Exponential and Logarithmic Functions -- ANSWERS -- Logarithms Practice Diploma ANSWERS 1
|
|
|
- Lewis Sanders
- 5 years ago
- Views:
Transcription
1 Eponenial and Logarihmic Funcions -- ANSWERS -- Logarihms racice Diploma ANSWERS
2 Logarihms Diploma Syle racice Eam Answers. C. D 9. A 7. C. A. C. B 8. D. D. C NR C 4. C NR. NR D. B. B. B. B 6. D 4. C. B. B NR. NR C. B 7. B. C 4. B 8. C 6. C NR NR B. C 9. A 8. D 6. B Logarihms racice Diploma ANSWERS
3 ) Graphical Soluion: The quesion ells you ha >, so you could graph y = and y = o see wha each case would look like. These graphs are symmerical wih respec o he y ais ( = ). = = Algeraic soluion: ( ) The answer is C. Compared o he original of, i s refleced in he y-ais. ) To graph y = log, change of ase is required since he calculaor only acceps ase logs. In your calculaor, you would use ) log log = log Change of Base log-log = Division Law log-log -log = Since log= -log log = Cancel ou he negaives log = log The answer is D. 4) Change of Base in reverse log y = and y 6 log =. The answer is A..46 = w.46(4).8 = w lug in =.8 and = = w Surac on oh sides.78.7 = w Cancel ou he negaives.78.7 = w = ( w ).7 Isolae w y raising each side o he reciprocal eponen w =.8.46(4) (You can also solve his equaion y graphing y =.8 and y, hen find he -value of he = w poin of inersecion.) Logarihms racice Diploma ANSWERS
4 ) log ( y z) log( y z = log yz yz ) = log y z 6) 7 = Raise o reciprocal eponens [] 4 = ( + ) 4 7 = + Conver fracional eponen o a radical = 4 7 The answer is D. NR #) log + log y ( y) = log + log Facor ou he = log y Muliplicaion Law = log 8 We know y=8 = () Evaluae log 8 using change of ase = The answer is. -- 7) Graph y = and y = in your calculaor and find he -value of he poin of inersecion. Rememer o keep your eponen in rackes! Logarihms racice Diploma ANSWERS 4
5 8) The iniial amoun A is. The final amoun A is 8. The lengh of ime is hours. The growh is ½. We wan o solve for. A = A 8 = 8 = = 4 = = = =. The half life is. hours, which is 6 minues. The answer is C NR #) a = a ( logc c ) a = ( alog c c) ower Law + 8 a = ( a) l ogc c = = + 8 Common Base = 8 = 4 The answer is 4.. 9) The poin (, a) can e ransformed o he poin (a, ) y drawing he inverse graph. The answer is A. Logarihms racice Diploma ANSWERS
6 ) Use he formula A = A A = he fuure score S A = he iniial score = elapsed days d = rae. Decreasing percenage; surac his decimal from. (.97) = he percenage loss is per day, so he period is. lug hese ino he formula o ge S = (.97) d The answer is D. ) db db db ( I ) ( I ) db = log db = log = I I = db db db db I = Common denominaor: = = I = The answer is C ) db I = I = I = I = I = NR #) log log ( ) log = The answer is Logarihms racice Diploma ANSWERS 6
7 ) Rewrie as: y = + 4 y Swap & y o ge: = + 4 Bring 4 o he lef side: 4= y Conver o log form (rememer a ase is always a ase) log 4 = y = ( ) Rewrie as f log 4 4) = y y log = log Solve for y y aking he log of oh sides log = ylog log y = log y = log Change of ase in reverse The negaive in fron indicaes a reflecion in he -ais. NR 4) k T () = Te.4 6 = 8(.78).4.6 = (.78) =.4 minues Solve y graphing & poin of inersecion. Keep eponen in rackes! ) Graph 4 f = and log g = log 4 = in your calculaor, and noice he reflecion line is y = log 4 = ( ) + 6) Rewrie y g as y = g 4 + o see ha he graph has een shifed 4 unis righ. Since a logarihm graph has a verical asympoe along he y-ais, he asympoe is shifed 4 unis righ o make he line = 4. Thus, he domain is > 4 Logarihms racice Diploma ANSWERS 7
8 7) If you are solving wo equaions graphically, he -value of he poin of inersecion is wha you require. B is he incorrec procedure. 8) 4 y = log 4 y = log ylog 4 = log log y = log4 Graphing his epression (rememer o keep he denominaor in rackes) gives graph D. 9) + f = 7a 7 + = a + = 7a + = a 7 + log = log a 7 log log 7 = ( + ) log a log log 7 = + log a log log 7 = log a The answer is A. Logarihms racice Diploma ANSWERS 8
9 ) ( a) log = 8a 7 a = 8 a = a = ( ) a = = NR #) Deermine he eponenial regression equaion for he daa. Don use he acual years for he regression - use elapsed ime. For eample, can e replaced wih since one full year has passed. Year 4 rofi $8 $ $4 $49 $6 y = 49.6(.) Deermine he profi in y plugging in =, since you wan he enh year. The resuling value is $767., or 8 housand dollars. Answer = 8 NR #6) Graph y = 49.6(.) and y = 8. The -value of he poin of inersecion gives he value 8.. Since he 8 h year is under $8, he firs ime he usiness will e over $8 is he 9 h year. Answer = 9 ) Graph oh equaions and find he -value of he poin of inersecion. This occurs a. years, so he second usiness will overake he firs usiness during he h year. The answer is B Logarihms racice Diploma ANSWERS 9
10 ) a4 = 4 4 a4 = a = 4 The answer is B ) log 6 Rewrie as log ( 6 ) log The numeraor is defined for < 6 The denominaor is defined for > The enire graph is defined eween and 6, wih he ecepion of = since ha makes he denominaor zero. The answer is C 4) y = logc a = logc a = logc a c = a = a = a The answer is B NR #7) Group like erms log + log = 8 log = 8 8 log = 8 = = 9.8 ) = = 6) loga + y = loga z y = loga z loga z y = loga 7) A = A A = 6 A = 88. 8) = (.) log = log (.) log = log + log(.) log = + log. log = log. log = log. The answer is D. The answer is 9.8 Logarihms racice Diploma ANSWERS
11 9) log + log + = log ( )( + ) = ( )( + ) = log log 4 = = =± ) log 6 6 log 6 6 log 6 log 6 6 = = 8 The answer is D. ) log c log c a a log c+ log c = a log c = a log c = a ) The graph has een moved down y hree unis. There is a horizonal asympoe a he line y = -, and he graph is aove his line. The range is y > - ) ( + ) + ( ) log log = log( + )( ) = = ( + )( ) = + = + = ( + 4)( ) = 4, Rejec -4 = Logarihms racice Diploma ANSWERS
12 Wrien Response : Domain > - Range y ε R Equaion of Asympoe = - -inercep (-, ) y-inercep (,.) y-value when =.6 log ( + ) Horizonal ranslaion of unis lef. The domain of a logarihmic epression can e found y seing wha is in he rackes greaer han zero. For he epression alog( + c) + d + c > > c c > Logarihms racice Diploma ANSWERS
13 Wrien Response : Solving Graphically Graph y = 7 8 y = 4 A window seing ha will le you see he poin of inersecion clearly is : [-,,.] y: [-.,.,.] The answer is =.7 Solve Using a Common Base 7 8 = 4 4 ( ) ( ) = ( ) = 4 = 4 = 4 = = 4 = Solve Using Logarihms 7 8 = 4 log 7 8 = log 4 log 7 + log8 = log 4 log 7 + ( ) log8 = log 4 log 7 + log8 log8 = log 4 log8+ log 4 = log8 log 7 ( log8+ log 4) = log8 log 7 log8 log 7 = log8+ log 4 =.7 Graph y = y = 4 log log Use a window of : [,, ] y: [-,, ] Answer: = 6 Logarihms racice Diploma ANSWERS
14 Solve for A= A A A = A log = log A log A log A = log log A log A log = ( ) = ( log A log A ) log Wrien Response : lug in your values and solve y graphing. (Or use he equaion derived aove.) A= A 9 = 6. = = 4.74 hours If he populaion of he own doules, he iniial amoun is A and he final amoun is A. Simplify and solve y graphing. A= A 8 A = A = 8 =. years If he ligh inensiy is 64% of he iniial amoun, Surac he rae from he iniial amoun is A and he final amoun since i is a decreasing is.64a. Simplify and solve y graphing. percen. Simplify and solve A= A y graphing. A= A.64A = A 4 A = A(.97).64 = 4 = (.97) =. m =.77 years Logarihms racice Diploma ANSWERS 4
Logarithms Practice Exam - ANSWERS
Logarihms racice Eam - ANSWERS Answers. C. D 9. A 9. D. A. C. B. B. D. C. B. B. C NR.. C. B. B. B. B 6. D. C NR. 9. NR. NR... C 7. B. C. B. C 6. C 6. C NR.. 7. B 7. D 9. A. D. C Each muliple choice & numeric
5.1 - Logarithms and Their Properties
Chaper 5 Logarihmic Funcions 5.1 - Logarihms and Their Properies Suppose ha a populaion grows according o he formula P 10, where P is he colony size a ime, in hours. When will he populaion be 2500? We
Note: For all questions, answer (E) NOTA means none of the above answers is correct.
Thea Logarihms & Eponens 0 ΜΑΘ Naional Convenion Noe: For all quesions, answer means none of he above answers is correc.. The elemen C 4 has a half life of 70 ears. There is grams of C 4 in a paricular
3 at MAC 1140 TEST 3 NOTES. 5.1 and 5.2. Exponential Functions. Form I: P is the y-intercept. (0, P) When a > 1: a = growth factor = 1 + growth rate
1 5.1 and 5. Eponenial Funcions Form I: Y Pa, a 1, a > 0 P is he y-inercep. (0, P) When a > 1: a = growh facor = 1 + growh rae The equaion can be wrien as The larger a is, he seeper he graph is. Y P( 1
UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Name: Par I Quesions UNIT #4 TEST REVIEW EXPONENTIAL AND LOGARITHMIC FUNCTIONS Dae: 1. The epression 1 is equivalen o 1 () () 6. The eponenial funcion y 16 could e rewrien as y () y 4 () y y. The epression
4.1 - Logarithms and Their Properties
Chaper 4 Logarihmic Funcions 4.1 - Logarihms and Their Properies Wha is a Logarihm? We define he common logarihm funcion, simply he log funcion, wrien log 10 x log x, as follows: If x is a posiive number,
Section 4.4 Logarithmic Properties
Secion. Logarihmic Properies 59 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
5.2. The Natural Logarithm. Solution
5.2 The Naural Logarihm The number e is an irraional number, similar in naure o π. Is non-erminaing, non-repeaing value is e 2.718 281 828 59. Like π, e also occurs frequenly in naural phenomena. In fac,
Chapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
Section 4.4 Logarithmic Properties
Secion. Logarihmic Properies 5 Secion. Logarihmic Properies In he previous secion, we derived wo imporan properies of arihms, which allowed us o solve some asic eponenial and arihmic equaions. Properies
Math 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
6. Solve by applying the quadratic formula.
Dae: Chaper 7 Prerequisie Skills BLM 7.. Apply he Eponen Laws. Simplify. Idenify he eponen law ha you used. a) ( c) ( c) ( c) ( y)( y ) c) ( m)( n ). Simplify. Idenify he eponen law ha you used. 8 w a)
MAC1105-College Algebra
. Inverse Funcions I. Inroducion MAC5-College Algera Chaper -Inverse, Eponenial & Logarihmic Funcions To refresh our undersnading of "Composiion of a funcion" or "Composie funcion" which refers o he comining
MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
![MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.](/thumbs/86/93221008.jpg "MATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.")
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
The average rate of change between two points on a function is d t
SM Dae: Secion: Objecive: The average rae of change beween wo poins on a funcion is d. For example, if he funcion ( ) represens he disance in miles ha a car has raveled afer hours, hen finding he slope
Math 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS TOTAL NAME DATE PERIOD DATE TOPIC ASSIGNMENT /19 10/22 10/23 10/24 10/25 10/26 10/29 10/30
NAME DATE PERIOD MATH ANALYSIS HONORS UNIT 6 EXPONENTIAL FUNCTIONS DATE TOPIC ASSIGNMENT 10 0 10/19 10/ 10/ 10/4 10/5 10/6 10/9 10/0 10/1 11/1 11/ TOTAL Mah Analysis Honors Workshee 1 Eponenial Funcions
Chapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
Solutionbank Edexcel AS and A Level Modular Mathematics
Page of 4 Soluionbank Edexcel AS and A Level Modular Mahemaics Exercise A, Quesion Quesion: Skech he graphs of (a) y = e x + (b) y = 4e x (c) y = e x 3 (d) y = 4 e x (e) y = 6 + 0e x (f) y = 00e x + 0
Solutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
Analyze patterns and relationships. 3. Generate two numerical patterns using AC
envision ah 2.0 5h Grade ah Curriculum Quarer 1 Quarer 2 Quarer 3 Quarer 4 andards: =ajor =upporing =Addiional Firs 30 Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 andards: Operaions and Algebraic Thinking
Exponential and Logarithmic Functions
Chaper 5 Eponenial and Logarihmic Funcions Chaper 5 Prerequisie Skills Chaper 5 Prerequisie Skills Quesion 1 Page 50 a) b) c) Answers may vary. For eample: The equaion of he inverse is y = log since log
!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)
"#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5
AP Calculus BC Chapter 10 Part 1 AP Exam Problems
AP Calculus BC Chaper Par AP Eam Problems All problems are NO CALCULATOR unless oherwise indicaed Parameric Curves and Derivaives In he y plane, he graph of he parameric equaions = 5 + and y= for, is a
Chapters 6 & 7: Trigonometric Functions of Angles and Real Numbers. Divide both Sides by 180
Algebra Chapers & : Trigonomeric Funcions of Angles and Real Numbers Chapers & : Trigonomeric Funcions of Angles and Real Numbers - Angle Measures Radians: - a uni (rad o measure he size of an angle. rad
Laplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
![1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.](/thumbs/71/65799522.jpg "1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.")
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18
A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly
Math 111 Midterm I, Lecture A, version 1 -- Solutions January 30 th, 2007
NAME: Suden ID #: QUIZ SECTION: Mah 111 Miderm I, Lecure A, version 1 -- Soluions January 30 h, 2007 Problem 1 4 Problem 2 6 Problem 3 20 Problem 4 20 Toal: 50 You are allowed o use a calculaor, a ruler,
Math 2214 Solution Test 1A Spring 2016
Mah 14 Soluion Tes 1A Spring 016 sec Problem 1: Wha is he larges -inerval for which ( 4) = has a guaraneed + unique soluion for iniial value (-1) = 3 according o he Exisence Uniqueness Theorem? Soluion
Notes 8B Day 1 Doubling Time
Noes 8B Day 1 Doubling ime Exponenial growh leads o repeaed doublings (see Graph in Noes 8A) and exponenial decay leads o repeaed halvings. In his uni we ll be convering beween growh (or decay) raes and
FITTING EQUATIONS TO DATA
TANTON S TAKE ON FITTING EQUATIONS TO DATA CURRICULUM TIDBITS FOR THE MATHEMATICS CLASSROOM MAY 013 Sandard algebra courses have sudens fi linear and eponenial funcions o wo daa poins, and quadraic funcions
Section 4.1 Exercises
Secion 4.1 Eponenial Funcions 459 Secion 4.1 Eercises For each able below, could he able represen a funcion ha is linear, eponenial, or neiher? 1. 1 2 3 4 f() 70 40 10-20 3. 1 2 3 4 h() 70 49 34.3 24.01
6. 6 v ; degree = 7; leading coefficient = 6; 7. The expression has 3 terms; t p no; subtracting x from 3x ( 3x x 2x)
70. a =, r = 0%, = 0. 7. a = 000, r = 0.%, = 00 7. a =, r = 00%, = 7. ( ) = 0,000 0., where = ears 7. ( ) = + 0.0, where = weeks 7 ( ) =,000,000 0., where = das 7 = 77. = 9 7 = 7 geomeric 0. geomeric arihmeic,
Chapter 2 Trigonometric Functions
Chaper Trigonomeric Funcions Secion.. 90 7 80 6. 90 70 89 60 70 9 80 79 60 70 70 09. 90 6 89 9 60 6 6 80 6 79 9 60 6 6 7. 9.. 0. 60 0 + 60 α is a quadran III angle coerminal wih an angle of measure 0..
KINEMATICS IN ONE DIMENSION
KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec
( ) ( ) ( ) ( u) ( u) = are shown in Figure =, it is reasonable to speculate that. = cos u ) and the inside function ( ( t) du
Porlan Communiy College MTH 51 Lab Manual The Chain Rule Aciviy 38 The funcions f ( = sin ( an k( sin( 3 38.1. Since f ( cos( k ( = cos( 3. Bu his woul imply ha k ( f ( = are shown in Figure =, i is reasonable
KEY. Math 334 Midterm I Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
1 KEY Mah 4 Miderm I Fall 8 secions 1 and Insrucor: Sco Glasgow Please do NOT wrie on his eam. No credi will be given for such work. Raher wrie in a blue book, or on our own paper, preferabl engineering
A Dynamic Model of Economic Fluctuations
CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model
3.6 Derivatives as Rates of Change
3.6 Derivaives as Raes of Change Problem 1 John is walking along a sraigh pah. His posiion a he ime >0 is given by s = f(). He sars a =0from his house (f(0) = 0) and he graph of f is given below. (a) Describe
( ) 2. Review Exercise 2. cos θ 2 3 = = 2 tan. cos. 2 x = = x a. Since π π, = 2. sin = = 2+ = = cotx. 2 sin θ 2+
Review Eercise sin 5 cos sin an cos 5 5 an 5 9 co 0 a sinθ 6 + 4 6 + sin θ 4 6+ + 6 + 4 cos θ sin θ + 4 4 sin θ + an θ cos θ ( ) + + + + Since π π, < θ < anθ should be negaive. anθ ( + ) Pearson Educaion
The Arcsine Distribution
The Arcsine Disribuion Chris H. Rycrof Ocober 6, 006 A common heme of he class has been ha he saisics of single walker are ofen very differen from hose of an ensemble of walkers. On he firs homework, we
Predator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
Check in: 1 If m = 2(x + 1) and n = find y when. b y = 2m n 2
7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions
Math 105 Second Midterm March 16, 2017
Mah 105 Second Miderm March 16, 2017 UMID: Insrucor: Iniials: Secion: 1. Do no open his exam unil you are old o do so. 2. Do no wrie your name anywhere on his exam. 3. This exam has 9 pages including his
Age (x) nx lx. Age (x) nx lx dx qx
Life Tables Dynamic (horizonal) cohor= cohor followed hrough ime unil all members have died Saic (verical or curren) = one census period (day, season, ec.); only equivalen o dynamic if populaion does no
Chapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
1.6. Slopes of Tangents and Instantaneous Rate of Change
1.6 Slopes of Tangens and Insananeous Rae of Change When you hi or kick a ball, he heigh, h, in meres, of he ball can be modelled by he equaion h() 4.9 2 v c. In his equaion, is he ime, in seconds; c represens
Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates
Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese
Math 36. Rumbos Spring Solutions to Assignment #6. 1. Suppose the growth of a population is governed by the differential equation.
Mah 36. Rumbos Spring 1 1 Soluions o Assignmen #6 1. Suppose he growh of a populaion is governed by he differenial equaion where k is a posiive consan. d d = k (a Explain why his model predics ha he populaion
10.1 EXERCISES. y 2 t 2. y 1 t y t 3. y e
66 CHAPTER PARAMETRIC EQUATINS AND PLAR CRDINATES SLUTIN We use a graphing device o produce he graphs for he cases a,,.5,.,,.5,, and shown in Figure 7. Noice ha all of hese curves (ecep he case a ) have
Exponential and Logarithmic Functions
Chaper 6 Eponenial and Logarihmic Funcions 6.R Chaper Review. f () = + 3 5 +3 = 5 = + 3 5 (5 )=+3 5 = + 3 5 = + 3 (5 )=+3 +3 = 5 f +3 ()= 5. f () = 3 + = 3 + = 3 + (3 + ) = 3 + = + = 3 ( +) = 3 = 3 + f
15. Vector Valued Functions
1. Vecor Valued Funcions Up o his poin, we have presened vecors wih consan componens, for example, 1, and,,4. However, we can allow he componens of a vecor o be funcions of a common variable. For example,
CHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
Topics covered in tutorial 01: 1. Review of definite integrals 2. Physical Application 3. Area between curves. 1. Review of definite integrals
MATH4 Calculus II (8 Spring) MATH 4 Tuorial Noes Tuorial Noes (Phyllis LIANG) IA: Phyllis LIANG Email: masliang@us.hk Homepage: hps://masliang.people.us.hk Office: Room 3 (Lif/Lif 3) Phone number: 3587453
3.1.3 INTRODUCTION TO DYNAMIC OPTIMIZATION: DISCRETE TIME PROBLEMS. A. The Hamiltonian and First-Order Conditions in a Finite Time Horizon
3..3 INRODUCION O DYNAMIC OPIMIZAION: DISCREE IME PROBLEMS A. he Hamilonian and Firs-Order Condiions in a Finie ime Horizon Define a new funcion, he Hamilonian funcion, H. H he change in he oal value of
ON THE DEGREES OF RATIONAL KNOTS
ON THE DEGREES OF RATIONAL KNOTS DONOVAN MCFERON, ALEXANDRA ZUSER Absrac. In his paper, we explore he issue of minimizing he degrees on raional knos. We se a bound on hese degrees using Bézou s heorem,
Chapter 7: Solving Trig Equations
Haberman MTH Secion I: The Trigonomeric Funcions Chaper 7: Solving Trig Equaions Le s sar by solving a couple of equaions ha involve he sine funcion EXAMPLE a: Solve he equaion sin( ) The inverse funcions
4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS
Funcions Modeling Change: A Preparaion for Calculus, 4h Ediion, 2011, Connally 4.1 INTRODUCTION TO THE FAMILY OF EXPONENTIAL FUNCTIONS Growing a a Consan Percen Rae Example 2 During he 2000 s, he populaion
Math 2214 Solution Test 1B Fall 2017
Mah 14 Soluion Tes 1B Fall 017 Problem 1: A ank has a capaci for 500 gallons and conains 0 gallons of waer wih lbs of sal iniiall. A soluion conaining of 8 lbsgal of sal is pumped ino he ank a 10 galsmin.
Section 5: Chain Rule
Chaper The Derivaive Applie Calculus 11 Secion 5: Chain Rule There is one more ype of complicae funcion ha we will wan o know how o iffereniae: composiion. The Chain Rule will le us fin he erivaive of
and v y . The changes occur, respectively, because of the acceleration components a x and a y
Week 3 Reciaion: Chaper3 : Problems: 1, 16, 9, 37, 41, 71. 1. A spacecraf is raveling wih a veloci of v0 = 5480 m/s along he + direcion. Two engines are urned on for a ime of 84 s. One engine gives he
, where P is the number of bears at time t in years. dt (a) If 0 100, lim Pt. Is the solution curve increasing or decreasing?
CALCULUS BC WORKSHEET 1 ON LOGISTIC GROWTH Work he following on noebook paper. Use your calculaor on 4(b) and 4(c) only. 1. Suppose he populaion of bears in a naional park grows according o he logisic
SMT 2014 Calculus Test Solutions February 15, 2014 = 3 5 = 15.
SMT Calculus Tes Soluions February 5,. Le f() = and le g() =. Compue f ()g (). Answer: 5 Soluion: We noe ha f () = and g () = 6. Then f ()g () =. Plugging in = we ge f ()g () = 6 = 3 5 = 5.. There is a
20. Applications of the Genetic-Drift Model
0. Applicaions of he Geneic-Drif Model 1) Deermining he probabiliy of forming any paricular combinaion of genoypes in he nex generaion: Example: If he parenal allele frequencies are p 0 = 0.35 and q 0
WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x
WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile
Answers to 1 Homework
Answers o Homework. x + and y x 5 y To eliminae he parameer, solve for x. Subsiue ino y s equaion o ge y x.. x and y, x y x To eliminae he parameer, solve for. Subsiue ino y s equaion o ge x y, x. (Noe:
Fishing limits and the Logistic Equation. 1
Fishing limis and he Logisic Equaion. 1 1. The Logisic Equaion. The logisic equaion is an equaion governing populaion growh for populaions in an environmen wih a limied amoun of resources (for insance,
Numerical Dispersion
eview of Linear Numerical Sabiliy Numerical Dispersion n he previous lecure, we considered he linear numerical sabiliy of boh advecion and diffusion erms when approimaed wih several spaial and emporal
This document was generated at 1:04 PM, 09/10/13 Copyright 2013 Richard T. Woodward. 4. End points and transversality conditions AGEC
his documen was generaed a 1:4 PM, 9/1/13 Copyrigh 213 Richard. Woodward 4. End poins and ransversaliy condiions AGEC 637-213 F z d Recall from Lecure 3 ha a ypical opimal conrol problem is o maimize (,,
Calculus Tricks #1. So if you understand derivatives, you ll understand the course material much better. a few preliminaries exponents
Calculus Tricks # Eric Doviak Calculus is no a pre-requisie or his course. However, he oundaions o economics are based on calculus, so wha we ll be discussing over he course o he semeser is he inuiion
Section 7.4 Modeling Changing Amplitude and Midline
488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves
Some Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
ACCUMULATION. Section 7.5 Calculus AP/Dual, Revised /26/2018 7:27 PM 7.5A: Accumulation 1
ACCUMULATION Secion 7.5 Calculus AP/Dual, Revised 2019 vie.dang@humbleisd.ne 12/26/2018 7:27 PM 7.5A: Accumulaion 1 APPLICATION PROBLEMS A. Undersand he quesion. I is ofen no necessary o as much compuaion
Unit Root Time Series. Univariate random walk
Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he
Radical Expressions. Terminology: A radical will have the following; a radical sign, a radicand, and an index.
Radical Epressions Wha are Radical Epressions? A radical epression is an algebraic epression ha conains a radical. The following are eamples of radical epressions + a Terminology: A radical will have he
MOMENTUM CONSERVATION LAW
1 AAST/AEDT AP PHYSICS B: Impulse and Momenum Le us run an experimen: The ball is moving wih a velociy of V o and a force of F is applied on i for he ime inerval of. As he resul he ball s velociy changes
Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.
Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers
Displacement ( x) x x x
Kinemaics Kinemaics is he branch of mechanics ha describes he moion of objecs wihou necessarily discussing wha causes he moion. 1-Dimensional Kinemaics (or 1- Dimensional moion) refers o moion in a sraigh
Answers to Algebra 2 Unit 3 Practice
Answers o Algebra 2 Uni 3 Pracice Lesson 14-1 1. a. 0, w, 40; (0, 40); {w w, 0, w, 40} 9. a. 40,000 V Volume c. (27, 37,926) d. 27 unis 2 a. h, 30 2 2r V pr 2 (30 2 2r) c. in. d. 3,141.93 in. 2 20 40 Widh
Position, Velocity, and Acceleration
rev 06/2017 Posiion, Velociy, and Acceleraion Equipmen Qy Equipmen Par Number 1 Dynamic Track ME-9493 1 Car ME-9454 1 Fan Accessory ME-9491 1 Moion Sensor II CI-6742A 1 Track Barrier Purpose The purpose
Instructor: Barry McQuarrie Page 1 of 5
Procedure for Solving radical equaions 1. Algebraically isolae one radical by iself on one side of equal sign. 2. Raise each side of he equaion o an appropriae power o remove he radical. 3. Simplify. 4.
On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature
On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check
NEWTON S SECOND LAW OF MOTION
Course and Secion Dae Names NEWTON S SECOND LAW OF MOTION The acceleraion of an objec is defined as he rae of change of elociy. If he elociy changes by an amoun in a ime, hen he aerage acceleraion during
Solutions from Chapter 9.1 and 9.2
Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is
6.2 Transforms of Derivatives and Integrals.
SEC. 6.2 Transforms of Derivaives and Inegrals. ODEs 2 3 33 39 23. Change of scale. If l( f ()) F(s) and c is any 33 45 APPLICATION OF s-shifting posiive consan, show ha l( f (c)) F(s>c)>c (Hin: In Probs.
Seminar 4: Hotelling 2
Seminar 4: Hoelling 2 November 3, 211 1 Exercise Par 1 Iso-elasic demand A non renewable resource of a known sock S can be exraced a zero cos. Demand for he resource is of he form: D(p ) = p ε ε > A a
Be able to sketch a function defined parametrically. (by hand and by calculator)
Pre Calculus Uni : Parameric and Polar Equaions (7) Te References: Pre Calculus wih Limis; Larson, Hoseler, Edwards. B he end of he uni, ou should be able o complee he problems below. The eacher ma provide
04. Kinetics of a second order reaction
4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius
Motion along a Straight Line
chaper 2 Moion along a Sraigh Line verage speed and average velociy (Secion 2.2) 1. Velociy versus speed Cone in he ebook: fer Eample 2. Insananeous velociy and insananeous acceleraion (Secions 2.3, 2.4)
AP CALCULUS AB/CALCULUS BC 2016 SCORING GUIDELINES. Question 1. 1 : estimate = = 120 liters/hr
AP CALCULUS AB/CALCULUS BC 16 SCORING GUIDELINES Quesion 1 (hours) R ( ) (liers / hour) 1 3 6 8 134 119 95 74 7 Waer is pumped ino a ank a a rae modeled by W( ) = e liers per hour for 8, where is measured
Best test practice: Take the past test on the class website
Bes es pracice: Take he pas es on he class websie hp://communiy.wvu.edu/~miholcomb/phys11.hml I have posed he key o he WebAssign pracice es. Newon Previous Tes is Online. Forma will be idenical. You migh
Constant Acceleration
Objecive Consan Acceleraion To deermine he acceleraion of objecs moving along a sraigh line wih consan acceleraion. Inroducion The posiion y of a paricle moving along a sraigh line wih a consan acceleraion
Honors Solutions. Honors Lesson 1. Honors Lesson = 2. substitute 3 for ( r = 90
Honors Lesson. A. Muliplying by : x x x x. subsiue for ( r s): + 8 9 + 9. since i is a square, we know all sides are equal, herefore: +9 9. A ( + 9) square unis using from # A ( + 9)( ) A square unis.
Sterilization D Values
Seriliaion D Values Seriliaion by seam consis of he simple observaion ha baceria die over ime during exposure o hea. They do no all live for a finie period of hea exposure and hen suddenly die a once,
Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution
Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his
Week 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A
Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In
72 Calculus and Structures
72 Calculus and Srucures CHAPTER 5 DISTANCE AND ACCUMULATED CHANGE Calculus and Srucures 73 Copyrigh Chaper 5 DISTANCE AND ACCUMULATED CHANGE 5. DISTANCE a. Consan velociy Le s ake anoher look a Mary s