Analysis 2 Paper 3

Question 1

(3D-P17-2b) f : [a,b] → R bounded if

Answer 1

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Question 2

(3D-P17-2bi) Define L, U Darboux Sums of f wrt a given partition of [a,b]

Answer 2

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Question 3

(3D-P17-2bii) Define L, U RI of f over [a,b]

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Question 4

(3D-P17-2biii) Definition of statement, function is RI on [a,b]. Define the RI

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Question 5

(3T-P11-5b) Prove Riemann’s criterion for integrability

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Question 6

(3D-P17-2c) Give definition of statement the function f : [a, b] → R is decreasing

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Question 7

(3D-P16-2a) Partition, Mesh

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Question 8

(3D-P16-3a) Define primitive of a function

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Question 9

(3D-P16-3b) Is Primitive unique?

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Question 10

(3D-P16-3d) Define what it means for a function F : I → R to be an indefinite integral of the locally Riemann integrable function f : I → R.

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Question 11

-(3E-P17-2f) Example of function f : [0, 1] → R which is not continuous but is Riemann integrable on [0, 1]

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Question 12

(3E-P16-3f) Find the derivative of the function G(x) = integral from sin x to cos x of e^t^2

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Question 13

(3E-P15-2b) Give an example of a bounded function f : [0,1] → R which has an infinite number of discontinuities but is Riemann integrable on [0,1]. Briefly justify your answer.

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Question 14

(3E-P14-2e) Look up. Prove this function is RI on [0,1]

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Question 15

(3E-P13-3f) Find the derivative of the function G(x) = integral from sin x to sin(x^2) of sin(t^3)

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Question 16

(3E-P12-5e) provide an example of a bounded function f : [−1, 1] → R which is not continuous on [−1, 1] but is Riemann integrable on [−1, 1]

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Question 17

(3P-P17-2d) Prove a decreasing function f : [a, b] → R is bounded

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Question 18

(3P-P17-2e) Prove a decreasing function f : [a, b] → R is RI on domain.

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Question 19

(3P-P16-2c) (look up), long proof with mesh. d) uses this result to want a proof of upper RI from 0 to 1 of x is equal to 1/2.

Answer 19

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Question 20

(3P-P15-2aiv) Suppose that for any ε > 0 there exists a partition P such that U(f,P)−L(f,P) < ε. Prove that f is Riemann integrable on [a, b]

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Question 21

(3P-P15-2avi) Let f : [a, b] → R be continuous. Using the result from (iv), prove that f is Riemann integrable on [a, b]

Answer 21

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Question 22

(3P-P15-3b) Let f : [a, b] → R be bounded and let {Pn} be a limiting sequence of partitions of the interval [a,b]. Using Darboux Theorem, prove that the lim of L(f, Pn) as n goes to infinity is LRI, and the opposite for Upper.

Answer 22

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Question 23

(3P-P15-3c) Look up. Proof with limits of partitions. Also d) is a show from cii question.

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Question 24

(3P-P13-2bc) Prove theorem about refinement of partition with extra point(s)

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Question 25

(3P-P13-2d) Prove that L(f,P) ≤ U(f,Q) for any pair of partitions P and Q.

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Question 26

(3P-P13-3e) Suppose that the interval I is open (i.e. the endpoints do not belong to the interval) and that the function f : I → R is continuous. Prove that any indefinite integral of f is a primitive of f.

Answer 26

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Question 27

(3P-P11-5aiii) Prove LRI less than or equal to URI

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Question 28

(3T-P16-3c) State and prove the FTC

Answer 28

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Question 29

(3T-P16-3e) State the theorem about the properties of the indefinite integral

Answer 29

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Question 30

(3T-P15-3a) State Darboux’s Theorem from the theory of Riemann integration

Answer 30

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