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# Calculus II MAC 2312

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This 30 page Class Notes was uploaded by Connor Abernathy II on Monday October 12, 2015. The Class Notes belongs to MAC 2312 at Florida International University taught by George Kafkoulis in Fall. Since its upload, it has received 9 views. For similar materials see /class/221706/mac-2312-florida-international-university in Calculus and Pre Calculus at Florida International University.

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Date Created: 10/12/15

FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkauli uaiu DM410B tel 3053482849 20 ASSIGNMENT 18 INTRODUCTION TO LIMITS OF SEQUENCES AND FUNCTIONS llmwvl 73 W a 44 2 0 ml m rI 5 3937 71 5 4 i m mv f39 39oa 6 4 527 m1 no If ngtm9 nnmgt2m 3 fm W ISMM flirt2 4 032 719on FIGURE 3 My 3 MM 2 my 04 6 n f 2 4m w e 5amp5 a jz 9 mazj w 6214 PM an Z 739 Mu a 5quotn3z g5 Jammy mg FIGURE 4 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online DrI George Kafkoulis ka couli u du DM410B tel 3053482849 12 ASSIGNMENT 11 APPLICATIONS OF THE FUNDAMENTAL THEOREMS OF CALCULUS PART II AREAS Problem 121 Calculate the following Shaded area quotltj amp vltr amp on FIGURE 1 Areal 18 Problem 122 Calculate the following shaded area 3k 52 gt6 amp on Problem 123 FIGURE 2 Area 2 Exercises 17147 from Exercise Set 71 of textbook 4 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 2 ASSIGNMENT 2 ON PROPERTIES OF THE SUMMATION NOTATION PART I Problem 21 3 k1 13k7 Problem 22 11 wk k2 k 91 7Tklkk139 Problem 23 X 6 91 Problem 24 n 2 xk xE k1 k2 k Problem 25 Compute the following sum in terms of 71x n 1 kz6kz7kz839 Problem 26 Compute the following sum 2 2sinegk i 2sinegk1 39 791 Problem 27 Compute each of the following sums 24 1999 23 33 1 2 7y T Z S1n5 3 i S1Dlt574 j11 j1 Problem 28 kl 13k7 3O FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 21 ASSIGNMENT 192 SERIES Problem 211 Study the convergence and divergence of the following series 211 212 213 214 215 216 217 218 219 2110 ii 2n n1 00 1 3k5 2111 2112 2113 2114 2115 2116 2117 2118 2119 31 3 i k k kl 7 4 f km 32 51 kl k 5k 00 2 2n S1n3n3 7 00 1 1 Z s1n7n6vg 7 7 n1 00 211755 k4 k5 k1 7 4 00 Sin nncosn 2 4n n1 00 2 003k k1 00 2 sink k1 1 6713 4712 1 FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 19 ASSIGNMENT 17 INTEGRATION TECHNIQUES PART 11 Problem 191 Calculate the following primitives 1 f 2 f 3 f FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 15 ASSIGNMENT 14 VOLUMES BY REVOLUTIONS PART 11 Problem 151 Compute the volumes by both the slicing and the cylin drical shells method of the solids obtained by revolving around the m axis by 27139 ie a full rotation the region between the graph of x3y7y2 and0 Problem 152 Compute the volumes by both the slicing and the cylin drical shells method of the solids obtained by revolving around the m axis by 27139 ie a full rotation the region between the graph of yxx2andyx 20 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 14 ASSIGNMENT 13 THE CYLINDRICAL SHELLS METHOD Problem 141 Compute the volumes ofthe solids obtained by revolving around the y axis by 27139 ie a full rotation the regions between the m axis and the graphs of each of the following functions f 1 f sinx for z E E7 2 f 3 7 2x2 7 z 1 for z E 07T 3 f cosx for z E 07T Problem 142 Compute the volumes by both the slicing and the cylin drical shells method of the solids obtained by revolving around the y axis by 27139 ie a full rotation the region between the graph of 11ix2 andyz2 32 mum mmmmomy nunmm me an 04 D am new wmrung mums H anSaAmMs rml 4 fquot 02 11101 1 P3 1 25 Z I39cn gt1 5 Z zvij 4 21quot 391 n 5 2quot Fmqu 5 8 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 6 ASSIGNMENT 62 RIEMANN INTEGRABLE FUNCTIONS Problem 61 Consider the following function f 2 1fze01 1 ifl f 3 ifs 6 12 75 ifze23 20 if3 Prove that the function is Riemann integrable over 07 4 and calculate its Riemann integral over 07 4 Hint Follow the technique presented in Lesson 6 on a similar function 6 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 4 ASSIGNMENT 4 INCREASING FUNCTION AND RIEMANN SUMS Problem 41 Give the precise de nition of the following concepts 1 f is a strictly increasing function on a set S which is a subset of the domain of f 2 f is an increasing function on a set S which is a subset of the domain of f 3 f is a strictly decreasing function on a set S which is a subset of the domain of f 4 f is a decreasing function on a set S which is a subset of the domain of f 5 U is the upper Riemann sum of an increasing function f over a partition P of an interval ab 6 L is the upper Riemann sum of an increasing function f over a partition P of an interval 1 b Problem 42 Write the following sum as an upper Riemann sum of an increasing function over some interval 6 TL 11k i e n Z k1 Problem 43 Write the following sum as an upper Riemann sum of an increasing function over some interval 5 5 i E 7 k7 n n k1 Problem 44 Give examples of two function which do not have neither a minimum nor a maximum in an interval and yet their values are between two xed numbers I and S ie I S f Sfor z 6 ab amp I 74 minbfw amp S7 maxbfz aSwS 10 FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 8 ASSIGNMENT 8 SUMS RIEMANN SUMS amp RIEMANN INTEGRALS Problem 81 Consider the following function Dlt i 3 if x E Q ie7 z is in the set of rational numbers 7 if z Z Q ie7 z is not in the set of rational numbers Calculate the upper and lower Riemann sums based on an arbitrary partition P x07 zn of 2 22 and then compute the upper and lower Riemann Integrals of f over 222 Problem 82 Given a Riemann Integrable function f7 calculate the values of A7 B7 0 in each of the following f3dz i 0 fd 27fz3dz 0 fz73dz Problem 83 Write the following sum as an upper Riemann sum of an increasing function over some interval 7 7L 7 1 1amp4 n n k1 Problem 84 Review and give the precise de nition of the following concepts 1 f is a strictly increasing function on a set S which is a subset of the domain of f 2 U is the upper Riemann sum of an increasing function f over a partition P of an interval cob 3 f is a Riemann integrable function over an interval cob Problem 85 Calculate the general formula for the following sums 81 k 15 82 k 1 83 genin 84 i 42 24 FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 16 ASSIGNMENT 15 ARCLENGTHS Problem 161 Exercises 3 14 Exercise set 74 of textbook of textbook Anton 8th Edition or Exercise Set 64 Anton 9th Edition the of cial course textbook FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 3 ASSIGNMENT 3 ON PROPERTIES OF THE SUMMATION NOTATION PART 11 Problem 31 Calculate the general formula for the following sums 239 Z k4 m 2 k5 k1 k1 Problem 32 Calculate the general formula for the following sums n n2 239 Zk52 2d 2k33 m 2k23k12 k1 km km2 Problem 33 Calculate the general formula for the following sums Zltilk22k 71k213k239 km k m This is a tricky7 but simple problem FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 13 ASSIGNMENT 12 THE SLICING METHOD PART 1 Problem 131 Compute the volumes ofthe solids obtained by revolving around the m axis by 27139 ie a full rotation the regions between the r axis and the graphs of each of the following functions f 1 f sinx for z E E7 2 f 3 for z E 07T 3 f cosx for z E 07T Problem 132 Compute the volumes of the solids obtained by revolving around the r axis by 27139 ie a full rotation the region between the graph ofy 17 2 and y 2 Problem 133 Compute the volumes ofthe solids obtained by revolving around the y axis by 27139 ie a full rotation the region between the graph ofy 17 2 and y 2 Problem 134 Compute the volumes ofthe solids obtained by revolving around the r axis by an angle or E 072 ie not a full rotation the regions between the r axis and the graphs of each of the following functions f 1 f sinx for z 6 7 2 f 3 for z E 07T 3 f cosx for z E 07T FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 11 ASSIGNMENT 10 APPLICATIONS OF THE FUNDAMENTAL THEOREMS OF CALCULUS PART 1 Problem 111 Calculate the following derivatives d zzicosw 39 t2 7 m IMO d 371 sinm t Problem 112 State precisely the First and second Fundamental The orems of Calculus Problem 113 Find the derivative of the following function with respect to x f 96 I s1nt4 dt 996 Where f f5 sint5dt and g f5 cost5dt 16 Problem 114 Find the following antiderivatives Problem 115 E3 f2 17 dz Find the following antiderivatives 4 1 fo15dz 5 f2 31 Ch 2 fx175d lt6 191212172 lt3 Izzmdz sm x x 4 m cm 7 fww7113dw 5 1744 d s 1 x f WW5 9 5m x 6 d 9 fcos2x4isin2zdx fCOS z z 17 Sim 7 fww1 dw 10 f 3Cosw3dx 8 f c0ng dz 11 9 fcoE 54isin8zdx 12 f gm dx 10 f735inm3d 13 f 1266195 11 f 95 14ft1t14dt 12 fmf fdz lt15gt1ltz21gt W 13 1in d 16 fz28z32723dz 10 95 f sinmcosm d sinmicosm S 95 15 flt22732d 18 f dx 1z2 1w23 lt19 2 FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 1 ASSIGNMENT 1 SUMMATION NOTATION Problem 11 Consider the sequence of natural numbers on 273117 for n E N Compute the following expressions 1 174 2 awaaa 3 aaai39 Problem 12 Write the precise de nition of the following 1 4amp7th is a sequence of reals 2 Give an example of a sequence of rational numbers 3 4 5 I Give the recursive de nition of Z 11 39 k 1 State the associativity property of the summation notation State the translation invariance property of the summation no tation Problem 13 Calculate A7 B in the following equations 1000 A B Zai 00 T 00 l2 Problem 14 Calculate ABC in the following equations 1000 A B 0 20 Zaizaiz 6 l60 l60 l70 l200 Problem 15 Calculate A7 B in the following equations 1000 B E ai5 ai iA l60 Problem 16 Calculate A7 B in the following equations 1000 B E ai5 E 01220 i60 iA Problem 17 Calculate A in terms of 239 in the following equations 1000 990 Z alti12 1A i60 i50 12 FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 9 ASSIGNMENT 8A PRACTICE PROBLEMS ON SUMS RIEMANN SUMS amp RIEMANN INTEGRALS Problem 91 Consider the following function Dlt 7 5 if z E Q ie7 z is in the set of rational numbers 7 6 if x g Q ie7 z is not in the set of rational numbers Calculate the upper and lower Riemann sums based on an arbitrary partition P 077n of 01 and then compute the upper and lower Riemann Integrals of f over 01 Problem 92 Consider the following function Dlt x if z E Q ie7 x is in the set of rational numbers z 71 if m g Q ie7 m is not in the set of rational numbers Calculate the upper and lower Riemann sums based on an arbitrary partition P x07 zn of 071 and then compute the upper and lower Riemann Integrals of f over 01 Hint The supremum of the function over xi7x141 is n1 Problem 93 Consider the following function Dlt em if z E Q ie7 z is in the set of rational numbers I if x g Q ie7 z is not in the set of rational numbers Calculate the upper and lower Riemann sums based on an arbitrary partition P 07n of 01 and then compute the upper and lower Riemann Integrals of f over 01 Problem 94 Given a Riemann Integrable function f7 calculate the values of A7 B7 C in each of the following 27f zdz Cijz z f4x3dx CAB f77x13dz ixihdx CABfx1dz 13 Problem 95 Generalize the previous Problem Given a Riemann lnte grable function f7 calculate the values of A7 B7 0 in terms of m k l in each of the following abfzdz OAB fozz dz b B fkzldz 0 faz d96 a A Problem 96 Write the following sum as an upper Riemann sum of an increasing function over some interval 7 71 7 Zn IX 71 n k1 Problem 97 Consider the following function 2 if 0 lt x g 1 fa 3 ifx0 Use a theorem about monotonic functions to conclude that the function is Riemann lntegrable Then pick the partition 1 Pnzi l zi0z forl0n n which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n 9 00 What is the value of the integral ls the value at z 0 relevant Problem 98 Consider the following function 2 if0ltz1 f1 ifx0 Use a theorem about monotonic functions to conclude that the function is Riemann lntegrable Then pick the partition 1 Pnzi l zi0z forl0n n which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n 9 00 What is the value of the integral ls the value at z 0 relevant Problem 99 Consider the following function 21f0zlt1 fa 3 ifac1 14 Use a theorem about monotonic functions to conclude that the function is Riemann Integrable Then pick the partition 1 Pnzi l zi0z forl07n n which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n a 00 What is the value of the integral Is the value at z 1 relevant Problem 910 Consider the following function 2 if0 xlt1 f1 ifx1 Use a theorem about monotonic functions to conclude that the function is Riemann lntegrable Then pick the partition 1 Pnxilxi0l forl0n n which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n e 00 What is the value of the integral ls the value at z 1 relevant Problem 911 Generalize the previous problems Consider the follow ing function z flt B if m b Use a theorem about monotonic functions to conclude that the function is Riemann lntegrable Then pick the partitions b a n which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n e 00 What is the value of the integral ls the value at z b relevant A ifa ltb7 Pnzilial forl0n Problem 912 Generalize the previous problems Consider the follow ing function A if a lt x g b7 fzB ifa Use a theorem about monotonic functions to conclude that the function is Riemann lntegrable Then pick the partition Pnzilial forl0n 15 which partitions the interval into 71 equal parts and calculate Uf7 P Using these UfPn compute the Riemann integral by letting n e 00 What is the value of the integral ls the value at x a relevant Problem 913 Review and give the precise de nition of the following concepts 1 f is a strictly increasing function on a set S which is a subset of the domain of f 2 U is the upper Riemann sum of an increasing function f over a partition P of an interval 17 3 f is a Riemann integrable function over an interval 1 b Problem 914 Calculate the general formula for the following sums 91 im 715 k1 92 Zak 10 93 271i2x5i 2 94 2 6k k1 95 96 ik12 k23 1 w H 97 iak 1 191 2k1 M 98 x w H 1 n 99 2 4kz10k1 km 12 FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 9 ASSIGNMENT 9 PROPERTIES OF RIEMANN INTEGRAL amp THE FUNDAMENTAL THEOREMS OF CALCULUS Problem 91 Prove that the function f C a constant function is Riemann integrable over Lab and calculate its Riemann integral over 1 b Hint Follow the technique presented in Lesson 7 on the function f 96 962 Problem 92 Suppose that we have a function with the following prop erty For any two 172 in an interval fx1 7 fm Mizlxgi prove that f is a continuous function Hint This is essential the rst half of the proof of 1st FTC Problem 93 Find by inspection the antiderivative of the polynomial f 3x5 7 4x4 7x3 2 7 z 7 40 and apply the 2nd FTC in order to calculate f fzdx FLORIDA INTERNATIONAL UNIVERSITY MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 17 ASSIGNMENT 16 INTEGRATION TECHNIQUES PART I Problem 171 Exercises 1 227 from Exercise Set 84 of textbook of textbook Anton 8th Edition or Exercise Set 74 Anton 9th Edition the of cial course textbook FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkouli uedu DM410B tel 3053482849 7 ASSIGNMENT 72 USING THE CRITERION FOR INTEGRABILITY Problem 71 Prove that the function fx z is Riemann integrable over cab and calculate its Riemann integral over 07b Hint Follow the technique presented in Lesson 7 on the function fx 2 Problem 72 Given a Riemann Integrable function f7 calculate the values of A7 B7 0 in each of the following 27f3dz CAB f7xdm 27fz71dx CABf7dz f3x 7 1dx CAB f2xdm FLORIDA INTERNATIONAL UNIVERSITY Spring 2008 MAC 2312 Online Dr George Kafkoulis kafkoull uedu DM410B tel 3053482849 5 ASSIGNMENT 5 COMPUTATIONS OF RIEMANN SUMS Problem 51 Given two different real numbers A7B7 consider the fol lowing function A f E Dltzgt T 9 Q B if z Z Q Calculate the upper and lower Riemann sums based on an arbitrary partition P 0 7 of ad and then compute the upper and lower Riemann Integrals of f over 07d Problem 52 Consider the function f7 where i m if m E 01 fa 7 952 ifz e 12 and take the partition P 07 57 81417 2 Compute the upper and lower Riemann sums of f for the partition P Problem 53 Consider the function f7 where x if z E 01 fa 7 2x ifz e 12 and take the partition P 07 57 81417 2 Compute the upper and lower Riemann sums of f for the partition P Problem 54 Consider the function f7 where x if z E 01 r 10 if z 1 27 ifr E 12 and take the partition P 07 57 81417 2 Compute the upper and lower Riemann sums of f for the partition P

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