# 1.3 The log laws. Inverse functions. We have defined logx as the index needed to write x as a power of 10. For example:

Save this PDF as:

Size: px
Start display at page:

Download "1.3 The log laws. Inverse functions. We have defined logx as the index needed to write x as a power of 10. For example:"

## Transcription

1 1.3 The log lws Inverse functions. We hve defined log s the inde needed to write s power of. For emple: since 1.8 = we hve log66.07 = 1.8 Note tht wht this sys is tht the functions f() = nd g() = log re inverse functions: mps 1.8 into nd log mps into 1.8. Putting this into digrm: : log: Tht s the ig ide: tht tking logs is the inverse of eponentition. nother wy to epress this is to write log =. If you put positive numer into your clcultor nd thn hit it with the LOG utton nd the utton, you ll get ck gin. This is lovely section. It hs ig ide (log nd ep re inverses) nd ig technique (crving log lws out of ep lws) nd when these re put together we get some cool formule. So there s lot going on nd tht cn e intimidting. Here s some good dvice: sty close to the numers. Notice how I hve done tht. Insted of using too mny symols right wy, I hve focused ttention on the reltionship etween the numers nd The log lws. Whenever new function comes into your life, one of the first things you wnt to know is how does it work? Wht cn I do with it? Wht properties does it hve? In prticulr, we re lwys interested in lgeric properties s these llow us to rewrite epressions nd solve equtions. For emple, re there ny nice lws round? Wht cn you sy out log(+y), log(y), log( r ), etc. Well, let s use the ig ide ove: log is the inverse of. We hve nice lgeric lws for the eponent lws. If we invert these we d surely get some nice lgeric lws for log. Let s try it. Emple 1. (The product lw) Tke the ddition lw for eponents: + = Wht log lw cn you etrct from tht? How should your thinking run? Well, you hve to keep hold of the ig picture. The two functions we re working with here, nd log, re inverses. Tht mens tht the inputs for one re going to e the outputs for the other. In the lw of eponents tht I wrote ove, the nd re the inputs for so they will e outputs for log. However, the log lws we re looking for will hve to involve the inputs of log nd these won t e nd, rther they ll e the outputs of, which re nd. So, to effectively use these s inputs, we hve to give them nmes nd those nmes won t e nd. Tht s the key first step, nd tht s how the following rgument hs to egin. This is n ecellent prolem to try to get the clss to tke on, ll working together with the techer coordinting from the front. Students find this tsk difficult. In mny wys, however, the prolem is essentilly nottionl. For emple if I strt them off using nd s rguments for, they ll wnt to use nd s rguments for log() s well, nd tht will surely give them grief. The key is to choose good nottion. log lws 9/13/007 1

2 The rgument. The ddition lw for eponents is + = nd t the right I hve put it in o. I like this o representtion ecuse it emphsizes the function cting s trnsformtion nd tht gets you thinking: if I do such nd such to the inputs, wht hppens to the outputs? In this cse, the o sys tht if the inputs re dded, the outputs re multiplied. Now the function log() is the inverse of, so it cn use the sme digrm ut the rrows should e reversed. Tht mens tht the ojects on the right re the inputs so we need to give them nmes. Let s set: = = B. The new version of the o is the second one t the right, nd underneth tht is the log() o with the rrows going the other wy. The first o sys tht the function mps sums into products nd the second o sys tht the function log() mps products into sums. Tht s the multipliction lw for logrithms: log(b) = log() + log(b) log( ) B B B B This two-o setup is wonderful. It encpsultes the ck nd forth etween the two functions, nd, for my money, offers just the right proof of the log-lws. Emple. (The eponent lw). If we tke the product lw for logs. nd set =B, we get significnt vrint. log() = log() + log() log( ) = log(). This generlizes redily to ny positive integer, n: log( n ) = nlog(). In fct n cn e ny rel numer r. The rgument is contined in the two oes t the right. The first o displys the eponentil power lw: r = ( ) r nd the second o presents the corresponding log lw: r r log( ) r r log( r ) = rlog(). The log lws log(y) = log() + log(y) log( r ) = rlog() log(/y) = log() log(y) log(1/) = log() The lst two, the quotient lw nd the reciprocl lw, cn e derived from the first two, ut they re stted for convenience. log lws 9/13/007

3 Emple 3. Solve the eqution: 0 = 0. Solution Tke logs of oth sides: log(0 ) = log0. log(0) = log0. log0 =.37 log0 We could hve solved this eqution nd those elow in the lst section, ut with more sic pproch converting everything to se. The power lw for logrithms gives us welcome short-cut. In the previous section we did not hve the log-lws ut we could still hve solved this eqution y writing everything s power of : log(0) = = log(0) = = Then the eqution ecomes: 1.30 ( ) = The eponentil power lw gives us: 1.30 = = Emple 4. Solve the eqution 3 = +1. Solution. Tke logs of oth sides: log3 = log +1 = log3 = (+1)log (log3 log)= log log log3 log Emple 5. popultion grows t the rte of 4% per dy. Over wht period will it grow y 40%? Solution. The 1-dy multiplier is The -dy multiplier will e (1.04). We wnt to choose so this is 1.40 (ecuse n increse of 40% is the sme multipliction y 1.40) = 1.40 Tke logs of oth sides: log(1.04 ) = log(1.40) log(1.04) = log(1.40) log1.40 = 8.58 log1.04 It will grow y 40% over period of 8.58 dys. I egin with n estimte. This will give me good check on my nswer t the end. If I put ten 4% increses together one fter the other I ll get 40% increse. Right? Wrong! Becuse of the compounding, I ll get n increse of more thn 40%. So tht tells me tht the time needed for n increse of ectly 40% is just it less thn dys nd our nswer is consistent with tht. log lws 9/13/007 3

4 Emple 6. The intensity of em of light pssing through murky wter decys with distnce trveled t constnt percentge rte. Suppose the em loses 5% every centimeter. Over wht distnce will its intensity e cut in hlf? Solution. loss of 5% is the sme s multipliction y 0.95, so this is the 1-cm multiplier. Then the -cm multiplier is (0.95). We wnt to choose so tht this equls (0.95) = log(0.95 ) = log0.50. log(0.95) = log0.50. = log0.50 log gin strt with n estimte. If I put ten 5% decreses together one fter the other I ll get 50% increse. Right? Wrong! Since the intensity is decresing, the loss over ech cm will decrese, nd over cm the loss will e less thn 50 cm. So tht tells me tht the distnce needed for decrese of 50% is more thn cm. The intensity will e cut in hlf over 13.5 cm. Emple 7. Solve the eqution Solution. Tke logs of oth sides: + 1 = 0. ( +1 ) log = log 0 log 0 +1 = log log = log log 0 = log I cn lmost gurntee tht hlf my clss will egin +1 = 400. Is tht vlid? Wht hve they done? They ll puse on tht, not ectly sure wht they hve done. I guess I ve squred oth sides. Relly? These prolems re esy if you follow the rules, ut follow the rules! This is good prolem in tht it forces the student to nlyze the structure of the epression. When my students first look t they see + 1 two nsties (or goodies?) squre root nd n eponentition. So there two things to e undone, ut which do we undo first? Do we squre oth sides or tke the log of oth sides? They look t me pledingly, witing to e told. I find the following conceptuliztion useful. Think of the epression + 1 s the effect of putting in sequence of oes, ech one contining the one efore. nd to identify the sequence, you imgine you + 1 hve entered in your clcultor, nd you hve to clculte y pushing the right sequence of uttons. Wht do you do? Well, you dd 1, nd then you tke the squre root nd then you eponentite with se. Ech utton corresponds to putting wht you lredy hve into new o. Now to solve the eqution you wnt to unwrp nd you do tht y opening the oes one t time. nd which o do you open first? the outside one, the lst one you closed up. So which step do you undo first? the eponentition. So the first thing to do is tke the log of oth sides. Prolems + 1 Constructing out of Step # Opertion Result 0 Begin 1 dd 1 +1 Squre root Ep (se ) + 1 log lws 9/13/007 4

5 1. Without the use of clcultor, simplify the following epressions. ll logs re se. log13 0 log13 log(0 13 ) (log0) 13 log(1/0) log4 + log5 log0 log 1/ 4 log + log(500 ). Solve the following equtions for. ll logs re se. log + 1 =. log ( +) = 1 = = 0 (log) 5 = 0 3. If log(178) =, wht is log(1.78) in terms of? If log(c) = 1.78, wht is 17.8 in terms of c? 4. The following identities re ll invlid. Mny rise from tking two trnsformtions nd pplying them one fter the other to, first in one order nd then in the other, nd setting the two results equl. This seldom works. However, it s lwys possile tht there re one or two vlues of for which the eqution just hppens to e true. In ech cse, find ll such vlues of. Your nswers should e written s simply s possile. () = () +3 = + 3 (c) 3 = 3 (d) log(3) = 3 log (e) / ( ) = / (f) = ( ) (g) log(3) = (log3)(log) (h) log(3+) = log3 + log (h) log( 3 ) = (log) 3 (i)* log( ) = log log lws 9/13/007 5

6 5.() n eponentilly growing popultion triples in size in 3 yers. How long does it tke to doule in size? () n eponentilly growing popultion doules in size every 7 dys. Wht is the dily percentge increse? How long does it tke to triple in size? To qudruple? 6. n eponentilly growing popultion increses in size y 4% every 4 hours. How long does it tke to increse in size y 8%? By %? Wht is the douling time (the time needed to doule in size)? 7. My tire hs slow lek nd loses pressure t constnt percentge rte. t noon it hs 400 kp nd t 1 pm this hs dropped to 360 kp. t wht time will it get to 0 kp? 8. ttery loses 5% of its power for every hour of use. It needs to e rechrged when its potency hs dropped to ¼ of its fully chrged vlue. For how mny hours cn it e used etween rechrgings? 9. The intensity of em of light pssing through murky wter decys with distnce trveled t constnt percentge rte. Suppose em loses 0% every meter. Over wht distnce will its intensity e cut y 90%?. In my discount wrehouse I cn sell 50 Trgiclly Hip disks per week t the regulr price of \$, nd the numer of sles will increse y 8% for every % decrese in price. How mny will I sell per week if I price them t 0% off? 11. Two sunflowers grow side y side, oth incresing in height t constnt percentge rte ech dy. Sm strts t height cm nd hs growth rte 5% per dy, nd Corl strts t height 5 cm. () How long until Sm is 1 meter tll? () fter 0 dys, Sm nd Corl re the sme height. Wht is Corl s percentge growth rte? 1. For yers I've hd n eponentilly decying popultion quietly eeking out the remins of its nturl life in the hollow trunk of my old ok tree. I rememer one frosty Christms some time go when the popultion decresed y % in one week nd then y 0 g during the very net week. But just this morning I went out nd found decrese of only 8 g since my lst mesurement 7 dys go. How deep is the snow this morning? 13. n eponentilly growing popultion requires two dys to increse y 0 grms, ut needs only one dy for the net 0 grm increse. How long will it need for the third 0 grm increse? 14. decreses in vlue t the rte of % per yer wheres B decreses in vlue t the rte of 5% per yer. If strts off twice s ig s B, how long until is hlf the size of B? 15. The vlue z of n le Colville pinting increses y % every 18 months. If I plot log(z) ginst time, I will get stright line. Find the slope of this line. [Mesure time in yers nd use se for the logrithm.] 16. My ike tire hs tiny hole nd I notice tht every dy it loses % of its pressure P. If I plot log(p) ginst time, I will get stright line. Find the slope of this line. [Mesure time in dys nd use se for the logrithm.] 17. My ike tire hs tiny hole nd I notice tht the pressure P is cut in hlf every 5 dys. If I plot log(p) ginst time, I will get stright line. Find the slope of this line. [Mesure time in dys nd use se for the logrithm.] log lws 9/13/007 6

7 Multipliction through ddition The logrithm Put yourself ck in the yer You re Johnnes Kepler nd you re ttempting to clculte the orit of the plnet Venus. You hve msses of oservtionl dt to work with, collected y the dedicted efforts of your predecessor Tycho Brhe, nd you re driven y your own ingenious nd wesomely simple theoreticl constructions. But for ll tht inspirtion, the huge ulk of your work is not the hppy reordering of eutiful conceptul structures, ut hour fter hour of tedious rithmeticl mnipultions, done with s much ccurcy s the given dt will llow. Now imgine tht collegue comes long nd gives you simple scientific clcultor, nd shows you how to multiply two 6-digit numers t the touch of utton. Imgine how you would feel, eing given such unimginle computtionl power. You d e lown wy, right out into the orit of one of your eloved plnets. Well tht didn t hppen, ut something just s spectculr, t lest to Kepler, did. In 1614 Scottish lnd ron nmed John Npier pulished n etensive tle which reduced the multipliction of two 7-digit numers to much simpler ddition. He ws t the time 64 yers old, nd he hd spent the pst 0 yers of his life on the construction of this tle. To Kepler, this tle ws mircle nd ccording to Lplce, it proly douled the mount of scientific work he ws le to do. The ide ehind Npier s tle ws simple enough. [Interestingly enough, the ide ehind lmost ll revolutionry dvnces is simple.] Tke numer c tht s just ove 1, nd consider the sequence: 1, c, c, c 3, c 4, c 5, c 6, c 7, c 8, c 9, c, c 11, Suppose you hd two numers in tht list, sy c 3 nd c 7, nd you wnted to multiply then together. Well, if you hd possession of the list you d simply hve to know the indices tht they elonged to, 3 nd 7, nd dd them together to get, nd then look up the numer c. So possession of the tle would llow you to multiply ny two numers in the tle essentilly y dding their indices. Now if the se numer c were very close to 1, the numers c r would grow slowly nd the list would contin firly dense pcking of numers, so if you strted with ny two numers which you wnted to multiply, you could find two numers in the list which were very close, nd multiply them. Tht s essentilly how Npier s tle looked nd the reson it took him 0 yers is ecuse he hd to do ll those clcultions y hnd. n interesting wy to look t wht hppened is tht one mn did huge numer of tedious clcultions ut recorded the results in systemtic wy tht llowed ny numer of others to do their clcultions much more esily. Within few yers n importnt simplifiction (using se nd frctionl eponents) ws introduced y n English geometer Henry Briggs nd the tle ws rewritten, giving us the form of the Tle of Common Logrithms which I ws required to worked with s high school student in the 1950 s, 300 yer fter the irth of Npier.. This invention of Npier cme out of the lue. Nothing seems to hve prefigured it. But once pulished, its use spred quickly throughout the world. Within few yers it ws relized tht the ide could e incorported into mechnicl device nd hence ws orn the slide rule, which ws the hnd clcultor which we ll used s university students in the 60 s. Mine cme in lether cse with slot through which elt could pss, nd thus we crried this device round with us s we went from clss to clss just s clcultors re crried round tody. There re still lots of slide rules round in old drwers nd cupords nd oes. sk your mth techer or your prent to find you one nd see if you cn figure out wht they hve to do with logrithms. We gve tht chllenge to our students nd they found it difficult. new ending to n old joke. God sid to the cretures in the Grden, Go forth nd multiply! to which the dders replied: We cn t, we re dders. nd so tht the dders might thrive, God creted Npier. log lws 9/13/007 7

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

XII. LINEAR ALGEBRA: SOLVING SYSTEMS OF EQUATIONS Tody we re going to tlk out solving systems of liner equtions. These re prolems tht give couple of equtions with couple of unknowns, like: 6= x + x 7=

### 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

### 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

### 10. AREAS BETWEEN CURVES

. AREAS BETWEEN CURVES.. Ares etween curves So res ove the x-xis re positive nd res elow re negtive, right? Wrong! We lied! Well, when you first lern out integrtion it s convenient fiction tht s true in

### Section 4: Integration ECO4112F 2011

Reding: Ching Chpter Section : Integrtion ECOF Note: These notes do not fully cover the mteril in Ching, ut re ment to supplement your reding in Ching. Thus fr the optimistion you hve covered hs een sttic

### 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

### Math 8 Winter 2015 Applications of Integration

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

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

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

### 5: The Definite Integral

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

Clculus Module C Ares Integrtion Copright This puliction The Northern Alert Institute of Technolog 7. All Rights Reserved. LAST REVISED Mrch, 9 Introduction to Ares Integrtion Sttement of Prerequisite

### set is not closed under matrix [ multiplication, ] and does not form a group.

Prolem 2.3: Which of the following collections of 2 2 mtrices with rel entries form groups under [ mtrix ] multipliction? i) Those of the form for which c d 2 Answer: The set of such mtrices is not closed

### The First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).

The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples

### 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

### 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.

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

MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27-233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.

### Worksheet A EXPONENTIALS AND LOGARITHMS PMT. 1 Express each of the following in the form log a b = c. a 10 3 = 1000 b 3 4 = 81 c 256 = 2 8 d 7 0 = 1

C Worksheet A Epress ech of the following in the form log = c. 0 = 000 4 = 8 c 56 = 8 d 7 0 = e = f 5 = g 7 9 = 9 h 6 = 6 Epress ech of the following using inde nottion. log 5 5 = log 6 = 4 c 5 = log 0

### 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

### 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.

### Name Ima Sample ASU ID

Nme Im Smple ASU ID 2468024680 CSE 355 Test 1, Fll 2016 30 Septemer 2016, 8:35-9:25.m., LSA 191 Regrding of Midterms If you elieve tht your grde hs not een dded up correctly, return the entire pper to

### 1 Nondeterministic Finite Automata

1 Nondeterministic Finite Automt Suppose in life, whenever you hd choice, you could try oth possiilities nd live your life. At the end, you would go ck nd choose the one tht worked out the est. Then you

### 5.1 Estimating with Finite Sums Calculus

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

### Homework 3 Solutions

CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.

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

Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given

### MTH 505: Number Theory Spring 2017

MTH 505: Numer Theory Spring 207 Homework 2 Drew Armstrong The Froenius Coin Prolem. Consider the eqution x ` y c where,, c, x, y re nturl numers. We cn think of \$ nd \$ s two denomintions of coins nd \$c

### This chapter will show you. What you should already know. 1 Write down the value of each of the following. a 5 2

1 Direct vrition 2 Inverse vrition This chpter will show you how to solve prolems where two vriles re connected y reltionship tht vries in direct or inverse proportion Direct proportion Inverse proportion

### The Trapezoidal Rule

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

### Project 6: Minigoals Towards Simplifying and Rewriting Expressions

MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

### CS12N: The Coming Revolution in Computer Architecture Laboratory 2 Preparation

CS2N: The Coming Revolution in Computer Architecture Lortory 2 Preprtion Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes

### Farey Fractions. Rickard Fernström. U.U.D.M. Project Report 2017:24. Department of Mathematics Uppsala University

U.U.D.M. Project Report 07:4 Frey Frctions Rickrd Fernström Exmensrete i mtemtik, 5 hp Hledre: Andres Strömergsson Exmintor: Jörgen Östensson Juni 07 Deprtment of Mthemtics Uppsl University Frey Frctions

### The Fundamental Theorem of Algebra

The Fundmentl Theorem of Alger Jeremy J. Fries In prtil fulfillment of the requirements for the Mster of Arts in Teching with Speciliztion in the Teching of Middle Level Mthemtics in the Deprtment of Mthemtics.

### Math 017. Materials With Exercises

Mth 07 Mterils With Eercises Jul 0 TABLE OF CONTENTS Lesson Vriles nd lgeric epressions; Evlution of lgeric epressions... Lesson Algeric epressions nd their evlutions; Order of opertions....... Lesson

### 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

### Homework Assignment 3 Solution Set

Homework Assignment 3 Solution Set PHYCS 44 6 Ferury, 4 Prolem 1 (Griffiths.5(c The potentil due to ny continuous chrge distriution is the sum of the contriutions from ech infinitesiml chrge in the distriution.

### Chapter 6 Continuous Random Variables and Distributions

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

### Boolean Algebra. Boolean Algebra

Boolen Alger Boolen Alger A Boolen lger is set B of vlues together with: - two inry opertions, commonly denoted y + nd, - unry opertion, usully denoted y ˉ or ~ or, - two elements usully clled zero nd

### CS 373, Spring Solutions to Mock midterm 1 (Based on first midterm in CS 273, Fall 2008.)

CS 373, Spring 29. Solutions to Mock midterm (sed on first midterm in CS 273, Fll 28.) Prolem : Short nswer (8 points) The nswers to these prolems should e short nd not complicted. () If n NF M ccepts

### Arithmetic & Algebra. NCTM National Conference, 2017

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

### 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

### Math 259 Winter Solutions to Homework #9

Mth 59 Winter 9 Solutions to Homework #9 Prolems from Pges 658-659 (Section.8). Given f(, y, z) = + y + z nd the constrint g(, y, z) = + y + z =, the three equtions tht we get y setting up the Lgrnge multiplier

### Linear Systems with Constant Coefficients

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

### Line and Surface Integrals: An Intuitive Understanding

Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of

### 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

### DIRECT CURRENT CIRCUITS

DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through

### STRAND B: NUMBER THEORY

Mthemtics SKE, Strnd B UNIT B Indices nd Fctors: Tet STRAND B: NUMBER THEORY B Indices nd Fctors Tet Contents Section B. Squres, Cubes, Squre Roots nd Cube Roots B. Inde Nottion B. Fctors B. Prime Fctors,

### Math 360: A primitive integral and elementary functions

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

### Lecture 1. Functional series. Pointwise and uniform convergence.

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

### along the vector 5 a) Find the plane s coordinate after 1 hour. b) Find the plane s coordinate after 2 hours. c) Find the plane s coordinate

L8 VECTOR EQUATIONS OF LINES HL Mth - Sntowski Vector eqution of line 1 A plne strts journey t the point (4,1) moves ech hour long the vector. ) Find the plne s coordinte fter 1 hour. b) Find the plne

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

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

### 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

### 8 factors of x. For our second example, let s raise a power to a power:

CH 5 THE FIVE LAWS OF EXPONENTS EXPONENTS WITH VARIABLES It s no time for chnge in tctics, in order to give us deeper understnding of eponents. For ech of the folloing five emples, e ill stretch nd squish,

Qudrtic recirocity Frncisc Bozgn Los Angeles Mth Circle Octoer 8, 01 1 Qudrtic Recirocity nd Legendre Symol In the eginning of this lecture, we recll some sic knowledge out modulr rithmetic: Definition

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

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

### dy ky, dt where proportionality constant k may be positive or negative

Section 1.2 Autonomous DEs of the form 0 The DE y is mthemticl model for wide vriety of pplictions. Some of the pplictions re descried y sying the rte of chnge of y(t) is proportionl to the mount present.

### Section 6.1 INTRO to LAPLACE TRANSFORMS

Section 6. INTRO to LAPLACE TRANSFORMS Key terms: Improper Integrl; diverge, converge A A f(t)dt lim f(t)dt Piecewise Continuous Function; jump discontinuity Function of Exponentil Order Lplce Trnsform

### Math Calculus with Analytic Geometry II

orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove x-xis) ( bove f under x-xis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem

### Regular Language. Nonregular Languages The Pumping Lemma. The pumping lemma. Regular Language. The pumping lemma. Infinitely long words 3/17/15

Regulr Lnguge Nonregulr Lnguges The Pumping Lemm Models of Comput=on Chpter 10 Recll, tht ny lnguge tht cn e descried y regulr expression is clled regulr lnguge In this lecture we will prove tht not ll

### 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 + \$

### 3 x x 3x x. 3x x x 6 x 3. PAKTURK 8 th National Interschool Maths Olympiad, h h

PAKTURK 8 th Ntionl Interschool Mths Olmpid,.9. Q: Evlute 6.9. 6 6 6... 8 8...... Q: Evlute bc bc. b. c bc.9.9b.9.9bc Q: Find the vlue of h in the eqution h 7 9 7.. bc. bc bc. b. c bc bc bc bc......9 h

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

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

### Section 3.1: Exponent Properties

Section.1: Exponent Properties Ojective: Simplify expressions using the properties of exponents. Prolems with exponents cn often e simplied using few sic exponent properties. Exponents represent repeted

### Section 6.1 Definite Integral

Section 6.1 Definite Integrl Suppose we wnt to find the re of region tht is not so nicely shped. For exmple, consider the function shown elow. The re elow the curve nd ove the x xis cnnot e determined

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

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

### 20 MATHEMATICS POLYNOMIALS

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

### CS 311 Homework 3 due 16:30, Thursday, 14 th October 2010

CS 311 Homework 3 due 16:30, Thursdy, 14 th Octoer 2010 Homework must e sumitted on pper, in clss. Question 1. [15 pts.; 5 pts. ech] Drw stte digrms for NFAs recognizing the following lnguges:. L = {w

### 13.3 CLASSICAL STRAIGHTEDGE AND COMPASS CONSTRUCTIONS

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

### MATH 573 FINAL EXAM. May 30, 2007

MATH 573 FINAL EXAM My 30, 007 NAME: Solutions 1. This exm is due Wednesdy, June 6 efore the 1:30 pm. After 1:30 pm I will NOT ccept the exm.. This exm hs 1 pges including this cover. There re 10 prolems.

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

Motion ccelertion Prt : Constnt ccelertion ccelertion ccelertion ccelertion is the rte of chnge of elocity. = - o t = Δ Δt ccelertion = = - o t chnge of elocity elpsed time ccelertion is ector, lthough

### expression simply by forming an OR of the ANDs of all input variables for which the output is

2.4 Logic Minimiztion nd Krnugh Mps As we found ove, given truth tle, it is lwys possile to write down correct logic expression simply y forming n OR of the ANDs of ll input vriles for which the output

### The Bernoulli Numbers John C. Baez, December 23, x k. x e x 1 = n 0. B k n = n 2 (n + 1) 2

The Bernoulli Numbers John C. Bez, December 23, 2003 The numbers re defined by the eqution e 1 n 0 k. They re clled the Bernoulli numbers becuse they were first studied by Johnn Fulhber in book published

### Improper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.

Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:

### What else can you do?

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

### Analytically, vectors will be represented by lowercase bold-face Latin letters, e.g. a, r, q.

1.1 Vector Alger 1.1.1 Sclrs A physicl quntity which is completely descried y single rel numer is clled sclr. Physiclly, it is something which hs mgnitude, nd is completely descried y this mgnitude. Exmples

### Shape and measurement

C H A P T E R 5 Shpe nd mesurement Wht is Pythgors theorem? How do we use Pythgors theorem? How do we find the perimeter of shpe? How do we find the re of shpe? How do we find the volume of shpe? How do

### 3.1 Review of Sine, Cosine and Tangent for Right Angles

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

### Section 7.1 Area of a Region Between Two Curves

Section 7.1 Are of Region Between Two Curves White Bord Chllenge The circle elow is inscried into squre: Clcultor 0 cm Wht is the shded re? 400 100 85.841cm White Bord Chllenge Find the re of the region

### 2. VECTORS AND MATRICES IN 3 DIMENSIONS

2 VECTORS AND MATRICES IN 3 DIMENSIONS 21 Extending the Theory of 2-dimensionl Vectors x A point in 3-dimensionl spce cn e represented y column vector of the form y z z-xis y-xis z x y x-xis Most of the

### SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics

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

### 4 VECTORS. 4.0 Introduction. Objectives. Activity 1

4 VECTRS Chpter 4 Vectors jectives fter studying this chpter you should understnd the difference etween vectors nd sclrs; e le to find the mgnitude nd direction of vector; e le to dd vectors, nd multiply

### 1.2 What is a vector? (Section 2.2) Two properties (attributes) of a vector are and.

Homework 1. Chpters 2. Bsis independent vectors nd their properties Show work except for fill-in-lnks-prolems (print.pdf from www.motiongenesis.com Textooks Resources). 1.1 Solving prolems wht engineers

### 5.5 The Substitution Rule

5.5 The Substitution Rule Given the usefulness of the Fundmentl Theorem, we wnt some helpful methods for finding ntiderivtives. At the moment, if n nti-derivtive is not esily recognizble, then we re in

### Topic 1 Notes Jeremy Orloff

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

### l 2 p2 n 4n 2, the total surface area of the

Week 6 Lectures Sections 7.5, 7.6 Section 7.5: Surfce re of Revolution Surfce re of Cone: Let C be circle of rdius r. Let P n be n n-sided regulr polygon of perimeter p n with vertices on C. Form cone

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

(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one

### MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35

MATH 101A: ALGEBRA I PART B: RINGS AND MODULES 35 9. Modules over PID This week we re proving the fundmentl theorem for finitely generted modules over PID, nmely tht they re ll direct sums of cyclic modules.

### INTRODUCTION TO LINEAR ALGEBRA

ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR AGEBRA Mtrices nd Vectors Prof. Dr. Bülent E. Pltin Spring Sections & / ME Applied Mthemtics for Mechnicl Engineers INTRODUCTION TO INEAR

### 8. Complex Numbers. We can combine the real numbers with this new imaginary number to form the complex numbers.

8. Complex Numers The rel numer system is dequte for solving mny mthemticl prolems. But it is necessry to extend the rel numer system to solve numer of importnt prolems. Complex numers do not chnge the

### Designing Information Devices and Systems I Fall 2016 Babak Ayazifar, Vladimir Stojanovic Homework 6. This homework is due October 11, 2016, at Noon.

EECS 16A Designing Informtion Devices nd Systems I Fll 2016 Bk Ayzifr, Vldimir Stojnovic Homework 6 This homework is due Octoer 11, 2016, t Noon. 1. Homework process nd study group Who else did you work

### 9.4. The Vector Product. Introduction. Prerequisites. Learning Outcomes

The Vector Product 9.4 Introduction In this section we descrie how to find the vector product of two vectors. Like the sclr product its definition my seem strnge when first met ut the definition is chosen

### Designing Information Devices and Systems I Spring 2018 Homework 8

EECS 16A Designing Informtion Devices nd Systems I Spring 2018 Homework 8 This homework is due Mrch 19, 2018, t 23:59. Self-grdes re due Mrch 22, 2018, t 23:59. Sumission Formt Your homework sumission

### Handout: Natural deduction for first order logic

MATH 457 Introduction to Mthemticl Logic Spring 2016 Dr Json Rute Hndout: Nturl deduction for first order logic We will extend our nturl deduction rules for sententil logic to first order logic These notes

### Lecture 7 notes Nodal Analysis

Lecture 7 notes Nodl Anlysis Generl Network Anlysis In mny cses you hve multiple unknowns in circuit, sy the voltges cross multiple resistors. Network nlysis is systemtic wy to generte multiple equtions

### Purpose of the experiment

Newton s Lws II PES 6 Advnced Physics Lb I Purpose of the experiment Exmine two cses using Newton s Lws. Sttic ( = 0) Dynmic ( 0) fyi fyi Did you know tht the longest recorded flight of chicken is thirteen

### Line Integrals. Partitioning the Curve. Estimating the Mass

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

### Chapter 4 Contravariance, Covariance, and Spacetime Diagrams

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

### Distance And Velocity

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

### UniversitaireWiskundeCompetitie. Problem 2005/4-A We have k=1. Show that for every q Q satisfying 0 < q < 1, there exists a finite subset K N so that

Problemen/UWC NAW 5/7 nr juni 006 47 Problemen/UWC UniversitireWiskundeCompetitie Edition 005/4 For Session 005/4 we received submissions from Peter Vndendriessche, Vldislv Frnk, Arne Smeets, Jn vn de

### LINEAR ALGEBRA APPLIED

5.5 Applictions of Inner Product Spces 5.5 Applictions of Inner Product Spces 7 Find the cross product of two vectors in R. Find the liner or qudrtic lest squres pproimtion of function. Find the nth-order

### STRAND J: TRANSFORMATIONS, VECTORS and MATRICES

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