Math 3B Final Review


 Gabriel Lawson
 4 years ago
 Views:
Transcription
1 Mth 3B Finl Review Written by Victori Kl 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 re locted in seprte documents. You will be expected to know the formule nd how to use them. I highly recommend completing your homework before the finl. Here re some things you should know how to do: Know how to evlute n integrl using integrtion by prts Know how to evlute n integrl filled with plethor of trigonometric functions Know how to evlute n integrl using trigonometric substitution Know how to evlute n integrl using prtil frction decomposition Know how to evlute n integrl by recognizing its form (strtegies for integrtion) Know how to evlute n improper integrl Know how to use the Comprison Theorem to decide if n integrl is convergent or divergent Know how to find the length of curve Know how to find the surfce re of curve Know how to find the Tylor Polynomil or Series of function Integrtion by Prts The following is the integrtion by prts formul: udv = uv vdu A helpful mnemonic device to remember this formul is the sying ultrviolet voodoo. To determine wht function you should choose for u is helpful by nother mnemonic LIATE: Logrithmic, Inverse Trig, Algebric, Trig, Exponentil. Whenever you hve n integrl of the form r(x)dx where is polynomil nd r(x) is repetitive function (i.e. e x, sin x, cos x, sinh x, cosh x, etc.), it is helpful to integrte by prts using the tbulr method. 1
2 7.2  Trigonometric Integrls Trig integrls re usully mixed. For exmple, you my see integrls with sine nd cosine, tngent nd secnt, cotngent nd cosecnt. Most of the time you will use u substitution nd simplify your integrl using trig identities. Note: whenever you hve sin n xdx or cos n xdx where n is n even number, you will need to use the hlf ngle identity. Here re some trig identities listed below: sin 2 x + cos 2 x = 1 tn 2 x + 1 = sec 2 x 1 + cot 2 x = csc 2 x sin 2x = 2 sin x cos x cos 2x = cos 2 x sin 2 x = 2 cos 2 x 1 = 1 2 sin 2 x cos 2 x = 1 (1 + cos 2x) 2 sin 2 x = 1 (1 cos 2x) 2 sin(a ± B) = sin A cos B ± sin B cos A cos(a ± B) = cos A cos B sin A sin B Trigonometric Substitution Trig substitutions re clled n inverse substitution. It my pper t first tht these substitutions mke the integrl more complicted, but by using trig identities, the integrl ctully becomes lot simpler. To use trig substitution, you need to choose x, then find dx, nd substitute both into your integrl. Here is summry of wht trig substitutions to use in wht scenrio: 2 x 2 = x = sin θ 2 + x 2 = x = tn θ x2 2 = x = sec θ 2
3 7.4  Integrtion of Rtionl Function by Prtil Frctions A rtionl function f(x) = q(x) where nd q(x) 0 re polynomils. To evlute n integrl of the form q(x) dx we need to use prtil frction decomposition on q(x). Steps for prtil frction decomposition on q(x) : 1. Mke sure deg < deg q(x). (If, however, deg deg q(x), use long division first to mke sure the degree of the top polynomil is less thn the degree of the bottom.) 2. Fctor q(x). (By the Fundmentl Theorem of Algebr, q(x) cn be fctored into liner or irreducible qudrtic fctors.) 3. Write out the prtil frction decomposition using the fctors of q(x) nd the cses below: I. If q(x) hs distinct liner fctors, i.e., q(x) = ( 1 x b 1 )( 2 x b 2 ) ( n x b n ) where n, then where A 1, A 2,..., A n re constnts. q(x) = A 1 A 2 A n x b 1 2 x b 2 n x b n II. If q(x) hs distinct irreducible qudrtic fctors, i.e. q(x) = ( 1 x 2 + b 1 x + c 1 )( 2 x 2 + b 2 x + c 2 ) ( n x 2 + b n x + c n ), then q(x) = A 1 x + B 1 A 2 x + B 2 A n x + B n 1 x b 1 x + c 1 2 x b 2 x + c 2 n x 2 + b n x + c n where A 1, A 2,..., A n, B 1, B 2,..., B n re constnts. III. If q(x) hs repeted liner fctors, i.e., q(x) = (x b) n, then where A 1, A 2,..., A n re constnts. q(x) = A 1 x b + A 2 (x b) A n (x b) n IV. If q(x) hs repeted qudrtic fctors, i.e., q(x) = (x 2 + bx + c) n, then q(x) = A 1x + B 1 x 2 + bx + c + A 2 x + B 2 (x 2 + bx + c) A nx + B n (x 2 + bx + c) n where A 1, A 2,..., A n, B 1, B 2,..., B n re constnts. V. q(x) my be mix of Cses I  IV. How fun! 4. Solve for the constnts in the numertors of the prtil frction decomposition. 5. Integrte q(x) dx using the prtil frction decomposition. Your nswer will most likely hve inverse tngent nd/or nturl log in your nswer. Note: You don t need to memorize exctly wht it is bove, but you should intuitively know wht to do when you see prtil frctions, i.e., if you see liner fctor just constnt goes on top, wheres if it is qudrtic fctor something like Ax + B goes on top. 3
4 7.5  Strtegies for Integrtion There is very helpful tble of integrtion formuls on pge 495 of your textbook. You should know formuls 112, 15, 16 by hert. The other formuls you cn derive using usubstitutions nd trig substitutions. Here re some helpful questions to sk yourself when deciding how to evlute n integrl: 1. Cn I simplify the integrl? Try distributing terms, see if stuff cncels out, etc. 2. Cn I use usubstitution? Try it out, nd if it doesn t work then try something else. 3. Does my integrl hve... () Two different types of functions multiplied (mybe with ln x, e x,...)? Try integrtion by prts. (b) Trig functions? Try rewriting the integrl in wy where you cn use usubstitution or hlf ngle identity. (c) Rdicls, irreducible qudrtics or both? Use trig substitution. This my involve completing the squre. (d) Rtionl frctions? Use prtil frction decomposition then try nd integrte. You my need to use trig substitution if it involves irreducible qudrtics. Sometimes your first ttempt t solving n integrl my not be successful, so try it gin nother wy. The more you prctice the clerer the steps t solving integrls will become Improper Integrls Improper integrls ren t tht different from wht we hve been studying, the only difference is tht we need to split them up in certin wy nd pply limit. There re two types of improper integrls: 1. Infinite Integrls () Rewrite f(x) dx s lim t t f(x) dx. (b) Rewrite f(x) dx s lim t f(x) dx. t (c) Rewrite f(x) dx s f(x) dx + f(x) dx where is ny rel number (you decide). Then pply () nd (b) bove. 2. Discontinuous Integrls () If f(x) is discontinuous t b, then rewrite b f(x) dx s lim t t b f(x) dx. (b) If f(x) is discontinuous t, then rewrite b f(x) dx s lim b t + f(x) dx. t (c) If f(x) is discontinuous t c where < c < b, then rewrite b f(x) dx s c f(x) dx + b f(x) dx. Then pply () nd (b) bove. c 4
5 If the limit exists (not infinite), then the integrl is convergent. If the limit does not exist (infinite), then the integrl is divergent. We cn lso mke use of the Comprison Theorem: If 0 g(x) f(x) where f nd g re continuous for x, then: () If (b) If f(x) dx is convergent, then g(x) dx is convergent. g(x) dx is divergent, then f(x) dx is divergent Arc Length If f is continuous on [, b], then the length of the curve y = f(x), x b is b b ( ) 2 L = 1 + [f dy (x)] 2 dx = 1 + dx dx Both of the bove equtions re the sme. It is similr for functions of y where x = g(y), c y d: d d L = 1 + [g (y)] 2 dy = Are of Surfce of Revolution c c ( ) 2 dx dy dy If we rotte curve bout the xxis, then the surfce re is given by S = 2πy ds where ds = 1 + (dy/dx) 2 dx or ds = 1 + (dx/dy) 2 dy. You pick ds depending on wht bounds you re given. For exmple, if you re told x b, then you would pick the first option. About the yxis, the surfce re is given by S = 2πx ds Tylor Series (k Tylor Polynomils) If you would like to red more bout Tylor Polynomils, I would suggest reding Section of your textbook nd/or the chpter the professor uploded on GuchoSpce. (Wrning: my red little bit hevy since you hven t covered much mteril on series.) 5
6 The Tylor Polynomil T n (x) of function f(x) centered bout is given by T n (x) = f() + f () 1! (x ) 1 + f () 2! (x ) f (n) () (x ) n n! The lrger the n, the better T n (x) will pproximte f(x). In fct, the Tylor Series is given by f(x) = n=0 We cn estimte the error using Tylor s inequlity: f (n) () (x ) n n! R n (x) M (n + 1)! x n+1 where f (n+1) M We cn lso use Binomil Series to estimte some functions. If k is ny rel number nd x < 1, then ( ) k (1 + x) k = x n k(k 1) = 1 + kx + x 2 k(k 1)(k 2) + x n 2! 3! n=0 6
Math 113 Exam 2 Practice
Mth Em Prctice Februry, 8 Em will cover sections 6.5, 7.7.5 nd 7.8. This sheet hs three sections. The first section will remind you bout techniques nd formuls tht you should know. The second gives number
More informationMath 3B Midterm Review
Mth 3B Midterm Review Writte by Victori Kl vtkl@mth.ucsb.edu SH 643u Office Hours: R 11:00 m  1:00 pm Lst updted /15/015 Here re some short otes o Sectios 7.17.8 i your ebook. The best idictio of wht
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 informationChapter 6 Techniques of Integration
MA Techniques of Integrtion Asst.Prof.Dr.Suprnee Liswdi Chpter 6 Techniques of Integrtion Recll: Some importnt integrls tht we hve lernt so fr. Tle of Integrls n+ n d = + C n + e d = e + C ( n ) d = ln
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 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 information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. sin 2 (θ) =
Review of some needed Trig. Identities for Integrtion. Your nswers should be n ngle in RADIANS. rccos( 1 ) = π rccos(  1 ) = 2π 2 3 2 3 rcsin( 1 ) = π rcsin(  1 ) = π 2 6 2 6 Cn you do similr problems?
More informationMA Exam 2 Study Guide, Fall u n du (or the integral of linear combinations
LESSON 0 Chpter 7.2 Trigonometric Integrls. Bsic trig integrls you should know. sin = cos + C cos = sin + C sec 2 = tn + C sec tn = sec + C csc 2 = cot + C csc cot = csc + C MA 6200 Em 2 Study Guide, Fll
More information. Doubleangle formulas. Your answer should involve trig functions of θ, and not of 2θ. cos(2θ) = sin(2θ) =.
Review of some needed Trig Identities for Integrtion Your nswers should be n ngle in RADIANS rccos( 1 2 ) = rccos(  1 2 ) = rcsin( 1 2 ) = rcsin(  1 2 ) = Cn you do similr problems? Review of Bsic Concepts
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 informationChapter 8: Methods of Integration
Chpter 8: Methods of Integrtion Bsic Integrls 8. Note: We hve the following list of Bsic Integrls p p+ + c, for p sec tn + c p + ln + c sec tn sec + c e e + c tn ln sec + c ln + c sec ln sec + tn + c ln
More informationf(x) dx, If one of these two conditions is not met, we call the integral improper. Our usual definition for the value for the definite integral
Improper Integrls Every time tht we hve evluted definite integrl such s f(x) dx, we hve mde two implicit ssumptions bout the integrl:. The intervl [, b] is finite, nd. f(x) is continuous on [, b]. If one
More informationImproper Integrals. Type I Improper Integrals How do we evaluate an integral such as
Improper Integrls Two different types of integrls cn qulify s improper. The first type of improper integrl (which we will refer to s Type I) involves evluting n integrl over n infinite region. In the grph
More informationThe Product Rule state that if f and g are differentiable functions, then
Chpter 6 Techniques of Integrtion 6. Integrtion by Prts Every differentition rule hs corresponding integrtion rule. For instnce, the Substitution Rule for integrtion corresponds to the Chin Rule for differentition.
More informationIf u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(g(x))g (x) dx = f(u) du
Integrtion by Substitution: The Fundmentl Theorem of Clculus demonstrted the importnce of being ble to find ntiderivtives. We now introduce some methods for finding ntiderivtives: If u = g(x) is differentible
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 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 informationSpring 2017 Exam 1 MARK BOX HAND IN PART PIN: 17
Spring 07 Exm problem MARK BOX points HAND IN PART 0 555=x5 0 NAME: Solutions 3 0 0 PIN: 7 % 00 INSTRUCTIONS This exm comes in two prts. () HAND IN PART. Hnd in only this prt. () STATEMENT OF MULTIPLE
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 informationTechniques of Integration
Chpter 8 Techniques of Integrtion 8. Integrtion by Prts Some Exmples of Integrtion Exmple 8... Use π/4 +cos4x. cos θ = +cosθ. Exmple 8... Find secx. The ide is to multiply secx+tnx both the numertor nd
More informationIf deg(num) deg(denom), then we should use longdivision of polynomials to rewrite: p(x) = s(x) + r(x) q(x), q(x)
Mth 50 The method of prtil frction decomposition (PFD is used to integrte some rtionl functions of the form p(x, where p/q is in lowest terms nd deg(num < deg(denom. q(x If deg(num deg(denom, then we should
More informationMath 100 Review Sheet
Mth 100 Review Sheet Joseph H. Silvermn December 2010 This outline of Mth 100 is summry of the mteril covered in the course. It is designed to be study id, but it is only n outline nd should be used s
More informationMath 107H Topics for the first exam. csc 2 x dx = cot x + C csc x cotx dx = csc x + C tan x dx = ln secx + C cot x dx = ln sinx + C e x dx = e x + C
Integrtion Mth 07H Topics for the first exm Bsic list: x n dx = xn+ + C (provided n ) n + sin(kx) dx = cos(kx) + C k sec x dx = tnx + C sec x tnx dx = sec x + C /x dx = ln x + C cos(kx) dx = sin(kx) +
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 informationImproper Integrals. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Improper Integrls MATH 2, Clculus II J. Robert Buchnn Deprtment of Mthemtics Spring 28 Definite Integrls Theorem (Fundmentl Theorem of Clculus (Prt I)) If f is continuous on [, b] then b f (x) dx = [F(x)]
More informationMath 142: Final Exam Formulas to Know
Mth 4: Finl Exm Formuls to Know This ocument tells you every formul/strtegy tht you shoul know in orer to o well on your finl. Stuy it well! The helpful rules/formuls from the vrious review sheets my be
More information1 Techniques of Integration
November 8, 8 MAT86 Week Justin Ko Techniques of Integrtion. Integrtion By Substitution (Chnge of Vribles) We cn think of integrtion by substitution s the counterprt of the chin rule for differentition.
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 informationFunctions of Several Variables
Functions of Severl Vribles Sketching Level Curves Sections Prtil Derivtives on every open set on which f nd the prtils, 2 f y = 2 f y re continuous. Norml Vector x, y, 2 f y, 2 f y n = ± (x 0,y 0) (x
More informationf a L Most reasonable functions are continuous, as seen in the following theorem:
Limits Suppose f : R R. To sy lim f(x) = L x mens tht s x gets closer n closer to, then f(x) gets closer n closer to L. This suggests tht the grph of f looks like one of the following three pictures: f
More informationIntegration Techniques
Integrtion Techniques. Integrtion of Trigonometric Functions Exmple. Evlute cos x. Recll tht cos x = cos x. Hence, cos x Exmple. Evlute = ( + cos x) = (x + sin x) + C = x + 4 sin x + C. cos 3 x. Let u
More informationUnit 5. Integration techniques
18.01 EXERCISES Unit 5. Integrtion techniques 5A. Inverse trigonometric functions; Hyperbolic functions 5A1 Evlute ) tn 1 3 b) sin 1 ( 3/) c) If θ = tn 1 5, then evlute sin θ, cos θ, cot θ, csc θ, nd
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 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 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 informationFinal Exam Review. [Top Bottom]dx =
Finl Exm Review Are Between Curves See 7.1 exmples 1, 2, 4, 5 nd exerises 133 (odd) The re of the region bounded by the urves y = f(x), y = g(x), nd the lines x = nd x = b, where f nd g re ontinuous nd
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 informationAQA Further Pure 2. Hyperbolic Functions. Section 2: The inverse hyperbolic functions
Hperbolic Functions Section : The inverse hperbolic functions Notes nd Emples These notes contin subsections on The inverse hperbolic functions Integrtion using the inverse hperbolic functions Logrithmic
More information0.1 Chapters 1: Limits and continuity
1 REVIEW SHEET FOR CALCULUS 140 Some of the topics hve smple problems from previous finls indicted next to the hedings. 0.1 Chpters 1: Limits nd continuity Theorem 0.1.1 Sndwich Theorem(F 96 # 20, F 97
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 informationImproper Integrals. Introduction. Type 1: Improper Integrals on Infinite Intervals. When we defined the definite integral.
Improper Integrls Introduction When we defined the definite integrl f d we ssumed tht f ws continuous on [, ] where [, ] ws finite, closed intervl There re t lest two wys this definition cn fil to e stisfied:
More informationIntroduction and Review
Chpter 6A Notes Pge of Introuction n Review Derivtives y = f(x) y x = f (x) Evlute erivtive t x = : y = x x= f f(+h) f() () = lim h h Geometric Interprettion: see figure slope of the line tngent to f t
More informationMath Calculus with Analytic Geometry II
orem of definite Mth 5.0 with Anlytic Geometry II Jnury 4, 0 orem of definite If < b then b f (x) dx = ( under f bove xxis) ( bove f under xxis) Exmple 8 0 3 9 x dx = π 3 4 = 9π 4 orem of definite Problem
More informationFinal Review, Math 1860 Thomas Calculus Early Transcendentals, 12 ed
Finl Review, Mth 860 Thoms Clculus Erly Trnscendentls, 2 ed 6. Applictions of Integrtion: 5.6 (Review Section 5.6) Are between curves y = f(x) nd y = g(x), x b is f(x) g(x) dx nd similrly for x = f(y)
More informationCalculus II: Integrations and Series
Clculus II: Integrtions nd Series August 7, 200 Integrls Suppose we hve generl function y = f(x) For simplicity, let f(x) > 0 nd f(x) continuous Denote F (x) = re under the grph of f in the intervl [,x]
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 informationMath 360: A primitive integral and elementary functions
Mth 360: A primitive integrl nd elementry functions D. DeTurck University of Pennsylvni October 16, 2017 D. DeTurck Mth 360 001 2017C: Integrl/functions 1 / 32 Setup for the integrl prtitions Definition:
More informationAP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals
AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into
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 informationFall 2017 Exam 1 MARK BOX HAND IN PART PIN: 17
Fll 7 Exm problem MARK BOX points HAND IN PART 35=x5 NAME: Solutions PIN: 7 % INSTRUCTIONS This exm comes in two prts. () HAND IN PART. Hnd in only this prt. () STATEMENT OF MULTIPLE CHOICE PROBLEMS.
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 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 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 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 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 informationPhysics 116C Solution of inhomogeneous ordinary differential equations using Green s functions
Physics 6C Solution of inhomogeneous ordinry differentil equtions using Green s functions Peter Young November 5, 29 Homogeneous Equtions We hve studied, especilly in long HW problem, second order liner
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 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 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 informationMath 118: Honours Calculus II Winter, 2005 List of Theorems. L(P, f) U(Q, f). f exists for each ǫ > 0 there exists a partition P of [a, b] such that
Mth 118: Honours Clculus II Winter, 2005 List of Theorems Lemm 5.1 (Prtition Refinement): If P nd Q re prtitions of [, b] such tht Q P, then L(P, f) L(Q, f) U(Q, f) U(P, f). Lemm 5.2 (Upper Sums Bound
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 information4181H Problem Set 11 Selected Solutions. Chapter 19. n(log x) n 1 1 x x dx,
48H Problem Set Selected Solutions Chpter 9 # () Tke f(x) = x n, g (x) = e x, nd use integrtion by prts; this gives reduction formul: x n e x dx = x n e x n x n e x dx. (b) Tke f(x) = (log x) n, g (x)
More informationNumerical Integration
Chpter 5 Numericl Integrtion Numericl integrtion is the study of how the numericl vlue of n integrl cn be found. Methods of function pproximtion discussed in Chpter??, i.e., function pproximtion vi the
More informationNUMERICAL INTEGRATION
NUMERICAL INTEGRATION How do we evlute I = f (x) dx By the fundmentl theorem of clculus, if F (x) is n ntiderivtive of f (x), then I = f (x) dx = F (x) b = F (b) F () However, in prctice most integrls
More informationn f(x i ) x. i=1 In section 4.2, we defined the definite integral of f from x = a to x = b as n f(x i ) x; f(x) dx = lim i=1
The Fundmentl Theorem of Clculus As we continue to study the re problem, let s think bck to wht we know bout computing res of regions enclosed by curves. If we wnt to find the re of the region below the
More informationMATH 222 Second Semester Calculus. Fall 2015
MATH Second Semester Clculus Fll 5 Typeset:August, 5 Mth nd Semester Clculus Lecture notes version. (Fll 5) This is self contined set of lecture notes for Mth. The notes were written by Sigurd Angenent,
More informationRAM RAJYA MORE, SIWAN. XI th, XII th, TARGET IITJEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII (PQRS) INDEFINITE INTERATION & Their Properties
M.Sc. (Mths), B.Ed, M.Phil (Mths) MATHEMATICS Mob. : 947084408 9546359990 M.Sc. (Mths), B.Ed, M.Phil (Mths) RAM RAJYA MORE, SIWAN XI th, XII th, TARGET IITJEE (MAIN + ADVANCE) & COMPATETIVE EXAM FOR XII
More information7. Indefinite Integrals
7. Indefinite Integrls These lecture notes present my interprettion of Ruth Lwrence s lecture notes (in Herew) 7. Prolem sttement By the fundmentl theorem of clculus, to clculte n integrl we need to find
More information1 The Riemann Integral
The Riemnn Integrl. An exmple leding to the notion of integrl (res) We know how to find (i.e. define) the re of rectngle (bse height), tringle ( (sum of res of tringles). But how do we find/define n re
More informationWe are looking for ways to compute the integral of a function f(x), f(x)dx.
INTEGRATION TECHNIQUES Introdction We re looking for wys to compte the integrl of fnction f(x), f(x)dx. To pt it simply, wht we need to do is find fnction F (x) sch tht F (x) = f(x). Then if the integrl
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationMATH 1080: Calculus of One Variable II Fall 2017 Textbook: Single Variable Calculus: Early Transcendentals, 7e, by James Stewart.
MATH 1080: Clculus of One Vrile II Fll 2017 Textook: Single Vrile Clculus: Erly Trnscendentls, 7e, y Jmes Stewrt Unit 2 Skill Set Importnt: Students should expect test questions tht require synthesis of
More information(www.grantstutoring.com) DO NOT RECOPY
CALCULUS (INTEGRATION nd APPLICATIONS) THE ENTIRE COURSE IN ONE BOOK Grnt Skene for Grnt s Tutoring (www.grntstutoring.com) DO NOT RECOPY Grnt s Tutoring is privte tutoring orgniztion nd is in no wy ffilited
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationpractice How would you find: e x + e x e 2x e x 1 dx 1 e today: improper integrals
prctice How would you find: dx e x + e x e 2x e x 1 dx e 2x 1 e x dx 1. Let u=e^x. Then dx=du/u. Ans = rctn ( e^x ) + C 2. Let u=e^x. Becomes u du / (u1), divide to get u/(u1)=1+1/(u1) Ans = e^x + ln
More informationMath 231E, Lecture 33. Parametric Calculus
Mth 31E, Lecture 33. Prmetric Clculus 1 Derivtives 1.1 First derivtive Now, let us sy tht we wnt the slope t point on prmetric curve. Recll the chin rule: which exists s long s /. = / / Exmple 1.1. Reconsider
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 informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
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 informationNotes on Calculus II Integral Calculus. Miguel A. Lerma
Notes on Clculus II Integrl Clculus Miguel A. Lerm November 22, 22 Contents Introduction 5 Chpter. Integrls 6.. Ares nd Distnces. The Definite Integrl 6.2. The Evlution Theorem.3. The Fundmentl Theorem
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 informationThe final exam will take place on Friday May 11th from 8am 11am in Evans room 60.
Mth 104: finl informtion The finl exm will tke plce on Fridy My 11th from 8m 11m in Evns room 60. The exm will cover ll prts of the course with equl weighting. It will cover Chpters 1 5, 7 15, 17 21, 23
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 2 MATH00040 SEMESTER /2018
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS MATH4 SEMESTER 7/8 DR. ANTHONY BROWN 4.. Introduction to Integrtion. 4. Integrl Clculus As ws the cse with the chpter on differentil clculus,
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 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 informationReversing the Chain Rule. As we have seen from the Second Fundamental Theorem ( 4.3), the easiest way to evaluate an integral b
Mth 32 Substitution Method Stewrt 4.5 Reversing the Chin Rule. As we hve seen from the Second Fundmentl Theorem ( 4.3), the esiest wy to evlute n integrl b f(x) dx is to find n ntiderivtive, the indefinite
More informationOrthogonal Polynomials
Mth 4401 Gussin Qudrture Pge 1 Orthogonl Polynomils Orthogonl polynomils rise from series solutions to differentil equtions, lthough they cn be rrived t in vriety of different mnners. Orthogonl polynomils
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 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 informationImproper Integrals. The First Fundamental Theorem of Calculus, as we ve discussed in class, goes as follows:
Improper Integrls The First Fundmentl Theorem of Clculus, s we ve discussed in clss, goes s follows: If f is continuous on the intervl [, ] nd F is function for which F t = ft, then ftdt = F F. An integrl
More informationntegration (p3) Integration by Inspection When differentiating using function of a function or the chain rule: If y = f(u), where in turn u = f(x)
ntegrtion (p) Integrtion by Inspection When differentiting using function of function or the chin rule: If y f(u), where in turn u f( y y So, to differentite u where u +, we write ( + ) nd get ( + ) (.
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 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 information13.4. Integration by Parts. Introduction. Prerequisites. Learning Outcomes
Integrtion by Prts 13.4 Introduction Integrtion by Prts is technique for integrting products of functions. In this Section you will lern to recognise when it is pproprite to use the technique nd hve the
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 informationMath 190 Chapter 5 Lecture Notes. Professor Miguel Ornelas
Mth 19 Chpter 5 Lecture Notes Professor Miguel Ornels 1 M. Ornels Mth 19 Lecture Notes Section 5.1 Section 5.1 Ares nd Distnce Definition The re A of the region S tht lies under the grph of the continuous
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 informationZ b. f(x)dx. Yet in the above two cases we know what f(x) is. Sometimes, engineers want to calculate an area by computing I, but...
Chpter 7 Numericl Methods 7. Introduction In mny cses the integrl f(x)dx cn be found by finding function F (x) such tht F 0 (x) =f(x), nd using f(x)dx = F (b) F () which is known s the nlyticl (exct) solution.
More informationA sequence is a list of numbers in a specific order. A series is a sum of the terms of a sequence.
Core Module Revision Sheet The C exm is hour 30 minutes long nd is in two sections. Section A (36 mrks) 8 0 short questions worth no more thn 5 mrks ech. Section B (36 mrks) 3 questions worth mrks ech.
More information