Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Thermal Diffusivity. Paul Hughes. Department of Physics and Astronomy The University of Manchester Manchester M13 9PL. Second Year Laboratory Report"

Transcription

1 Therml iffusivity Pul Hughes eprtment of Physics nd Astronomy The University of nchester nchester 3 9PL Second Yer Lbortory Report Nov 4 Abstrct We investigted the therml diffusivity of cylindricl block of resin by observing the temperture s function of time t it s centre with respect to n outside chnge of temperture. This ws done in two different wys; step method (involving single chnge from ice wter to boiling wter) nd periodic method (involving multiple chnges from ice wter to boiling wter). The result ws ( 9.45 ±.) m s.

2 . Introduction In n electricl conductor, the resistnce of the conductor, mongst other things, depends upon the shpe of the conductor nd the resistivity of the mteril from which it is mde from. This is kin to the properties displyed by therml conductor. However the conductivity of the therml conductor is nlogous to the resistivity of the electricl conductor. But neither of these situtions depend on time. To include time dependence into the sitution we must define new quntity; diffusivity. The study of therml diffusivity is useful in wide rnge of fields including building science, mteril science, meteorology, the design of het flu sensors nd hs even been linked to the effect dentures hve on some people s sense of tste.. Theory..Step ethod The conduction Eqution is ; t θ θ Where θ is the temperture s function of position nd time. By epnding this in cylindricl polr coordintes nd neglecting φ nd z θ θ θ θ ) dependence; + t r r r (i.e. θ ( r,t) The solution to this long the is of the cylinder is series of eponentil terms: θ n (, t) θ + ( θ + θ ) e n n λ t Where λ n re the positive roots of the Bessel function, i.e. λ. 45, λ 5. 5, λ etc. By substituting these vlues it is observed tht close pproimtion is to neglect the n > terms: θ θ + ( θ + θ ) e t e 3.47t e t... This pproimtion gives; θ θ e t

3 θ θ i.e θ λ t e [Eqution ] Where : the therml diffusivity of the block t : time/ s : rdius of the cylindricl block/ m λ.45 Eqution cn be rerrnged to; θ θ θe Hence using nturl logrithms : ln λ t ( θ θ) ln θ + ln e ln θ λ t λ t [Eqution ] Hence plotting the logrithm of the difference of the current temperture with the eternl temperture ginst time will yield liner plot with grdient m; λ m [Eqution 3] This cn be rerrnged to find the diffusivity of the block; m [Eqution 4] λ However, using vernier cllipers gives mesurement of the dimeter not the rdius, hence substituting for the dimeter, d, gives: md [Eqution 5] 4λ

4 .. Periodic ethod Using similr set up to the Step ethod the resin is cooled to be ºC nd then heted to ºC. However this time the resin is spends certin mount of time in the ice nd certin mount of time in the boiling wter. This cycle of ice nd boiling wter is repeted few times to give periodic chnge in the temperture of the resin. Using this method the diffusivity of the resin cn be found. 4 As the temperture is function of rdius nd the periodic chnge of the eternl iωt temperture; θ ( r, t) f ( r) e Substituting this into the conduction eqution; Chnging the vrible to f iω f r iω f f z r gives; + + f z z z f + r r The solution to this eqution is the Kelvin function, ( ). As our periodic chnge of eternl temperture is ctully squre wve (i.e. it is not sinusoidl chnge in θ r,t cn be epnded s Fourier series to give; temperture), ( ) (, t) (, t) ( ) ω θ [Eqution 6] θ This cn be rerrnged to give two different equtions, one concerning the mplitudes of the il temperture nd the eternl temperture: ( ) 4 θ θ [Eqution 7] π B And the other concerning the phse difference between the il temperture nd the eternl temperture; t φ π rg[ ( ) ] T [Eqution 7b] Where; π T [Eqution 8] Notice tht is unitless quntity. Where B : The difference between the mim temperture nd the minim temperture : The Kelvin function θ : The temperture of the boiling wter θ : The temperture of the ice wter φ : The phse difference between the il nd eternl tempertures t : The time difference between pek of the il nd eternl tempertures Thus by plotting the Kelvin function, the vlue of the diffusivity cn be found.

5 3. ethod 3.3. Step ethod To find the diffusivity of the smple of resin, it ws cooled using ice to the point where it s temperture ws minimum. Inside the resin is temperture probe which gives plot of the il temperture (the temperture t the centre of the resin) with time. At this point the il temperture ws ssumed to be ºC. The resin ws then plced in boiling wter nd the il temperture incresed until it reched mim, this temperture ws ssumed to be ºC Periodic ethod Using the sme setup s for the Step ethod, the resin ws cooled to ºC nd ws then plced in boiling wter. However for the periodic method, the resin ws not llowed to rech mimum il temperture, but insted the resin ws left in the boiling wter for set mount of time. At the end of this time, the resin ws once gin plced in the ice wter for the sme mount of time s before. This process ws repeted until few periods of the il temperture hd been completed.

6 4. Results 4.. Results from the Step ethod 4... Tril For this tril the resin ws cooled to the minimum il temperture nd ws then left in the boiling wter for 9 minutes to give the following plot; 6 Step ethod - Tril 4 y , m dev:.395, r E-4, b.5 ln( θ) Time/ s Figure Results of Tril of the Step ethod Using Eqution 5;.6 m s Using the stndrd methods 3 the vlue of Chi Squred is found to be; χ 88.. This is high vlue but cn esily be rectified by removing the first point. This point hs tht smllest error nd so will hve the gretest effect on reducing the vlue of Chi Squred. 6 Step ethod - Tril (with first point removed 4 y , m dev:.8, r.99.9e-4, b.4 ln( θ) Time/ s Figure Results of Tril of the Step ethod (with first point removed) Removing this point, reduces the vlue of Chi Squred to. 537, which is fr more cceptble. This gives; m.87s 3.7 m s. (By Eqution 5)

7 4... Tril Unlike the first tril, the resin ws heted to the mimum il temperture of ºC in the boiling wter nd then plced in the ice wter to observe how the il temperture vries s the resin is cooled for minutes to give the following plot; 5 Step ethod - Tril y , m dev:.56, r.979.5e-4, b ln( θ) From Eqution 5 we find tht: Time/ s Figure 3 Results of Tril of the Step ethod 5.6 m s Using the stndrd methods 3 the vlue of Chi Squred is found to be; χ 387. However this vlue of Chi squred is too lrge, it cn be decresed to more cceptble vlue by discounting the first five nd lst si points from the eperiment. 4 Step ethod - Tril (with first five nd lst si results omitted) y , m dev:.76, r.988.e-4, b ln( θ) Time/ s Figure 4 Results of Tril of the Step ethod (with first five nd lst si points removed) Removing these points gives more cceptble vlue of for Chi Squred. This gives: m.35s 5.5 m s (By Eqution 5)

8 4.. Results from the Periodic ethod Amplitude 4... etermining the il temperture For the periodic method the plot oscillted between the mimum nd minimum il tempertures. At the end of the eperiment (fter few periods hd been completed), the il temperture ws llowed to rech ºC nd then ºC. This ws done so tht the mimum nd minimum tempertures could be determined. The equivlent temperture for ech of the smllest divisions is mrked on ech plot Tril For the first tril time period of 36s (i.e. leving the resin in the ice wter for 3 minutes then in the boiling wter for 3 minutes, then repeting) The following ws recorded: θ C θ C B C T 36s ( ). 94 (By Eqution 7) To determine the vlue of the diffusivity, plot of ginst the Kelvin function ws used; 3.5 Step ethod - Kelvin Plot for Tril 3. y , m dev:.767, r.99.33, b.468 () Figure 5 Kelvin plot for tril of the Periodic ethod - Amplitude From this plot we find tht π 3.69 (By Eqution 8) T.6 m s

9 4..3. Tril For this second tril time period of s (i.e. leving the resin in the ice wter for minute then in the boiling wter for minute, then repeting) The following ws recorded: θ C θ B 6.8 C T s C ( ) (By Eqution 7) As before plot of the Kelvin function is required: 7 Step ethod - Kelvin plot for Tril 6 y , m dev:.43, r.99.48, b. () Figure 6 Kelvin plot for tril of the Periodic ethod - Amplitude From this we find tht π 4.49 (By Eqution 8) T 3.6 m s

10 4.3. Results from the Periodic ethod Phse ifference Tril As for the first tril of the Amplitude method, time period of 36s ws used first; The following ws recorded: T 36s t 6s φ π.47 rg[ ( ) ] (By Eqution 7b) 3 To determine the vlue of the diffusivity, plot of ginst the rgument of the Kelvin function ws used;.4 Argument of Kelvin Function plot. y , m dev:.3, r , b.4 rg [ ()] Figure 7 Argument of Kelvin plot for tril of the Periodic ethod - Phse From this plot we find tht π.7 (By Eqution 8) T 33.8 m s

11 4.3.. Tril Agin for this second tril time period of s ws used The following ws recorded: The following ws recorded: T s t 6s [ ( ) ] φ π 3.4 rg (By Eqution 7b) As before plot of the Kelvin function is used: 3.5 Argument of Kelvin Function plot 3.5 y , m dev:.3, r , b.985 rg [ ()] Figure 8 Argument of Kelvin plot for tril of the Periodic ethod - Phse From this we find tht π 5. (By Eqution 8) T 8.95 m s

12 4.4. Errors There re two types of errors 3 ; Systemtic errors; which re flws in the eperiment. The systemtic errors of this eperiment include:. The ssumption tht the resin would be cooled to ºC or heted to ºC, for this to ctully hppen the temperture of the ice wter would need to be less thn ºC nd the temperture of the boiling wter would need to be greter thn.ºc. The ssumption tht the het density in the ice wter nd the boiling wter were uniform. As they were not, the il temperture would vry long the is of the resin. The wy to eliminte these systemtic errors would be to use more ccurte equipment nd to use different mediums to cool nd to het the resin. For emple lower rnge of tempertures could be chieved if liquid nitrogen were used in plce of the ice wter. Rndom errors; which re imperfect redings due to the error of the equipment we cn use nd the humn error involved. These cn e ccounted for. From our bse mesurements we evluted the errors to be: o Error on dimeter (s given by Vernier Cllipers): ±.5mm o Error on temperture reding ±.5 smll squres this is different for the two trils s the squre equte to differing tempertures Clcultion of errors for the Step ethod Using the stndrd methods 3 the error for the vlue of the diffusivity s given by the Step ethod is given by; md [Eqution 5] 4λ m m + 4 d d + 4 m d [Eqution 9] m d For Tril ; For Tril :.57 m s.9 m s

13 4.4.. Clcultion of errors for the Periodic ethod - Amplitude The error for the vlue of the diffusivity s given by the Periodic ethod is given by 3 ; ( ) 4 θ θ [Eqution 7] π B θ θ B θ θ + 4 B θ θ + 4 θ θ B B [Eqution ] This gives tht. 5 for Tril nd. 3 for Tril. This error in the vlue given by the Kelvin function cn now be clculted using the Kelvin plots (see Figures 5 nd 6) with the outcome being the error in the vlue of For Tril : ( + ) ( ) Hence we cn find men error; For Tril : ( + ) ( ) Agin tking men error; As is unitless quntity, so too is the error on. By substituting nd rerrnging Eqution 8; d π [Eqution ] T

14 It is found tht 3 ; d d This rerrnges to; + d [Eqution ] d For Tril ; For Tril :. m s.5 m s Clcultion of errors for the Periodic ethod Phse ifference The error for the vlue of the diffusivity s given by the Periodic ethod is given by 3 ; t φ π rg[ ( ) ] [Eqution 7b] T rg rg [ ] [ ] t t T + T rg + t T t T [ ] rg[ ] [Eqution 3] Given the following for both trils; ± 3s t ±.5s T This gives tht [ ]. 54 for Tril nd [ ]. 57 for Tril. rg rg This error in the vlue given by the rgument of the Kelvin function cn now be clculted using the rgument Kelvin plots (see Figures 7 nd 8) with the outcome being the error in the vlue of For Tril : ( rg[ ] +.54) ( rg[ ].54) Hence we cn find men error; +. 47

15 For Tril : ( rg[ ] +.57) ( rg[ ].57) Agin tking men error; +. 5 As is unitless quntity, so too is the error on. As for the Amplitude method; + d [Eqution ] d For Tril ; For Tril :.. m s.35 m s 4.5. Finl Answer From the four trils (two in ech method), it ws found tht; Step ethod Tril ( ) 3.7 ±.57 m s Step ethod Tril ( ) 5.5 ±.9 m s Periodic ethod Amplitude Tril ( ).6 ±.3 m s Periodic ethod Amplitude Tril ( ) 3.6 ±.49 m s Periodic ethod Phse Tril ( ) 33.8 ±.. m s Periodic ethod Phse Tril ( ) 8.95 ±.35 m s Using these results weighted men 3 cn be used to rrive t finl nswer for the diffusivity; The weighted men of the diffusivity: 7.86 m s The error ssocited with this weighted men;. m s Hence the finl nswer is ( ) m 9.45 ±. s

16 5. Conclusion From the eperiment it ws found tht the vlue of the diffusivity of our cylindricl piece of resin is ( 9.45 ±.) m s A liner reltionship ws demonstrted, the grph of which hd negtive grdient, this showed greement with Eqution. In the future, the errors could be reduced to show this reltionship more clerly in number of wys. Firstly we could repet the sme method, lthough this would require much time to return ny significnt improvement in the vlue of the diffusivity. The method could be modified such tht the mesurements re more ccurte. For emple greter rnge of tempertures could be chieved by mking use of different cooling methods. Alterntively the diffusivity of the smple could be clculted if the therml conductivity, the density nd the specific het of the resin could be found using the reltionship 3 ; K ρ C 6. References. G. Stephenson, An Introduction to Prtil ifferentil Equtions for Science Students, Longmn, Esse, 988. Young & Freedmn, University Physics, Person Addison Wesley, P John R. Tylor, An Introduction to Error Anlysis: The Study of Uncertinties in Physicl esurements nd edition, University Science Books, Chrles Kittel & Herbert Kroemer, Therml Physics nd Edition, W. H. Freemn, 98

A LEVEL TOPIC REVIEW. factor and remainder theorems

A LEVEL TOPIC REVIEW. factor and remainder theorems A LEVEL TOPIC REVIEW unit C fctor nd reminder theorems. Use the Fctor Theorem to show tht: ) ( ) is fctor of +. ( mrks) ( + ) is fctor of ( ) is fctor of + 7+. ( mrks) +. ( mrks). Use lgebric division

More information

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives

Properties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums - 1 Riemnn

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite

Goals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite

More information

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student)

A-Level Mathematics Transition Task (compulsory for all maths students and all further maths student) A-Level Mthemtics Trnsition Tsk (compulsory for ll mths students nd ll further mths student) Due: st Lesson of the yer. Length: - hours work (depending on prior knowledge) This trnsition tsk provides revision

More information

MA Lesson 21 Notes

MA Lesson 21 Notes MA 000 Lesson 1 Notes ( 5) How would person solve n eqution with vrible in n eponent, such s 9? (We cnnot re-write this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this

More information

Scientific notation is a way of expressing really big numbers or really small numbers.

Scientific notation is a way of expressing really big numbers or really small numbers. Scientific Nottion (Stndrd form) Scientific nottion is wy of expressing relly big numbers or relly smll numbers. It is most often used in scientific clcultions where the nlysis must be very precise. Scientific

More information

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below.

Duality # Second iteration for HW problem. Recall our LP example problem we have been working on, in equality form, is given below. Dulity #. Second itertion for HW problem Recll our LP emple problem we hve been working on, in equlity form, is given below.,,,, 8 m F which, when written in slightly different form, is 8 F Recll tht we

More information

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).

2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ). AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following

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

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

Unit 1 Exponentials and Logarithms

Unit 1 Exponentials and Logarithms HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10)

More information

(i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation

(i) b b. (ii) (iii) (vi) b. P a g e Exponential Functions 1. Properties of Exponents: Ex1. Solve the following equation P g e 30 4.2 Eponentil Functions 1. Properties of Eponents: (i) (iii) (iv) (v) (vi) 1 If 1, 0 1, nd 1, then E1. Solve the following eqution 4 3. 1 2 89 8(2 ) 7 Definition: The eponentil function with se

More information

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION

THERMAL EXPANSION COEFFICIENT OF WATER FOR VOLUMETRIC CALIBRATION XX IMEKO World Congress Metrology for Green Growth September 9,, Busn, Republic of Kore THERMAL EXPANSION COEFFICIENT OF WATER FOR OLUMETRIC CALIBRATION Nieves Medin Hed of Mss Division, CEM, Spin, mnmedin@mityc.es

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

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation

Sections 1.3, 7.1, and 9.2: Properties of Exponents and Radical Notation Sections., 7., nd 9.: Properties of Eponents nd Rdicl Nottion Let p nd q be rtionl numbers. For ll rel numbers nd b for which the epressions re rel numbers, the following properties hold. i = + p q p q

More information

Chapter 1: Logarithmic functions and indices

Chapter 1: Logarithmic functions and indices Chpter : Logrithmic functions nd indices. You cn simplify epressions y using rules of indices m n m n m n m n ( m ) n mn m m m m n m m n Emple Simplify these epressions: 5 r r c 4 4 d 6 5 e ( ) f ( ) 4

More information

than 1. It means in particular that the function is decreasing and approaching the x-

than 1. It means in particular that the function is decreasing and approaching the x- 6 Preclculus Review Grph the functions ) (/) ) log y = b y = Solution () The function y = is n eponentil function with bse smller thn It mens in prticulr tht the function is decresing nd pproching the

More information

Pressure Wave Analysis of a Cylindrical Drum

Pressure Wave Analysis of a Cylindrical Drum Pressure Wve Anlysis of Cylindricl Drum Chris Clrk, Brin Anderson, Brin Thoms, nd Josh Symonds Deprtment of Mthemtics The University of Rochester, Rochester, NY 4627 (Dted: December, 24 In this pper, hypotheticl

More information

Designing Information Devices and Systems I Spring 2018 Homework 7

Designing Information Devices and Systems I Spring 2018 Homework 7 EECS 16A Designing Informtion Devices nd Systems I Spring 2018 omework 7 This homework is due Mrch 12, 2018, t 23:59. Self-grdes re due Mrch 15, 2018, t 23:59. Sumission Formt Your homework sumission should

More information

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors

Higher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rel-life exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector

More information

SUMMER KNOWHOW STUDY AND LEARNING CENTRE

SUMMER KNOWHOW STUDY AND LEARNING CENTRE SUMMER KNOWHOW STUDY AND LEARNING CENTRE Indices & Logrithms 2 Contents Indices.2 Frctionl Indices.4 Logrithms 6 Exponentil equtions. Simplifying Surds 13 Opertions on Surds..16 Scientific Nottion..18

More information

Lecture 13 - Linking E, ϕ, and ρ

Lecture 13 - Linking E, ϕ, and ρ Lecture 13 - Linking E, ϕ, nd ρ A Puzzle... Inner-Surfce Chrge Density A positive point chrge q is locted off-center inside neutrl conducting sphericl shell. We know from Guss s lw tht the totl chrge on

More information

Logarithmic Functions

Logarithmic Functions Logrithmic Functions Definition: Let > 0,. Then log is the number to which you rise to get. Logrithms re in essence eponents. Their domins re powers of the bse nd their rnges re the eponents needed to

More information

DERIVATIVES NOTES HARRIS MATH CAMP Introduction

DERIVATIVES NOTES HARRIS MATH CAMP Introduction f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we

More information

Interpreting Integrals and the Fundamental Theorem

Interpreting Integrals and the Fundamental Theorem Interpreting Integrls nd the Fundmentl Theorem Tody, we go further in interpreting the mening of the definite integrl. Using Units to Aid Interprettion We lredy know tht if f(t) is the rte of chnge of

More information

Chapter 0. What is the Lebesgue integral about?

Chapter 0. What is the Lebesgue integral about? Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous

More information

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?

1 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum? Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? frequency speed (in vcuum) decreses decreses decreses remins constnt

More information

AP Calculus. Fundamental Theorem of Calculus

AP Calculus. Fundamental Theorem of Calculus AP Clculus Fundmentl Theorem of Clculus Student Hndout 16 17 EDITION Click on the following link or scn the QR code to complete the evlution for the Study Session https://www.surveymonkey.com/r/s_sss Copyright

More information

CHAPTER 20: Second Law of Thermodynamics

CHAPTER 20: Second Law of Thermodynamics CHAER 0: Second Lw of hermodynmics Responses to Questions 3. kg of liquid iron will hve greter entropy, since it is less ordered thn solid iron nd its molecules hve more therml motion. In ddition, het

More information

Chapter 6 Notes, Larson/Hostetler 3e

Chapter 6 Notes, Larson/Hostetler 3e Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn

More information

Physics 1402: Lecture 7 Today s Agenda

Physics 1402: Lecture 7 Today s Agenda 1 Physics 1402: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

Physics 1502: Lecture 7 Today s Agenda

Physics 1502: Lecture 7 Today s Agenda 1 Physics 1502: Lecture 7 Tody s gend nnouncements: Lectures posted on: www.phys.uconn.edu/~rcote/ HW ssignments, solutions etc. Homework #2: On Msterphysics tody: due Fridy Go to msteringphysics.com Ls:

More information

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.

Logarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100. Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the

More information

Name Class Date. Match each phrase with the correct term or terms. Terms may be used more than once.

Name Class Date. Match each phrase with the correct term or terms. Terms may be used more than once. Exercises 341 Flow of Chrge (pge 681) potentil difference 1 Chrge flows when there is between the ends of conductor 2 Explin wht would hppen if Vn de Grff genertor chrged to high potentil ws connected

More information

Math 124A October 04, 2011

Math 124A October 04, 2011 Mth 4A October 04, 0 Viktor Grigoryn 4 Vibrtions nd het flow In this lecture we will derive the wve nd het equtions from physicl principles. These re second order constnt coefficient liner PEs, which model

More information

Exponents and Logarithms Exam Questions

Exponents and Logarithms Exam Questions Eponents nd Logrithms Em Questions Nme: ANSWERS Multiple Choice 1. If 4, then is equl to:. 5 b. 8 c. 16 d.. Identify the vlue of the -intercept of the function ln y.. -1 b. 0 c. d.. Which eqution is represented

More information

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve

4 7x =250; 5 3x =500; Read section 3.3, 3.4 Announcements: Bell Ringer: Use your calculator to solve Dte: 3/14/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: Use your clcultor to solve 4 7x =250; 5 3x =500; HW Requests: Properties of Log Equtions

More information

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac

ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b

More information

Chapter 9 Definite Integrals

Chapter 9 Definite Integrals Chpter 9 Definite Integrls In the previous chpter we found how to tke n ntiderivtive nd investigted the indefinite integrl. In this chpter the connection etween ntiderivtives nd definite integrls is estlished

More information

Fundamentals of Analytical Chemistry

Fundamentals of Analytical Chemistry Homework Fundmentls of nlyticl hemistry hpter 9 0, 1, 5, 7, 9 cids, Bses, nd hpter 9(b) Definitions cid Releses H ions in wter (rrhenius) Proton donor (Bronsted( Lowry) Electron-pir cceptor (Lewis) hrcteristic

More information

Math 153: Lecture Notes For Chapter 5

Math 153: Lecture Notes For Chapter 5 Mth 5: Lecture Notes For Chpter 5 Section 5.: Eponentil Function f()= Emple : grph f ) = ( if = f() 0 - - - - - - Emple : Grph ) f ( ) = b) g ( ) = c) h ( ) = ( ) f() g() h() 0 0 0 - - - - - - - - - -

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

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available.

A formula sheet and table of physical constants is attached to this paper. Linear graph paper is available. DEPARTMENT OF PHYSICS AND ASTRONOMY Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Liner grph pper is vilble. Spring Semester 2015-2016 PHYSICS 1 HOUR Answer questions

More information

Operations with Polynomials

Operations with Polynomials 38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: How to identify the leding coefficients nd degrees of polynomils How to dd nd subtrct polynomils How to multiply polynomils

More information

Chapter 3 Exponential and Logarithmic Functions Section 3.1

Chapter 3 Exponential and Logarithmic Functions Section 3.1 Chpter 3 Eponentil nd Logrithmic Functions Section 3. EXPONENTIAL FUNCTIONS AND THEIR GRAPHS Eponentil Functions Eponentil functions re non-lgebric functions. The re clled trnscendentl functions. The eponentil

More information

and that at t = 0 the object is at position 5. Find the position of the object at t = 2.

and that at t = 0 the object is at position 5. Find the position of the object at t = 2. 7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we

More information

11.1 Exponential Functions

11.1 Exponential Functions . Eponentil Functions In this chpter we wnt to look t specific type of function tht hs mny very useful pplictions, the eponentil function. Definition: Eponentil Function An eponentil function is function

More information

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin Trditionl 10 6 Conclusion 1 Introduction An Egyptin frction

More information

Read section 3.3, 3.4 Announcements:

Read section 3.3, 3.4 Announcements: Dte: 3/1/13 Objective: SWBAT pply properties of exponentil functions nd will pply properties of rithms. Bell Ringer: 1. f x = 3x 6, find the inverse, f 1 x., Using your grphing clcultor, Grph 1. f x,f

More information

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. is a real number.

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. is a real number. The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points on grph, the pproprite clcultor

More information

3.1 Exponential Functions and Their Graphs

3.1 Exponential Functions and Their Graphs . Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.

More information

Probabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods

Probabilistic Investigation of Sensitivities of Advanced Test- Analysis Model Correlation Methods Probbilistic Investigtion of Sensitivities of Advnced Test- Anlysis Model Correltion Methods Liz Bergmn, Mtthew S. Allen, nd Dniel C. Kmmer Dept. of Engineering Physics University of Wisconsin-Mdison Rndll

More information

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer.

The semester B examination for Algebra 2 will consist of two parts. Part 1 will be selected response. Part 2 will be short answer. ALGEBRA B Semester Em Review The semester B emintion for Algebr will consist of two prts. Prt will be selected response. Prt will be short nswer. Students m use clcultor. If clcultor is used to find points

More information

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS

12 TRANSFORMING BIVARIATE DENSITY FUNCTIONS 1 TRANSFORMING BIVARIATE DENSITY FUNCTIONS Hving seen how to trnsform the probbility density functions ssocited with single rndom vrible, the next logicl step is to see how to trnsform bivrite probbility

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

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS

ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS ADVANCEMENT OF THE CLOSELY COUPLED PROBES POTENTIAL DROP TECHNIQUE FOR NDE OF SURFACE CRACKS F. Tkeo 1 nd M. Sk 1 Hchinohe Ntionl College of Technology, Hchinohe, Jpn; Tohoku University, Sendi, Jpn Abstrct:

More information

13: Diffusion in 2 Energy Groups

13: Diffusion in 2 Energy Groups 3: Diffusion in Energy Groups B. Rouben McMster University Course EP 4D3/6D3 Nucler Rector Anlysis (Rector Physics) 5 Sept.-Dec. 5 September Contents We study the diffusion eqution in two energy groups

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

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations.

Lecture 3. In this lecture, we will discuss algorithms for solving systems of linear equations. Lecture 3 3 Solving liner equtions In this lecture we will discuss lgorithms for solving systems of liner equtions Multiplictive identity Let us restrict ourselves to considering squre mtrices since one

More information

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007

A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007 A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus

More information

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction

UNIT 1 FUNCTIONS AND THEIR INVERSES Lesson 1.4: Logarithmic Functions as Inverses Instruction Lesson : Logrithmic Functions s Inverses Prerequisite Skills This lesson requires the use of the following skills: determining the dependent nd independent vribles in n exponentil function bsed on dt from

More information

Recitation 3: More Applications of the Derivative

Recitation 3: More Applications of the Derivative Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech

More information

Hung problem # 3 April 10, 2011 () [4 pts.] The electric field points rdilly inwrd [1 pt.]. Since the chrge distribution is cylindriclly symmetric, we pick cylinder of rdius r for our Gussin surfce S.

More information

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1

Algebra Readiness PLACEMENT 1 Fraction Basics 2 Percent Basics 3. Algebra Basics 9. CRS Algebra 1 Algebr Rediness PLACEMENT Frction Bsics Percent Bsics Algebr Bsics CRS Algebr CRS - Algebr Comprehensive Pre-Post Assessment CRS - Algebr Comprehensive Midterm Assessment Algebr Bsics CRS - Algebr Quik-Piks

More information

25 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum?

25 Which of the following summarises the change in wave characteristics on going from infra-red to ultraviolet in the electromagnetic spectrum? PhysicsndMthsTutor.com 25 Which of the following summrises the chnge in wve chrcteristics on going from infr-red to ultrviolet in the electromgnetic spectrum? 972//M/J/2 frequency speed (in vcuum) decreses

More information

Using air lines as references for VNA phase measurements

Using air lines as references for VNA phase measurements Using ir lines s references for VNA phse mesurements Stephen Protheroe nd Nick Ridler Electromgnetics Tem, Ntionl Physicl Lbortory, UK Emil: Stephen.protheroe@npl.co.uk Abstrct Air lines re often used

More information

Main topics for the Second Midterm

Main topics for the Second Midterm Min topics for the Second Midterm The Midterm will cover Sections 5.4-5.9, Sections 6.1-6.3, nd Sections 7.1-7.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the

More information

Lesson 1: Quadratic Equations

Lesson 1: Quadratic Equations Lesson 1: Qudrtic Equtions Qudrtic Eqution: The qudrtic eqution in form is. In this section, we will review 4 methods of qudrtic equtions, nd when it is most to use ech method. 1. 3.. 4. Method 1: Fctoring

More information

SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive)

SESSION 2 Exponential and Logarithmic Functions. Math 30-1 R 3. (Revisit, Review and Revive) Mth 0-1 R (Revisit, Review nd Revive) SESSION Eponentil nd Logrithmic Functions 1 Eponentil nd Logrithmic Functions Key Concepts The Eponent Lws m n 1 n n m m n m n m mn m m m m mn m m m b n b b b Simplify

More information

Math 135, Spring 2012: HW 7

Math 135, Spring 2012: HW 7 Mth 3, Spring : HW 7 Problem (p. 34 #). SOLUTION. Let N the number of risins per cookie. If N is Poisson rndom vrible with prmeter λ, then nd for this to be t lest.99, we need P (N ) P (N ) ep( λ) λ ln(.)

More information

Appendix 3, Rises and runs, slopes and sums: tools from calculus

Appendix 3, Rises and runs, slopes and sums: tools from calculus Appendi 3, Rises nd runs, slopes nd sums: tools from clculus Sometimes we will wnt to eplore how quntity chnges s condition is vried. Clculus ws invented to do just this. We certinly do not need the full

More information

ES.182A Topic 32 Notes Jeremy Orloff

ES.182A Topic 32 Notes Jeremy Orloff ES.8A Topic 3 Notes Jerem Orloff 3 Polr coordintes nd double integrls 3. Polr Coordintes (, ) = (r cos(θ), r sin(θ)) r θ Stndrd,, r, θ tringle Polr coordintes re just stndrd trigonometric reltions. In

More information

fractions Let s Learn to

fractions Let s Learn to 5 simple lgebric frctions corne lens pupil retin Norml vision light focused on the retin concve lens Shortsightedness (myopi) light focused in front of the retin Corrected myopi light focused on the retin

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

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17

Discrete Mathematics and Probability Theory Spring 2013 Anant Sahai Lecture 17 EECS 70 Discrete Mthemtics nd Proility Theory Spring 2013 Annt Shi Lecture 17 I.I.D. Rndom Vriles Estimting the is of coin Question: We wnt to estimte the proportion p of Democrts in the US popultion,

More information

Math 113 Exam 2 Practice

Math 113 Exam 2 Practice Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.-7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number

More information

Lecture 3 Gaussian Probability Distribution

Lecture 3 Gaussian Probability Distribution Introduction Lecture 3 Gussin Probbility Distribution Gussin probbility distribution is perhps the most used distribution in ll of science. lso clled bell shped curve or norml distribution Unlike the binomil

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

p-adic Egyptian Fractions

p-adic Egyptian Fractions p-adic Egyptin Frctions Tony Mrtino My 7, 20 Theorem 9 negtiveorder Theorem 11 clss2 Contents 1 Introduction 1 2 Trditionl Egyptin Frctions nd Greedy Algorithm 2 3 Set-up 3 4 p-greedy Algorithm 5 5 p-egyptin

More information

The Moving Center of Mass of a Leaking Bob

The Moving Center of Mass of a Leaking Bob The Moving Center of Mss of Leking Bob rxiv:1002.956v1 [physics.pop-ph] 21 Feb 2010 P. Arun Deprtment of Electronics, S.G.T.B. Khls College University of Delhi, Delhi 110 007, Indi. Februry 2, 2010 Abstrct

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

Exponentials & Logarithms Unit 8

Exponentials & Logarithms Unit 8 U n i t 8 AdvF Dte: Nme: Eponentils & Logrithms Unit 8 Tenttive TEST dte Big ide/lerning Gols This unit begins with the review of eponent lws, solving eponentil equtions (by mtching bses method nd tril

More information

First Semester Review Calculus BC

First Semester Review Calculus BC First Semester Review lculus. Wht is the coordinte of the point of inflection on the grph of Multiple hoice: No lcultor y 3 3 5 4? 5 0 0 3 5 0. The grph of piecewise-liner function f, for 4, is shown below.

More information

Name Solutions to Test 3 November 8, 2017

Name Solutions to Test 3 November 8, 2017 Nme Solutions to Test 3 November 8, 07 This test consists of three prts. Plese note tht in prts II nd III, you cn skip one question of those offered. Some possibly useful formuls cn be found below. Brrier

More information

Bridging the gap: GCSE AS Level

Bridging the gap: GCSE AS Level Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions

More information

MATHEMATICS AND STATISTICS 1.2

MATHEMATICS AND STATISTICS 1.2 MATHEMATICS AND STATISTICS. Apply lgebric procedures in solving problems Eternlly ssessed 4 credits Electronic technology, such s clcultors or computers, re not permitted in the ssessment of this stndr

More information

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses

Chapter 9: Inferences based on Two samples: Confidence intervals and tests of hypotheses Chpter 9: Inferences bsed on Two smples: Confidence intervls nd tests of hypotheses 9.1 The trget prmeter : difference between two popultion mens : difference between two popultion proportions : rtio of

More information

Loudoun Valley High School Calculus Summertime Fun Packet

Loudoun Valley High School Calculus Summertime Fun Packet Loudoun Vlley High School Clculus Summertime Fun Pcket We HIGHLY recommend tht you go through this pcket nd mke sure tht you know how to do everything in it. Prctice the problems tht you do NOT remember!

More information

CONTRIBUTION TO THE EXTENDED DYNAMIC PLANE SOURCE METHOD

CONTRIBUTION TO THE EXTENDED DYNAMIC PLANE SOURCE METHOD CONTRIBUTION TO THE EXTENDED DYNAMIC PLANE SOURCE METHOD Svetozár Mlinrič Deprtment of Physics, Fculty of Nturl Sciences, Constntine the Philosopher University, Tr. A. Hlinku, SK-949 74 Nitr, Slovki Emil:

More information

University of Alabama Department of Physics and Astronomy. PH126: Exam 1

University of Alabama Department of Physics and Astronomy. PH126: Exam 1 University of Albm Deprtment of Physics nd Astronomy PH 16 LeClir Fll 011 Instructions: PH16: Exm 1 1. Answer four of the five questions below. All problems hve equl weight.. You must show your work for

More information

CHM Physical Chemistry I Chapter 1 - Supplementary Material

CHM Physical Chemistry I Chapter 1 - Supplementary Material CHM 3410 - Physicl Chemistry I Chpter 1 - Supplementry Mteril For review of some bsic concepts in mth, see Atkins "Mthemticl Bckground 1 (pp 59-6), nd "Mthemticl Bckground " (pp 109-111). 1. Derivtion

More information

Tests for the Ratio of Two Poisson Rates

Tests for the Ratio of Two Poisson Rates Chpter 437 Tests for the Rtio of Two Poisson Rtes Introduction The Poisson probbility lw gives the probbility distribution of the number of events occurring in specified intervl of time or spce. The Poisson

More information

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that

( dg. ) 2 dt. + dt. dt j + dh. + dt. r(t) dt. Comparing this equation with the one listed above for the length of see that Arc Length of Curves in Three Dimensionl Spce If the vector function r(t) f(t) i + g(t) j + h(t) k trces out the curve C s t vries, we cn mesure distnces long C using formul nerly identicl to one tht we

More information

Measuring Electron Work Function in Metal

Measuring Electron Work Function in Metal n experiment of the Electron topic Mesuring Electron Work Function in Metl Instructor: 梁生 Office: 7-318 Emil: shling@bjtu.edu.cn Purposes 1. To understnd the concept of electron work function in metl nd

More information

Math 1B, lecture 4: Error bounds for numerical methods

Math 1B, lecture 4: Error bounds for numerical methods Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the

More information

MEASUREMENTS AND UNCERTAINTIES

MEASUREMENTS AND UNCERTAINTIES 978--7-49575-3 Physics for the IB Diplom Exm Preprtion Guide MEASUREMENTS AND UNCERTAINTIES This chpter covers the following topics: Fundmentl nd derived units Significnt figures nd scientific nottion

More information

Recitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications

Recitation 3: Applications of the Derivative. 1 Higher-Order Derivatives and their Applications Mth 1c TA: Pdric Brtlett Recittion 3: Applictions of the Derivtive Week 3 Cltech 013 1 Higher-Order Derivtives nd their Applictions Another thing we could wnt to do with the derivtive, motivted by wht

More information

Mathematics Number: Logarithms

Mathematics Number: Logarithms plce of mind F A C U L T Y O F E D U C A T I O N Deprtment of Curriculum nd Pedgogy Mthemtics Numer: Logrithms Science nd Mthemtics Eduction Reserch Group Supported y UBC Teching nd Lerning Enhncement

More information