fr a vectr that pints 50 degrees east f suth? What is

More information

Review for the final exam (Math 127)

Review for the final exam (Math 127) . Evaluate 3 tan tan 4 3 (b) (c) cs cs 4 7 3 sec cs 4 4 (d) cs tan 3 Review fr the final eam (Math 7). If sec, and 7 36, find cs, sin, tan, ct, csc tan (b) If, evaluate cs, sin 7 36 (c) Write the csc in

More information

Math Foundations 10 Work Plan

Math Foundations 10 Work Plan Math Fundatins 10 Wrk Plan Units / Tpics 10.1 Demnstrate understanding f factrs f whle numbers by: Prime factrs Greatest Cmmn Factrs (GCF) Least Cmmn Multiple (LCM) Principal square rt Cube rt Time Frame

More information

You need to be able to define the following terms and answer basic questions about them:

You need to be able to define the following terms and answer basic questions about them: CS440/ECE448 Sectin Q Fall 2017 Midterm Review Yu need t be able t define the fllwing terms and answer basic questins abut them: Intr t AI, agents and envirnments Pssible definitins f AI, prs and cns f

More information

https://goo.gl/eaqvfo SUMMER REV: Half-Life DUE DATE: JULY 2 nd

https://goo.gl/eaqvfo SUMMER REV: Half-Life DUE DATE: JULY 2 nd NAME: DUE DATE: JULY 2 nd AP Chemistry SUMMER REV: Half-Life Why? Every radiistpe has a characteristic rate f decay measured by its half-life. Half-lives can be as shrt as a fractin f a secnd r as lng

More information

Activity Guide Loops and Random Numbers

Activity Guide Loops and Random Numbers Unit 3 Lessn 7 Name(s) Perid Date Activity Guide Lps and Randm Numbers CS Cntent Lps are a relatively straightfrward idea in prgramming - yu want a certain chunk f cde t run repeatedly - but it takes a

More information

Experiment #3. Graphing with Excel

Experiment #3. Graphing with Excel Experiment #3. Graphing with Excel Study the "Graphing with Excel" instructins that have been prvided. Additinal help with learning t use Excel can be fund n several web sites, including http://www.ncsu.edu/labwrite/res/gt/gt-

More information

1 PreCalculus AP Unit G Rotational Trig (MCR) Name:

1 PreCalculus AP Unit G Rotational Trig (MCR) Name: 1 PreCalculus AP Unit G Rtatinal Trig (MCR) Name: Big idea In this unit yu will extend yur knwledge f SOH CAH TOA t wrk with btuse and reflex angles. This extensin will invlve the unit circle which will

More information

READING STATECHART DIAGRAMS

READING STATECHART DIAGRAMS READING STATECHART DIAGRAMS Figure 4.48 A Statechart diagram with events The diagram in Figure 4.48 shws all states that the bject plane can be in during the curse f its life. Furthermre, it shws the pssible

More information

CONSTRUCTING STATECHART DIAGRAMS

CONSTRUCTING STATECHART DIAGRAMS CONSTRUCTING STATECHART DIAGRAMS The fllwing checklist shws the necessary steps fr cnstructing the statechart diagrams f a class. Subsequently, we will explain the individual steps further. Checklist 4.6

More information

Chapter 2 GAUSS LAW Recommended Problems:

Chapter 2 GAUSS LAW Recommended Problems: Chapter GAUSS LAW Recmmended Prblems: 1,4,5,6,7,9,11,13,15,18,19,1,7,9,31,35,37,39,41,43,45,47,49,51,55,57,61,6,69. LCTRIC FLUX lectric flux is a measure f the number f electric filed lines penetrating

More information

4. Find a, b, and c. 6. Find x and y.

4. Find a, b, and c. 6. Find x and y. Grace Brethren Christian Schl Entering Trig/Analysis: Page f Summer Packet fr Students entering Trig/Analysis Review prblems frm Gemetry: Shw yur wrk!. Twice the cmplement f angle A is 35 less than the

More information

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers

LHS Mathematics Department Honors Pre-Calculus Final Exam 2002 Answers LHS Mathematics Department Hnrs Pre-alculus Final Eam nswers Part Shrt Prblems The table at the right gives the ppulatin f Massachusetts ver the past several decades Using an epnential mdel, predict the

More information

Solving Inequalities: Multiplying or Dividing by a Negative Number

Solving Inequalities: Multiplying or Dividing by a Negative Number 11 Slving Inequalities: Multiplying r Dividing by a Negative Number We slve inequalities the same way we slve equatins, with ne exceptin. When we divide r multiply bth sides f an inequality by a negative

More information

Preparation work for A2 Mathematics [2017]

Preparation work for A2 Mathematics [2017] Preparatin wrk fr A2 Mathematics [2017] The wrk studied in Y12 after the return frm study leave is frm the Cre 3 mdule f the A2 Mathematics curse. This wrk will nly be reviewed during Year 13, it will

More information

EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017

EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017 OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 FALLF, 2017 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a

More information

AP Statistics Practice Test Unit Three Exploring Relationships Between Variables. Name Period Date

AP Statistics Practice Test Unit Three Exploring Relationships Between Variables. Name Period Date AP Statistics Practice Test Unit Three Explring Relatinships Between Variables Name Perid Date True r False: 1. Crrelatin and regressin require explanatry and respnse variables. 1. 2. Every least squares

More information

CHM112 Lab Graphing with Excel Grading Rubric

CHM112 Lab Graphing with Excel Grading Rubric Name CHM112 Lab Graphing with Excel Grading Rubric Criteria Pints pssible Pints earned Graphs crrectly pltted and adhere t all guidelines (including descriptive title, prperly frmatted axes, trendline

More information

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter).

Three charges, all with a charge of 10 C are situated as shown (each grid line is separated by 1 meter). Three charges, all with a charge f 0 are situated as shwn (each grid line is separated by meter). ) What is the net wrk needed t assemble this charge distributin? a) +0.5 J b) +0.8 J c) 0 J d) -0.8 J e)

More information

Preparation work for A2 Mathematics [2018]

Preparation work for A2 Mathematics [2018] Preparatin wrk fr A Mathematics [018] The wrk studied in Y1 will frm the fundatins n which will build upn in Year 13. It will nly be reviewed during Year 13, it will nt be retaught. This is t allw time

More information

Chem 163 Section: Team Number: ALE 24. Voltaic Cells and Standard Cell Potentials. (Reference: 21.2 and 21.3 Silberberg 5 th edition)

Chem 163 Section: Team Number: ALE 24. Voltaic Cells and Standard Cell Potentials. (Reference: 21.2 and 21.3 Silberberg 5 th edition) Name Chem 163 Sectin: Team Number: ALE 24. Vltaic Cells and Standard Cell Ptentials (Reference: 21.2 and 21.3 Silberberg 5 th editin) What des a vltmeter reading tell us? The Mdel: Standard Reductin and

More information

AP Statistics Notes Unit Two: The Normal Distributions

AP Statistics Notes Unit Two: The Normal Distributions AP Statistics Ntes Unit Tw: The Nrmal Distributins Syllabus Objectives: 1.5 The student will summarize distributins f data measuring the psitin using quartiles, percentiles, and standardized scres (z-scres).

More information

EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016

EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016 OBJECTIVES 1. Ft Pressure EXAM #1 PHYSICAL SCIENCE 103 Spring, 2016 Determine the surface area f an bject. Given the weight and surface area, calculate the pressure. 2. Measuring Vlume & Mass Prvided a

More information

Weathering. Title: Chemical and Mechanical Weathering. Grade Level: Subject/Content: Earth and Space Science

Weathering. Title: Chemical and Mechanical Weathering. Grade Level: Subject/Content: Earth and Space Science Weathering Title: Chemical and Mechanical Weathering Grade Level: 9-12 Subject/Cntent: Earth and Space Science Summary f Lessn: Students will test hw chemical and mechanical weathering can affect a rck

More information

M thematics. National 5 Practice Paper B. Paper 1. Duration 1 hour. Total marks 40

M thematics. National 5 Practice Paper B. Paper 1. Duration 1 hour. Total marks 40 M thematics Natinal 5 Practice Paper B Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain

More information

Pre-Calculus Individual Test 2017 February Regional

Pre-Calculus Individual Test 2017 February Regional The abbreviatin NOTA means Nne f the Abve answers and shuld be chsen if chices A, B, C and D are nt crrect. N calculatr is allwed n this test. Arcfunctins (such as y = Arcsin( ) ) have traditinal restricted

More information

MATHEMATICS Higher Grade - Paper I

MATHEMATICS Higher Grade - Paper I Higher Mathematics - Practice Eaminatin D Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS

More information

We can see from the graph above that the intersection is, i.e., [ ).

We can see from the graph above that the intersection is, i.e., [ ). MTH 111 Cllege Algebra Lecture Ntes July 2, 2014 Functin Arithmetic: With nt t much difficulty, we ntice that inputs f functins are numbers, and utputs f functins are numbers. S whatever we can d with

More information

How can standard heats of formation be used to calculate the heat of a reaction?

How can standard heats of formation be used to calculate the heat of a reaction? Answer Key ALE 28. ess s Law and Standard Enthalpies Frmatin (Reerence: Chapter 6 - Silberberg 4 th editin) Imprtant!! Fr answers that invlve a calculatin yu must shw yur wrk neatly using dimensinal analysis

More information

DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2.

DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2010/2011 Paper 1 Assessing Units 1 & 2. INGWLL EMY Mathematics Higher Prelim Eaminatin 00/0 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0

More information

CHAPTER 6 -- ENERGY. Approach #2: Using the component of mg along the line of d:

CHAPTER 6 -- ENERGY. Approach #2: Using the component of mg along the line of d: Slutins--Ch. 6 (Energy) CHAPTER 6 -- ENERGY 6.) The f.b.d. shwn t the right has been prvided t identify all the frces acting n the bdy as it mves up the incline. a.) T determine the wrk dne by gravity

More information

Lab 1 The Scientific Method

Lab 1 The Scientific Method INTRODUCTION The fllwing labratry exercise is designed t give yu, the student, an pprtunity t explre unknwn systems, r universes, and hypthesize pssible rules which may gvern the behavir within them. Scientific

More information

Chapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion.

Chapter 2. Kinematics in One Dimension. Kinematics deals with the concepts that are needed to describe motion. Chapter Kinematics in One Dimensin Kinematics deals with the cncepts that are needed t describe mtin. Dynamics deals with the effect that frces have n mtin. Tgether, kinematics and dynamics frm the branch

More information

CHAPTER 8b Static Equilibrium Units

CHAPTER 8b Static Equilibrium Units CHAPTER 8b Static Equilibrium Units The Cnditins fr Equilibrium Slving Statics Prblems Stability and Balance Elasticity; Stress and Strain The Cnditins fr Equilibrium An bject with frces acting n it, but

More information

ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels

ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION. Instructions: If asked to label the axes please use real world (contextual) labels ANSWER KEY FOR MATH 10 SAMPLE EXAMINATION Instructins: If asked t label the axes please use real wrld (cntextual) labels Multiple Chice Answers: 0 questins x 1.5 = 30 Pints ttal Questin Answer Number 1

More information

Medium Scale Integrated (MSI) devices [Sections 2.9 and 2.10]

Medium Scale Integrated (MSI) devices [Sections 2.9 and 2.10] EECS 270, Winter 2017, Lecture 3 Page 1 f 6 Medium Scale Integrated (MSI) devices [Sectins 2.9 and 2.10] As we ve seen, it s smetimes nt reasnable t d all the design wrk at the gate-level smetimes we just

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm

Faculty of Engineering and Department of Physics Engineering Physics 131 Midterm Examination February 27, 2006; 7:00 pm 8:30 pm Faculty f Engineering and Department f Physics Engineering Physics 131 Midterm Examinatin February 27, 2006; 7:00 pm 8:30 pm N ntes r textbks allwed. Frmula sheet is n the last page (may be remved). Calculatrs

More information

ES201 - Examination 2 Winter Adams and Richards NAME BOX NUMBER

ES201 - Examination 2 Winter Adams and Richards NAME BOX NUMBER ES201 - Examinatin 2 Winter 2003-2004 Adams and Richards NAME BOX NUMBER Please Circle One : Richards (Perid 4) ES201-01 Adams (Perid 4) ES201-02 Adams (Perid 6) ES201-03 Prblem 1 ( 12 ) Prblem 2 ( 24

More information

M thematics. National 5 Practice Paper C. Paper 1. Duration 1 hour. Total marks 40

M thematics. National 5 Practice Paper C. Paper 1. Duration 1 hour. Total marks 40 N5 M thematics Natinal 5 Practice Paper C Paper 1 Duratin 1 hur Ttal marks 40 Yu may NOT use a calculatr Attempt all the questins. Use blue r black ink. Full credit will nly be given t slutins which cntain

More information

Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN

Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Professor and Chair Mechanical Engineering Department Christian Brothers University 650 East Parkway South Memphis, TN Yeu-Sheng Paul Shiue, Ph.D 薛宇盛 Prfessr and Chair Mechanical Engineering Department Christian Brthers University 650 East Parkway Suth Memphis, TN 38104 Office: (901) 321-3424 Rm: N-110 Fax : (901) 321-3402

More information

DIFFERENTIATION RULES

DIFFERENTIATION RULES 3 DIFFERENTIATION RULES DIFFERENTIATION RULES If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. However,

More information

**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!**

**DO NOT ONLY RELY ON THIS STUDY GUIDE!!!** Tpics lists: UV-Vis Absrbance Spectrscpy Lab & ChemActivity 3-6 (nly thrugh 4) I. UV-Vis Absrbance Spectrscpy Lab Beer s law Relates cncentratin f a chemical species in a slutin and the absrbance f that

More information

5 th grade Common Core Standards

5 th grade Common Core Standards 5 th grade Cmmn Cre Standards In Grade 5, instructinal time shuld fcus n three critical areas: (1) develping fluency with additin and subtractin f fractins, and develping understanding f the multiplicatin

More information

CHE 105 EXAMINATION III November 11, 2010

CHE 105 EXAMINATION III November 11, 2010 CHE 105 EXAMINATION III Nvember 11, 2010 University f Kentucky Department f Chemistry READ THESE DIRECTIONS CAREFULLY BEFORE STARTING THE EXAMINATION! It is extremely imprtant that yu fill in the answer

More information

has five more sides than regular polygon P has half the number of sides as regular polygon P, when both are in centimeters.

has five more sides than regular polygon P has half the number of sides as regular polygon P, when both are in centimeters. egular plygn has five mre sides than regular plygn egular plygn has half the number f sides as regular plygn The perimeter f plygn is twenty mre than the perimeter f regular plygn, when bth are in centimeters

More information

Instructional Plan. Representational/Drawing Level

Instructional Plan. Representational/Drawing Level Instructinal Plan Representatinal/Drawing Level Name f Math Skill/Cncept: Divisin Prcess and Divisin with Remainders Prerequisite Skills Needed: 1.) Mastery f dividing cncrete bjects int equal grups. 2.)

More information

WYSE Academic Challenge Regional Mathematics 2007 Solution Set

WYSE Academic Challenge Regional Mathematics 2007 Solution Set WYSE Academic Challenge Reginal Mathematics 007 Slutin Set 1. Crrect answer: C. ( ) ( ) 1 + y y = ( + ) + ( y y + 1 ) = + 1 1 ( ) ( 1 + y ) = s *1/ = 1. Crrect answer: A. The determinant is ( 1 ( 1) )

More information

EEO 401 Digital Signal Processing Prof. Mark Fowler

EEO 401 Digital Signal Processing Prof. Mark Fowler EEO 401 Digital Signal Prcessing Prf. Mark Fwler Intrductin Nte Set #1 ading Assignment: Ch. 1 f Prakis & Manlakis 1/13 Mdern systems generally DSP Scenari get a cntinuus-time signal frm a sensr a cnt.-time

More information

Function notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property

Function notation & composite functions Factoring Dividing polynomials Remainder theorem & factor property Functin ntatin & cmpsite functins Factring Dividing plynmials Remainder therem & factr prperty Can d s by gruping r by: Always lk fr a cmmn factr first 2 numbers that ADD t give yu middle term and MULTIPLY

More information

Exam Review Trigonometry

Exam Review Trigonometry Exam Review Trignmetry (Tyler, Chris, Hafsa, Nasim, Paniz,Tng) Similar Triangles Prving Similarity (AA, SSS, SAS) ~ Tyler Garfinkle 3 Types f Similarities: 1. Side Side Side Similarity (SSS) If three pairs

More information

A - LEVEL MATHEMATICS 2018/2019

A - LEVEL MATHEMATICS 2018/2019 A - LEVEL MATHEMATICS 2018/2019 STRUCTURE OF THE COURSE Yur maths A-Level Maths curse cvers Pure Mathematics, Mechanics and Statistics. Yu will be eamined at the end f the tw-year curse. The assessment

More information

Chapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms

Chapter Summary. Mathematical Induction Strong Induction Recursive Definitions Structural Induction Recursive Algorithms Chapter 5 1 Chapter Summary Mathematical Inductin Strng Inductin Recursive Definitins Structural Inductin Recursive Algrithms Sectin 5.1 3 Sectin Summary Mathematical Inductin Examples f Prf by Mathematical

More information

PRE-CALCULUS B FINAL REVIEW NAME Work out problems in your notebook or on a separate piece of paper.

PRE-CALCULUS B FINAL REVIEW NAME Work out problems in your notebook or on a separate piece of paper. PRE-CALCULUS B FINAL REVIEW NAME Wrk ut prblems in yur ntebk r n a separate piece f paper. CHAPTER 5 Simplify each t ne trig wrd r a number: tan (1) sec sec () sin ct + 1 + cs + ( sin cs ) () sin - cs

More information

Precalculus A. Semester Exam Review

Precalculus A. Semester Exam Review Precalculus A 015-016 MCPS 015 016 1 The semester A eaminatin fr Precalculus cnsists f tw parts. Part 1 is selected respnse n which a calculatr will NOT be allwed. Part is shrt answer n which a calculatr

More information

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System

Flipping Physics Lecture Notes: Simple Harmonic Motion Introduction via a Horizontal Mass-Spring System Flipping Physics Lecture Ntes: Simple Harmnic Mtin Intrductin via a Hrizntal Mass-Spring System A Hrizntal Mass-Spring System is where a mass is attached t a spring, riented hrizntally, and then placed

More information

MATHEMATICS Higher Grade - Paper I

MATHEMATICS Higher Grade - Paper I Higher Mathematics - Practice Eaminatin B Please nte the frmat f this practice eaminatin is different frm the current frmat. The paper timings are different and calculatrs can be used thrughut. MATHEMATICS

More information

Department: MATHEMATICS

Department: MATHEMATICS Cde: MATH 022 Title: ALGEBRA SKILLS Institute: STEM Department: MATHEMATICS Curse Descriptin: This curse prvides students wh have cmpleted MATH 021 with the necessary skills and cncepts t cntinue the study

More information

I understand the new topics for this unit if I can do the practice questions in the textbook/handouts

I understand the new topics for this unit if I can do the practice questions in the textbook/handouts 1 U n i t 6 11U Date: Name: Sinusidals Unit 6 Tentative TEST date Big idea/learning Gals In this unit yu will learn hw trignmetry can be used t mdel wavelike relatinships. These wavelike functins are called

More information

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007

CS 477/677 Analysis of Algorithms Fall 2007 Dr. George Bebis Course Project Due Date: 11/29/2007 CS 477/677 Analysis f Algrithms Fall 2007 Dr. Gerge Bebis Curse Prject Due Date: 11/29/2007 Part1: Cmparisn f Srting Algrithms (70% f the prject grade) The bjective f the first part f the assignment is

More information

MODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards:

MODULE FOUR. This module addresses functions. SC Academic Elementary Algebra Standards: MODULE FOUR This mdule addresses functins SC Academic Standards: EA-3.1 Classify a relatinship as being either a functin r nt a functin when given data as a table, set f rdered pairs, r graph. EA-3.2 Use

More information

Math 0310 Final Exam Review Problems

Math 0310 Final Exam Review Problems Math 0310 Final Exam Review Prblems Slve the fllwing equatins. 1. 4dd + 2 = 6 2. 2 3 h 5 = 7 3. 2 + (18 xx) + 2(xx 1) = 4(xx + 2) 8 4. 1 4 yy 3 4 = 1 2 yy + 1 5. 5.74aa + 9.28 = 2.24aa 5.42 Slve the fllwing

More information

Lab 11 LRC Circuits, Damped Forced Harmonic Motion

Lab 11 LRC Circuits, Damped Forced Harmonic Motion Physics 6 ab ab 11 ircuits, Damped Frced Harmnic Mtin What Yu Need T Knw: The Physics OK this is basically a recap f what yu ve dne s far with circuits and circuits. Nw we get t put everything tgether

More information

CHAPTER 6 WORK AND ENERGY

CHAPTER 6 WORK AND ENERGY CHAPTER 6 WORK AND ENERGY CONCEPTUAL QUESTIONS 16. REASONING AND SOLUTION A trapeze artist, starting rm rest, swings dwnward n the bar, lets g at the bttm the swing, and alls reely t the net. An assistant,

More information

DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2009/2010 Paper 1 Assessing Units 1 & 2. Time allowed - 1 hour 30

DINGWALL ACADEMY NATIONAL QUALIFICATIONS. Mathematics Higher Prelim Examination 2009/2010 Paper 1 Assessing Units 1 & 2. Time allowed - 1 hour 30 INGWLL EMY Mathematics Higher Prelim Eaminatin 009/00 Paper ssessing Units & NTIONL QULIFITIONS Time allwed - hur 0 minutes Read carefull alculatrs ma NOT be used in this paper. Sectin - Questins - 0 (0

More information

Relationships Between Frequency, Capacitance, Inductance and Reactance.

Relationships Between Frequency, Capacitance, Inductance and Reactance. P Physics Relatinships between f,, and. Relatinships Between Frequency, apacitance, nductance and Reactance. Purpse: T experimentally verify the relatinships between f, and. The data cllected will lead

More information

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y )

[COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t o m a k e s u r e y o u a r e r e a d y ) (Abut the final) [COLLEGE ALGEBRA EXAM I REVIEW TOPICS] ( u s e t h i s t m a k e s u r e y u a r e r e a d y ) The department writes the final exam s I dn't really knw what's n it and I can't very well

More information

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle? Physics 0 Public Exam Questins Unit 1: Circular Mtin NAME: August 009---------------------------------------------------------------------------------------------------------------------- 1. Which describes

More information

More Tutorial at

More Tutorial at Answer each questin in the space prvided; use back f page if extra space is needed. Answer questins s the grader can READILY understand yur wrk; nly wrk n the exam sheet will be cnsidered. Write answers,

More information

Activity 2 Dimensional Analysis

Activity 2 Dimensional Analysis Activity 2 Dimensinal Analysis Gals! Develp cnversin factrs frm cmmn equalities.! Use cnversin factrs t cnvert between different units f measure.! Apply the cncept f dimensinal analysis t string tgether

More information

3. Classify the following Numbers (Counting (natural), Whole, Integers, Rational, Irrational)

3. Classify the following Numbers (Counting (natural), Whole, Integers, Rational, Irrational) After yu cmplete each cncept give yurself a rating 1. 15 5 2 (5 3) 2. 2 4-8 (2 5) 3. Classify the fllwing Numbers (Cunting (natural), Whle, Integers, Ratinal, Irratinal) a. 7 b. 2 3 c. 2 4. Are negative

More information

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle?

14. Which shows the direction of the centripetal force acting on a mass spun in a vertical circle? Physics 3204 Public Exam Questins Unit 1: Circular Mtin NAME: August 2009---------------------------------------------------------------------------------------------------------------------- 12. Which

More information

Exam #1. A. Answer any 1 of the following 2 questions. CEE 371 October 8, Please grade the following questions: 1 or 2

Exam #1. A. Answer any 1 of the following 2 questions. CEE 371 October 8, Please grade the following questions: 1 or 2 CEE 371 Octber 8, 2009 Exam #1 Clsed Bk, ne sheet f ntes allwed Please answer ne questin frm the first tw, ne frm the secnd tw and ne frm the last three. The ttal ptential number f pints is 100. Shw all

More information

Homology groups of disks with holes

Homology groups of disks with holes Hmlgy grups f disks with hles THEOREM. Let p 1,, p k } be a sequence f distinct pints in the interir unit disk D n where n 2, and suppse that fr all j the sets E j Int D n are clsed, pairwise disjint subdisks.

More information