First midterm topics Second midterm topics End of quarter topics. Math 3B Review. Steve. 18 March 2009


 Prosper Baker
 5 years ago
 Views:
Transcription
1 Mth 3B Review Steve 18 Mrch 2009
2 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 Riemnn sums
3 Integrtion The ide behind integrtion is to slice wht we re interested in finding (i.e., re) into little pieces tht we cn mnge. We then dd up ll of the little pieces nd get the result. If we do this for the function f (x) where we slice between x = nd x = b we get the following interprettion. b [ ] re bove f (x) dx = xxis [ re below xxis ]. Things to wtch out for In cses where our re is mde of tringles, squres nd/or circles it is usully better to find the integrl by finding the pproprite re.
4 Properties of integrls b b f (x) dx = 0. f (x) dx = f (x) dx = c b f (x) dx. b f (x) dx + f (x) dx. c
5 Differentition nd integrtion re closely connected. Fundmentl theorem of clculus I If f is continuous on [, b] nd F (x) = x f (u) du then F (x) = f (x). Fundmentl theorem of clculus II If f is continuous on [, b] then b f (x) dx = F(b) F(), where F(x) is ny ntiderivtive of f (x).
6 Appliction of FTC I: Leibniz s rule Combining the fundmentl theorem of clculus with the chin rule we hve ( d h(x) ) f (u) du = f ( h(x) ) h (x) f ( g(x) ) g (x). dx g(x) Exmple Find lim x 0 sin x x e u du x 4.
7 Appliction of FTC II: Esier integrls If we know n ntiderivtive of f (x) we cn now find integrls. Functions tht we know ntiderivtives for include: x k, e x, sin x, cos x, sec 2 x, sec x tn x, x 2 Things to try when looking for ntiderivtive Expnding polynomils. Using trigonometric identities. Breking into sum of simpler pieces.
8 Applictions of integrtion: Are When finding the re between curves y = f (x) nd y = g(x), if f (x) g(x) between x = nd x = b then the re is given by Are = b ( f (x) g(x) ) dx. Things to wtch out for If no bounds re given then we need to solve for intersection, i.e., f (x) = g(x). If the curves intersect or severl curves bound the region then brek the problem into smller pieces. Sometimes it is esier to do the integrl with respect to y.
9 Applictions of integrtion: Volume We find volume by dding up the re of cross sections. The only cross sections tht we looked t (i.e., could be tested on) re circles, formed by solids of revolution. So if f (x) g(x) 0 nd we revolve the region between these curves nd x = nd x = b round the xxis, the resulting volume is b ( Volume = π (f (x)) 2 (g(x)) 2) dx. Things to wtch out for Don t forget to squre the functions nd be creful when simplifying. Volume (s with re) should never be negtive.
10 Applictions of integrtion: Averge vlue The verge vlue, f vg of the function on the intervl x b is the vlue so tht the rectngle with width (b ) nd height f vg hs the sme re s under the curve, i.e., Averge vlue = f vg = b f (x) dx b Things to wtch out for Averge vlue cn be negtive (it cn even be 0). Men vlue theorem tells us tht for continuous function on [, b] there is some c between nd b so tht f (c) = f vg.
11 Substitution rule Rules for integrtion come from rules for differentition. The most importnt rule for integrtion is the substitution rule which comes from the chin rule. f ( g(x) ) g (x) dx = f (u) du where u = g(x). For definite integrls we hve b f ( g(x) ) g (x) dx = g(b) g() f (u) du Things to wtch out for Look for functions inside of functions. If you don t see substitution to mke, try rewriting the integrl in different form.
12 Integrtion by prts The product rule for integrtion becomes integrtion by prts. u dv = uv v du or b u dv = uv b b v du. This technique is often used when there is product of two functions; lso when there is some function which is hrd to integrte but esy to differentite (i.e., ln x or rctn x). Things to wtch out for Alwys try substitution before integrtion by prts. Sometimes must be done severl times. When doing definite integrl sometimes esier to do the indefinite integrl to find ntiderivtive nd then t the end use FTC to evlute.
13 Prtil frctions A rtionl function is polynomil over polynomil if we hve rtionl function then we cn use the method of prtil frctions. First check degrees. If the degree on top is greter thn or equl to tht on the bottom do long division. Next fctor the denomintor nd use prtil frction techniques to brek it into smll chunks. Integrte ech chunk. Things to wtch out for If it is not polynomil over polynomil don t use prtil frctions. The denomintor will lwys fctor s liner nd irreducible qudrtics.
14 Prtil frctions the tricky step ( ) (x + b) k ( ) = C 1 x + b + C 2 (x + b) C k (x + b) k + ( ) (x 2 + bx + c) l ( ) = D 1x + E 1 D l x + E l x bx + c (x 2 + bx + c) l + To solve for coefficients we first cler the denomintors then choose nice vlues for x; or group coefficients nd set them equl. (Note: the number of coefficients tht need to be solved for is the sme s the degree of the denomintor.)
15 Improper integrls An improper integrl is one tht involves infinity in some wy; either bounds involving / or verticl symptotes (such s division by 0 or ln 0). The wy to del with improper integrl is to pproximte it by n integrl which is not improper nd then tke limit, i.e., b Things to wtch out for f (x) dx = lim t b t f (x) dx. Wtch out for verticl symptotes ( sneky wy to hide n improper integrl). It might need to be broken into severl prts. Sometimes better to first do integrl s n indefinite integrl L Hospitl s rule is useful for evluting limits.
16 Tylor polynomils We cn pproximte functions by polynomils. To find the best polynomils round x = we use informtion bout derivtives, we cll these Tylor polynomils. P n (x) = f ()+f ()(x )+ f () 2 (x )2 + + f (n) () (x ) n. n! We cn use these polynomils to pproximte the function for points ner x =. Things to wtch out for Mke sure tht the coefficients re numbers, i.e., don t forget to evlute. Given function we cn find the Tylor series, but we lso cn tke Tylor series nd sy something bout the function.
17 Differentil equtions Differentil eqution reltes how dependent vrible is chnging (i.e., the derivtive) in terms of the dependent vrible nd the independent vrible. These re very importnt in physics, biology, chemistry, economics, nd getting good grde in Mth 3B. We restrict ourselves to solving seprble differentil equtions, Method to solve: dy dx = f (x)g(y) with y(x 0) = y 0. 1 Seprte, put ll y terms on one side, x on the other. 2 Integrte, integrte ech side. 3 Simplify, solve for the constnt nd (perhps) solve for dependent vrible.
18 (more) Differentil equtions Things to wtch out for We cn solve for the constnt nd then simplify for the dependent vrible or vicevers We cn use the differentil eqution to solve for x tht y will hit certin vlue; or we cn see wht the vlue of y will be t time x; nd so on. If dy/dx = g(y) then n equilibrium solution is solution to g(y) = 0. A stble equilibrium is one tht ttrcts points nerby while n unstble equilibrium is one tht repels points nerby. There re two methods to test stbility of equilibrium solution ŷ: g (ŷ) < 0 is stble; g (ŷ) > 0 is unstble Drw picture.
19 Vectors A vector x = [x 1, x 2,..., x n ] cn be used to indicte direction in spce. Given points A = ( 1, 2,..., n ) nd B = (b 1, b 2,..., b n ) the vector tht strts t A nd points to B is found by AB = [b 1 1, b 2 2,..., b n n ]. The length (or mgnitude) of vector is found by x = x1 2 + x x n 2. A vector which hs length 1 is clled unit vector (these re often used when deling with direction).
20 Dot product Given vectors x = [x 1, x 2,..., x n ] nd y = [y 1, y 2,..., y n ] the dot product is x y = x 1 y 1 + x 2 y x n y n = x y cos θ, where θ is the ngle between the two vectors (if they emnted from the sme point). We cn use this to solve for the ngle between vectors. Two vectors re perpendiculr if the ngle they form right ngle. Equivlently two vectors re perpendiculr if their dot product is 0.
21 Lines To find line we need point (ˆx 1, ˆx 2,... ˆx n ) nd direction which will be provided for us by vector [ 1, 2,..., n ]. Then points on the line stisfy Vector form: x 1. x n = ˆx 1. ˆx n + t 1. n or Prmetric form: x 1 = ˆx 1 + t 1. x n = ˆx n + t n
22 Plnes A plne cn be thought of s flt 2dimensionl surfce embedded in three dimensionl spce. The most importnt prt bout the plne is norml vector, or vector which is perpendiculr to ll points in the plne. If plne hs the point (x 0, y 0, z 0 ) nd norml vector n = [, b, c] then the set of points (x, y, z) which re in the plne re those which stisfy Things to wtch out for (x x 0 ) + b(y y 0 ) + c(z z 0 ) = 0. Norml vectors re not unique, they cn chnge by scling by constnt. Two plnes re prllel if their norml vectors re prllel (i.e., sclr multiples of ech other). When reding coefficients to get the norml vector mke sure ll the coefficients re on the sme side.
23 Functions of severl vribles The domin of the function is the set of ll possible inputs. Generlly when looking for domin we void problems, the type of problems tht we might hve include 0, < 0, ln( 0). The rnge of the function is the set of ll possible outputs. Generlly speking this cn be very hrd to find (one of the resons Clculus ws invented ws to help nswer such questions). To understnd function we often look t slices in either x, y or z direction. In the z direction this mens f (x, y) = C which re curves in the plne clled level curves or contour lines.
24 Limits lim f (x, y) = L (x,y) (x 0,y 0 ) Limits don t tell us wht does hppen, rther they tell us wht should hppen bsed on wht is hppening nerby. For most functions we cn plug in the limit point nd evlute; if we don t get 0/0 then we re done. If we do get 0/0 then more work is hed. If limit exists then we should lwys get the sme nswer regrdless of which pth we tke to get to the limit point. So if long two different curves we get two different vlues then the limit does not exist. (Best curves re usully stright lines or curves of the form y = x k, x = y l.)
25 Continuity A function is continuous t point if wht should hppen is wht does hppen, i.e., lim f (x, y) = f (x 0, y 0 ). (x,y) (x 0,y 0 ) All polynomils re continuous, nd composition of continuous functions re continuous.
26 Prtil derivtives For multivrible functions we often use prtil derivtives (denoted insted of d ). Ide is to tret ll but one vrible s constnt nd then tke derivtives. f x (x, y) = f f (x + h, y) f (x, y) (x, y) = lim x h 0 h f y (x, y) = f f (x, y + h) f (x, y) (x, y) = lim y h 0 h We cn lso tke higher order derivtives, i.e., f xx, f xy, f yx, f yy nd so on. For nice functions (i.e., functions in our clss), 2 f (x, y) = 2 x y f (x, y). y x
27 Tngent plnes A tngent plne is higher dimensionl nlog of tngent line. For function f (x, y) the tngent plne t point (x 0, y 0 ) is given by z = f (x 0, y 0 ) + f (x 0, y 0 ) x (x x 0 ) + f (x 0, y 0 ) (y y 0 ). y Since tngent plnes mke good pproximtions loclly we cn use tngent plnes to pproximte the function. Things to wtch out for Mke sure to evlute the prtil derivtives.
28 Grdients The grdient vector is vector where the entries re prtil derivtives. f (x, y) f (x, y) = x f (x, y) y We cn use the grdient to find directionl derivtives t point in direction u ( unit vector). Nmely, D u f (x, y) = f (x, y) u. Things to wtch out for Mke sure u is unit vector!
29 Properties of grdients At point (x, y) we hve, f (x, y) D u f (x, y) f (x, y). So the grdient bounds the directionl derivtives! The grdient points in the direction of gretest increse. Opposite the grdient points in the direction of gretest decrese. The grdient vector f (x, y) is perpendiculr to the level curve tht psses through the point (x, y).
30 Locl mximum/minimum A locl mximum (or minimum) is point which is greter thn (less thn) ll other points nerby. Locl mximum/minimum occur t criticl points which re where f (x, y) = 0. To determine if it is mximum, minimum or sddle we use the second derivtive test. Let D(x, y) = f xx (x, y)f yy (x, y) ( f xy (x, y) ) 2 > 0 nd f xx < 0 then mximum, > 0 nd f xx > 0 then minimum, then D(x, y) < 0 then sddle, = 0 inconclusive.
31 Lgrnge multipliers When mximizing/minimizing f (x, y) given constrint g(x, y) = c then mximum/minimum occur when f (x, y) = λ g(x, y) nd g(x, y) = c. Equivlently, the criticl points of F(x, y, λ) = f (x, y) λ ( g(x, y) c ). Find the points, then plug them in to find the extreme vlues. Things to wtch out for The equtions re lmost lwys nonliner (i.e., tricky to solve). If nothing else, solve for λ nd get reltionships mong vribles.
Definition 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 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 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 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 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 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 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 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 informationPartial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the righthnd side limit equls to the lefthnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationRecitation 3: More Applications of the Derivative
Mth 1c TA: Pdric Brtlett Recittion 3: More Applictions of the Derivtive Week 3 Cltech 2012 1 Rndom Question Question 1 A grph consists of the following: A set V of vertices. A set E of edges where ech
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 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 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 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 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 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 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 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 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 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 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 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 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 informationCalculus III Review Sheet
Clculus III Review Sheet 1 Definitions 1.1 Functions A function is f is incresing on n intervl if x y implies f(x) f(y), nd decresing if x y implies f(x) f(y). It is clled monotonic if it is either incresing
More informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 2013 Outline 1 Riemnn Sums 2 Riemnn Integrls 3 Properties
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 informationHigher Checklist (Unit 3) Higher Checklist (Unit 3) Vectors
Vectors Skill Achieved? Know tht sclr is quntity tht hs only size (no direction) Identify rellife exmples of sclrs such s, temperture, mss, distnce, time, speed, energy nd electric chrge Know tht vector
More 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 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 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 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 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 informationRiemann Sums and Riemann Integrals
Riemnn Sums nd Riemnn Integrls Jmes K. Peterson Deprtment of Biologicl Sciences nd Deprtment of Mthemticl Sciences Clemson University August 26, 203 Outline Riemnn Sums Riemnn Integrls Properties Abstrct
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 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 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 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 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 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 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 informationStudent Handbook for MATH 3300
Student Hndbook for MATH 3300 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 0.5 0 0.5 0.5 0 0.5 If people do not believe tht mthemtics is simple, it is only becuse they do not relize how complicted life is. John Louis
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 informationES 111 Mathematical Methods in the Earth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry and basic calculus
ES 111 Mthemticl Methods in the Erth Sciences Lecture Outline 1  Thurs 28th Sept 17 Review of trigonometry nd bsic clculus Trigonometry When is it useful? Everywhere! Anything involving coordinte systems
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 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 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 informationThings to Memorize: A Partial List. January 27, 2017
Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors  Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved
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 informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
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 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 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 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 informationMath 1102: Calculus I (Math/Sci majors) MWF 3pm, Fulton Hall 230 Homework 2 solutions
Mth 1102: Clculus I (Mth/Sci mjors) MWF 3pm, Fulton Hll 230 Homework 2 solutions Plese write netly, nd show ll work. Cution: An nswer with no work is wrong! Do the following problems from Chpter III: 6,
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 informationLine and Surface Integrals: An Intuitive Understanding
Line nd Surfce Integrls: An Intuitive Understnding Joseph Breen Introduction Multivrible clculus is ll bout bstrcting the ides of differentition nd integrtion from the fmilir single vrible cse to tht of
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 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 informationStuff You Need to Know From Calculus
Stuff You Need to Know From Clculus For the first time in the semester, the stuff we re doing is finlly going to look like clculus (with vector slnt, of course). This mens tht in order to succeed, you
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 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 informationMath 1431 Section M TH 4:00 PM 6:00 PM Susan Wheeler Office Hours: Wed 6:00 7:00 PM Online ***NOTE LABS ARE MON AND WED
Mth 43 Section 4839 M TH 4: PM 6: PM Susn Wheeler swheeler@mth.uh.edu Office Hours: Wed 6: 7: PM Online ***NOTE LABS ARE MON AND WED t :3 PM to 3: pm ONLINE Approimting the re under curve given the type
More informationThe Fundamental Theorem of Calculus
The Fundmentl Theorem of Clculus Professor Richrd Blecksmith richrd@mth.niu.edu Dept. of Mthemticl Sciences Northern Illinois University http://mth.niu.edu/ richrd/mth229. The Definite Integrl We define
More informationNew Expansion and Infinite Series
Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06073 HIKARI Ltd, www.mhikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationWe partition C into n small arcs by forming a partition of [a, b] by picking s i as follows: a = s 0 < s 1 < < s n = b.
Mth 255  Vector lculus II Notes 4.2 Pth nd Line Integrls We begin with discussion of pth integrls (the book clls them sclr line integrls). We will do this for function of two vribles, but these ides cn
More informationand that at t = 0 the object is at position 5. Find the position of the object at t = 2.
7.2 The Fundmentl Theorem of Clculus 49 re mny, mny problems tht pper much different on the surfce but tht turn out to be the sme s these problems, in the sense tht when we try to pproimte solutions we
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 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 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 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 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 information