0.1 Chapters 1: Limits and continuity


 Ambrose Gregory
 2 years ago
 Views:
Transcription
1 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 Sndwich Theorem(F 96 # 20, F 97 # 12) If f(x) g(x) h(x) (or f(x) g(x) h(x)) on n intervl contining c nd x c f(x) x c h(x) b, then x c g(x) b. Definition A function f(x) is defined on n intervl contining c is continuous t c if nd only if x c f(x) f(c). Theorem If f(x) nd g(x) re both continuous t c, then so re the functions i) f(x) ± g(x), ii) f(x) g(x) nd iii) f(x)/g(x), in cse iii), provided tht g(c) 0. If f(x) is continuous t b nd g(x) is continuous t c, with g(c) b, then f(g(x)) is continuous t c. Polynomils re continuous t every point, ll trigonometric functions re continuous t ll points where the definition does not involve division by zero, nd by the previous theorem, so ll sums products, differences, quotients (excluding division by 0), nd compositions of these. Once we hve differentition t our disposl, there is the following theorem bout continuity. Theorem If f(x) is differentible t c, then f is continuous t c. Asymptotes(S 97 # 15) Given polynomils, f(x) n x n nd g(x) b m x m + + b 0, n 0 nd b m 0. f(x) x g(x) i) n /b n if m n, ii)0 if n<m, iii) ± if n>m f(x) iv) x g(x) ( n x)c< if m n 1. b n 1 When the function f(x) is grphed, cses i) ndii) correspond to horizontl symptotes y g(x) n bn nd y 0, respectively, nd cse iv) corresponds to n oblique symptote y n b n 1 x + c. Theorem Intermedite Vlue Theorem If f(x) is continuous on the intervl [, b] nd d is some number between f() nd f(b), then there exist number c in the intervl such tht f(c) d. Theorem Extreme Vlue Theorem If f(x) is continuous on the intervl [, b], then there exist numbers c, d in the intervl such tht f(c) f(x) f(d) for ll x in the intervl.
2 2 0.2 Chpters 23 nd prts of Chpter 6:Differentition Rules for derivtives(f 96 #14, F 97 #3, S 98 #20) 1. Product rule: [f(x)g(x)] f (x)g(x)+f(x)g (x). 2. Quotient rule: [ ] f(x) g(x) f (x)g(x) f(x)g (x). (g(x)) 2 3. Chin rule: [f(g(x))] f (g(x))g (x). Used in relted rtes problems. 4. Some derivtives: (x ) x 1, (e x ) e x, for ll rel 0,nd( x ) ln() x, [ln(x)] 1, [log x (x)] 1 for >0. ln()x Tngent line nd liner pproximtion (F 98 # 5, F 98 #20, F 99 #6, S 2000 #16) If the function f(x) is differentible t x, the tngent line to the grph y f(x) t the point (, f()) is y f()+f () (x ). The function l(x) f()+f () (x ) is clled the liner pproximtion to f(x) tx. The expression df f () is clled the differentil. Substituting x, the differentil is the liner pproximtion to f f(x) f(). Implicit differentition(f 96 #6, S 97 #19, F 97 #13, S 98 #15, S 99 #4) The curve described by f(x, y) 0 hs tngent line t the point (, b) givenbythe eqution y b + y () (x ), where y () solves the eqution given by differentiting the originl eqution with respect to x, treting y s function of x nd using the stndrd rules, nd finlly substituting x, y b. For exmple, the curve x 2 + xy + y 3 7 hs tngent line t the point (2, 1) given by the eqution y 1 (x 2) 3 x, s we see from differentiting the eqution to get 2x + y + xy +3y 2 y 0, substituting x 2,y 1 nd solving for y (2), which gives y (2) 1. Theorem Men Vlue Theorem If f(x) is differentible on the open intervl (, b) nd continuous on the closed intervl [, b], then there is number c in the intervl such tht f (c) [f(b) f()]/(b ). Some useful consequences of the Men Vlue Theorem Let f(x) be differentible function defined on the intervl I 1. If f (x) 0onI, thenf(x) is constnt. 2. If f (x) > 0onI, thenf(x) is incresing. 3. If f (x) < 0onI, thenf(x) is decresing.( S 97 #14) 4. If f (x) > 0onI, then the grph of f(x) is concve up.(f 96 #18) 5. If f (x) < 0onI, then the grph of f(x) is concve down.(s 98 #7. F 99 #7)
3 0.2. CHAPTERS 23 AND PARTS OF CHAPTER 6:DIFFERENTIATION 3 6. If f (c) 0ndf (c) > 0, then c is locl minimum. (F 98 #14, F 96 #4, Remember to check endpoints when looking for mxmin on [,b]) 7. If f (c) 0ndf (c) < 0, then c is locl mximum. (S 98 #18) 8. If f (c) 0, nd f (x) is of opposite sign for x<cnd x>c,thenc is point of inflection.(s 98 #19) Theorem l Hôpitl s rule (Chpter 6) (S 98 #2, F 98 # 1,2,4, F 99 #16) If x f(x) x g(x) 0or both its equl, nd g(x) 0on n intervl contining, then f(x) x g(x) f (x) x g (x) Some exmples: 1. x 0 sin(x)/x x 0 cos(x)/ x 0 x ln(x) ln(x)/x [1/x]/[( )x 1 1 ] x 0 x 0 x 0 ( ) x 0forll>0. x x n differentiting ntimes ex x n! 0 for ll positive integers n. ex 4. x 0 (1 + x) 1 x x (1 + x )x e, for ll. The third exmple is proved using the following useful fct If x ln(f(x)) L, then x f(x) e L. Exponentil growth nd decy: ( F 97 # 19, S 98 # 3 (growth), F 99 #14) A(t) A 0 e kt, where A 0 is the initil mount. For rdioctive decy, k ln(2) where T is the hlflife. T Derivtive of the inverse function(f 96 #23, F 97 #16, S 97 #6, S 99 #15) If f(x) is onetoone on the intervl I, then there is n inverse function on the imge of the intervl, f(i). If f(x) is differentible t the point in I nd f () 0, then the inverse function f 1 (x) is differentible nd [f 1 (x)] xf() 1/[f ()].
4 4 0.3 Chpters 47 : Integrtion Riemnn sums: (F 96 #11) If f(x) is continuous on the intervl [0, 1], then n n i1 1 n f( i 1 n ) 0 f(x). Initil vlue problems (S 98 #16) The initil vlue problem y f(x), y() b is solved by first finding the indefinite integrl (set of ntiderivtives) of f(x) nd then choosing the free constnt so tht the vlue of the ntiderivtive t x is b. Exmple: y 2x 3 + x, y(1) 2. The indefinite integrl of 2x 3 + x is 1 2 (x4 + x 2 )+c. Setting x 1,weseetht1+c 2,soc 1. The solution is y 1 2 (x4 + x 2 )+1. Integrtion by substitution or reding the chin rule bckwrds(mny exmples, S 97 #3, F 97 #10, S 98 #6, F 98 #19, S 99, #11, F 99 #10, 11) b u(b) f(u(x))u (x) f(u)du, or s definite integrl f(u(x))u (x) f(u)du. Integrtion by prts or reding the product rule bckwrds (F 96 #13, S 97 #1) udv uv vdu, s definite integrl b u() b u(x)v (x) u(b)v(b) u()v() v(x)u (x). Averge vlue of f(x) min <x<b f(x) 1 b f(x) mx <x<b f(x). b Theorem Fundmentl Theorem of Clculus (F 97 #6,S 97 #11) ( d x ) f(t)dt) f(x). Note: To differentite u(x) f(t)dt with respect to x, when the upper it is function u(x), use the chin rule. Are between curves(f 97#15, F 99 #9) If f(x) ndg(x) re continuous on [, b] ndf(x) g(x) on tht intervl, then the re of the region bounded by the grphs y f(x), y g(x) nd the verticl lines x, x b is given by the definite integrl b [f(x) g(x)]. If the region is between curves x h(y) n k(y) which re grphs over the xxis with h(y) k(y) on the intervl c y d then the re of the region bounded by the grphs nd the horizontl lines y c nd y d is d c [k(y) h(y)]dy. A region my stisfy both conditions. For exmple the region between the curves y 2x nd y x 2 lies over the intervl 0 x 2, but it is lso the region between x y nd 2 x y over the intervl 0 y 4. The re is clculted either by 2 0 [2x x 2 ] or [ y y 2 ]dy
5 0.3. CHAPTERS 47 : INTEGRATION 5 Volumes by slicing( F 96 #16,19, F 97 B2, F 98 #9) The volume of known crosssectionl re A(x) fromx to x b is given by the integrl b A(x). ThisgivestheWsher formul for the volume of solid of revolution given by rotting the region between the curves y f(x) ndy g(x) over the intervl [, b] round the xxis, (where f(x) g(x) on[, b]) : V b π[f(x) 2 g(x) 2 ]. If the region is lso described s being between the curves x k(y) n h(y) over the intervl c y d, but we still rotte it round the xxis, then the volume is given by the method of Cylindricl shells: V d c 2πy[k(y) h(y)]dy. Note tht the formul in the book on pge 389 uses f(y) insted of k(y) h(y). These both men the height of the cylindricl shell. In the wsher formul the generting segment is perpendiculr to the xis of rottion, wheres, in the cylindricl shell formul the generting segment is prllel to the xis. See pge 391 in the book. Prtil frctions (S 97 #4, F 97#9,10 F 98 #6) To integrte rtionl function f(x)/g(x), first mke sure the expression is in reduced form with deg(f) < deg(g) (dividing if necessry). Then fctor the denomintor g(x) into product of liner fctors x r corresponding to rel roots nd qudrtic fctors x 2 + bx + c corresponding to compex roots. Then expnd the expression in prtil frctions s sum of C terms,wherem runs from 1 to the highest power of x r in the fctoriztion of g(x) (x r) m Ax+B nd terms,wherek is the highest power of x 2 + bx + c in the fctoriztion of g(x). (x 2 +bx+c) k In most of the exmples, the highest power for both cses is 1. Remember tht when x r occurs just once s fctor in g(x), then the coefficient C in C is esily clculted by cncelling the term x r in g(x) nd substituting x r in the (x r) remining terms of f(x) ndg(x). For exmple: 1/[(x 1)(x 2)(x 3)] + b + c. So 1/[( 1)( 2)], (x 1) (x 2) (x 3) b 1/[(1)( 1)] nd c 1/[(2)(1)]. /[(x 1)(x 2)(x 3)] 1/2 (x 1) + 1 (x 2) + 1/2 (x 3) b 1 2 ln x 1 ln x ln x 3 2 ln( (x 1)(x 3)/ x 2 ). Improper integrls (S 97 #9, F 98 # 8, S 98 # 5, S 99 #24) If f(x) is continuous on the hlf open intervl (, b] nd x + f(x) ± then the f(x) is clled n improper integrl nd it is defined s it: b b f(x) f(x) c + c
6 6 Similrly, if f(x) is continuous on the hlf open intervl [, b) nd x b f(x) ± b c f(x) f(x) c b Integrls over n infinite intervl re lso improper integrls, nd re defined s its b f(x) f(x), b b f(x) b f(x). Exmples: b x k b { x k 1 b 1 k x1 k b b 1 1 k b1 k 1 1 k 1 k for k 1 b lnb ln for k 1. { 1 k /(k 1) for k>1 for k 1. b 0 b x k 0 + { x k k b1 k 1 1 k 1 k for k lnb ln for k 1. { b 1 k /(1 k) fork<1 for k 1 Another exmple (using the clcultion from the prtil frctions exmple given bove) /[(x 1)(x 2)(x 3)] 4 ln( (b 1)(b 3)/ b 2 ) ln( (4 1)(4 3)/ 4 2 ) ln( 3/2), b becuse b ( (b 1)(b 3)/ b 2 ) 1ndln(1) 0. However, if we consider the improper integrl 3 /[(x 1)(x 2)(x 3)], then we need to pick some midpoint where there is no discontinuity, for exmple, x 4 nd consider its t both endpoints 4 b /[(x 1)(x 2)(x 3)] /[(x 1)(x 2)(x 3)]+ /[(x 1)(x 2)(x 3)] b 4 We just clculted he second it nd it is finite, but the first it 4 /[(x 1)(x 2)(x 3)] [ln( 3/2) ln( ( 1)( 3)/ 2 )] is infinite, so the improper integrl 3 /[(x 1)(x 2)(x 3)] diverges.
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& 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 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 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 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 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 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 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 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 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 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 information1. Find the derivative of the following functions. a) f(x) = 2 + 3x b) f(x) = (5 2x) 8 c) f(x) = e2x
I. Dierentition. ) Rules. *product rule, quotient rule, chin rule MATH 34B FINAL REVIEW. Find the derivtive of the following functions. ) f(x) = 2 + 3x x 3 b) f(x) = (5 2x) 8 c) f(x) = e2x 4x 7 +x+2 d)
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 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 informationHandout I  Mathematics II
Hndout I  Mthemtics II The im of this hndout is to briefly summrize the most importnt definitions nd theorems, nd to provide some smple exercises. The topics re discussed in detil t the lectures nd seminrs.
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 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 informationTest 3 Review. Jiwen He. I will replace your lowest test score with the percentage grade from the final exam (provided it is higher).
Test 3 Review Jiwen He Test 3 Test 3: Dec. 46 in CASA Mteril  Through 6.3. No Homework (Thnksgiving) No homework this week! Hve GREAT Thnksgiving! Finl Exm Finl Exm: Dec. 1417 in CASA You Might Be Interested
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 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 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 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 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 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 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 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 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 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 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 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 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 informationTopics for final
Topics for 161.01 finl.1 The tngent nd velocity problems. Estimting limits from tbles. Instntneous velocity is limit of verge velocity. Slope of tngent line is limit of slope of secnt lines.. The limit
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 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 informationMath Bootcamp 2012 Calculus Refresher
Mth Bootcmp 0 Clculus Refresher Exponents For ny rel number x, the powers of x re : x 0 =, x = x, x = x x, etc. Powers re lso clled exponents. Remrk: 0 0 is indeterminte. Frctionl exponents re lso clled
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationOverview of Calculus
Overview of Clculus June 6, 2016 1 Limits Clculus begins with the notion of limit. In symbols, lim f(x) = L x c In wors, however close you emn tht the function f evlute t x, f(x), to be to the limit L
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 informationF (x) dx = F (x)+c = u + C = du,
35. The Substitution Rule An indefinite integrl of the derivtive F (x) is the function F (x) itself. Let u = F (x), where u is new vrible defined s differentible function of x. Consider the differentil
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 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 informationDERIVATIVES NOTES HARRIS MATH CAMP Introduction
f DERIVATIVES NOTES HARRIS MATH CAMP 208. Introduction Reding: Section 2. The derivtive of function t point is the slope of the tngent line to the function t tht point. Wht does this men, nd how do we
More informationlim f(x) does not exist, such that reducing a common factor between p(x) and q(x) results in the agreeable function k(x), then
AP Clculus AB/BC Formul nd Concept Chet Sheet Limit of Continuous Function If f(x) is continuous function for ll rel numers, then lim f(x) = f(c) Limits of Rtionl Functions A. If f(x) is rtionl function
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 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 informationx 2 1 dx x 3 dx = ln(x) + 2e u du = 2e u + C = 2e x + C 2x dx = arcsin x + 1 x 1 x du = 2 u + C (t + 2) 50 dt x 2 4 dx
. Compute the following indefinite integrls: ) sin(5 + )d b) c) d e d d) + d Solutions: ) After substituting u 5 +, we get: sin(5 + )d sin(u)du cos(u) + C cos(5 + ) + C b) We hve: d d ln() + + C c) Substitute
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 informationMath 116 Calculus II
Mth 6 Clculus II Contents 5 Exponentil nd Logrithmic functions 5. Review........................................... 5.. Exponentil functions............................... 5.. Logrithmic functions...............................
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 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 information38 Riemann sums and existence of the definite integral.
38 Riemnn sums nd existence of the definite integrl. In the clcultion of the re of the region X bounded by the grph of g(x) = x 2, the xxis nd 0 x b, two sums ppered: ( n (k 1) 2) b 3 n 3 re(x) ( n These
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 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 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 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 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 informationMath 0230 Calculus 2 Lectures
Mth Clculus Lectures Chpter 7 Applictions of Integrtion Numertion of sections corresponds to the text Jmes Stewrt, Essentil Clculus, Erly Trnscendentls, Second edition. Section 7. Ares Between Curves Two
More informationIMPORTANT THEOREMS CHEAT SHEET
IMPORTANT THEOREMS CHEAT SHEET BY DOUGLAS DANE Howdy, I m Bronson s dog Dougls. Bronson is still complining bout the textbook so I thought if I kept list of the importnt results for you, he might stop.
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 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 informationChapter 6. Riemann Integral
Introduction to Riemnn integrl Chpter 6. Riemnn Integrl WonKwng Prk Deprtment of Mthemtics, The College of Nturl Sciences Kookmin University Second semester, 2015 1 / 41 Introduction to Riemnn integrl
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 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 information7.2 Riemann Integrable Functions
7.2 Riemnn Integrble Functions Theorem 1. If f : [, b] R is step function, then f R[, b]. Theorem 2. If f : [, b] R is continuous on [, b], then f R[, b]. Theorem 3. If f : [, b] R is bounded nd continuous
More informationMath 113 Exam 2 Practice
Mth 3 Exm Prctice Februry 8, 03 Exm will cover 7.4, 7.5, 7.7, 7.8, 8.3 nd 8.5. Plese note tht integrtion skills lerned in erlier sections will still be needed for the mteril in 7.5, 7.8 nd chpter 8. This
More informationMath 554 Integration
Mth 554 Integrtion Hndout #9 4/12/96 Defn. A collection of n + 1 distinct points of the intervl [, b] P := {x 0 = < x 1 < < x i 1 < x i < < b =: x n } is clled prtition of the intervl. In this cse, we
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 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 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 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 informationDEFINITE INTEGRALS. f(x)dx exists. Note that, together with the definition of definite integrals, definitions (2) and (3) define b
DEFINITE INTEGRALS JOHN D. MCCARTHY Astrct. These re lecture notes for Sections 5.3 nd 5.4. 1. Section 5.3 Definition 1. f is integrle on [, ] if f(x)dx exists. Definition 2. If f() is defined, then f(x)dx.
More informationWe know that if f is a continuous nonnegative function on the interval [a, b], then b
1 Ares Between Curves c 22 Donld Kreider nd Dwight Lhr We know tht if f is continuous nonnegtive function on the intervl [, b], then f(x) dx is the re under the grph of f nd bove the intervl. We re going
More informationFINALTERM EXAMINATION 2011 Calculus &. Analytical GeometryI
FINALTERM EXAMINATION 011 Clculus &. Anlyticl GeometryI Question No: 1 { Mrks: 1 )  Plese choose one If f is twice differentible function t sttionry point x 0 x 0 nd f ''(x 0 ) > 0 then f hs reltive...
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 informationChapter 1  Functions and Variables
Business Clculus 1 Chpter 1  Functions nd Vribles This Acdemic Review is brought to you free of chrge by preptests4u.com. Any sle or trde of this review is strictly prohibited. Business Clculus 1 Ch 1:
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 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 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 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 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 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 informationAPPLICATIONS OF THE DEFINITE INTEGRAL
APPLICATIONS OF THE DEFINITE INTEGRAL. Volume: Slicing, disks nd wshers.. Volumes by Slicing. Suppose solid object hs boundries extending from x =, to x = b, nd tht its crosssection in plne pssing through
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 informationMAT137 Calculus! Lecture 20
officil website http://uoft.me/mat137 MAT137 Clculus! Lecture 20 Tody: 4.6 Concvity 4.7 Asypmtotes Net: 4.8 Curve Sketching 4.5 More Optimiztion Problems MVT Applictions Emple 1 Let f () = 3 27 20. 1 Find
More information