Precalculus Spring 2017


 Sydney Stone
 6 years ago
 Views:
Transcription
1 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 complex frctions Rewrite difference quotients Section 4.1 Find the domin of rtionl function. Find the verticl symptotes of rtionl function. Find the horizontl symptotes of rtionl function. Section 4.2 Sketch the grph of rtionl function. Find slnt symptote of rtionl function. Sketch the grph of rtionl function with slnt symptote. Section 9.4 Divide polynomils using polynomil long division. Fctor polynomil. Find the prtil frction decomposition of rtionl expression tht contins distinct or repeted liner fctors nd/or distinct or repeted qudrtic fctors. Section 4.3 Write the eqution for prbol, ellipse, or hyperbol with center or vertex (0,0). Grph circle, prbol, ellipse, or hyperbol with center or vertex (0,0) Find ll the pertinent informtion (vertices, foci, endpoints, vertex, p, symptotes, directrix) bout prbol, ellipse, or hyperbol from the grph or the eqution with center/vertex (0,0). Section 4.4 Write the eqution for prbol, ellipse, or hyperbol in stndrd form by completing the squre. Determine the type of conic section once the conic is in stndrd form. Write the eqution for prbol, ellipse, or hyperbol with center or vertex (h,k). Grph prbol, ellipse, or hyperbol with center or vertex (h, k). Find ll the pertinent informtion (vertices, foci, endpoints, vertex, p, symptotes, directrix) bout prbol, ellipse, or hyperbol from the grph or the eqution with center/vertex (h,k) Section P.2 Use properties of exponents Use scientific nottion to represent rel numbers Use properties of rdicls Simplify nd combine rdicls Rtionlize numertors nd denomintors Use properties of rtionl exponents Section 5.1 Recognize n exponentil function Grph n exponentil function with bse or bse e Evlute n exponentil function with bse or bse e Solve rel world problem using n exponentil function Section 5.2 Recognize logrithmic function Grph logrithmic function with bse or bse e Use the properties of logrithms nd nturl logrithms to simplify Evlute logrithmic function with bse or bse e Solve rel world problem using logrithmic function
2 P.5 Rtionl Expressions I Domin Domin: Rtionl expressions : Finding domin. polynomils: b. Rdicls: keep it rel! i. sqrt(x2) x>=2 [2, inf) ii. cubert(x2) ll rels since cube rootscn be positive nd negtive c. rtionl expressions: cn t divide by zero (undefined) so look t denomintor nd set it equl to zero. i. x+ 1 ( x 2)( x 4) ii. 2 x x x+ 3 II Simplifying rtionl expressions *simplifying rtionl expression cn hve n impct on domin helps determine grphs in this course nd clculus. #36 2 x x 2 x x III Opertoins with Rtionl Expressions remember frction opertions from P.1 notes A) Multiply B) Dividing
3 C) Combining rtionl expressions + + IV Complex Frctions Complex frction: seprte frctions in numertor nd/or denomintor EX x x 2 methods: stright division, multiply numertor nd denomintor by LCD Fctoring out negtive exponents / / sign 2 wys: fctor out smllest exponent, multiply top/bottom by the smllest exponent with the opposite V Difference Quotients: gol is to eliminte the originl denomintor Ex 11 x+ h x h rtionlize numertor
4 I Intro 4.1 Rtionl Functions nd Asymptotes Rtionl function: = given tht nd re polynomils. Ex1 Domin of rtionl function = II Verticl nd Horizontl Asymptotes Definitions p.333 blue box 1) The line = is verticl Asymptote of the grph of if or. 2) The line = is horizontl symptote of the grph of if s or. Find the domin (from p. 333) = f(x)= f(x)= = P. 334 Verticl nd Horizontl Asymptotes of Rtionl Function f(x)= = with no common fctors 1) The grph of hs verticl symptotes t the zeros of 2) The grph of hs one or no Horizontl Asymptote determined by compring the degrees of nd. If, the grph of hs the line (xxis) s horizontl symptote. b. If, the grph of hs the line (rtio of leding coefficients) s horizontl symptote c. If, the grph of hs horizontl symptote. Find the symptotes Ex 2: = 2b) = Ex 3 f(x)= =
5 Ex 4 & 5 re Appliction problems only going to do ex 5 For person with sensitive skin, the mount of time hours the person cn be exposed to the sun with miniml burning cn be modeled by =.. 0< 120 where is the Sun sore Scle Reding (bsed on intensity of Ultrviolet rys). d. Find the mounts of time person with sensitive skin cn be exposed to the sun with miniml burning when =10, =25,& =100. e. If the model were vlid for ll >0, wht would be the horizontl symptote of this function, nd wht would it represent? Clss Discussion: cn horizontl symptote be crossed? How do we find tht? 1) = 2) = 3) h= I Anlyzing grphs of rtionl functions 4.2 Grphs of Rtionl functions Guidelines Let = /, where nd re polynomils Steps to grphing rtionl functions: Grph the following functions Ex1 = #26 =
6 Ex2 = #20 = Ex3 = #37 = = #40 =
7 II Slnt Asymptotes Only occurs when the degree of is bigger thn the degree of. Use to find the slnt symptote Ex 5 = Similr: #58 = III Appliction finding minimum re Ex6 A rectngulr pge is designed to contin 48 squre inches of print. The mrgins t the top nd bottom of the pge re 1 inch deep. The mrgins on ech side re 1 inches wide. Wht should the dimensions of the pge be so tht the lest mount of pper is used?
8 9.4 Prtil Frctions Decomposition into prtil Frctions (p. 690): 1) If degree of numertor is gret thn denomintor, the divide first, then use reminder to pply steps 2 nd 3 2) Fctor denomintor (over the integers) to get fctors of or + + 3) Liner fctors: for ech fctor of the form, the prticl decomposition must include the following sum of fctors: ) Qudrtic fctors: for ech fctor of the form, the prticl decomposition must include the following sum of fctors: Ex 1: Ex 3: Ex 4:
9 4.3A Prbols (p ) Imge credit: From previous sections: = + + cn become = h + If >0 then opens If <0 then opens Conic section formt: h =4 verticl =4 h horizontl Focus: equidistnt from this point, point Directrix: equidistnt from this line, line P: the distnce from the vertex to the focus, number Vertex: is hlf wy between the focus nd directrix, point D of O: direction of opening (up/down/left/right if >0 then it opens right/up A of S: xis of symmetry, line FD: focl dimeter, found by finding the bsolute vlue of 4 Bsic Verticl Prbol =4 : D of O: Vertex: A of S: Focus: Directrix: FD: Focl dimeter is 8; so t the focus we know tht ech side there is point 8/2=4 wy. Then points re (4,2) nd (4,2) to help us get the shpe of the prbol. Bsic Horizontl Prbol =4 : D of O: Vertex: A of S: Focus: Directrix: FD: Exmple = 2 D of O: Vertex: A of S: Focus: Directrix: FD: Exmple: Find the eqution of prbol tht hs vertex t 0,0 nd its focus t 5,0 D of O: Vertex: A of S: Focus: Directrix: FD: Exmple: A prbol s vertex is (0,0) nd its FD=10 nd it opens verticlly. Wht is its eqution? D of O: Vertex: A of S: Focus: Directrix: FD: Exmple 6 =0 D of O: Vertex: A of S: Focus: Directrix: FD: Exmple: Find the eqution of prbol tht hs its vertex t (0,0) nd the directrix is =6 D of O: Vertex: A of S: Focus: Directrix: FD:
10 4.3B Ellipses nd circles Circles (p. 349) h + = mens the circle hs its center t (h,k) nd its rdius is r. Ex: =1 Ellipses (p ) Assume > Horizontl + =1 Center: (0,0) Foci,0 Vertices,0 Mjor xis 2 Minor xis 2 To get : = Imges credit: Verticl + =1 Center: (0,0) Foci 0, Vertices 0, Mjor xis 2 Minor xis 2 To get : = Eccentricity of Ellipse is how stretched it is. We mesure it by =. If is 0, then the reltion is circle. If is 1, then it is very elongted/stretched. Exmple: =100 Stndrd form: Center = b= c= Vertices Foci Mjor Axis Minor Axis Eccentricity Sketch Exmple: 9 +4 =1 Stndrd form: Center = b= c= Vertices Foci Mjor Axis Minor Axis Eccentricity Sketch Exmple: 9 +9 =81 Stndrd form: Center = b= c= Vertices Foci Mjor Axis Minor Axis Eccentricity Sketch Exmple: The mjor xis length is 6; the minor xis length is 4. The foci re on the xxis. Wht is the eqution of the ellipse? Exmple: The foci re 8,0 nd the eccentricity is 0.8; wht is the eqution of the ellipse?
11 4.3C Hyperbols Imge credit: The first term indictes the type of opening. Find c: = + Horizontl =1 Verticl =1 Center (0,0) Foci,0 Vertices,0 Brnches (this is the hyperbol itself) Trnsverse xis (vertex to vertex distnce) = 2 Conjugte xis =2b Asymptotes = Center (0,0) Foci 0, Vertices 0, Brnches (this is the hyperbol itself) Trnsverse xis (vertex to vertex distnce) = 2 Conjugte xis =2b Asymptotes = Exmples: 9 16 =144 Stndrd form: Center Find c: Foci Vertices Trnsverse xis Conjugte xis Asymptotes Sketch Exmple: 9 +9=0 Stndrd form: Center Find c: Foci Vertices Trnsverse xis Conjugte xis Asymptotes Sketch Exmple: Wht is the eqution of the hyperbol with 0, 6 nd symptotes =?
12 Circle h + = with center (h,k) Ellipse + =1 With center (h,k) Vertices t h, Foci h, h + =1 With center (h,k) Vertices h, Foci h, 4.4 Moving Conic Sections Hyperbol h =1 Center (h,k) Foci h, Vertices h, Asymptotes = h =1 Center (h,k) Foci h, Vertices h, Asymptotes = h Prbol h =4 verticl Center (h,k) Focus (h, k+p) =4 h horizontl Center (h,k) Focus (h+p, k Conic generl eqution =0 (Usully B is zero) 1. If or is zero, then it s usully 2. If A nd C hve the sme sign then. A=C mens it s b. mens it s n 3. If A nd C hve different signs then it s Nme tht conic section! Conic: Center: =16 Notble chrcteristics/sketch: Stndrd form: =0 Center: Notble chrcteristics/sketch:
13 =0 Conic: Stndrd form: Center: Notble chrcteristics/sketch: Conic: Stndrd form: =0 Center: Notble chrcteristics/sketch: Degenerte conics: give reson why ech conic is degenerte = = =1
14 Exponent Rules ) Properties (p. 15) m n m+ n = m n n n 0 1 = n 1 = n = 1 m ( b) = b ( ) = m n m m n = = b b m m m* n m m P.2 Exponent properties EX 3 & c 4 3 ( 3 b )(4 b ) 12x y x y b) Scientific nottion n extension of the exponent rules (cn tke ny number nd crete it s power of 10) = 4 x 10^4 b. # lwys between 1 & 10 Divide by Exponentil Functions nd their grphs I Exponentil Functions The exponentil function with bse is denoted by where >0, 1, nd is ny rel number. =2 for x=2 =2 for x=2 =0.6 for x=2/3 II Grphs of exponentil functions EX2 = =2 EX3b = =4 Mke tble nd plot points (3 to 3)
15 Generl conclusion on pre 382 see if we cn get these from students! = = Domin rnge yint Incresing/decresing Horizontl Asymptote Continuous These re lso onetoone Onetoone functions help us relize tht they hve nd tells us tht we cn! EX4 9=3 EX4b =8 Trnsformtions of exponentils : =2 +5 = 5 +1 III The nturl bse know the first 5 digits minimum! EX 6: use clcultor to evlute = t = 2,& =.25 EX 7 Grphing =2. =. p. 384 for exct picture
16 IV Appliction: Continuously compounded interest compounded once yer Yerly = 1+ p. 386 N compoundings per yer = 1+ P = principl Continuous = nnul interest rte r (Proof on pge 385) Ex 8,c A totl of $12000 is invested t n nnul interest rte of 9%. Find the blnce fter 5 yers if it is compounded () qurterly (c) continuously Rdioctive decy : Hlflife uses bse of where h is the number of yers in hlflife for prticulr element. The generl eqution is =. Exmple: If n element hs n initil vlue of 25 grms nd its hlflife is 1599 yers, how much of the smple if left fter 2000 yers? Extr one to one property to solve equtions = =81 2 =8 =16
17 5.2 Logrithmic Functions nd their Grphs Logrithmic Functions The logrithmic function of bse is the inverse of the exponentil function with bse. For x> 0, > 0, nd 1, y= log xifndonlyif x= The function given by f( x) = log x is the logrithmic function with bse. y *The logrithmic function with bse 10 is clled the common logrithmic function. It is denoted most often by log. The Nturl Logrithmic Function f( x) = log x= ln xx, > 0 e The nturl logrithm is the inverse function of the nturl exponentil function Evluting Logrithms f( x) = log 1. 4x, when x = 1 64 f( x) = log x f( x) = log, when x = x, when x =49 Properties of Logrithms/ Properties of Nturl Logrithms 1. log 1 0 = becuse 0 = 1. / ln1 0 0 = becuse e = log = 1 becuse 1 1 =. / lne= 1 becuse e = e. x 3. log = x nd log x = x. (Inverse Properties) / lne x = x nd lnx e = x. (Inverse Properties) 4. If log x= log y, then x = y. (OneToOne Property) / Iflnx= lny, then x = y. (OnetoOne Property) Simplify or solve using properties of logrithms. log713 log log lne 8. 7ln1 9. ln 10. e 2 ln8 e
18 11. log 4(5x 9) = log 4(15x+ 29) 12. log(2 x ) = log Grphs of Logrithmic Functions *Logrithmic functions re the inverse of exponentil functions. It is possible to obtin the grph of logrithmic function by finding its inverse nd then reflecting it over the line y = x. Trnsformtion of logrithmic function: f( x) = k± log ( x h) Prent function: f( x) = log x * The verticl symptote trvels with the horizontl shift. 13. Sketch the following grphs. (Use n x/y chrt nd shifting from the prent function.) ) y = log x b) f( x) = log 2( x+ 3) c) g( x) = 4+ log3x d) h( x) = log( x 3) + 5
19 Trnsformtion of the Nturl Logrithmic Function: y = ln (x h) + k Prent Function: y = ln x Grph nd find the domin of ech function. 14. y = ln x 15. y = ln (x+6)  1 Appliction of Logrithmic Functions 16. Students prticipting in psychology experiment ttended severl lectures on subject nd were given n exm. Every month for yer fter the exm, the students were retested to see how much of the mteril they remembered. The verge scores for the grph re given by the humn memory model f(t) = 75 6ln(t+1), 0 t 12, where t is time in months. ) Wht ws the verge score on the originl exm? b) Wht ws the verge score t the end of 4 months? c) Wht ws the verge score t the end of 8 months?
ARITHMETIC OPERATIONS. The real numbers have the following properties: a b c ab ac
REVIEW OF ALGEBRA Here we review the bsic rules nd procedures of lgebr tht you need to know in order to be successful in clculus. ARITHMETIC OPERATIONS The rel numbers hve the following properties: b b
More 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 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 informationI. Equations of a Circle a. At the origin center= r= b. Standard from: center= r=
11.: Circle & Ellipse I cn Write the eqution of circle given specific informtion Grph circle in coordinte plne. Grph n ellipse nd determine ll criticl informtion. Write the eqution of n ellipse from rel
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 informationMATH 115: Review for Chapter 7
MATH 5: Review for Chpter 7 Cn you stte the generl form equtions for the circle, prbol, ellipse, nd hyperbol? () Stte the stndrd form eqution for the circle. () Stte the stndrd form eqution for the prbol
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 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 informationThe graphs of Rational Functions
Lecture 4 5A: The its of Rtionl Functions s x nd s x + The grphs of Rtionl Functions The grphs of rtionl functions hve severl differences compred to power functions. One of the differences is the behvior
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 informationChapter 1: Fundamentals
Chpter 1: Fundmentls 1.1 Rel Numbers Types of Rel Numbers: Nturl Numbers: {1, 2, 3,...}; These re the counting numbers. Integers: {... 3, 2, 1, 0, 1, 2, 3,...}; These re ll the nturl numbers, their negtives,
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 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 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 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 information( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet  Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
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 information( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
More information3.1 Exponential Functions and Their Graphs
. Eponentil Functions nd Their Grphs Sllbus Objective: 9. The student will sketch the grph of eponentil, logistic, or logrithmic function. 9. The student will evlute eponentil or logrithmic epressions.
More 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 information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
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 information5.2 Exponent Properties Involving Quotients
5. Eponent Properties Involving Quotients Lerning Objectives Use the quotient of powers property. Use the power of quotient property. Simplify epressions involving quotient properties of eponents. Use
More informationAntiderivatives/Indefinite Integrals of Basic Functions
Antiderivtives/Indefinite Integrls of Bsic Functions Power Rule: In prticulr, this mens tht x n+ x n n + + C, dx = ln x + C, if n if n = x 0 dx = dx = dx = x + C nd x (lthough you won t use the second
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 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 informationAlgebra II Notes Unit Ten: Conic Sections
Syllus Ojective: 10.1 The student will sketch the grph of conic section with centers either t or not t the origin. (PARABOLAS) Review: The Midpoint Formul The midpoint M of the line segment connecting
More informationChapter 7 Notes, Stewart 8e. 7.1 Integration by Parts Trigonometric Integrals Evaluating sin m x cos n (x) dx...
Contents 7.1 Integrtion by Prts................................... 2 7.2 Trigonometric Integrls.................................. 8 7.2.1 Evluting sin m x cos n (x)......................... 8 7.2.2 Evluting
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 informationAdding and Subtracting Rational Expressions
6.4 Adding nd Subtrcting Rtionl Epressions Essentil Question How cn you determine the domin of the sum or difference of two rtionl epressions? You cn dd nd subtrct rtionl epressions in much the sme wy
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 information( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
More informationI do slope intercept form With my shades on MartinGay, Developmental Mathematics
AATA Dte: 1//1 SWBAT simplify rdicls. Do Now: ACT Prep HW Requests: Pg 49 #1745 odds Continue Vocb sheet In Clss: Complete Skills Prctice WS HW: Complete Worksheets For Wednesdy stmped pges Bring stmped
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 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 informationRational Parents (pp. 1 of 4)
Rtionl Prents (pp of 4) Unit: 08 Lesson: 0 The grphs below describe two prent functions, ech of which is referred to s rtionl function Why do you think they re clled rtionl functions? From the grphs, provide
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 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 informationWe divide the interval [a, b] into subintervals of equal length x = b a n
Arc Length Given curve C defined by function f(x), we wnt to find the length of this curve between nd b. We do this by using process similr to wht we did in defining the Riemnn Sum of definite integrl:
More informationMORE FUNCTION GRAPHING; OPTIMIZATION. (Last edited October 28, 2013 at 11:09pm.)
MORE FUNCTION GRAPHING; OPTIMIZATION FRI, OCT 25, 203 (Lst edited October 28, 203 t :09pm.) Exercise. Let n be n rbitrry positive integer. Give n exmple of function with exctly n verticl symptotes. Give
More information12.1 Introduction to Rational Expressions
. Introduction to Rtionl Epressions A rtionl epression is rtio of polynomils; tht is, frction tht hs polynomil s numertor nd/or denomintor. Smple rtionl epressions: 0 EVALUATING RATIONAL EXPRESSIONS To
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 informationJim Lambers MAT 169 Fall Semester Lecture 4 Notes
Jim Lmbers MAT 169 Fll Semester 200910 Lecture 4 Notes These notes correspond to Section 8.2 in the text. Series Wht is Series? An infinte series, usully referred to simply s series, is n sum of ll of
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 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 informationDefinition of Continuity: The function f(x) is continuous at x = a if f(a) exists and lim
Mth 9 Course Summry/Study Guide Fll, 2005 [1] Limits Definition of Limit: We sy tht L is the limit of f(x) s x pproches if f(x) gets closer nd closer to L s x gets closer nd closer to. We write lim f(x)
More informationMath 113 Fall Final Exam Review. 2. Applications of Integration Chapter 6 including sections and section 6.8
Mth 3 Fll 0 The scope of the finl exm will include: Finl Exm Review. Integrls Chpter 5 including sections 5. 5.7, 5.0. Applictions of Integrtion Chpter 6 including sections 6. 6.5 nd section 6.8 3. Infinite
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 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 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 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 informationMath 154B Elementary Algebra2 nd Half Spring 2015
Mth 154B Elementry Alger nd Hlf Spring 015 Study Guide for Exm 4, Chpter 9 Exm 4 is scheduled for Thursdy, April rd. You my use " x 5" note crd (oth sides) nd scientific clcultor. You re expected to know
More informationB Veitch. Calculus I Study Guide
Clculus I Stuy Guie This stuy guie is in no wy exhustive. As stte in clss, ny type of question from clss, quizzes, exms, n homeworks re fir gme. There s no informtion here bout the wor problems. 1. Some
More information1.) King invests $11000 in an account that pays 3.5% interest compounded continuously.
DAY 1 Chpter 4 Exponentil nd Logrithmic Functions 4.3 Grphs of Logrithmic Functions Converting between exponentil nd logrithmic functions Common nd nturl logs The number e Chnging bses 4.4 Properties of
More informationSAINT IGNATIUS COLLEGE
SAINT IGNATIUS COLLEGE Directions to Students Tril Higher School Certificte 0 MATHEMATICS Reding Time : 5 minutes Totl Mrks 00 Working Time : hours Write using blue or blck pen. (sketches in pencil). This
More informationPART 1 MULTIPLE CHOICE Circle the appropriate response to each of the questions below. Each question has a value of 1 point.
PART MULTIPLE CHOICE Circle the pproprite response to ech of the questions below. Ech question hs vlue of point.. If in sequence the second level difference is constnt, thn the sequence is:. rithmetic
More informationLesson 2.4 Exercises, pages
Lesson. Exercises, pges A. Expnd nd simplify. ) + b) ( ) () 0  ( ) () 0 c) 7 + d) (7) ( ) 7  + 8 () ( 8). Expnd nd simplify. ) b)  7  + 7 7( ) ( ) ( ) 7( 7) 8 (7) P DO NOT COPY.. Multiplying nd Dividing
More informationMath Sequences and Series RETest Worksheet. Short Answer
Mth 0 Nme: Sequences nd Series RETest Worksheet Short Answer Use n infinite geometric series to express 353 s frction [ mrk, ll steps must be shown] The popultion of community ws 3 000 t the beginning
More informationA. Limits  L Hopital s Rule ( ) How to find it: Try and find limits by traditional methods (plugging in). If you get 0 0 or!!, apply C.! 1 6 C.
A. Limits  L Hopitl s Rule Wht you re finding: L Hopitl s Rule is used to find limits of the form f ( x) lim where lim f x x! c g x ( ) = or lim f ( x) = limg( x) = ". ( ) x! c limg( x) = 0 x! c x! c
More information( ) Same as above but m = f x = f x  symmetric to yaxis. find where f ( x) Relative: Find where f ( x) x a + lim exists ( lim f exists.
AP Clculus Finl Review Sheet solutions When you see the words This is wht you think of doing Find the zeros Set function =, fctor or use qudrtic eqution if qudrtic, grph to find zeros on clcultor Find
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 informationChapter 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 informationEquations and Inequalities
Equtions nd Inequlities Equtions nd Inequlities Curriculum Redy ACMNA: 4, 5, 6, 7, 40 www.mthletics.com Equtions EQUATIONS & Inequlities & INEQUALITIES Sometimes just writing vribles or pronumerls in
More informationThe discriminant of a quadratic function, including the conditions for real and repeated roots. Completing the square. ax 2 + bx + c = a x+
.1 Understnd nd use the lws of indices for ll rtionl eponents.. Use nd mnipulte surds, including rtionlising the denomintor..3 Work with qudrtic nd their grphs. The discriminnt of qudrtic function, including
More informationMath 3B Final Review
Mth 3B Finl Review Written by Victori Kl vtkl@mth.ucsb.edu SH 6432u Office Hours: R 9:4510:45m SH 1607 Mth Lb Hours: TR 12pm Lst updted: 12/06/14 This is continution of the midterm review. Prctice problems
More informationMath 520 Final Exam Topic Outline Sections 1 3 (Xiao/Dumas/Liaw) Spring 2008
Mth 520 Finl Exm Topic Outline Sections 1 3 (Xio/Dums/Liw) Spring 2008 The finl exm will be held on Tuesdy, My 13, 25pm in 117 McMilln Wht will be covered The finl exm will cover the mteril from ll of
More informationMath 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationk ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
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 information20 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
More informationPolynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
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 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 informationdifferent 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
More informationYear 12 Mathematics Extension 2 HSC Trial Examination 2014
Yer Mthemtics Etension HSC Tril Emintion 04 Generl Instructions. Reding time 5 minutes Working time hours Write using blck or blue pen. Blck pen is preferred. Bordpproved clcultors my be used A tble of
More informationLesson 25: Adding and Subtracting Rational Expressions
Lesson 2: Adding nd Subtrcting Rtionl Expressions Student Outcomes Students perform ddition nd subtrction of rtionl expressions. Lesson Notes This lesson reviews ddition nd subtrction of frctions using
More informationSample Problems for the Final of Math 121, Fall, 2005
Smple Problems for the Finl of Mth, Fll, 5 The following is collection of vrious types of smple problems covering sections.8,.,.5, nd.8 6.5 of the text which constitute only prt of the common Mth Finl.
More information1 ELEMENTARY ALGEBRA and GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE
ELEMENTARY ALGEBRA nd GEOMETRY READINESS DIAGNOSTIC TEST PRACTICE Directions: Study the exmples, work the prolems, then check your nswers t the end of ech topic. If you don t get the nswer given, check
More informationPreCalculus TMTA Test 2018
. For the function f ( x) ( x )( x )( x 4) find the verge rte of chnge from x to x. ) 70 4 8.4 8.4 4 7 logb 8. If logb.07, logb 4.96, nd logb.60, then ).08..867.9.48. For, ) sec (sin ) is equivlent to
More informationMath& 152 Section Integration by Parts
Mth& 5 Section 7.  Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible
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 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 informationBridging the gap: GCSE AS Level
Bridging the gp: GCSE AS Level CONTENTS Chpter Removing rckets pge Chpter Liner equtions Chpter Simultneous equtions 8 Chpter Fctors 0 Chpter Chnge the suject of the formul Chpter 6 Solving qudrtic equtions
More informationREVIEW Chapter 1 The Real Number System
Mth 7 REVIEW Chpter The Rel Number System In clss work: Solve ll exercises. (Sections. &. Definition A set is collection of objects (elements. The Set of Nturl Numbers N N = {,,,, 5, } The Set of Whole
More informationAdd and Subtract Rational Expressions. You multiplied and divided rational expressions. You will add and subtract rational expressions.
TEKS 8. A..A, A.0.F Add nd Subtrct Rtionl Epressions Before Now You multiplied nd divided rtionl epressions. You will dd nd subtrct rtionl epressions. Why? So you cn determine monthly cr lon pyments, s
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 informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
More informationSection 7.1 Integration by Substitution
Section 7. Integrtion by Substitution Evlute ech of the following integrls. Keep in mind tht using substitution my not work on some problems. For one of the definite integrls, it is not possible to find
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
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 informationChapters Five Notes SN AA U1C5
Chpters Five Notes SN AA U1C5 Nme Period Section 5: Fctoring Qudrtic Epressions When you took lger, you lerned tht the first thing involved in fctoring is to mke sure to fctor out ny numers or vriles
More informationMain topics for the Second Midterm
Min topics for the Second Midterm The Midterm will cover Sections 5.45.9, Sections 6.16.3, nd Sections 7.17.7 (essentilly ll of the mteril covered in clss from the First Midterm). Be sure to know the
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 informationENGI 3424 Engineering Mathematics Five Tutorial Examples of Partial Fractions
ENGI 44 Engineering Mthemtics Five Tutoril Exmples o Prtil Frctions 1. Express x in prtil rctions: x 4 x 4 x 4 b x x x x Both denomintors re liner nonrepeted ctors. The coverup rule my be used: 4 4 4
More informationProperties of Integrals, Indefinite Integrals. Goals: Definition of the Definite Integral Integral Calculations using Antiderivatives
Block #6: Properties of Integrls, Indefinite Integrls Gols: Definition of the Definite Integrl Integrl Clcultions using Antiderivtives Properties of Integrls The Indefinite Integrl 1 Riemnn Sums  1 Riemnn
More 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 informationExample 1: Express as a sum of logarithms by using the Product Rule. (By the definition of log)
Section 5. Properties of Logrithmic Functions Section 5. Properties of Logrithmic Functions This section covers some properties of rithmic function tht re very similr to the rules for exponents. Properties
More informationHow can we approximate the area of a region in the plane? What is an interpretation of the area under the graph of a velocity function?
Mth 125 Summry Here re some thoughts I ws hving while considering wht to put on the first midterm. The core of your studying should be the ssigned homework problems: mke sure you relly understnd those
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 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 information