Shorter questions on this concept appear in the multiplechoice sections. As always, look over as many questions of this kind from past exams.


 Baldric Briggs
 2 years ago
 Views:
Transcription
1 22 TYPE PROBLEMS The AP clculus exms contin fresh crefully thought out often clever questions. This is especilly true for the freeresponse questions. The topics nd style of the questions re similr from yer to yer. By style I men whether the stem presents the given informtion s n eqution, grph or tble of vlues. Most of the topics cn be nd hve been presented in ech of the styles. With tht in mind this chpter will discuss the common types of questions on the exms. Some will be discussed bsed on their style nd other by the clculus topic being tested. Some of the questions test topics tht re usully found in one or severl contiguous sections of most textbook. These include the revolume questions, differentil equtions, nd in BC the prmetric equtionvector question, nd the power series question. The others tend to drw questions from diverse prts of the book tught t different times of the yer, yet bsed on the sme stem. Students need to be wre of this nd be redy to shift gers in the middle of the question. They my not be used to this since textbook questions tend to be bout the topic in tht section nd not drwn from previous sections. Using relesed exm questions during the yer nd especilly s prt of the review t the end of the yer will help students drw their knowledge together nd do their best on the exm. Another wy of developing this skill in students is to mke ll tests nd quizzes cumultive from the beginning of the yer. Alwys include few from previous units on ech test. 246
2 TYPE PROBLEMS 247 Keep in mind tht ech prt of freeresponse question cn be rewritten s multiplechoice question. You cn lso use them s shorter open end questions on your tests. Wht follows is discussion of the 10 types of questions. Type 1: Rte & Accumultion These questions re often in rel context with lot of words describing sitution in which some things re chnging. There re usully two rtes given cting in opposite wys. These re lso known s inout questions.students re sked bout the chnge tht the rtes produce over some time intervl either seprtely or together. The rtes my be given in n eqution, grph or tble. The question lmost lwys ppers on the clcultor llowed prt of the freeresponse exm nd the rtes re often firly complicted functions. If they re on the clcultor llowed section, students should store the equtions in the eqution editor of their clcultor nd use their clcultor to do ny eqution solving, integrtion, or differentition tht my be necessry. Shorter questions on this concept pper in the multiplechoice sections. As lwys, look over s mny questions of this kind from pst exms. Be redy to red nd pply; often these problems contin lot of writing which needs to be crefully red nd interepreted. Recognize tht the word rte mens tht derivtive is given even though the word derivtive my not pper. Recognize rte from the units given without the words rte or derivtive. b 8 f l^th dt = f^bh f^h. nitil mount to the mount t f^th = f^t0h + 7 f l^xh dx, where t = t 0 is the initil time, nd f^t 0 h is the t0 initil mount. Understnd the question. It is often not necessry to s much computtion integrnd is the derivtive of the integrl. Explin the mening of derivtive or its vlue in terms of the context of the problem. Explin the mening of context of the problem. Justify your nswer. (FR) 2007 AB 2 (FR) 2009 AB 3 (FR) 2010 AB 1 BC 1 (MC) 2012 AB 81 BC 81 (FR) 2013 AB 1 BC 1
3 248 CHAPTER 22 Store functions in their clcultor recll them to do computtions on their clcultor. If the rtes re given in tble, be redy to pproximte n integrl using Riemnn sum or by trpezoids. Do mx/min or incresing/decresing nlysis. These topics re discussed in chpters 7, 14 nd 20 of this book. These questions my give the position, the velocity, or the ccelertion long with n initil condition. Students my be sked bout the motion of the prticle: its direction, when it chnges direction, when it is frthest left or right, when it turns round, how fr it trvels its position t certin time, etc. Speed, the bsolute vlue of velocity, is lso common topic. The prticle my be prticle, person, cr, etc. The position, velocity, or ccelertion my be given s n eqution, grph or tble. There re lot of different question is verstile wy to test vriety of clculus concepts. (FR) 2003 AB 5 (FR) 2006 AB 4 (FR) 2008 AB 4, BC 4 (FR) 2009 AB 1, BC 1 (MC) 2012 AB 16, 28, 79, 83, 89 BC 2, 89 (FR) 2013 AB 2 Solve n initil vlue differentil eqution problems: given the velocity (or Distinguish between position t some time, nd the totl distnce trveled during the time (displcement). The totl distnce trveled is the b vlue of velocity): 7 v^h t dt The net distnce (displ b 7 v ^h t dt nl position is the initil position plus the net chnge in distnce t from x = to x = t: s^th= s^h+7 v^xh dx Notice tht this is n ccumultion function eqution. verge vlue of function) Determine the speed nd whether it is incresing or decresing. If t some time, the velocity nd ccelertion hve the sme sign then the speed is incresing. If they hve different signs the speed is decresing. If the velocity grph is moving wy from (towrds) the txis the speed is incresing (decresing). Use difference quotient to pproximte derivtive from grph or tble. (Show quotient even if the denomintor is 1.) Approximte velocity or ccelertion from grph of tble.
4 TYPE PROBLEMS 249 Approximte n integrl using Riemnn sum or trpezoid sum with vlues from tble. Determine units of mesure. Interpret mening of derivtive or problem. Justify your nswer. These topics re discussed in chpter 9 with ides from chpters 11, 14 nd 20 of this book. Type 3: Interpreting Grphs The long nme is Here s the grph of the derivtive, tell me things bout the function. There is no eqution given nd it is not expected tht students will write the eqution; rther, students re expected to determine importnt fetures of the function or its grph directly from the grph of the derivtive. They my be sked for the loction of extreme vlues, intervls where the function is incresing or decresing, concvity, etc. They my be sked for function vlues t points, found by using the res of the regions on the derivtive s grph. The grph my be given in context nd student will be sked bout tht context. sked bout the motion nd position. Less often the function s grph my be given nd students will be sked bout its derivtives. Red informtion bout the function from the grph of the derivtive. This my be pproched using derivtive techniques or techniques using the intervl test, etc.) Write n eqution of tngent line. Evlute Riemnn or trpezoidl sums from geometry of the grph or from tble Evlute integrls from res of regions on the grph. Understnd tht the function, g^xh given grph is the grph of the integrnd, f^h. t So students should know x tht if g^xh g^h f^th dt, then gl^xh= f^xh nd gll^xh= f l^xh = +7 Justify your nswer. (FR) 2005 AB 5 (FR) 2010 AB 5 (FR) 2011 AB 4, BC 4 (MC) 2012 AB 15, 17, 76, 80, 85, 87 BC 15, 18, 76, 78, 80, 88 (FR) 2012 AB 3, BC 3 (FR) 2013 AB 4, BC 4
5 250 CHAPTER 22 The ides nd concepts tht cn be tested with this type question re numerous. The type ppers on the multiplechoice exms s well s the freeresponse. They hve ccounted for bout 20% of the points vilble on recent tests. It is very importnt tht students re fmilir with ll of the ins nd outs of this sitution. These topics re discussed minly in chpters 9 12, 14 nd 20 of this book. Type 4: Are Volume the volume of the solid formed when the region is revolved round line, or used s bse of solid with regulr crosssections. These stndrd pplictions of the integrl the BC exm. If this questions ppers on the clcultor ctive section: Students should write the its vlue fter it. It is not required to give the ntiderivtive nd if students is (somehow) correct. Students should enter the equtions in the equtions editor nd then recll them for computtions. use them s the limits of integrtion. Students should write the coordintes of the intersection(s) on their pper nd ssign letter to ech. They my them use the letter s limit of integrtion from then on. Students should store nd recll them on the clcultor for computtions. This voids copy errors nd round off errors. Don t round. If rounded or truncted limits of integrtion result in (FR) 2008 AB 1, BC 1 (FR) 2009 AB 4 (FR) 2010 AB 5 (MC) 2012 AB 10, 92, BC 87 (FR) 2012 AB 2 re of the region between the grph nd the xis or between two grphs. n xis, by the disk/wsher method. (Shell method is never necessry, but is eligible for full credit if properly used.) or volume of the region in hlf. This involves setting up nd solving n
6 TYPE PROBLEMS 251 integrl eqution where the limit is the vrible for which the eqution is re of region bounded by polr curves. These topics re discussed minly in chpters 11, 12 nd 15 of this book. Type 5: Tble Questions Tbles my be used to test vriety of ides in clculus including nlysis of functions, ccumultion, positionvelocityccelertion, theory, nd theorems mong problems re how this is tested. derivtive using difference quotient. Use the two vlues closest to the Use Riemnn sums (left, right, midpoint) or trpezoidl pproximtion to pproximte the vlue of (typiclly with subintervls of uneven length). The Trpezoidl Rule, per se, is not tested; it is expected tht students will dd the res of severl trpezoids without reference to formul. verge vlue, verge rte of chnge, Rolle s Theorem, the Men Vlue AB 3 which hs men score of less thn 1out of 9 points) Use tble vlues to evlute derivtives by the product, quotient nd chin rules. If the question is presented in some context nswers should be in tht context. Do s nd Don ts Do: Remember tht you do not know wht hppens between the vlues in the tble unless some other informtion is given. Don t ssume tht the lrgest number in the tble is the mximum vlue of the function, or the smllest vlue is the minimum. Do: Show wht you re doing even if you cn do it in your hed. If you re becuse you re required to show your work. A bld nswer, one with no work, even if correct my not receive credit. Don t do rithmetic. A long expression consisting entirely of numbers such ny wy. If you simplify correct nswer incorrectly, you will lose credit. It is oky to leve things like cos ^h 2 or even Do not: Use to nswer prts of the question. While regression is good mthemtics, regression equtions re not one of the four things students my do with (FR) 2007 AB 3 (FR) 2010 AB 2, BC 2 (FR) 2011 AB 2, BC 2 (MC) 2012 AB 8, 86, 91, BC 8, 81, 86 (FR) 2013 AB 3, BC 3 (FR) 2014 AB 4, BC 4
7 252 CHAPTER 22 their clcultor. Regression gives only n pproximtion of our function. The exm is testing whether students cn work with numbers. Shorter questions on this concept pper in the multiplechoice sections. As lwys, look over s mny questions of this kind from pst exms. These topics re discussed in chpters 7 15 of this book. Type 6: Differentil Equtions The ctul solving of the differentil eqution is usully the min prt of the pproximtion. BC students my lso be sked to pproximte using Euler s Method. (FR) 2002 BC 5 (FR) 2007 AB 4b (FR) 2008 AB 5, BC 6 (MC) 2008 AB 22, 27 BC 24, 27 (FR) 2010 AB 6 (MC) 2012 AB 23, 25 BC 12, 16, 23 (FR) 2012 AB 5, BC 5 (FR) 2013 AB 6, BC 5 generl solution of differentil eqution using the method of seprtion of vribles (this is the only method tested). prticulr solution using the initil condition to evlute the constnt of integrtion initil vlue problem (IVP). Understnd tht proposed solution of differentil eqution is function (not number) nd if it nd its derivtive re substituted into the given differentil eqution the resulting eqution is true. This my be prt of doing the problem even if solving the differentil eqution is not required Growthdecy problems. extreme vlues fter seprting the vribles. mening, etc. The exms hve never sked students to ctully solve logistic eqution IVP. Lrge prts of the BC questions re often suitble for AB students nd contribute to the AB subscore of the BC exm. AB techers my dpt these for their students. Shorter questions on this concept pper in the multiplechoice sections. These topics re discussed minly in chpter 16 of this book.
8 TYPE PROBLEMS 253 Type 7: Occsionl Topics These two topics pper now nd then on the freeresponse exms. Sometimes they pper s full questions, other times s prt of question. Implicit Differentition to show tht it is correct. (This is becuse without the correct derivtive the rest of the question cnnot be done.) The followup is to nswer some question bout the function horizontl or verticl. Other times implicit differentition my be only prt of question. Implicit differentition questions lso pper on the multiplechoice sections reltion using the product rule, quotient rule, the Chin rule, etc. derivtive. coordintes of the point nd the vlue of the derivtive there. (Note: If ll tht is needed is the numericl vlue of the derivtive then the substitution 2 is often esier if done before solving for dy dx or d y dx 2 ) Anlyze the derivtive to determine where the reltion hs horizontl nd/ or verticl tngents. extreme vlues. It my lso be necessry to show tht the point where the derivtive is zero is ctully on the grph. (FR) 2004 AB 4 (FR) 2008 AB 6 (form B) (MC) 2012 AB 27 These topics re discussed minly in chpter 8 of this book. Relted Rtes Derivtives re rtes nd when more thn one vrible is involved the reltionships mong the rtes cn be found by differentiting with respect to time. The time vrible my not pper in the equtions. These questions pper occsionlly on the freeresponse sections often s prt of longer question. A simple question my pper in the multiplechoice sections. Set up nd solve relted rte problems. Know how to differentite with respect to time Interpret the nswer including units. These topics re discussed minly in chpter 10 of this book. (FR) 2001 AB 5 (FR) 2002 AB 6 (FR) 2003 AB 5, BC 5 (FR) 2008 AB 3 (MC) 2012 AB 88 BC 88
9 254 CHAPTER 22 Type 8: Prmetric nd Vector Questions ( BC topic) Rther thn the AB question bout prticle moving on line (Type 2), the BC exms hve things moving in the plne. The position or velocity is given s pir of prmetric equtions or vector. Students re sked questions bout the motion, the velocity, ccelertion, nd speed. Vectors my be written using prentheses, ( ), or pointed brckets,, or even i  j form. The pointed brckets seem to be the most populr right now, but ny nottion is llowed. (FR) 2010 BC 3 (FR) 2011 BC 1 (MC) 2012 BC 4 (FR) 2012 BC 2 (FR) 2014 BC 5c prticulr time. Given the velocity or or position. This is type of IVP differentil eqution question. Determine when the prticle is moving left or right (yl^th = 0). Determine when the prticle is moving up or down (xl^th = 0). t. Use the distnce trveled. Shorter questions on these ides pper in the multiplechoice sections. As These topics re discussed minly in chpter 19 of this book. Type 9: Polr Equtions ( BC topic) Every few yers question bout polr equtions ppers s freeresponse on the exm. The question concerns clculus concepts, so knowing the nmes or grphs of the vrious curves is not necessry. The grphs re usully given. If the topic is not on on the multiplechoice section. (FR) 2005 BC 2 (FR) 2007 BC 3 (MC) 2012 BC 91 (FR) 2013 BC 2 re enclosed by grph or between two grphs using the formul. Use the formuls x^ih= r^ih cos ^ih nd y^ih= r^ih sin ^ih to
10 TYPE PROBLEMS 255 Convert from polr to rectngulr form, Clculte the coordintes of point on the grph, nd dy Clculte d i nd dx (using the product rule). di Discuss the motion of prticle moving on the grph by discussing the mening of dr d i (motion towrds or wy from the pole), dy (motion in the di verticl direction) or dx (motion in the horizontl direction). d i dy dy/ di slope dx = t point on the grph. dx/ di These topics re discussed in chpter 19 of this book. Type 10: Sequences nd Series ( BC topic) Convergence tests for sequences pper on both sections of the BC Clculus exm. In the multiplechoice section students my be sked to sy if series converges or which of severl series converge. On the freeresponse section there is usully one full question devoted to power series. The Rtio test is used most often to determine the rdius of convergence nd the other tests to determine if the series convergences t the end points of the intervl of convergence. Students should be fmilir with nd ble to write severl terms nd the generl or given series. They my do this by substituting into series, differentiting it or integrting it. Use convergence tests to determine if series converges. The test to be used is rrely given so students need to know when to use ech of the common tests. This my be prt the endpoint nlysis for the intervl of convergence. Write the terms of Tylor or Mclurin series by clculting the Distinguish between the Tylor series for function nd the function itself. Do NOT sy tht the Tylor polynomil is equl to the function; sy it is pproximtely equl. Write series by substituting into known series, by differentiting or integrting known series, or by some other lgebric mnipultion of series. Know (memorize) the Mclurin series for sin ^xh, cos ^xh, e x nd x m. (FR) 2010 BC 6 (FR) 2011 BC 6 (FR) 2012 BC 4d, 6 (MC) 2012 BC 5, 9, 13, 17, 22, 27, 79, 90 (FR) 2014 BC 6
11 256 CHAPTER 22 convergence. This is usully done by using the Rtio test nd checking the endpoints. Be fmilir with geometric series, its rdius of convergence, nd the be ble S 3 = r. Rewriting rtionl expression s the sum of geometric series nd then writing the series. Be fmilir with the hrmonic series (diverges) nd lternting hrmonic series (converges). function t point in the intervl of convergence. Determine the error bound for convergent series (Alternting series error bound nd Lgrnge error bound). function (e.g. extreme vlues). This list is long, but only few of these items cn be sked in ny given yer. The series question on the exm is usully quite strightforwrd. As I hve suggested before, look t nd work s mny pst exm questions to get n ide of wht is sked These topics re discussed minly in chpters 17 nd 18 of this book. (FR) 2009 AB 3, BC 3 it for them, if you like. It contins hints nd suggestions to help them get redy for the exms. Much of this ws mentioned previously for you in this book; it is given here, in one plce, for you nd your students.
Prep Session Topic: Particle Motion
Student Notes Prep Session Topic: Prticle Motion Number Line for AB Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position,
More informationStudent Session Topic: Particle Motion
Student Session Topic: Prticle Motion Prticle motion nd similr problems re on the AP Clculus exms lmost every yer. The prticle my be prticle, person, cr, etc. The position, velocity or ccelertion my be
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 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 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 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 informationAP Calculus Multiple Choice: BC Edition Solutions
AP Clculus Multiple Choice: BC Edition Solutions J. Slon Mrch 8, 04 ) 0 dx ( x) is A) B) C) D) E) Divergent This function inside the integrl hs verticl symptotes t x =, nd the integrl bounds contin this
More informationGoals: Determine how to calculate the area described by a function. Define the definite integral. Explore the relationship between the definite
Unit #8 : The Integrl Gols: Determine how to clculte the re described by function. Define the definite integrl. Eplore the reltionship between the definite integrl nd re. Eplore wys to estimte the definite
More informationDisclaimer: This Final Exam Study Guide is meant to help you start studying. It is not necessarily a complete list of everything you need to know.
Disclimer: This is ment to help you strt studying. It is not necessrily complete list of everything you need to know. The MTH 33 finl exm minly consists of stndrd response questions where students must
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 informationThe Fundamental Theorem of Calculus, Particle Motion, and Average Value
The Fundmentl Theorem of Clculus, Prticle Motion, nd Averge Vlue b Three Things to Alwys Keep In Mind: (1) v( dt p( b) p( ), where v( represents the velocity nd p( represents the position. b (2) v ( dt
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 informationReview of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 200910 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
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 informationMain topics for the First Midterm
Min topics for the First Midterm The Midterm will cover Section 1.8, Chpters 23, Sections 4.14.8, nd Sections 5.15.3 (essentilly ll of the mteril covered in clss). Be sure to know the results of the
More informationObjectives. Materials
Techer Notes Activity 17 Fundmentl Theorem of Clculus Objectives Explore the connections between n ccumultion function, one defined by definite integrl, nd the integrnd Discover tht the derivtive of the
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 informationA REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H. Thomas Shores Department of Mathematics University of Nebraska Spring 2007
A REVIEW OF CALCULUS CONCEPTS FOR JDEP 384H Thoms Shores Deprtment of Mthemtics University of Nebrsk Spring 2007 Contents Rtes of Chnge nd Derivtives 1 Dierentils 4 Are nd Integrls 5 Multivrite Clculus
More informationa < a+ x < a+2 x < < a+n x = b, n A i n f(x i ) x. i=1 i=1
Mth 33 Volume Stewrt 5.2 Geometry of integrls. In this section, we will lern how to compute volumes using integrls defined by slice nlysis. First, we recll from Clculus I how to compute res. Given the
More information5.1 How do we Measure Distance Traveled given Velocity? Student Notes
. How do we Mesure Distnce Trveled given Velocity? Student Notes EX ) The tle contins velocities of moving cr in ft/sec for time t in seconds: time (sec) 3 velocity (ft/sec) 3 A) Lel the xxis & yxis
More informationChapters 4 & 5 Integrals & Applications
Contents Chpters 4 & 5 Integrls & Applictions Motivtion to Chpters 4 & 5 2 Chpter 4 3 Ares nd Distnces 3. VIDEO  Ares Under Functions............................................ 3.2 VIDEO  Applictions
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 informationOverview of Calculus I
Overview of Clculus I Prof. Jim Swift Northern Arizon University There re three key concepts in clculus: The limit, the derivtive, nd the integrl. You need to understnd the definitions of these three things,
More informationThe Fundamental Theorem of Calculus. The Total Change Theorem and the Area Under a Curve.
Clculus Li Vs The Fundmentl Theorem of Clculus. The Totl Chnge Theorem nd the Are Under Curve. Recll the following fct from Clculus course. If continuous function f(x) represents the rte of chnge of F
More information4.4 Areas, Integrals and Antiderivatives
. res, integrls nd ntiderivtives 333. Ares, Integrls nd Antiderivtives This section explores properties of functions defined s res nd exmines some connections mong res, integrls nd ntiderivtives. In order
More informationMath 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
More informationMath 8 Winter 2015 Applications of Integration
Mth 8 Winter 205 Applictions of Integrtion Here re few importnt pplictions of integrtion. The pplictions you my see on n exm in this course include only the Net Chnge Theorem (which is relly just the Fundmentl
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 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 informationf(a+h) f(a) x a h 0. This is the rate at which
M408S Concept Inventory smple nswers These questions re openended, nd re intended to cover the min topics tht we lerned in M408S. These re not crnkoutnnswer problems! (There re plenty of those in the
More informationMA 124 January 18, Derivatives are. Integrals are.
MA 124 Jnury 18, 2018 Prof PB s oneminute introduction to clculus Derivtives re. Integrls re. In Clculus 1, we lern limits, derivtives, some pplictions of derivtives, indefinite integrls, definite integrls,
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 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 information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Tody we provide the connection
More informationMA123, Chapter 10: Formulas for integrals: integrals, antiderivatives, and the Fundamental Theorem of Calculus (pp.
MA123, Chpter 1: Formuls for integrls: integrls, ntiderivtives, nd the Fundmentl Theorem of Clculus (pp. 27233, Gootmn) Chpter Gols: Assignments: Understnd the sttement of the Fundmentl Theorem of Clculus.
More informationAPPROXIMATE INTEGRATION
APPROXIMATE INTEGRATION. Introduction We hve seen tht there re functions whose ntiderivtives cnnot be expressed in closed form. For these resons ny definite integrl involving these integrnds cnnot be
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 informationTable of Contents. 1. Limits The Formal Definition of a Limit The Squeeze Theorem Area of a Circle
Tble of Contents INTRODUCTION 5 CROSS REFERENCE TABLE 13 1. Limits 1 1.1 The Forml Definition of Limit 1. The Squeeze Theorem 34 1.3 Are of Circle 43. Derivtives 53.1 Eploring Tngent Lines 54. Men Vlue
More informationTopics Covered AP Calculus AB
Topics Covered AP Clculus AB ) Elementry Functions ) Properties of Functions i) A function f is defined s set of ll ordered pirs (, y), such tht for ech element, there corresponds ectly one element y.
More information7.2 The Definite Integral
7.2 The Definite Integrl the definite integrl In the previous section, it ws found tht if function f is continuous nd nonnegtive, then the re under the grph of f on [, b] is given by F (b) F (), where
More information7.1 Integral as Net Change Calculus. What is the total distance traveled? What is the total displacement?
7.1 Integrl s Net Chnge Clculus 7.1 INTEGRAL AS NET CHANGE Distnce versus Displcement We hve lredy seen how the position of n oject cn e found y finding the integrl of the velocity function. The chnge
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 informationx = b a n x 2 e x dx. cdx = c(b a), where c is any constant. a b
CHAPTER 5. INTEGRALS 61 where nd x = b n x i = 1 (x i 1 + x i ) = midpoint of [x i 1, x i ]. Problem 168 (Exercise 1, pge 377). Use the Midpoint Rule with the n = 4 to pproximte 5 1 x e x dx. Some quick
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 informationDistance And Velocity
Unit #8  The Integrl Some problems nd solutions selected or dpted from HughesHllett Clculus. Distnce And Velocity. The grph below shows the velocity, v, of n object (in meters/sec). Estimte the totl
More informationMath 116 Final Exam April 26, 2013
Mth 6 Finl Exm April 26, 23 Nme: EXAM SOLUTIONS Instructor: Section:. Do not open this exm until you re told to do so. 2. This exm hs 5 pges including this cover. There re problems. Note tht the problems
More informationSection 6: Area, Volume, and Average Value
Chpter The Integrl Applied Clculus Section 6: Are, Volume, nd Averge Vlue Are We hve lredy used integrls to find the re etween the grph of function nd the horizontl xis. Integrls cn lso e used to find
More information1 The fundamental theorems of calculus.
The fundmentl theorems of clculus. The fundmentl theorems of clculus. Evluting definite integrls. The indefinite integrl new nme for ntiderivtive. Differentiting integrls. Theorem Suppose f is continuous
More informationFinal Exam  Review MATH Spring 2017
Finl Exm  Review MATH 5  Spring 7 Chpter, 3, nd Sections 5.5.5, 5.7 Finl Exm: Tuesdy 5/9, :37:pm The following is list of importnt concepts from the sections which were not covered by Midterm Exm or.
More information5: The Definite Integral
5: The Definite Integrl 5.: Estimting with Finite Sums Consider moving oject its velocity (meters per second) t ny time (seconds) is given y v t = t+. Cn we use this informtion to determine the distnce
More informationUnit #9 : Definite Integral Properties; Fundamental Theorem of Calculus
Unit #9 : Definite Integrl Properties; Fundmentl Theorem of Clculus Gols: Identify properties of definite integrls Define odd nd even functions, nd reltionship to integrl vlues Introduce the Fundmentl
More informationChapter 0. What is the Lebesgue integral about?
Chpter 0. Wht is the Lebesgue integrl bout? The pln is to hve tutoril sheet ech week, most often on Fridy, (to be done during the clss) where you will try to get used to the ides introduced in the previous
More informationImproper Integrals, and Differential Equations
Improper Integrls, nd Differentil Equtions October 22, 204 5.3 Improper Integrls Previously, we discussed how integrls correspond to res. More specificlly, we sid tht for function f(x), the region creted
More informationSection 5.1 #7, 10, 16, 21, 25; Section 5.2 #8, 9, 15, 20, 27, 30; Section 5.3 #4, 6, 9, 13, 16, 28, 31; Section 5.4 #7, 18, 21, 23, 25, 29, 40
Mth B Prof. Audrey Terrs HW # Solutions by Alex Eustis Due Tuesdy, Oct. 9 Section 5. #7,, 6,, 5; Section 5. #8, 9, 5,, 7, 3; Section 5.3 #4, 6, 9, 3, 6, 8, 3; Section 5.4 #7, 8,, 3, 5, 9, 4 5..7 Since
More informationTime in Seconds Speed in ft/sec (a) Sketch a possible graph for this function.
4. Are under Curve A cr is trveling so tht its speed is never decresing during 1second intervl. The speed t vrious moments in time is listed in the tle elow. Time in Seconds 3 6 9 1 Speed in t/sec 3 37
More informationLecture 14: Quadrature
Lecture 14: Qudrture This lecture is concerned with the evlution of integrls fx)dx 1) over finite intervl [, b] The integrnd fx) is ssumed to be relvlues nd smooth The pproximtion of n integrl by numericl
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 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 informationROB EBY Blinn College Mathematics Department
ROB EBY Blinn College Mthemtics Deprtment Mthemtics Deprtment 5.1, 5.2 Are, Definite Integrls MATH 2413 Rob EbyFll 26 Weknowthtwhengiventhedistncefunction, wecnfindthevelocitytnypointbyfindingthederivtiveorinstntneous
More informationMATH , Calculus 2, Fall 2018
MATH 362, 363 Clculus 2, Fll 28 The FUNdmentl Theorem of Clculus Sections 5.4 nd 5.5 This worksheet focuses on the most importnt theorem in clculus. In fct, the Fundmentl Theorem of Clculus (FTC is rgubly
More informationNUMERICAL INTEGRATION. The inverse process to differentiation in calculus is integration. Mathematically, integration is represented by.
NUMERICAL INTEGRATION 1 Introduction The inverse process to differentition in clculus is integrtion. Mthemticlly, integrtion is represented by f(x) dx which stnds for the integrl of the function f(x) with
More informationINTRODUCTION TO INTEGRATION
INTRODUCTION TO INTEGRATION 5.1 Ares nd Distnces Assume f(x) 0 on the intervl [, b]. Let A be the re under the grph of f(x). b We will obtin n pproximtion of A in the following three steps. STEP 1: Divide
More informationIntegrals  Motivation
Integrls  Motivtion When we looked t function s rte of chnge If f(x) is liner, the nswer is esy slope If f(x) is nonliner, we hd to work hrd limits derivtive A relted question is the re under f(x) (but
More informationSection 4.8. D v(t j 1 ) t. (4.8.1) j=1
Difference Equtions to Differentil Equtions Section.8 Distnce, Position, nd the Length of Curves Although we motivted the definition of the definite integrl with the notion of re, there re mny pplictions
More informationFirst midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009
Mth 3B Review Steve 18 Mrch 2009 About the finl Fridy Mrch 20, 3pm6pm, Lkretz 110 No notes, no book, no clcultor Ten questions Five review questions (Chpters 6,7,8) Five new questions (Chpters 9,10) No
More informationThe First Fundamental Theorem of Calculus. If f(x) is continuous on [a, b] and F (x) is any antiderivative. f(x) dx = F (b) F (a).
The Fundmentl Theorems of Clculus Mth 4, Section 0, Spring 009 We now know enough bout definite integrls to give precise formultions of the Fundmentl Theorems of Clculus. We will lso look t some bsic emples
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 informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2019
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH00030 SEMESTER 208/209 DR. ANTHONY BROWN 7.. Introduction to Integrtion. 7. Integrl Clculus As ws the cse with the chpter on differentil
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 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 informationDefinite integral. Mathematics FRDIS MENDELU
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová Brno 1 Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function defined on [, b]. Wht is the re of the
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 informationChapter 8.2: The Integral
Chpter 8.: The Integrl You cn think of Clculus s doulewide triler. In one width of it lives differentil clculus. In the other hlf lives wht is clled integrl clculus. We hve lredy eplored few rooms in
More information5.7 Improper Integrals
458 pplictions of definite integrls 5.7 Improper Integrls In Section 5.4, we computed the work required to lift pylod of mss m from the surfce of moon of mss nd rdius R to height H bove the surfce of the
More informationUnit Six AP Calculus Unit 6 Review Definite Integrals. Name Period Date NONCALCULATOR SECTION
Unit Six AP Clculus Unit 6 Review Definite Integrls Nme Period Dte NONCALCULATOR SECTION Voculry: Directions Define ech word nd give n exmple. 1. Definite Integrl. Men Vlue Theorem (for definite integrls)
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 113 Exam 1Review
Mth 113 Exm 1Review September 26, 2016 Exm 1 covers 6.17.3 in the textbook. It is dvisble to lso review the mteril from 5.3 nd 5.5 s this will be helpful in solving some of the problems. 6.1 Are Between
More informationMath 131. Numerical Integration Larson Section 4.6
Mth. Numericl Integrtion Lrson Section. This section looks t couple of methods for pproimting definite integrls numericlly. The gol is to get good pproimtion of the definite integrl in problems where n
More informationDefinite integral. Mathematics FRDIS MENDELU. Simona Fišnarová (Mendel University) Definite integral MENDELU 1 / 30
Definite integrl Mthemtics FRDIS MENDELU Simon Fišnrová (Mendel University) Definite integrl MENDELU / Motivtion  re under curve Suppose, for simplicity, tht y = f(x) is nonnegtive nd continuous function
More informationNumerical Analysis. 10th ed. R L Burden, J D Faires, and A M Burden
Numericl Anlysis 10th ed R L Burden, J D Fires, nd A M Burden Bemer Presenttion Slides Prepred by Dr. Annette M. Burden Youngstown Stte University July 9, 2015 Chpter 4.1: Numericl Differentition 1 ThreePoint
More informationAP Calculus AB Unit 5 (Ch. 6): The Definite Integral: Day 12 Chapter 6 Review
AP Clculus AB Unit 5 (Ch. 6): The Definite Integrl: Dy Nme o Are Approximtions Riemnn Sums: LRAM, MRAM, RRAM Chpter 6 Review Trpezoidl Rule: T = h ( y + y + y +!+ y + y 0 n n) **Know how to find rectngle
More informationThe practical version
Roerto s Notes on Integrl Clculus Chpter 4: Definite integrls nd the FTC Section 7 The Fundmentl Theorem of Clculus: The prcticl version Wht you need to know lredy: The theoreticl version of the FTC. Wht
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 informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
More information7.1 Integral as Net Change and 7.2 Areas in the Plane Calculus
7.1 Integrl s Net Chnge nd 7. Ares in the Plne Clculus 7.1 INTEGRAL AS NET CHANGE Notecrds from 7.1: Displcement vs Totl Distnce, Integrl s Net Chnge We hve lredy seen how the position of n oject cn e
More informationBig idea in Calculus: approximation
Big ide in Clculus: pproximtion Derivtive: f (x) = df dx f f(x +h) f(x) =, x h rte of chnge is pproximtely the rtio of chnges in the function vlue nd in the vrible in very short time Liner pproximtion:
More informationWeek 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
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 information2 b. , a. area is S= 2π xds. Again, understand where these formulas came from (pages ).
AP Clculus BC Review Chpter 8 Prt nd Chpter 9 Things to Know nd Be Ale to Do Know everything from the first prt of Chpter 8 Given n integrnd figure out how to ntidifferentite it using ny of the following
More informationP 3 (x) = f(0) + f (0)x + f (0) 2. x 2 + f (0) . In the problem set, you are asked to show, in general, the n th order term is a n = f (n) (0)
1 Tylor polynomils In Section 3.5, we discussed how to pproximte function f(x) round point in terms of its first derivtive f (x) evluted t, tht is using the liner pproximtion f() + f ()(x ). We clled this
More informationp(t) dt + i 1 re it ireit dt =
Note: This mteril is contined in Kreyszig, Chpter 13. Complex integrtion We will define integrls of complex functions long curves in C. (This is bit similr to [relvlued] line integrls P dx + Q dy in R2.)
More informationAP * Calculus Review
AP * Clculus Review The Fundmentl Theorems of Clculus Techer Pcket AP* is trdemrk of the College Entrnce Emintion Bord. The College Entrnce Emintion Bord ws not involved in the production of this mteril.
More information63. Representation of functions as power series Consider a power series. ( 1) n x 2n for all 1 < x < 1
3 9. SEQUENCES AND SERIES 63. Representtion of functions s power series Consider power series x 2 + x 4 x 6 + x 8 + = ( ) n x 2n It is geometric series with q = x 2 nd therefore it converges for ll q =
More informationExam 2, Mathematics 4701, Section ETY6 6:05 pm 7:40 pm, March 31, 2016, IH1105 Instructor: Attila Máté 1
Exm, Mthemtics 471, Section ETY6 6:5 pm 7:4 pm, Mrch 1, 16, IH115 Instructor: Attil Máté 1 17 copies 1. ) Stte the usul sufficient condition for the fixedpoint itertion to converge when solving the eqution
More information2008 Mathematical Methods (CAS) GA 3: Examination 2
Mthemticl Methods (CAS) GA : Exmintion GENERAL COMMENTS There were 406 students who st the Mthemticl Methods (CAS) exmintion in. Mrks rnged from to 79 out of possible score of 80. Student responses showed
More informationChapter 5. Numerical Integration
Chpter 5. Numericl Integrtion These re just summries of the lecture notes, nd few detils re included. Most of wht we include here is to be found in more detil in Anton. 5. Remrk. There re two topics with
More informationAP CALCULUS AB & BC. Course Description
AP CALCULUS AB & BC Course Description Clculus is n dvnced mthemtics course tht uses meningful problems nd pproprite technology to develop concepts nd pplictions relted to continuity nd discontinuity of
More informationIndefinite Integral. Chapter Integration  reverse of differentiation
Chpter Indefinite Integrl Most of the mthemticl opertions hve inverse opertions. The inverse opertion of differentition is clled integrtion. For exmple, describing process t the given moment knowing the
More information5.2 Volumes: Disks and Washers
4 pplictions of definite integrls 5. Volumes: Disks nd Wshers In the previous section, we computed volumes of solids for which we could determine the re of crosssection or slice. In this section, we restrict
More informationSection Areas and Distances. Example 1: Suppose a car travels at a constant 50 miles per hour for 2 hours. What is the total distance traveled?
Section 5.  Ares nd Distnces Exmple : Suppose cr trvels t constnt 5 miles per hour for 2 hours. Wht is the totl distnce trveled? Exmple 2: Suppose cr trvels 75 miles per hour for the first hour, 7 miles
More information