3.1 Exponential Functions and Their Graphs


 Alban Carroll
 6 years ago
 Views:
Transcription
1 . 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. Eponentil Function: function of the form f b ;, b, b : initil vlue (intercept); f b b: bse Evluting n Eponentil Function E: Determine the vlue for f 4. Note: A common mistke is multipling the nd 4. Remember the order of opertions (eponents before multipliction).. f b. f Grphs of Eponentil Functions E: Plot points nd grph the functions. Eponentil Growth Domin: Rnge: Intercepts: Asmptotes: End Behvior: Incresing/decresing Continuous? Eponentil Dec Domin: Rnge: Intercepts: Asmptotes: End Behvior: Incresing/decresing Continuous? Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
2 Grphs of Eponentil Functions: Grph is INCREASING when b & ; grph is DECREASING when b &. Trnsformtions of Eponentil Functions: h b k Verticl Stretch: Verticl Shrink: Reflection over is: Reflection over is: Horizontl Trnsltion: h Verticl Trnsltion: k Horizontl Asmptote: k Investigte the following trnsformtions: f( ) vs. f( ) 4 f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) f( ) vs. f( ) E: Grph the eponentil function Trnsformtion of.. Rewrite s. Horizontl Trnsltion: Reflect: Verticl Trnsltion: intercept: f Horizontl Asmptote: E4: Use clcultor to evlute the function f ( ) e when = E5: Sketch the grph of the nturl eponentil function. f ( ) e.4 Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
3 Properties of Eponents: E. 6.) b.) c.) d.) e.) ( b ) f.) ( ) b g.) Eponentil Model: b Eponentil Growth: & b ; b is the growth fctor Eponentil Dec: & b ; b is the dec fctor Applictions of the Eponentil Model E7: CCSD s student popultion went from,4 in 956 to 9,5 students in 5. Write n eponentil function tht represents the student popultion. Predict the popultion in. Let represent the er 95 nd represent the number of students. Substitute the given vlues into the eponentil model b nd solve for nd b. r Formul for Compound Interest: A P n nt A = blnce r = nnul interest rte P = principl t = time in ers n = number of times interest is compounded ech er Note: Annull = time per er Seminnull = times per er Qurterl = 4 times per er Monthl = times per er Weekl = 5 times per er Interest Compounded Continuousl: A Pe rt A P E8: Clculte the blnce if $ is invested for ers t 6% compounded weekl. r n nt A? P t r n Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
4 The vlues of re pproching.78. This is the pproimte vlue of the trnscendentl number e. Nturl Bse e: e lim.7888 Interest Compounded Continuousl: A Pe rt Nturl Eponentil Function: k f e Grph of e : E. 9 A totl of $, is invested t n nnul interest rte of %. Find the blnce fter 4 ers if the interest is compounded continuousl. You Tr: Describe the trnsformtions needed to drw the grph of grph. f 4. Sketch the QOD: Using tble of vlues, how cn ou determine whether the hve n eponentil reltionship? Pge 4 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
5 . Logrithmic Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of n eponentil, logistic or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions. 9.5 The student will grph the inverse of n eponentil or logrithmic function. Review: Solve the following for Eplortion: Use our clcultor to write 997 s power of. Tr Note: 4, , 584.9, 995.(close!) Now use our clcultor to find LOG 997. log (Compre to bove) A LOGARITHM IS AN EXPONENT. Logrithm (bse b): b. log b log b (Red s log bse b of. ) for, b nd b if nd onl if b Note: You cnnot tke the log of negtive number! Common Logrithm: Given positive number p, the solution to p is clled the bse logrithm of p, epressed s log p, or simpl s log p. (When no bse is specified, it is understood to be bse.) Nturl Logrithmic Function: logrithm bse e, log e, written ln Nottions:. log is used to represent log (common logrithm). ln is used to represent log e (nturl logrithm) Pge 5 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
6 E: Rewrite ech eqution (eponentil form) to logrithmic form.. b. Bse (b) = Eponent = 4 6 Bse (b) = Eponent = E: Rewrite ech eqution (logrithmic form) to eponentil form.. log5 5 Bse (b) = Eponent = b. log, 5 Bse (b) = Eponent = Evluting Logrithms: logrithm is n eponent E: Evlute the logrithms.. log 8 b. f, f log 4 6 c. log5 d. f., f log f log. Clcultor: Grphing Logrithmic Functions: logrithmic function is the inverse of n eponentil function E4: Use tble of vlues to sketch the grph of log grph nd compre the grph to the grph of.. Discuss the chrcteristics of the Note: To crete the tble, it is helpful to rewrite the function s nd choose vlues for. Domin: Rnge: Intercepts: Asmptote: Incresing: End Behvior: lim f lim f The function log is the inverse of, so its grph is the reflection of over the line. Pge 6 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
7 Trnsformtions of log b : k log b h Verticl Stretch:, Horizontl Stretch:, Reflection over is: Reflection over is: Horizontl Trnsltion: h Verticl Trnsltion: k E5: Describe the trnsformtions used to grph the function log 5. Then sketch the grph. Properties of Logrithms: A.) log becuse B.) log becuse C.) log D.) If log Note: Grph in red is the grph of log. becuse log, then Evluting Logrithmic Epressions: E6: Evlute.5 log4. log nd..5 log Let log.5. Rewrite in eponentil form: log4. Let log4.. Rewrite in logrithmic form: Note: B the properties of inverses, we could hve evluted the bove emples without rewriting using the following: log nd log Solving Logrithmic Equtions: E7: Solve the equtions for.. log log 6 The bses re equl, so 7 7 Note: Both solutions work in the originl eqution.. log9 Rewrite eponentill:. 4 ln e Rewrite eponentill: Note: Remember tht the bse of ln is e. You Tr: Sketch the grph of function on the sme grid with its inverse. f ln Pge 7 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
8 Pge 8 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
9 . Properties of Logrithms Sllbus Objective: 9. The student will ppl the properties of logrithms to evlute epressions, chnge bses, nd reepress dt. Eplortion: Use our clcultor to find log nd log. Evlute the following logrithms on our clcultor, then speculte how ou could clculte them using onl the vlues of log nd/or log. log. nd log.477. log6 log log8 log8.9. log log.76 Recll: Properties of Eponents b b b Becuse logrithm is n eponent, the rules re the sme! Properties of Logrithms: Condensed Epnded logb RS logb R logb S R logb logb R logb S S log b c R c log R b E: Use the properties of logrithms to epnd the following epressions. b b b b b.) log 5 b.) ln c.) log d.) z log E: Use the properties of logrithms to condense the following epressions..) log log4 b.) log log c.) ln ln ln Pge 9 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
10 Evluting Logrithmic Epressions with Bse b Let log b. Rewrite in eponentil form: b. Tke the log of both sides. log b log Use the properties of logs to solve for. log b log Note: This will work for logrithm of n bse, including the nturl log. log log b Chnge of Bse Formul: log b log log b ln ln b E: Evlute the epression log5 8. B the chnge of bse formul,. Using the clcultor, E4: Evlute the epression log64. B the chnge of bse formul,. Using the clcultor, Note This did NOT require the use of clcultor! We know tht So log64 6 Grphing Logrithmic Functions on the Clcultor E5: Grph log nd log 5 on the sme grid on the grphing clcultor. Some clcultors cnnot tpe in log bse into the clcultor, so we must rewrite the functions using the chnge of bse formul. Our Clcultor: Tpe green lph f, no need to use chnge of bse. log log log log 5 5 log log Cution: The grph creted b the clcultor is misleding t the smptote! You Tr: Epnd the epression using the properties of logrithms. ln z. QOD: When is it pproprite to use the chnge of bse formul? Eplin how to evlute logrithm of bse b without the chnge of bse formul. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
11 .4 Solving Eponentil nd Logrithmic Equtions Sllbus Objectives: 9.4 The student will solve eponentil, logrithmic nd logistic equtions nd inequlities. 9.6 The student will compre equivlent logrithmic nd eponentil equtions. Strtegies for Solving n Eponentil Eqution: Rewrite both sides with the sme bse Tke the log of both sides fter isolting the eponentil E: Solve the following eponentil equtions..) 4 Rewrite with bse : Both sides hve the sme bse, so the eponents must be equl: b.) 8 6 We cnnot rewrite both sides with the sme bse, so tke the log of both sides. c.) 7 e 5 Isolte the eponentil: Tke the nturl log of both sides: Note: We could hve determined tht immeditel using the eqution e. Strtegies for Solving Logrithmic Eqution: Condense n logrithms with the sme bse using the properties of logs Rewrite the eqution in eponentil form Check for etrneous solutions E: Solve the following logrithmic equtions..) ln ln5 Condense: Rewrite in eponentil form: Solve for : Check: b.) log 65 4 Rewrite in ep. form: Tke the log of both sides: Rewrite in eponentil form: Note: We could hve determined tht 5 immeditel using the eqution root of ech side. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter 4 65, b tking the 4 th
12 c.) log log 8 Condense: Rewrite in eponentil form: Solve for : Check: (remember cn t tke the log of negtive!) Note: You must check ever possible solution for etrneous solutions. All negtive nswers re not necessril etrneous! E: Solve the eqution log5 4. Rewrite in eponentil form: Solve for : Check: E4: Solve the eqution Rewrite using properties of logs: log 6. Rewrite in eponentil form: Check: Chllenge Problems: Use our rsenl of eponentil nd logrithmic properties! E5: Solve the eqution 6 7. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
13 E6: Solve the eqution Tke the log of both sides: 4 7. Rewrite: Solve for : Check You Tr: Solve the eqution e 5. QOD: Compre nd contrst the methods for solving eponentil nd logrithmic equtions. Pge of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
14 .5 Eponentil nd Logrithmic Models Sllbus Objective: 9.7 The student will solve ppliction problems involving eponentil nd logrithmic functions. kt Newton s Lw of Cooling: The temperture T of n object t time t is T t Ts T TS e, where T s is the surrounding temperture nd T is the initil temperture of the object. E: A 5 F potto is left out in 7 F room for minutes, nd its temperture dropped to 5 F. How mn more minutes will it tke to rech F? Solve for k using the given informtion: T t Ts T TS e kt Use k to solve for t: It tkes bout minutes for the potto to cool to F. This is minutes longer. Compound Interest: A P r n nt A = blnce mount P = principl (beginning) mount r = nnul interest rte (deciml) n = # of times compounded in er t = time in ers E: How long will it tke for n investment of $, t 6% compounded seminnull to rech $5? A P r n nt It will tke bout ers. Pge 4 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
15 A Pe E: How long will it tke for n investment of $, t 6% compounded continuousl to rech $5? rt It will tke bout ers. Annul Percentge Yield (APY): the rte, compounded nnull ( t ), tht would ield the sme return For A P r n nt n r, APY n E4: An mount of $4 is invested for 8 ers t 5% compounded qurterl. Wht is the equivlent APY? You Tr: Determine the mount of mone tht should be invested t 9% interest compounded monthl to produce blnce of $, in 5 ers..6 Nonliner Models QOD: Wh is using the nnul percentge ield more fir w to compre investments? Pge 5 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
16 Sllbus Objectives: 9.7 The student will solve ppliction problems involving eponentil nd logrithmic functions. 9.4 The student will solve eponentil, logrithmic nd logistic equtions nd inequlities. t Eponentil Model: P t Pb P t : popultion t time t P : initil popultion Growth Model: b r; b is clled the Growth Fctor Dec Model: b r; b is clled the Dec Fctor E: Write n eponentil function tht models the popultion of Smllville if the initil popultion ws,85, nd it is decresing b.% ech er. Predict how long it will tke for the popultion to fll to. P r, so b Pt Solve for t when Pt t It will tke bout ers. E: The popultion of nts is incresing eponentill such tht on d there re nts, nd on d 4 there re nts. How mn nts re there on d 5? t Use P t Pb nd write sstem of equtions with the given informtion: Pb Pb Solve the sstem b dividing the equtions: P b P P P t 5 P t P 5 nts Note: Students could hve clled d t to come up with the sme solution. 4 Eponentil Regression Pge 6 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
17 E: Find n eponentil regression for the popultion of Ls Vegs using the tble below. Then predict the popultion in ,4 4,64 5,787 58,95 54,847 9: Rdioctive Dec: the process in which the number of toms of specific element chnge from rdioctive stte to nonrdioctive stte HlfLife: the time it tkes for hlf of smple of rdioctive substnce to chnge its stte E4: The hlflife of rdioctive Strontium is 8 ds. Write n eqution nd predict the mount of 5 grm smple tht remins fter ds. t Use P t Pb : Solve for b when P, P.5, nd t 8. c Logistic Function: f c (constnt): limit to growth (mimum) b The logistic function is used for popultions tht will be limited in their bilit to grow due to limited resources or spce. Think About It: Wht would limit popultion growth? Grph of Logistic Function: Domin: f e Rnge: Alws Incresing Horizontl Asmptotes: Pge 7 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
18 E5: Estimte the mimum popultion for Dlls nd find the popultion for the er 8,,64 given the function Pt () tht models the popultion from t.6e Mimum popultion: c,,64 Popultion in 8: t P (8) E6: Sketch the grph of the function. Stte the intercept nd horizontl smptotes..4 intercept: H.A.: Writing Logistic Function f E5: Find logistic function tht stisfies the given conditions: Initil vlue = 4; limit to growth = ; psses through the point,. c b c, 4,, b? You Tr: Complete the tble. Isotope HlfLife (ers) Initil Quntit Amount After Yers 4 C 57 5 grms 4 C 57.7 grm 6 R 6 grms Reflection: Eplin how to determine if n eponentil function is growth or dec model. Pge 8 of 8 Preclculus Grphicl, Numericl, Algebric Lrson Chpter
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 nonlgebric functions. The re clled trnscendentl functions. The eponentil
More informationMath 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(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 informationSESSION 2 Exponential and Logarithmic Functions. Math 301 R 3. (Revisit, Review and Revive)
Mth 01 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 information3.1 EXPONENTIAL FUNCTIONS & THEIR GRAPHS
. EXPONENTIAL FUNCTIONS & THEIR GRAPHS EXPONENTIAL FUNCTIONS EXPONENTIAL nd LOGARITHMIC FUNCTIONS re nonlgebric. These functions re clled TRANSCENDENTAL FUNCTIONS. DEFINITION OF EXPONENTIAL FUNCTION The
More informationUnit 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 informationthan 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 informationThe 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 informationSections 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 information4.1 OnetoOne Functions; Inverse Functions. EX) Find the inverse of the following functions. State if the inverse also forms a function or not.
4.1 OnetoOne Functions; Inverse Functions Finding Inverses of Functions To find the inverse of function simply switch nd y vlues. Input becomes Output nd Output becomes Input. EX) Find the inverse of
More informationAdvanced Functions Page 1 of 3 Investigating Exponential Functions y= b x
Advnced Functions Pge of Investigting Eponentil Functions = b Emple : Write n Eqution to Fit Dt Write n eqution to fit the dt in the tble of vlues. 0 4 4 Properties of the Eponentil Function =b () The
More information3.1 Exponential Functions and Their Graphs
.1 Eponential Functions and Their Graphs Sllabus Objective: 9.1 The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential or logarithmic
More informationapproaches 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 informationMATHS 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 informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllabus Objective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationExponentials & 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 informationMA 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 rewrite this eqution esil with the sme bse.) A nottion ws developed so tht equtions such s this
More informationExponential and logarithmic functions
5 Eponentil nd logrithmic functions 5A Inde lws 5B Negtive nd rtionl powers 5C Indicil equtions 5D Grphs of eponentil functions 5E Logrithms 5F Solving logrithmic equtions 5G Logrithmic grphs 5H Applictions
More informationFUNCTIONS: Grade 11. or y = ax 2 +bx + c or y = a(x x1)(x x2) a y
FUNCTIONS: Grde 11 The prbol: ( p) q or = +b + c or = ( 1)( ) The hperbol: p q The eponentil function: b p q Importnt fetures: intercept : Let = 0 intercept : Let = 0 Turning points (Where pplicble)
More informationLATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS ASSIGNMENT DUE
Trig/Mth Anl Nme No LATE AND ABSENT HOMEWORK IS ACCEPTED UP TO THE TIME OF THE CHAPTER TEST ON HW NO. SECTIONS ASSIGNMENT DUE LG 0/0 Prctice Set E #,, 9,, 7,,, 9,, 7,,, 9, Prctice Set F #9 odd Prctice
More informationOperations 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 informationUnit 2 Exponents Study Guide
Unit Eponents Stud Guide 7. Integer Eponents Prt : Zero Eponents Algeric Definition: 0 where cn e n nonzero vlue 0 ecuse 0 rised to n power less thn or equl to zero is n undefined vlue. Eple: 0 If ou
More informationExponents 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 informationLogarithmic 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 informationf 0 ab a b: base f
Precalculus Notes: Unit Eponential and Logarithmic Functions Sllaus Ojective: 9. The student will sketch the graph of a eponential, logistic, or logarithmic function. 9. The student will evaluate eponential
More informationMCR 3U Exam Review. 1. Determine which of the following equations represent functions. Explain. Include a graph. 2. y x
MCR U MCR U Em Review Introduction to Functions. Determine which of the following equtions represent functions. Eplin. Include grph. ) b) c) d) 0. Stte the domin nd rnge for ech reltion in question.. If
More informationPrecalculus Spring 2017
Preclculus Spring 2017 Exm 3 Summry (Section 4.1 through 5.2, nd 9.4) Section P.5 Find domins of lgebric expressions Simplify rtionl expressions Add, subtrct, multiply, & divide rtionl expressions Simplify
More informationTO: Next Year s AP Calculus Students
TO: Net Yer s AP Clculus Students As you probbly know, the students who tke AP Clculus AB nd pss the Advnced Plcement Test will plce out of one semester of college Clculus; those who tke AP Clculus BC
More informationLogarithms. 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 informationBefore we can begin Ch. 3 on Radicals, we need to be familiar with perfect squares, cubes, etc. Try and do as many as you can without a calculator!!!
Nme: Algebr II Honors PreChpter Homework Before we cn begin Ch on Rdicls, we need to be fmilir with perfect squres, cubes, etc Try nd do s mny s you cn without clcultor!!! n The nth root of n n Be ble
More informationSUMMER 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 informationARITHMETIC 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 informationLoudoun 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( ) 1. 1) Let f( x ) = 10 5x. Find and simplify f( 2) and then state the domain of f(x).
Mth 15 Fettermn/DeSmet Gustfson/Finl Em Review 1) Let f( ) = 10 5. Find nd simplif f( ) nd then stte the domin of f(). ) Let f( ) = +. Find nd simplif f(1) nd then stte the domin of f(). ) Let f( ) = 8.
More informationChapter 6 Notes, Larson/Hostetler 3e
Contents 6. Antiderivtives nd the Rules of Integrtion.......................... 6. Are nd the Definite Integrl.................................. 6.. Are............................................ 6. Reimnn
More information4 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 informationIntroduction. Definition of Hyperbola
Section 10.4 Hperbols 751 10.4 HYPERBOLAS Wht ou should lern Write equtions of hperbols in stndrd form. Find smptotes of nd grph hperbols. Use properties of hperbols to solve rellife problems. Clssif
More informationExponential and Logarithmic Functions
5 Chpter Eponentil nd Logrithmic Functions 5. Eponentil Functions nd Their Grphs 5. Applictions of Eponentil Functions 5. Logrithmic Functions nd Their Grphs 5. Properties nd Applictions of Logrithms 5.5
More informationRead 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 informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Algebr Opertions nd Epressions Common Mistkes Division of Algebric Epressions Eponentil Functions nd Logrithms Opertions nd their Inverses Mnipulting
More informationALevel Mathematics Transition Task (compulsory for all maths students and all further maths student)
ALevel 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 informationMath 31S. Rumbos Fall Solutions to Assignment #16
Mth 31S. Rumbos Fll 2016 1 Solutions to Assignment #16 1. Logistic Growth 1. Suppose tht the growth of certin niml popultion is governed by the differentil eqution 1000 dn N dt = 100 N, (1) where N(t)
More informationPreSession Review. Part 1: Basic Algebra; Linear Functions and Graphs
PreSession Review Prt 1: Bsic Algebr; Liner Functions nd Grphs A. Generl Review nd Introduction to Algebr Hierrchy of Arithmetic Opertions Opertions in ny expression re performed in the following order:
More informationExponential and logarithmic. functions. Topic: Exponential and logarithmic functions and applications
MQ Mths B Yr Ch 07 Pge 7 Mondy, October 9, 00 7: AM 7 Eponentil nd logrithmic functions syllbus ref efer erence ence Topic: Eponentil nd logrithmic functions nd pplictions In this ch chpter pter 7A Inde
More informationLesson 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 informationPrecalculus Due Tuesday/Wednesday, Sept. 12/13th Mr. Zawolo with questions.
Preclculus Due Tuesd/Wednesd, Sept. /th Emil Mr. Zwolo (isc.zwolo@psv.us) with questions. 6 Sketch the grph of f : 7! nd its inverse function f (). FUNCTIONS (Chpter ) 6 7 Show tht f : 7! hs n inverse
More informationAdvanced Algebra & Trigonometry Midterm Review Packet
Nme Dte Advnced Alger & Trigonometry Midterm Review Pcket The Advnced Alger & Trigonometry midterm em will test your generl knowledge of the mteril we hve covered since the eginning of the school yer.
More informationChapter 5 1. = on [ 1, 2] 1. Let gx ( ) e x. . The derivative of g is g ( x) e 1
Chpter 5. Let g ( e. on [, ]. The derivtive of g is g ( e ( Write the slope intercept form of the eqution of the tngent line to the grph of g t. (b Determine the coordinte of ech criticl vlue of g. Show
More informationLesson 5.3 Graph General Rational Functions
Copright Houghton Mifflin Hrcourt Publishing Compn. All rights reserved. Averge cost ($) C 8 6 4 O 4 6 8 Number of people number of hits.. number of times t bt.5 n n 4 b. 4.5 4.5.5; No, btting verge of.5
More informationMathematics Extension 1
04 Bored of Studies Tril Emintions Mthemtics Etension Written by Crrotsticks & Trebl. Generl Instructions Totl Mrks 70 Reding time 5 minutes. Working time hours. Write using blck or blue pen. Blck pen
More information11.1 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 informationThe Trapezoidal Rule
_.qd // : PM Pge 9 SECTION. Numericl Integrtion 9 f Section. The re of the region cn e pproimted using four trpezoids. Figure. = f( ) f( ) n The re of the first trpezoid is f f n. Figure. = Numericl Integrtion
More informationChapter 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 informationMAC 1105 Final Exam Review
1. Find the distnce between the pir of points. Give n ect, simplest form nswer nd deciml pproimtion to three plces., nd, MAC 110 Finl Em Review, nd,0. The points (, ) nd (, ) re endpoints of the dimeter
More informationUNIT 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 informationName Date. In Exercises 1 6, tell whether x and y show direct variation, inverse variation, or neither.
1 Prctice A In Eercises 1 6, tell whether nd show direct vrition, inverse vrition, or neither.. 7. 6. 10. 8 6. In Eercises 7 10, tell whether nd show direct vrition, inverse vrition, or neither. 8 10 8.
More informationThe use of a so called graphing calculator or programmable calculator is not permitted. Simple scientific calculators are allowed.
ERASMUS UNIVERSITY ROTTERDAM Informtion concerning the Entrnce exmintion Mthemtics level 1 for Interntionl Bchelor in Communiction nd Medi Generl informtion Avilble time: 2 hours 30 minutes. The exmintion
More informationMATH 115: Review for Chapter 7
MATH 5: Review for Chpter 7 Cn ou stte the generl form equtions for the circle, prbol, ellipse, nd hperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationA. Limits  L Hopital s Rule. x c. x c. f x. g x. x c 0 6 = 1 6. D. 1 E. nonexistent. ln ( x 1 ) 1 x 2 1. ( x 2 1) 2. 2x x 1.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( ) lim where lim f or lim f limg. c g = c limg( ) = c = c = c How to find it: Try nd find limits by
More informationMATH SS124 Sec 39 Concepts summary with examples
This note is mde for students in MTH124 Section 39 to review most(not ll) topics I think we covered in this semester, nd there s exmples fter these concepts, go over this note nd try to solve those exmples
More informationSCHEME OF WORK FOR IB MATHS STANDARD LEVEL
Snnrpsgymnsiet Lott Hydén Mthemtics, Stndrd Level Curriculum SCHEME OF WORK FOR IB MATHS STANDARD LEVEL Min resource: Mthemtics for the interntionl student, Mthemtics SL, Hese PART 1 Sequences nd Series
More informationExponential and logarithmic. functions. Areas of study Unit 2 Functions and graphs Algebra
Eponentil nd logrithmic functions VCE co covverge Ares of study Unit Functions nd grphs Algebr In this ch chpter pter A Inde lws B Negtive nd rtionl powers C Indicil equtions D Grphs of eponentil functions
More informationQuotient Rule: am a n = am n (a 0) Negative Exponents: a n = 1 (a 0) an Power Rules: (a m ) n = a m n (ab) m = a m b m
Formuls nd Concepts MAT 099: Intermedite Algebr repring for Tests: The formuls nd concepts here m not be inclusive. You should first tke our prctice test with no notes or help to see wht mteril ou re comfortble
More informationA 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 informationPrecalculus Chapter P.2 Part 1 of 3. Mr. Chapman Manchester High School
Preclculus Chpter P. Prt of Mr. Chpmn Mnchester High School Eponents Scientific Nottion Recll: ( ) () 5 ( )( )( ) ()()()() Consider epression n : Red s to the nth power. is clled the bse n is clled the
More informationFunctions and transformations
Functions nd trnsformtions A Trnsformtions nd the prbol B The cubic function in power form C The power function (the hperbol) D The power function (the truncus) E The squre root function in power form
More informationSECTION 94 Translation of Axes
94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 feet bove the verte.
More informationObj: SWBAT Recall the many important types and properties of functions
Obj: SWBAT Recll the mny importnt types nd properties of functions Functions Domin nd Rnge Function Nottion Trnsformtion of Functions Combintions/Composition of Functions OnetoOne nd Inverse Functions
More informationGrade 10 Math Academic Levels (MPM2D) Unit 4 Quadratic Relations
Grde 10 Mth Acdemic Levels (MPMD) Unit Qudrtic Reltions Topics Homework Tet ook Worksheet D 1 Qudrtic Reltions in Verte Qudrtic Reltions in Verte Form (Trnsltions) Form (Trnsltions) D Qudrtic Reltions
More informationNAME: MR. WAIN FUNCTIONS
NAME: M. WAIN FUNCTIONS evision o Solving Polnomil Equtions i one term in Emples Solve: 7 7 7 0 0 7 b.9 c 7 7 7 7 ii more thn one term in Method: Get the right hnd side to equl zero = 0 Eliminte ll denomintors
More informationWarmup for Honors Calculus
Summer Work Assignment Wrmup for Honors Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Honors Clculus in the fll of 018. Due Dte: The
More informationReview Exercises for Chapter 4
_R.qd // : PM Pge CHAPTER Integrtion Review Eercises for Chpter In Eercises nd, use the grph of to sketch grph of f. To print n enlrged cop of the grph, go to the wesite www.mthgrphs.com... In Eercises
More informationMATH 144: Business Calculus Final Review
MATH 144: Business Clculus Finl Review 1 Skills 1. Clculte severl limits. 2. Find verticl nd horizontl symptotes for given rtionl function. 3. Clculte derivtive by definition. 4. Clculte severl derivtives
More informationSummary Information and Formulae MTH109 College Algebra
Generl Formuls Summry Informtion nd Formule MTH109 College Algebr Temperture: F = 9 5 C + 32 nd C = 5 ( 9 F 32 ) F = degrees Fhrenheit C = degrees Celsius Simple Interest: I = Pr t I = Interest erned (chrged)
More informationTHE DISCRIMINANT & ITS APPLICATIONS
THE DISCRIMINANT & ITS APPLICATIONS The discriminnt ( Δ ) is the epression tht is locted under the squre root sign in the qudrtic formul i.e. Δ b c. For emple: Given +, Δ () ( )() The discriminnt is used
More informationMATHEMATICS 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 informationWorksheet 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
More informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More informationIdentify graphs of linear inequalities on a number line.
COMPETENCY 1.0 KNOWLEDGE OF ALGEBRA SKILL 1.1 Identify grphs of liner inequlities on number line.  When grphing firstdegree eqution, solve for the vrible. The grph of this solution will be single point
More information1 Part II: Numerical Integration
Mth 4 Lb 1 Prt II: Numericl Integrtion This section includes severl techniques for getting pproimte numericl vlues for definite integrls without using ntiderivtives. Mthemticll, ect nswers re preferble
More informationReview of basic calculus
Review of bsic clculus This brief review reclls some of the most importnt concepts, definitions, nd theorems from bsic clculus. It is not intended to tech bsic clculus from scrtch. If ny of the items below
More informationdu = C dy = 1 dy = dy W is invertible with inverse U, so that y = W(t) is exactly the same thing as t = U(y),
29. Differentil equtions. The conceptul bsis of llometr Did it occur to ou in Lecture 3 wh Fiboncci would even cre how rpidl rbbit popultion grows? Mbe he wnted to et the rbbits. If so, then he would be
More informationCHAPTER 9. Rational Numbers, Real Numbers, and Algebra
CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers
More informationHIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time allowed Two hours (Plus 5 minutes reading time)
HIGHER SCHOOL CERTIFICATE EXAMINATION 998 MATHEMATICS 3 UNIT (ADDITIONAL) AND 3/4 UNIT (COMMON) Time llowed Two hours (Plus 5 minutes reding time) DIRECTIONS TO CANDIDATES Attempt ALL questions ALL questions
More informationThe Trapezoidal Rule
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 Approimte
More information13.3. The Area Bounded by a Curve. Introduction. Prerequisites. Learning Outcomes
The Are Bounded b Curve 3.3 Introduction One of the importnt pplictions of integrtion is to find the re bounded b curve. Often such n re cn hve phsicl significnce like the work done b motor, or the distnce
More informationMAT187H1F Lec0101 Burbulla
Chpter 6 Lecture Notes Review nd Two New Sections Sprint 17 Net Distnce nd Totl Distnce Trvelled Suppose s is the position of prticle t time t for t [, b]. Then v dt = s (t) dt = s(b) s(). s(b) s() is
More informationAP 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 informationSEE the Big Idea. Cost of Fuel (p. 397) Galapagos Penguin (p. 382) Lightning Strike (p. 371) 3D Printer (p. 369) Volunteer Project (p.
7 Rtionl Functions 7. Inverse Vrition 7. Grphing Rtionl Functions 7.3 Multipling nd Dividing Rtionl Epressions 7. Adding nd Subtrcting Rtionl Epressions 7. Solving Rtionl Equtions Cost of Fuel (p. 397)
More informationReview Factoring Polynomials:
Chpter 4 Mth 0 Review Fctoring Polynomils:. GCF e. A) 5 5 A) 4 + 9. Difference of Squres b = ( + b)( b) e. A) 9 6 B) C) 98y. Trinomils e. A) + 5 4 B) + C) + 5 + Solving Polynomils:. A) ( 5)( ) = 0 B) 4
More informationL35Wed23Nov2016Sec55PropertiesofLogsHW36MoodleQ29
L35Wed3Nov016Sec55PropertiesofLogsHW36MoodleQ9 pge 49 L35Wed3Nov016Sec55PropertiesofLogsHW36MoodleQ9 We hve looked t severl chrcteristics of the log function. Now, we will look
More informationfractions 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 informationAB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
More informationCalculus AB. For a function f(x), the derivative would be f '(
lculus AB Derivtive Formuls Derivtive Nottion: For function f(), the derivtive would e f '( ) Leiniz's Nottion: For the derivtive of y in terms of, we write d For the second derivtive using Leiniz's Nottion:
More informationPrerequisites CHAPTER P
CHAPTER P Prerequisites P. Rel Numers P.2 Crtesin Coordinte System P.3 Liner Equtions nd Inequlities P.4 Lines in the Plne P.5 Solving Equtions Grphiclly, Numericlly, nd Algericlly P.6 Comple Numers P.7
More informationCalculus Module C21. Areas by Integration. Copyright This publication The Northern Alberta Institute of Technology All Rights Reserved.
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
More information4.6 Numerical Integration
.6 Numericl Integrtion 5.6 Numericl Integrtion Approimte definite integrl using the Trpezoidl Rule. Approimte definite integrl using Simpson s Rule. Anlze the pproimte errors in the Trpezoidl Rule nd Simpson
More information