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Im going to take math 252 exam which is Calculus II on March. 20. 2:45pm. I can take a picture of questions. Solve those for me with steps!
Exam is going to be like:
•
Chapter 4
: (only Section 4.8) Short review of derivatives and practic
e with antiderivatives.
•
Chapter 5
: Introduction of definite, indefinite, and improper integra
ls, beginning with Riemann sums and the Fundamental Theorem of Calculus, followed by standard integration
techniques: substitution, integration by parts, and trigo
nometric manipulation/substitution.
•
Chapter 6
: Applications of integrals to geometry and physics, including areas of enclosed regions, volumes of solids, average values, centers of mass, c
entroids, force, work, and hydrostatic
pressure.
•
Chapter 7
: Introduction to differential equations. This will include
modeling, separable equations, direction fields, Euler’s method, exponential growt
h/decay, and logistics equations.

Im going to take math 252 exam which is Calculus II

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THE UNIVERSITY OF SYDNEY MATH3066 ALGEBRA AND LOGIC Semester 1 2014 Second Assignment This assignment comprises a total of 60 marks, and is worth 15% of the overall assessment. It should be completed, accompanied by a signed cover sheet, and handed in at the lecture on Wednesday 28 May. Acknowledge any sources or assistance. Please note that the ﬁrst question is about the Proposition Calculus (not the Predicate Calculus). You should ﬁnd part (a) straightforward. Part (b) is diﬃcult and optional. Students that complete it successfully may be awarded bonus marks, and there may be a prize for the best correct answer. 1. A positive wellformed formula (positive wﬀ) in the Propositional Calculus is a wellformed formula that avoids all use of the negation symbol ∼ . (a) Use induction on the length of a wﬀ to prove that if W = W (P1 , . . . , Pn ) is a positive wﬀ in terms of propositional variables P1 , . . . , Pn , then V (P1 ) = . . . = V (Pn ) = T implies V (W ) = T . (5 marks) (b) Prove that if W = W (P1 , . . . , Pn ) is any wﬀ in the Propositional Calculus such that V (P1 ) = . . . = V (Pn ) = T implies V (W ) = T , then W is logically equivalent to a positive wﬀ. (optional, bonus marks) 2. Use the rules of deduction in the Predicate Calculus to ﬁnd formal proofs for the following sequents (without invoking sequent or theorem introduction): (a) (∃x)(∃y)(∀z) K(y, x, z) ⊢ (∀z)(∃y)(∃x) K(y, x, z) (b) (∀x)(G(x) ⇒ F (x)) (c) (∀x)(∀y)(∃z) R(x, z) ∧ R(y, z) (d) (∀x)(∀y)(∀z) R(x, y) ∧ R(y, z) ⇒ R(x, z) , ⊢ (∃x) ∼ F (x) ⇒ (∃x) ∼ G(x) ⊢ (∀x)(∃y) R(x, y) (∀x)(∀y)(∃z) R(x, z) ∧ R(z, y) ⊢ (∀x) R(x, x) (21 marks) 3. Consider the following wellformed formulae in the Predicate Calculus: W1 W2 W3 = = = (∃x)(∃y) R(x, y) (∀x)(∀y) R(x, y) ⇒ ∼ R(y, x) (∀x)(∀y) R(x, y) ⇒ (∃z) R(z, x) ∧ R(y, z) Prove that any model in which W1 , W2 and W3 are all true must have at least 3 elements. Find one such model with 3 elements. (6 marks) 4. Let R = Z[x] and I = 2Z + xZ[x] , the subset of R consisting of polynomials with integer coeﬃcients with even constant terms. Verify that I is an ideal of R. Show that I not a principal ideal. (8 marks) 5. Let R = Z3 [x]/(x2 − x − 1)Z3 [x], so we may write R = { 0 , 1 , 2 , x , x + 1 , x + 2 , 2x , 2x + 1 , 2x + 2 } , where we identify equivalence classes with remainders after division by the polynomial x2 − x − 1. Then R is a commutative ring with identity. Construct the multiplication table for R and use it to explain why R is a ﬁeld. Now ﬁnd a primitive element, that is, an element a ∈ R such that all nonzero elements of R are powers of a. (8 marks) 6. In each case below, if it helps, you may identify the ring with remainders after division by x2 + x + 1, so that the elements become linear expressions of the form a + bx where a, b come from Z3 in part (a) or from R in part (b). (a) Explain why R = Z3 [x]/(x2 + x + 1)Z3 [x] is not a ﬁeld. (b) Prove that F = R[x]/(x2 + x + 1)R[x] is isomorphic to C, the ﬁeld of complex numbers. (12 marks)
Deduction Rules for the Propositional and Predicate Calculi
The Ten Rules of Deduction for the Propositional Calculus:
(1) Rule of Assumptions (A): Any wﬀ may be written down as an assumption, depending only on itself.
(2) Modus Ponens (MP): Given V and V ⇒ W , we may deduce W , depending on the pooled assumptions for V and V ⇒ W .
(3) Modus Tollens (MT): Given ∼ W and V ⇒ W , we may deduce ∼ V , depending on the pooled assumptions for ∼ W and V ⇒ W .
(4) Double Negation (DN): Given ∼∼ W , we may deduce W , and given W we may deduce ∼∼ W , in each case depending on the same underlying assumptions.
(5) Conditional Proof (CP): Given V , introduced at some earlier step by Rule of Assumptions, and given W , relying on V as an underlying assumption, we may deduce V ⇒ W , discharging the assumption V , but relying on any remaining assumptions used to deduce W from V .
(6) ∧Introduction (∧ I): Given V and W , we may deduce V ∧ W , relying on the pooled assumptions for V and W .
(7) ∧Elimination (∧ E): Given V ∧ W , we may deduce V or deduce W , relying on assumptions for V ∧ W .
(8) ∨Introduction (∨ I): Given V , we may deduce V ∨ W or deduce W ∨ V for any W , relying on the assumptions for V .
(9) ∨Elimination (∨ E): Given V ∨ W and two deductions of

C , firstly

from V , introduced by Rule of Assumptions, and secondly from

W , intro

duced by Rule of Assumptions, we may deduce C again, but from V ∨ W , discharging the assumptions V and W , but pooling any assumptions for
V ∨ W and any assumptions used to deduce C from V and C from W .
(10) Reductio ad Absurdum (RAA): Given V , introduced at some earlier step by Rule of Assumptions, and given the contradiction W ∧ ∼ W , relying on V as an underlying assumption, we may deduce ∼ V , discharging the assumption V , but relying on any remaining assumptions used to deduce
W ∧ ∼ W from V .
The Four Extra Rules of Deduction for the Predicate Calculus:
(11) ∀Introduction (∀ I): Given a wﬀ W (B) , where B is a constant symbol that occurs at least once, we may deduce (∀X) W (X) , where X is a new variable that does not appear in W (B) and replaces B uniformly throughout W (B) , relying on the assumptions for W (B) , provided the symbol B does not appear in any wﬀ in this list of underlying assumptions.
(12) ∀Elimination (∀ E): Given a wﬀ (∀X) W (X) , we may deduce W (B) , where B is a constant symbol replacing X uniformly throughout W (X) , relying on assumptions for (∀X) W (X) .
(13) ∃Introduction (∃ I): Given a wﬀ W (B) , where B is a constant symbol that occurs at least once, we may deduce (∃X) W (X) , where W (X) results from W (B) by replacing at least one occurrence of B by X, relying on the assumptions for W (B) .
(14) ∃Elimination (∃ E): Given a wﬀ (∃X) W (X) and a deduction of C from W (B), introduced by Rule of Assumptions, where B is a new constant symbol that replaces X uniformly throughout W (X) , we may deduce C again, but from (∃X) W (X) , discharging the assumption W (B) , but pooling any assumptions for (∃X) W (X) and any assumptions used to deduce C from W (B), provided B does not appear in C or in any of these underlying assumptions.

MATH3066 ALGEBRA AND LOGIC

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Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the course text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. Article: Roger Smith. (1993, Sept.). Beyond Tocqueville, Myrdal and Hartz: The multiple traditions in America. American Political Science Review, 8, (3), 54966. You can access the article via the web resources link or directly from the online library.  This article is a study of the the US political culture for the period 18701920. It illustrates the conflictual and contradictory product of multiple political traditions. Alexis de Tocqueville's "Democracy in America," Gunnar Myrdal's "American Dilemma" and Louis Hartz's "Liberal Tradition in America" are critiqued. Upon review, post (and respond to postings) under Discussion Question #1 your thoughts and analysis of the article as it relates to module 1 topics. Questions to consider are (These are only suggestions, not required): 1. Does the article address the market model of governance? If so how? 2. How is the concept of "polis" described in the article? What are the "common problems"? 3. Is income redistribution a factor in the article? if so, how? 4. Were resources equally distributed? Were deviations legitimate in terms of other social goals? Utilize external resources as necessary to support your opinions and claims.
Discussion Question #2 ( 51 messages  46 unread ) Post New ThreadPlease respond to the following question.Hide Full Description
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week?Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: Moon, Jae M. & DeLeon, Peter. (2001, July). Municipal reinvention: Managerial values and diffusion among municipalities. Journal of Public Administration Research and Theory, 11, (3), 327352. Upon review, post (and respond to postings) under Discussion Question #2 your thoughts and analysis of the article as it relates to Module 2 topics. Utilize external resources as necessary to support your opinions and claims. Questions to consider are: 1. How does the article relate to the concept of efficiency within the context of the public policy? 2. Is there an equalityefficiency tradeoff? 3. What are the dimensions of need outlined in the article? 4. Is there a securityefficiency tradeoff?
Forum 3
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: Lemley, Ellen C. & Russell, Gregory D. (2002). Implementing restorative justice by "groping along:" A case study in program evolutionary implementation. Justice System Journal, 23, (2), 157192. Upon review, post (and respond to postings) under Discussion Question #3 your thoughts and analysis of the article as it relates to Module 3 topics. Utilize external resources as necessary to support your opinions and claims. Consider the following questions in your postings: 1. Are there concepts of liberty in the article? If so, what is the libertysecurity tradeoff? 2. Are there symbolic representations of a policy problem? If so, are they symbolic devices of narrative stories, synecdoche, metaphors or ambiguous?
Discussion Question #4
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: : Behn, Robert D. (2003, SepOct). Why measure performance? Different purposes require different measures. Public Administration Review, 63, (5), 586. Upon review, post (and respond to postings) under Discussion Question #4 your thoughts and analysis of the article as it relates to module 4 topics. Utilize external resources as necessary to support your opinions and claims. Questions to consider in your postings are: 1. In this article, are numbers evaluated as metaphors? 2. Can these numbers be analyzed as norms and symbols? 3. How is the assessment measured in the polis? 4. Is there a causal story? If so, what is the problem definition?
Discussion Question #5
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: Bianco, William T. (1998, Oct). Different paths to the same result: Rational choice, political psychology and impression formation in campaigns. American Journal of Political Science, 42, (4), 10611082. Upon review, post (and respond to postings) under Discussion Question #5 your thoughts and analysis of the article as it relates to module 5 topics. Utilize external resources as necessary to support your opinions and claims. Consider the following questions in your postings: 1. What is of "interest" in the article? How are these interests mobilized in the polis? 2. How do issues and interests define each other in the article? 3. Does the concept of rational decision apply to the article? 4. What is decisionmaking as it relates to the polis? 5. Is there a decisionanalysis strategy?
Forum 6
The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: Pierce, W. David, Banko, Katherine M., & So, Sylvia. (2003, Fall). Positive effects of rewards and performance standards on intrinsic motivation. The Psychological Record, 53, (4), 561. Upon review, post (and respond to postings) under Discussion Question #6 your thoughts and analysis of the article as it relates to module 6 topics. Utilize external resources as necessary to support your opinions and claims. Consider the following questions in your postings: 1. How are inducements used in the polis from this article? Is there a difference between inducements in the rationality model and the polis model? 2. Are there examples of good rules? How does it apply to rulemaking in the polis?
Forum 7
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer.
Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors.
The following criteria are used to grade your discussion group submissions.
� Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.)
� Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?)
� Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?)
� Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?)
The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus.
This discussion involves Journal Article:
Cobb, Michael D. & Kuklinski, James H. (1997, Jan). Changing Minds: Political arguments and political persuasion. American Journal of Political Science
Upon review, post (and respond to postings) under Discussion Question #7 your thoughts and analysis of the article as it relates to module 7 topics. Utilize external resources as necessary to support your opinions and claims.
Consider the following questions:
1. Are there two faces of persuasion in the article?
2. Are there examples of information and propaganda?
3. How can the use of persuasion within the context of the rationaldemocratic model, the totalitarian model and the polis model be described in this article?
4. What are the concepts of rights in the polis as it relates to the topic?
5. Can rights be evaluated as policy instruments?
Forum 8
Articles for the Discussion Question are assigned from professional journals and are available via the Online Research Center. Students must provide a critical review of these articles and substantially reply to the contributions of at least one peer. Individual postings should include an evaluation of the content of the article and explain how it relates to the concepts in the text and other external resources. The postings should be analytic in nature and include comparisons/contrasts and examples that can bolster your argument. Postings and replies should be 250  300 words in length and free of any spelling or grammar errors. The following criteria are used to grade your discussion group submissions. � Timeliness (Did the student contribute throughout the week? Post early to allow fellow students ample time to review and respond to your postings.) � Use of External Resources, Citations, and Resources (Did student perform independent research and cite sources from scholarly academic journals or recognized research institutions?) � Quality (Is the submission professional, focused, pertinent, and free of grammar and spelling errors?) � Participation and Classroom Interaction (Did the student substantially respond and interact with other student postings in a collegial and/or critical manner?) The Discussion Groups are designed to enhance (and evaluate) student participation and interaction during the course. Accordingly, your interaction with other students will be part of the grading calculus. This discussion involves Journal Article: Acemoglu, Daron & Robinson, James A. (2001, Sep). Inefficient redistribution. The American Political Science Review, 95, (3), 649662. Upon review, post (and respond to postings) under Discussion Question #8 your thoughts and analysis of the article as it relates to module 8 topics. Utilize external resources as necessary to support your opinions and claims. Consider the following questions: 1. Does the impulse to restructure authority (constitutional engineering) apply in this article? Why/Why not? 2. What is the impact of redefining membership in the polis? 3. What are the arguments for centralization versus decentralization? 4. Is there value to political reasoning as strategic representation?

Current Event Analysis #1

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A. Create a realworld science problem that requires the use of differentiation to solve by doing the following:
1. Provide a description of each of the following within the context of the selected real world application using the appropriate
units:
a. Dependent variable
b. Independent variable
c. Domain
d. Range
2. Explain the specific issue that the created problem is addressing.
3. The created problem should involve taking the first and second derivative of a selected functional relationship f(x), which
includes the following components:
a. Describe how f (x) describes the behavior of f(x) within the context of the real world application.
b. Describe how f(x) describes the behavior of f(x) and f(x) within the context of the realworld application.
4. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.
B. Create a realworld science problem that requires the use of integration to solve by doing the following:
1. Provide a description of each of the following within the context of the selected real world application using the appropriate units:
a. Dependent variable
b. Independent variable
c. Domain
d. Range
2. Explain the specific issue that the created problem is addressing.
3. The created problem should involve calculating the volume of a solid of revolution obtained by revolving a planar curve about a straight line in the plane (e.g. xaxis or yaxis), which includes the following components:
a. Describe how to calculate the volume of the solid of revolution using either the disk method or the washer method.
b. Describe how to calculate the volume of the solid of revolution using the shell method.
4. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.
Note: Demonstrate that you can generate the same answer for volume by using two different integration methods.
C. Compare the methods that were used within both of the problems you have created for parts A and B to explain the
relationship between differentiation and integration.
D. Create a realworld science problem that requires the use of both parts of the Fundamental Theorem of Calculus to solve by doing the following:
1. Provide a description of each of the following within the context of the selected real world application using the appropriate
units:
a. Dependent variable
b. Independent variable
c. Domain
d. Range
2. Explain the specific issue that the created problem is addressing.
3. The created problem should require use of the first part of the Fundamental Theorem of Calculus to solve that includes
the following components:
State the first part of the Fundamental Theorem of Calculus. a. Explain how the first part of the Fundamental Theorem of Calculus was used to solve part of the created problem.
4. The created problem should require use of the second part of the Fundamental Theorem of Calculus to solve that includes the following components:
State the second part of the Fundamental Theorem of Calculus. a. Explain how the second part of the Fundamental Theorem of Calculus was used to solve part of the created problem.
5. Provide a solution to the created problem using both parts of the Fundamental Theorem of Calculus.
6. Provide an answer key that includes all relevant mathematical justifications for each step within the solution to the problem.

Create a realworld science problem
