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Patent Cooperation Treaty (PCT)Done at Washington on June 19, 1970,amended on September 28, 1979,modified on February 3, 1984, and on October 3, 2001Editor’s Note:For details concerning amendments and modifications to the Patent Cooperation Treaty(PCT),and for access to decisions of the Assembly of the International Patent Cooperation Union(PCT Assembly)concerning their entry into force and transitional arrangements,reference should be made to the relevant reports of the PCT Assembly available from the International Bureau or via the WIPO website at: www.wipo.int/pct/en/meetings/assemblies/reports.htm.Patent Cooperation TreatyDone at Washington on June19,1970,amended on September28,1979,modified on February3,1984,and on October3,2001TABLE OF CONTENTS*PreambleIntroductory ProvisionsArticle1Establishment of a UnionArticle2DefinitionsChapter I:International Application and International SearchArticle3The International ApplicationArticle4The RequestArticle5The DescriptionArticle6The ClaimsArticle7The DrawingsArticle8Claiming PriorityArticle9The ApplicantArticle10The Receiving OfficeArticle11Filing Date and Effects of the International ApplicationArticle12Transmittal of the International Application to theInternational Bureau and the International SearchingAuthorityArticle13Availability of Copy of the International Application toDesignated OfficesArticle14Certain Defects in the International ApplicationArticle15The International SearchArticle16The International Searching AuthorityArticle17Procedure before the International Searching AuthorityArticle18The International Search ReportArticle19Amendment of the Claims before the International Bureau Article20Communication to Designated OfficesArticle21International PublicationArticle22Copy,Translation,and Fee,to Designated OfficesArticle23Delaying of National Procedure*Table of Contents and Editor’s Notes are added for the convenience of the reader;they do not form part of the Treaty.Article24Possible Loss of Effect in Designated StatesArticle25Review by Designated OfficesArticle26Opportunity to Correct before Designated OfficesArticle27National RequirementsArticle28Amendment of the Claims,the Description,and theDrawings,before Designated OfficesArticle29Effects of the International PublicationArticle30Confidential Nature of the International Application Chapter II:International Preliminary ExaminationArticle31Demand for International Preliminary ExaminationArticle32The International Preliminary Examining AuthorityArticle33The International Preliminary ExaminationArticle34Procedure before the International Preliminary Examining AuthorityArticle35The International Preliminary Examination ReportArticle36Transmittal,Translation,and Communication,of theInternational Preliminary Examination Report Article37Withdrawal of Demand or ElectionArticle38Confidential Nature of the International PreliminaryExaminationArticle39Copy,Translation,and Fee,to Elected OfficesArticle40Delaying of National Examination and Other Processing Article41Amendment of the Claims,the Description,and theDrawings,before Elected OfficesArticle42Results of National Examination in Elected Offices Chapter III:Common ProvisionsArticle43Seeking Certain Kinds of ProtectionArticle44Seeking Two Kinds of ProtectionArticle45Regional Patent TreatiesArticle46Incorrect Translation of the International ApplicationArticle47Time LimitsArticle48Delay in Meeting Certain Time LimitsArticle49Right to Practice before International AuthoritiesChapter IV:Technical ServicesArticle50Patent Information ServicesArticle51Technical AssistanceArticle52Relations with Other Provisions of the Treaty Chapter V:Administrative ProvisionsArticle53AssemblyArticle54Executive CommitteeArticle55International BureauArticle56Committee for Technical CooperationArticle57FinancesArticle58RegulationsChapter VI:DisputesArticle59DisputesChapter VII:Revision and AmendmentArticle60Revision of the TreatyArticle61Amendment of Certain Provisions of the Treaty Chapter VIII:Final ProvisionsArticle62Becoming Party to the TreatyArticle63Entry into Force of the TreatyArticle64ReservationsArticle65Gradual ApplicationArticle66DenunciationArticle67Signature and LanguagesArticle68Depositary FunctionsArticle69NotificationsThe Contracting States,Desiring to make a contribution to the progress of science and technology, Desiring to perfect the legal protection of inventions,Desiring to simplify and render more economical the obtaining of protection for inventions where protection is sought in several countries,Desiring to facilitate and accelerate access by the public to the technical information contained in documents describing new inventions,Desiring to foster and accelerate the economic development of developing countries through the adoption of measures designed to increase the efficiency of their legal systems,whether national or regional,instituted for the protection of inventions by providing easily accessible information on the availability of technological solutions applicable to their special needs and by facilitating access to the ever expanding volume of modern technology,Convinced that cooperation among nations will greatly facilitate the attainment of these aims,Have concluded the present Treaty.INTRODUCTORY PROVISIONSArticle1Establishment of a Union(1)The States party to this Treaty(hereinafter called“the Contracting States”)constitute a Union for cooperation in the filing,searching,and examination,of applications for the protection of inventions,and for rendering special technical services.The Union shall be known as the International Patent Cooperation Union.(2)No provision of this Treaty shall be interpreted as diminishing the rights under the Paris Convention for the Protection of Industrial Property of any national or resident of any country party to that Convention.Article2DefinitionsFor the purposes of this Treaty and the Regulations and unless expressly stated otherwise:(i)“application”means an application for the protection of an invention;references to an“application”shall be construed as references to applications for patents for inventions,inventors’certificates,utility certificates, utility models,patents or certificates of addition,inventors’certificates of addition,and utility certificates of addition;(ii)references to a“patent”shall be construed as references to patents for inventions,inventors’certificates,utility certificates,utility models,patents or certificates of addition,inventors’certificates of addition,and utility certificates of addition;(iii)“national patent”means a patent granted by a national authority;(iv)“regional patent”means a patent granted by a national or an intergovernmental authority having the power to grant patents effective in more than one State;(v)“regional application”means an application for a regional patent;(vi)references to a“national application”shall be construed as references to applications for national patents and regional patents,other than applications filed under this Treaty;(vii)“international application”means an application filed under this Treaty;(viii)references to an“application”shall be construed as references to international applications and national applications;(ix)references to a“patent”shall be construed as references to national patents and regional patents;(x)references to“national law”shall be construed as references to the national law of a Contracting State or,where a regional application or a regional patent is involved,to the treaty providing for the filing of regional applications or the granting of regional patents;(xi)“priority date,”for the purposes of computing time limits,means:(a)where the international application contains a priority claim under Article8,the filing date of the application whose priority is so claimed;(b)where the international application contains several priority claims under Article8,the filing date of the earliest application whose priority is so claimed;(c)where the international application does not contain any priority claim under Article8,the international filing date of such application;(xii)“national Office”means the government authority of a Contracting State entrusted with the granting of patents;references to a“national Office”shall be construed as referring also to any intergovernmental authority which several States have entrusted with the task of granting regional patents, provided that at least one of those States is a Contracting State,and provided that the said States have authorized that authority to assume the obligations and exercise the powers which this Treaty and the Regulations provide for in respect of national Offices;(xiii)“designated Office”means the national Office of or acting for the State designated by the applicant under Chapter I of this Treaty;(xiv)“elected Office”means the national Office of or acting for the State elected by the applicant under Chapter II of this Treaty;(xv)“receiving Office”means the national Office or the intergovernmental organization with which the international application has been filed;(xvi)“Union”means the International Patent Cooperation Union;(xvii)“Assembly”means the Assembly of the Union;(xviii)“Organization”means the World Intellectual Property Organization;(xix)“International Bureau”means the International Bureau of the Organization and,as long as it subsists,the United International Bureaux for the Protection of Intellectual Property(BIRPI);(xx)“Director General”means the Director General of the Organization and,as long as BIRPI subsists,the Director of BIRPI.CHAPTER IINTERNATIONAL APPLICATION AND INTERNATIONAL SEARCHArticle3The International Application(1)Applications for the protection of inventions in any of the Contracting States may be filed as international applications under this Treaty.(2)An international application shall contain,as specified in this Treaty and the Regulations,a request,a description,one or more claims,one or more drawings(where required),and an abstract.(3)The abstract merely serves the purpose of technical information and cannot be taken into account for any other purpose,particularly not for the purpose of interpreting the scope of the protection sought.(4)The international application shall:(i)be in a prescribed language;(ii)comply with the prescribed physical requirements;(iii)comply with the prescribed requirement of unity of invention;(iv)be subject to the payment of the prescribed fees.Article4The Request(1)The request shall contain:(i)a petition to the effect that the international application be processed according to this Treaty;(ii)the designation of the Contracting State or States in which protection for the invention is desired on the basis of the international application (“designated States”);if for any designated State a regional patent is available and the applicant wishes to obtain a regional patent rather than a national patent,the request shall so indicate;if,under a treaty concerning a regional patent,the applicant cannot limit his application to certain of the States party to that treaty,designation of one of those States and the indication of the wish to obtain the regional patent shall be treated as designation of all the States party to that treaty;if,under the national law of the designated State,the designation of that State has the effect of an application for a regional patent,thedesignation of the said State shall be treated as an indication of the wish to obtain the regional patent;(iii)the name of and other prescribed data concerning the applicant and the agent(if any);(iv)the title of the invention;(v)the name of and other prescribed data concerning the inventor where the national law of at least one of the designated States requires that these indications be furnished at the time of filing a national application.Otherwise, the said indications may be furnished either in the request or in separate notices addressed to each designated Office whose national law requires the furnishing of the said indications but allows that they be furnished at a time later than that of the filing of a national application.(2)Every designation shall be subject to the payment of the prescribed fee within the prescribed time limit.(3)Unless the applicant asks for any of the other kinds of protection referred to in Article43,designation shall mean that the desired protection consists of the grant of a patent by or for the designated State.For the purposes of this paragraph,Article2(ii)shall not apply.(4)Failure to indicate in the request the name and other prescribed data concerning the inventor shall have no consequence in any designated State whose national law requires the furnishing of the said indications but allows that they be furnished at a time later than that of the filing of a national application.Failure to furnish the said indications in a separate notice shall have no consequence in any designated State whose national law does not require the furnishing of the said indications.Article5The DescriptionThe description shall disclose the invention in a manner sufficiently clear and complete for the invention to be carried out by a person skilled in the art.Article6The ClaimsThe claim or claims shall define the matter for which protection is sought. Claims shall be clear and concise.They shall be fully supported by the description.Article7The Drawings(1)Subject to the provisions of paragraph(2)(ii),drawings shall be required when they are necessary for the understanding of the invention.(2)Where,without being necessary for the understanding of the invention, the nature of the invention admits of illustration by drawings:(i)the applicant may include such drawings in the international application when filed,(ii)any designated Office may require that the applicant file such drawings with it within the prescribed time limit.Article8Claiming Priority(1)The international application may contain a declaration,as prescribed in the Regulations,claiming the priority of one or more earlier applications filed in or for any country party to the Paris Convention for the Protection of Industrial Property.(2)(a)Subject to the provisions of subparagraph(b),the conditions for,and the effect of,any priority claim declared under paragraph(1)shall be as provided in Article4of the Stockholm Act of the Paris Convention for the Protection of Industrial Property.(b)The international application for which the priority of one or more earlier applications filed in or for a Contracting State is claimed may contain the designation of that State.Where,in the international application,the priority of one or more national applications filed in or for a designated State is claimed,or where the priority of an international application having designated only one State is claimed,the conditions for,and the effect of,the priority claim in that State shall be governed by the national law of that State.Article9The Applicant(1)Any resident or national of a Contracting State may file an international application.(2)The Assembly may decide to allow the residents and the nationals of any country party to the Paris Convention for the Protection of Industrial Property which is not party to this Treaty to file international applications.(3)The concepts of residence and nationality,and the application of those concepts in cases where there are several applicants or where the applicants are not the same for all the designated States,are defined in the Regulations.Article10The Receiving OfficeThe international application shall be filed with the prescribed receiving Office,which will check and process it as provided in this Treaty and the Regulations.Article11Filing Date and Effects of the International Application(1)The receiving Office shall accord as the international filing date the date of receipt of the international application,provided that that Office has found that,at the time of receipt:(i)the applicant does not obviously lack,for reasons of residence or nationality,the right to file an international application with the receiving Office,(ii)the international application is in the prescribed language,(iii)the international application contains at least the following elements:(a)an indication that it is intended as an international application,(b)the designation of at least one Contracting State,(c)the name of the applicant,as prescribed,(d)a part which on the face of it appears to be a description,(e)a part which on the face of it appears to be a claim or claims.(2)(a)If the receiving Office finds that the international application did not, at the time of receipt,fulfill the requirements listed in paragraph(1),it shall,as provided in the Regulations,invite the applicant to file the required correction.(b)If the applicant complies with the invitation,as provided in the Regulations,the receiving Office shall accord as the international filing date the date of receipt of the required correction.(3)Subject to Article64(4),any international application fulfilling the requirements listed in items(i)to(iii)of paragraph(1)and accorded an international filing date shall have the effect of a regular national application in each designated State as of the international filing date,which date shall be considered to be the actual filing date in each designated State.(4)Any international application fulfilling the requirements listed in items(i)to(iii)of paragraph(1)shall be equivalent to a regular national filing within the meaning of the Paris Convention for the Protection of Industrial Property.Article12Transmittal of the International Application to the International Bureau and the International Searching Authority(1)One copy of the international application shall be kept by the receiving Office(“home copy”),one copy(“record copy”)shall be transmitted to the International Bureau,and another copy(“search copy”)shall be transmitted to the competent International Searching Authority referred to in Article16,as provided in the Regulations.(2)The record copy shall be considered the true copy of the international application.(3)The international application shall be considered withdrawn if the record copy has not been received by the International Bureau within the prescribed time limit.Article13Availability of Copy of theInternational Application to Designated Offices(1)Any designated Office may ask the International Bureau to transmit to ita copy of the international application prior to the communication provided for in Article20,and the International Bureau shall transmit such copy to the designated Office as soon as possible after the expiration of one year from the priority date.(2)(a)The applicant may,at any time,transmit a copy of his international application to any designated Office.(b)The applicant may,at any time,ask the International Bureau to transmit a copy of his international application to any designated Office,and the International Bureau shall transmit such copy to the designated Office as soon as possible.(c)Any national Office may notify the International Bureau that it does not wish to receive copies as provided for in subparagraph(b),in which case that subparagraph shall not be applicable in respect of that Office.Article14Certain Defects in the International Application(1)(a)The receiving Office shall check whether the international application contains any of the following defects,that is to say:(i)it is not signed as provided in the Regulations;(ii)it does not contain the prescribed indications concerning the applicant;(iii)it does not contain a title;(iv)it does not contain an abstract;(v)it does not comply to the extent provided in the Regulations with the prescribed physical requirements.(b)If the receiving Office finds any of the said defects,it shall invite the applicant to correct the international application within the prescribed time limit,failing which that application shall be considered withdrawn and the receiving Office shall so declare.(2)If the international application refers to drawings which,in fact,are not included in that application,the receiving Office shall notify the applicant accordingly and he may furnish them within the prescribed time limit and,if he does,the international filing date shall be the date on which the drawings are received by the receiving Office.Otherwise,any reference to the said drawings shall be considered non-existent.(3)(a)If the receiving Office finds that,within the prescribed time limits, the fees prescribed under Article3(4)(iv)have not been paid,or no fee prescribed under Article4(2)has been paid in respect of any of the designated States,the international application shall be considered withdrawn and the receiving Office shall so declare.(b)If the receiving Office finds that the fee prescribed under Article4(2) has been paid in respect of one or more(but less than all)designated States within the prescribed time limit,the designation of those States in respect of which it has not been paid within the prescribed time limit shall be considered withdrawn and the receiving Office shall so declare.(4)If,after having accorded an international filing date to the international application,the receiving Office finds,within the prescribed time limit,that any of the requirements listed in items(i)to(iii)of Article11(1)was not complied with at that date,the said application shall be considered withdrawn and the receiving Office shall so declare.Article15The International Search(1)Each international application shall be the subject of international search.(2)The objective of the international search is to discover relevant prior art.(3)International search shall be made on the basis of the claims,with due regard to the description and the drawings(if any).(4)The International Searching Authority referred to in Article16shall endeavor to discover as much of the relevant prior art as its facilities permit,and shall,in any case,consult the documentation specified in the Regulations.(5)(a)If the national law of the Contracting State so permits,the applicant who files a national application with the national Office of or acting for such State may,subject to the conditions provided for in such law,request that a search similar to an international search(“international-type search”)be carried out on such application.(b)If the national law of the Contracting State so permits,the national Office of or acting for such State may subject any national application filed with it to an international-type search.(c)The international-type search shall be carried out by the International Searching Authority referred to in Article16which would be competent for an international search if the national application were an international application and were filed with the Office referred to in subparagraphs(a)and(b).If the national application is in a language which the International Searching Authority considers it is not equipped to handle,the international-type search shall be carried out on a translation prepared by the applicant in a language prescribed for international applications and which the International Searching Authority has undertaken to accept for international applications.The national application and the translation,when required,shall be presented in the form prescribed for international applications.Article16The International Searching Authority(1)International search shall be carried out by an International Searching Authority,which may be either a national Office or an intergovernmental organization,such as the International Patent Institute,whose tasks include the establishing of documentary search reports on prior art with respect to inventions which are the subject of applications.(2)If,pending the establishment of a single International Searching Authority,there are several International Searching Authorities,each receiving Office shall,in accordance with the provisions of the applicable agreement referred to in paragraph(3)(b),specify the International Searching Authority or Authorities competent for the searching of international applications filed with such Office.(3)(a)International Searching Authorities shall be appointed by the Assembly.Any national Office and any intergovernmental organization satisfying the requirements referred to in subparagraph(c)may be appointed as International Searching Authority.(b)Appointment shall be conditional on the consent of the national Office or intergovernmental organization to be appointed and the conclusion of an agreement,subject to approval by the Assembly,between such Office or organization and the International Bureau.The agreement shall specify the rights and obligations of the parties,in particular,the formal undertaking by the said Office or organization to apply and observe all the common rules of international search.(c)The Regulations prescribe the minimum requirements,particularly as to manpower and documentation,which any Office or organization must satisfy before it can be appointed and must continue to satisfy while it remains appointed.(d)Appointment shall be for a fixed period of time and may be extended for further periods.(e)Before the Assembly makes a decision on the appointment of any national Office or intergovernmental organization,or on the extension of its appointment,or before it allows any such appointment to lapse,the Assembly shall hear the interested Office or organization and seek the advice of the Committee for Technical Cooperation referred to in Article56once that Committee has been established.Article17Procedure before the International Searching Authority(1)Procedure before the International Searching Authority shall be governed by the provisions of this Treaty,the Regulations,and the agreement which the International Bureau shall conclude,subject to this Treaty and the Regulations,with the said Authority.(2)(a)If the International Searching Authority considers(i)that the international application relates to a subject matter whichthe International Searching Authority is not required,under theRegulations,to search,and in the particular case decides not tosearch,or(ii)that the description,the claims,or the drawings,fail to comply with the prescribed requirements to such an extent that ameaningful search could not be carried out,the said Authority shall so declare and shall notify the applicant and the International Bureau that no international search report will be established.(b)If any of the situations referred to in subparagraph(a)is found to exist in connection with certain claims only,the international search report shall so indicate in respect of such claims,whereas,for the other claims,the said report shall be established as provided in Article18.(3)(a)If the International Searching Authority considers that the international application does not comply with the requirement of unity of invention as set forth in the Regulations,it shall invite the applicant to pay additional fees.The International Searching Authority shall establish the international search report on those parts of the international application which relate to the invention first mentioned in the claims(“main invention”)and, provided the required additional fees have been paid within the prescribed time limit,on those parts of the international application which relate to inventions in respect of which the said fees were paid.(b)The national law of any designated State may provide that,where the national Office of that State finds the invitation,referred to in subparagraph(a), of the International Searching Authority justified and where the applicant has not paid all additional fees,those parts of the international application which consequently have not been searched shall,as far as effects in that State are concerned,be considered withdrawn unless a special fee is paid by the applicant to the national Office of that State.Article18The International Search Report(1)The international search report shall be established within the prescribed time limit and in the prescribed form.。

What is prompt engineering?Prompt engineering is the craft of designing and refining inputs (prompts) to elicit the desired output from AI language models. It requires a blend of creativity, understanding of the model’s capabilities, and strategic structuring of the question or statement to guide the AI towards providing accurate, relevant, and useful responses. Prompt engineering improves communication between humans and machines, ensuring the resulting interaction is efficient and effective.Why is prompt engineering important?Prompt engineering is crucial because it influences the performance and utility of AI language models. The quality of the input determines the relevance and accuracy of the AI’s response, making prompt engineering a pivotal skill for anyone looking to harness the full potential of these powerful tools. Prompt engineering is not only for prompt engineers. By effectively communicating with AI, anyone can unlock insights, generate ideas, and solve problems more efficiently.Here are several reasons why prompt engineering is important: •Improves accuracy: Well-crafted prompts lead to more precise answers, reducing the likelihood of misinterpretation orirrelevant responses from the AI.•Saves time: Prompt engineering streamlines interactions with the AI by getting the desired information in fewer attempts,saving valuable time for users.•Facilitates complex tasks: Complex tasks require complex understanding; good prompts translate intricate questions intoa form that AI can process effectively.•Improves user experience: A user’s experience with an AI system can greatly improve when the prompts lead to clear,concise, and contextually appropriate answers.•Enables better outcomes: In areas such as coding, content creation, and data analysis, well-engineered prompts can leadto higher-quality outcomes by leveraging AI’s capabilities tothe fullest.•Drives innovation: As we better understand how tocommunicate with AI, we can push the boundaries of what’spossible, leading to innovative applications and solutions.10 Prompt engineering best practicesCrafting effective prompts for AI can improve the quality and relevance of the responses you receive. This expertise requires a nuanced understanding of how AI interprets and processes natural language inputs. Ahead, we explore ten AI prompt engineering best practices to help you communicate with AI more effectively:1. Be as specific as possibleSpecificity is key to obtaining the most accurate and relevant information from an AI when writing prompts. A specific prompt minimizes ambiguity, allowing the AI to understand the request’s context and nuance, preventing it from providing overly broad or unrelated responses. To achieve this, include as many relevant details as possible without overloading the AI with superfluous information. This balance ensures that the AI has just enough guidance to produce the specific outcome you’re aiming for.When creating the best prompts for an AI, ask for the following specifics:•Detailed context: Provide the AI with enough background information to understand the scenario you’re inquiring about.This includes the subject matter, scope, and any relevantconstraints.•Desired format: Clearly specify the format in which you want the information to be presented, whether it’s a list, a detailedreport, bullet points, or a summary. Mention any structuralpreferences, such as headings, subheadings, or paragraphlimits.•Output length: Detail how long you want the AI’s response, whether “3paragraphs” or “250words.”•Level of detail: Indicate the level of detail required for the response, from high-level overviews to in-depth analysis, toensure the model’s output matches your informational needs. •Tone and style: Request the preferred tone and style, whether it’s formal, conversational, persuasive, or informational, tomake sure the output aligns with your intended audience orpurpose.•Examples and comparisons: Ask the AI to include examples, analogies, or comparisons to clarify complex concepts or make the information more relatable and easily understood.Prompt Example:Please provide an outline for a comprehensive report that analyzes the current trends in social media marketing for technology companies, focusing on the developments from 2020 onward.The outline should include an introduction, three main sections addressing different aspects of social media trends, and a conclusion summarizing the findings. Please suggest the types of graphs that could illustrate user engagement trends and list bullet points that summarize key marketing strategies in each section. 2. Supply the AI with examplesIncorporating examples into your prompts is a powerful technique to steer the AI’s responses in the desired direction. By providing examples as you write prompts, you set a precedent for the type of information or response you expect. This practice is particularly useful for complex tasks where the desired output might be ambiguous or for creative tasks with more than one correct answer. When you supply the AI with examples, ensure they represent the quality and style of your desired result. This strategy clarifies your expectations and helps the AI model its responses after the examples provided, leading to more accurate and tailored outputs. Here are some example types you could provide to an AI to help guide it toward generating the best response possible:•Sample texts: Share excerpts reflecting the style, tone, and content you want the AI to replicate.•Data formats: To guide the AI’s output, provide specific data structures, such as table layouts or spreadsheet formats.•Templates for documents: Offer templates to ensure the AI’s response follows a desired structure and format.•Code snippets: Provide code examples if you need help with programming tasks to ensure correct syntax and logic. •Graphs and charts examples: If you’re asking the AI to create similar graphics, share samples of visual data representation.•Marketing copy: If you’re crafting marketing content, present ad copy that aligns with your brand’s voice for the AI to mimic. Prompt Example:Create a comparison table for two project management tools, Tool A and Tool B.Include the following categories: Price, Key Features, User Reviews, and Support Options. For instance, under Key Features, list things like ‘Task Assignment’,‘Time Tracking’, and ‘File Sharing’.The format should mirror something like this:Please ensure the table is concise and suitable for inclusion in a business report.3. Get better answers by providing dataIncorporating specific and relevant data into your prompts significantly enhances the quality of AI-generated responses, providing a solid foundation for the AI to understand the context and craft precise answers. Providing data that includes numerical values, dates, or categories, organized in a clear and structured way, allows for detailed analysis and decision-making. It’s essential to give context to the data and, when possible, to cite its source, lending credibility and clarity to the specific task, whether for quantitative analysis or comparisons.To ensure the AI delivers the most relevant and insightful answers, always use updated and well-organized information, and if comparisons are needed, establish clear parameters. Supplying the AI with concrete, contextualized data transforms raw figures into intelligible and actionable insights. Data-driven prompts are particularly valuable in tasks requiring a deep dive into numbers, trends, or patterns, enabling the AI to generate outputs that can effectively inform business strategies or research conclusions.Prompt Example:Please analyze the sales data from the first quarter of 2024 provided in the attached PDF document. I need a summary that identifies our best-selling product, the overall sales trend, and any notable patterns in customer purchases.The PDF contains detailed monthly sales units for three products: Product A, Product B, and Product C. After reviewing the data, summarize your findings in a concise paragraph that is suitable for a business meeting. Highlight significant increases or decreases in sales and offer insights into potential factors driving these trends. 4. Specify your desired outputWhen engaging with AI, articulate the precise format and structure you expect in the response. Specify whether you require a detailedreport, a summary, bullet points, or a narrative form to ensure the AI tailors its output to your needs.Indicate any preferences such as tone, style, and the inclusion of certain elements like headings or subheadings. By clearly defining your desired output, you guide the AI to deliver information that aligns seamlessly with your intended use.Prompt Example:Create a comprehensive overview of the key milestones in the history of software development. The output should be structured as a timeline with bullet points, each bullet including the year, the milestone event, and a brief description of its significance. Start from the 1980s. The tone should be educational. Please limit the overview to ten major milestones to maintain conciseness.5. Provide instructions on what to do instead of what not to doWhen constructing prompts for AI, it’s more effective to direct the system toward the desired action rather than detailing what it should avoid. This positive instruction approach reduces ambiguity and focuses the AI’s processing power on generating constructive outcomes.Negative instructions often require the AI to interpret and invert them, increasing the cognitive load and potential for misunderstanding. By clearly stating the intended actions, you enable the AI to apply its capabilities directly to fulfilling the task at hand, improving the efficiency and accuracy of the response.Prompt Examples:•Avoid: “Don’t write too much detail. → Use Instead: “Please provide a concise summary.”•Avoid: “Avoid using technical jargon.”→ Use Instead: “Use clear and simple language accessible to a general audience.”•Avoid: “Don’t give examples from before the year 2000.”→ Use Instead: “Provide examples from the year 2000 onwards.”6. Give the model a persona or frame of referenceAssigning a persona or a specific frame of reference to an AI model can significantly enhance the relevance and precision of its output. By doing so, you get more relevant responses, aligned with a particular perspective or expertise, ensuring that the information provided meets the unique requirements of your query.This approach is especially beneficial in business contexts where domain-specific knowledge is pivotal, as it guides the AI to utilize a tone and terminology appropriate for the given scenario. The persona also helps set the right expectations and can make interactions with the AI more relatable and engaging for the end user.Prompt Example:Imagine you are a seasoned marketing consultant. Please draft an email to a new startup client outlining three digital marketing strategies tailored for their upcoming product launch (see attached PDF for details).Include key performance indicators (KPIs) for each strategy that will help track their campaign’s success. Ensure the tone is encouraging and professional, imparting confidence in your expertise.7. Try chain of thought promptingChain of thought prompting is a technique that elicits a more deliberate and explanatory response from an AI by specifically asking it to detail the reasoning behind its answer. By prompting the AI to articulate the steps it takes to reach a conclusion, users can better understand the logic employed and the reliability of the response.This approach is particularly useful when tackling complex problems or when the reasoning process itself is as important as the answer. It ensures a deeper level of problem-solving and provides a learning opportunity for the user to see a modeled approach to reasoning.Prompt Example:Imagine you are a software engineer tasked with optimizing this piece of software for performance:[Insert code block]Use the following chain of thought to guide your approach:•Performance profiling: Start with how you would profile the software to identify current performance bottlenecks.•Optimization techniques: Discuss the specific techniques you would consider to address the identified bottlenecks, such asalgorithm optimization, code refactoring, or hardwareacceleration.•Testing and validation: Describe your method for testing the optimized software to ensure that the changes have had thedesired effect and have not introduced new issues.•Implementation strategy: Finally, outline how you would safely implement the optimized code into the production environment, ensuring minimal disruption.Conclude with a summary of the key steps in the optimization process and how you would document and maintain the improvements over time.8. Split complex tasks into simpler onesWhen dealing with complex tasks, breaking them into simpler, more manageable components can make them more approachable for an AI. Using step by step instructions helps prevent the AI frombecoming overwhelmed and ensures that each part of the task is handled with attention to detail.Additionally, this approach allows for easier monitoring and adjustment of each step, facilitating better quality control throughout the process. By compartmentalizing tasks, the AI can also use its resources more efficiently, allocating the necessary attention where it’s most needed, resulting in a more effective problem-solving strategy.Prompt Example:Avoid a single broad prompt:•“Write a 1500-word article on the impact of AI on remote work.”Try an initial prompt and follow-up prompts instead: •“Develop a detailed outline for a 1500-word article titled ‘Revolutionizing Remote Work: The Role of AI for TechProfessionals.’ The outline should include an engagingintroduction, three main sections titled ‘Enhancing Productivity with AI Tools,’‘AI-Driven Communication Optimization,’ and‘Advanced Project Management through AI,’ plus a conclusion that offers a perspective on future developments.”•“Compose a detailed introduction for the article‘Revolutio nizing Remote Work: The Role of AI for TechProfessionals.’ The introduction should be 150-200 words,setting the stage for how AI is changing the game for remoteworkers in the tech industry, and providing a hook that willentice tech professionals to continue reading.”9. Understand the model’s shortcomingsIn crafting prompts for an AI, recognize the model’s limitations to set realistic expectations. Prompting AI to perform tasks it’s not designed for, such as interacting with external databases or providing real-time updates, will lead to ineffective and potentially misleading outputs called AI hallucinations.Here are some known shortcomings of AI models:•Lack of real-time data processing, as the knowledge is up-to-date only until the last training cut-off.•Inability to access or retrieve personal data unless it has been shared during the interaction.•No direct interaction with external software, databases, or live web content.•Potential bias in the data, as AI models can inadvertently learn and replicate biases present in their training data.•Limited understanding of context can lead to less nuanced responses in complex or ambiguous situations.•The absence of personal experiences or emotions means the AI cannot form genuine, empathetic connections or offerpersonal anecdotes.10. Take an experimental approach to promptingPrompt engineering is an emergent field that necessitates an experimental mindset. As you navigate this new territory, use an iterative process to test various prompts, paying careful attention to how slight modifications can significantly alter the AI’s responses. You’ll only learn how models respond by testing them.While maintaining a commitment to AI privacy and ethical standards is key, don’t hesitate to explore diverse phrasings and structures to discover the most effective prompts. This trial-and-error process can yield better results and contribute to a broader understanding of how large language models interpret and act on different types of instructions.。

以下是根据⼴⼤学员咨询的怎么申请成为⼀个合格的微软认证获得MCT证书的要求和步骤，任何在微软相关技术和演讲⽅⾯有⼀定能⼒和兴趣的⼈员都有机会成为微软认证(MCT)。

如果您⼀直致⼒于研究或使⽤微软产品和解决⽅案、拥有相当的实际经验并具备良好的表达能⼒和整体形象，您可以按照以下5步完成微软认证(以下简称MCT)的申请。

第1步：获得微软认证专家(MCP)或微软商业解决⽅案(MBS)资格证书 根据微软公司全球统⼀标准，在申请成为 MCT 之前，您必须获得下列证书之⼀。

此项标准的⽬的是为了核实 MCT申请者具备经过证实的技术实⼒和教学⼯作能⼒。

⼀旦获得某项微软认证，即表明您已经在与您的证书相关的领域或⼯作岗位中展⽰了您运⽤微软产品和技术开展⼯作的熟练程度。

Microsoft Certified Desktop Support Techniciann (MCDST) Microsoft Certified Systems Administrator (MCSA) on Microsoftn Windows 2000 Microsoft Certified Systems Administrator (MCSA) on Microsoftn Windows Server 2003 Microsoft Certified Systems Engineer (MCSE) onn Microsoft Windows 2000 Microsoft Certified Systems Engineer (MCSE) onn Microsoft Windows Server 2003 Microsoft Certified Application Developern (MCAD) for Microsoft .NET Microsoft Certified Professional Developern (MCPD) Microsoft Certified Solution Developer (MCSD) on Visual Studion 6.0 Microsoft Certified Solution Developer (MCSD) for Microsoft .NETn n Microsoft Certified Database Administrator (MCDBA) on Microsoft SQL Server 2000 Microsoft Certified IT Professional (MCITP)n Microsoft Officen Specialist (MOS) – pass at least one of the following exams: • Word 2003 Expert • Excel 2003 Expert • PowerPoint 2003 • Access 2003 • Outlook 2003 Microsoft Certifiedn Application Specialist – pass at least one of the following exams: • Exam 70-601: Using Microsoft Office Word 2007 • Exam 70-602: Using Microsoft Office Excel 2007 • Exam 70-603: Using Microsoft Office PowerPoint 2007 • Exam 70-604: Using Microsoft Office Outlook 2007 • Exam 70-605: Using Microsoft Office Access 2007 • Exam 70-600: Using Microsoft Windows Vista Microsoft Certifiedn Professional: Exam 70-301, Managing, Organizing, and Delivering IT Projects by Using the Microsoft Solutions Framework Microsoft Certified Businessn Management Solutions Professional – Application for Microsoft Dynamics GP Microsoft Certified Business Management Solutions Professional – Installation & Configuration for Microsoft Dynamics GP Microsoft Certified Businessn Management Solutions Professional – Application for Microsoft Dynamics SL Microsoft Certified Business Management Solutions Professional – Installation & Configuration for Microsoft Dynamics SL Microsoft Certified Businessn Management Solutions Professional – Application for Microsoft Dynamics CRM Microsoft Certified Business Management Solutions Professional – Installation & Configuration for Microsoft Dynamics CRM Microsoft Certifiedn Business Management Solutions Professional – Developer for Microsoft Dynamics CRM Microsoft Certified Business Management Solutions Professional –n Application for Microsoft Dynamics NAV Microsoft Certified Businessn Management Solutions Professional – Installation & Configuration for Microsoft Dynamics NAV Microsoft Certified Business Management Solutionsn Professional – Developer for Microsoft Dynamics NAV Microsoft Certifiedn Business Management Solutions Professional – Application for Microsoft Dynamics AX Microsoft Certified Business Management Solutions Professional –n Installation & Configuration for Microsoft Dynamics AX Microsoftn Certified Business Management Solutions Professional—Developer for Microsoft Dynamics AX Microsoft Retail Management System MCT Designationn。

2024年6月GESP编程能力认证C++等级考试七级真题(含答案)一、单选题（每题2分，共30分）。

1. 下列C++代码的输出结果是（）。

2. 对于如下图的二叉树，说法正确的是（）。

A. 先序遍历是132。

B. 中序遍历是123。

C. 后序遍历是312。

D. 先序遍历和后序遍历正好是相反的。

3. 已知两个序列s1={1,3,4,5,6,7,7,8,1}、s2={3,5,7,4,8,2,9,5,1}，则它们的最长公共子序列是（）。

4. 关于序列{2,7,1,5,6,4,3,8,9}，以下说法错误的是（）。

A. {2,5,6,8,9}是它的最长上升子序列。

B. {1,5,6,8,9}是它的最长上升子序列。

C. {7,5,4,3}是它的最长下降子序列。

D. {1,5,6,8,9}是它的唯一最长上升子序列。

5. 关于图的深度优先搜索和广度优先搜索，下列说法错误的是（）。

A. 二叉树是也是一种图。

B. 二叉树的前序遍历和后序遍历都是深度优先搜索的一种。

C. 深度优先搜索可以从任意根节点开始。

D. 二叉树的后序遍历也是广度优先搜索的一种。

6. 对于如下二叉树，下面访问顺序说法错误的是（）。

A. HDEBFIGCA不是它的后序遍历序列B. ABCDEFGHI是它的广度优先遍历序列C. ABDHECFGI是它的深度优先遍历序列D. ABDHECFGI是它的先序遍历序列7. 以下哪个方案不能合理解决或缓解哈希表冲突（）。

A. 丢弃发生冲突的新元素。

B. 在每个哈希表项处，使用不同的哈希函数再建立一个哈希表，管理该表项的冲突元素。

C. 在每个哈希表项处，建立二叉排序树，管理该表项的冲突元素。

D. 使用不同的哈希函数建立额外的哈希表，用来管理所有发生冲突的元素。

8. 在C++中，关于运算符&，下面说法正确的是（）。

9. 下面关于图的说法正确的是（）。

A. 在无向图中，环是指至少包含三个不同顶点，并且第一个顶点和最后一个顶点是相同的路径。

初一英语作文范文15篇关于回信The ability to effectively communicate through written correspondence is an invaluable skill that can open doors to new opportunities and foster meaningful connections. As students in junior high school, the task of crafting a well-structured and thoughtful reply to a letter can be both exciting and challenging. In this essay, we will explore 15 sample English essays that provide a framework for composing compelling responses to letters.1. Expressing Gratitude: When receiving a letter that conveys kindness, support, or appreciation, it is important to respond with heartfelt gratitude. In this essay, the student expresses sincere thankfulness for the recipient's thoughtfulness and the positive impact it has had on their life.2. Offering Condolences: Letters that convey news of loss or tragedy require a delicate and empathetic response. This essay demonstrates how to offer sincere condolences, provide comfort, and express solidarity during a difficult time.3. Responding to Requests: Letters often include requests for assistance, information, or favors. In this essay, the student articulates a clear and helpful response, addressing the request directly and offering practical solutions or suggestions.4. Addressing Concerns: When a letter raises issues or problems, it is crucial to respond with understanding and a willingness to address the concerns. This essay shows how to acknowledge the recipient's perspective, offer reassurance, and propose constructive resolutions.5. Maintaining Connections: Replying to letters from friends or family members can help strengthen relationships and keep lines of communication open. This essay demonstrates how to maintain a warm and engaging tone while updating the recipient on personal developments and shared experiences.6. Providing Advice: Letters may sometimes seek guidance or counsel on personal matters. In this essay, the student offers thoughtful advice, drawing upon their own experiences and perspectives to provide meaningful insights and support.7. Expressing Congratulations: When a letter conveys news of an achievement, promotion, or significant life event, a response filled with genuine congratulations and well-wishes can be deeply meaningful. This essay showcases how to craft a celebratory anduplifting reply.8. Clarifying Misunderstandings: Occasionally, letters may contain misunderstandings or require further explanation. This essay illustrates how to address such situations with clarity, empathy, and a desire to resolve the issue effectively.9. Declining Requests: While it can be challenging to refuse a request made in a letter, this essay demonstrates how to do so in a respectful and considerate manner, offering alternative solutions or suggestions when appropriate.10. Responding to Invitations: Letters that extend invitations to events, gatherings, or opportunities require a prompt and gracious response. This essay shows how to accept or politely decline an invitation, while expressing appreciation for the thoughtfulness.11. Providing Updates: When corresponding with friends, family, or acquaintances, letters can serve as a means to share updates onone's life, experiences, and personal growth. This essay highlights how to craft an engaging and informative reply that keeps the recipient engaged and connected.12. Expressing Sympathy: In the face of loss or hardship, a heartfelt letter of sympathy can provide comfort and support. This essaydemonstrates how to convey sincere condolences, offer emotional support, and provide practical assistance during a difficult time.13. Responding to Inquiries: Letters that pose questions or seek information require a clear and comprehensive response. This essay illustrates how to address the recipient's inquiries thoroughly, while providing relevant and helpful details.14. Maintaining Professional Relationships: In the context of academic or professional correspondence, replying to letters with a formal and polished tone is essential. This essay showcases how to craft a professional and courteous response that upholds the appropriate tone and etiquette.15. Fostering Meaningful Connections: Ultimately, the art of letter-writing is about forging and sustaining meaningful connections with others. This final essay emphasizes the importance of crafting thoughtful, empathetic, and engaging responses that deepen the bond between the writer and the recipient.By exploring these 15 sample English essays on replying to letters, students in junior high school can develop a deeper understanding of the nuances and strategies involved in effective written communication. Through practice and refinement, they can honetheir skills in crafting thoughtful, well-structured, and impactful responses that leave a lasting impression on the recipient.。

0-based, best in class practice -回复“0based, best in class practice”是指一种在实践中被广泛接受且被认为是最佳实践的方法或模式。

本文将深入探讨该主题，一步一步解释它的定义、应用和优势，以及如何实施和评估。

第一部分：定义与背景“0based, best in class practice”是指以零为基准，通过选择和采用业界最佳实践来优化业务流程或提高绩效的一种方法。

它强调了持续改进和创新，并倡导不断超越常规标准，成为行业中的佼佼者。

这种方法主要用于制定策略、规划项目和优化流程等方面。

通过深入了解业界最佳实践，组织可以确保在实施新项目或进行业务改进时获得最佳结果。

第二部分：应用“0based, best in class practice”可以在许多不同的领域应用。

以下是一些常见的应用领域：1. 组织与管理：组织管理是一个关键领域，可以通过采用最佳实践来提高工作效率、降低成本并提升员工满意度。

这些最佳实践可能涉及人力资源管理、绩效评估和培训发展。

2. 项目管理：在规划和实施项目过程中，采用最佳实践可以帮助组织确保项目按时交付、高质量完成并在预算范围内。

3. 市场营销：市场营销是一个变化迅速的领域，采用最佳实践可以帮助企业与时俱进，调整营销策略以适应不同的消费者需求和市场趋势。

4. 供应链管理：在供应链管理中，采用最佳实践可以提高物流效率、降低库存成本以及减少延误和废品。

第三部分：优势“0based, best in class practice”的好处是多方面的。

以下是一些与采用这种方法相关的优势：1. 高效性：通过采用业界最佳实践，组织可以避免重复劳动，提高工作流程的效率和质量。

2. 创新性：从业界最佳实践中汲取灵感可以激发创新思维，促使组织寻找新的解决方案和改进业务流程。

3. 竞争优势：通过成为行业中的佼佼者，组织可以获得竞争优势，更好地满足客户需求并获得市场份额。

软考高级架构师系统设计40题1. In a system design, which of the following is the most important consideration for scalability?A. Hardware performanceB. Software architectureC. Network bandwidthD. User interface design答案：B。

解析：软件架构对于系统的可扩展性至关重要。

硬件性能在一定程度上影响，但不是最关键的。

网络带宽主要影响数据传输，对可扩展性的直接影响较小。

用户界面设计与系统的可扩展性关系不大。

2. When designing a system, which principle should be followed to ensure high availability?A. RedundancyB. Minimization of componentsC. Simple architectureD. Low cost答案：A。

解析：冗余是确保高可用性的重要原则。

减少组件可能会降低复杂性，但不一定能保证高可用性。

简单架构有助于理解和维护，但不是保证高可用性的关键。

低成本通常不是高可用性设计的首要考虑因素。

3. Which of the following is a key factor in determining theperformance of a system?A. The number of usersB. The algorithm usedC. The color scheme of the interfaceD. The brand of the hardware答案：B。

解析：算法的优劣直接决定了系统的性能。

用户数量会影响系统负载，但不是决定性能的根本因素。

界面的颜色方案与性能无关。

硬件品牌对性能有一定影响，但算法的影响更为关键。

Turn over*P46734A0112*P46734A©2016 Pearson Education Ltd.1/1/1/1/1/1Setting up time1. Open a new project in the music production software using 16 bit/44.1kHz sample rate.2. Save the project as unit4_your candidate number (e.g. unit4_1234) in the folder designatedby your centre.3. Set the metronome to 128 bpm.4. Import “drums.wav” from the CD ROM to a stereo audio track in the music production software,aligned with the beginning of bar 1.5. Ensure that the drums are audible and play in time with the metronome. The drums beginplaying in bar 2.Instructions• Useblack ink or ball-point pen.• Fill in the boxes at the top of this page with your name, centre number and candidate number.• Answerall questions.• Write your answers to Section A in the spaces provided in this question paper.• You must save your exported audio files for Questions 2 & 3 in Section A, and Question 5 in Section B to your project folder within the 2 hour examination time.• You must ensure that the left and right earpieces of your headphones are worn correctly.• Access to the internet or local network is not permitted.Information• The total mark for this paper is 80.• The marks for each question are shown in brackets–use this as a guide as to how much time to spend on each question.• Questions labelled with an asterisk (*) are ones where the quality of your written communication will be assessed– you should take particular care on these questions with your spelling, punctuation and grammar,as well as the clarity of expression.Advice• Read each question carefully before you start to answer it.• Try to answer every question.• Check your answers if you have time at the end.2DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA Import “synth chords.wav” from the CD ROM to a new stereo track in your music*P46734A0212*3 DONOTWRITEINTHISAREADONOTWRITEINTHISAREADONOTWRITEINTHISAREA*P46734A0312*Turn over4DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA*P46734A0412*5 DONOTWRITEINTHISAREADONOTWRITEINTHISAREADONOTWRITEINTHISAREA*P46734A0512*Turn over6DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA(d) (i) On the graph below, draw the waveform identified in part (c).(1)(ii)Labeltheaxes.(2)(iii) Label the amplitude of the wave.(1)(iv) Label the wavelength of the wave.(1)(e) On the blank graph below, draw the same wave with the phase inverted.(1)(f) Describe what would happen if the waveforms from parts (d) and (e) were addedtogether.(1) ....................................................................................................................................................................................................................................................................................*P46734A0612*7 DONOTWRITEINTHISAREADONOTWRITEINTHISAREADONOTWRITEINTHISAREA*P46734A0712*Turn over8DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA DO NOT WRITE IN THIS AREA*P46734A0812*9 DONOTWRITEINTHISAREADONOTWRITEINTHISAREADONOTWRITEINTHISAREA............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................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for Question 4 = 16 marks)TOTAL FOR SECTION A = 62 MARKS*P46734A0912*Turn over10DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA SECTION B5You should now have the following tracks imported on the computer: drums, synth chords, bass and vocals.Follow the instructions below to produce a final stereo mix.(a) Apply automated panning to the drums.• Onlybars 4 and 5 should be affected; all other bars should be panned to the centre.• The hand clap in bar 4 should be panned hard left.• The hand clap in bar 5 should be panned hard right.• The bass drum should be panned centre throughout.(3)(b) Listen to the automated EQ on the vocals in bars 18–25. Recreate that EQ inbars 2–9. You will need to cut and paste bars 2–9 on to a new track becauseapplying EQ may reintroduce the unwanted tone.(3)(c) Compressthesynth chords.• Onlybars 18–25 should be affected.• The drums should trigger the side-chain of the compressor so that the synth chords part pumps in time with the bass drum.• The side-chained compression should suit the style of the music.(3)(d) Apply a mono delay effect to the synth chords.• Use a crotchet synced delay.• The delay should fill the gaps in the introduction.• The delay should be clearly audible.• Ensure that the delay is not intrusive.(3)*P46734A01012*11 DONOTWRITEINTHISAREADONOTWRITEINTHISAREADONOTWRITEINTHISAREA*P46734A01112*12DO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREADO NOT WRITE IN THIS AREA BLANK PAGE*P46734A01212*。

2023年12月GESP编程能力认证C++等级考试四级真题(含答案)一、单选题（每题 2 分，共 30 分）。

1.下面有关函数参数的说法，正确的是( )。

A. 函数参数传递时，主函数当中采用值传递方式将参数传递给子函数时，若子函数将参数值改变，主函数当中的参数值不变。

B. 函数参数传递时，主函数当中采用值传递方式将参数传递给子函数时，若子函数将参数值改变，主函数当中的参数值将随子函数一样改变而改变。

C. 函数参数传递时，主函数如果将参数的地址传递给子函数，若子函数将参数值改变，主函数当中的参数值将不改变。

D. 函数参数传递可以不满足子函数的参数个数要求。

正确答案：A。

2.下面C++代码执行后，输出的是( )。

A. chenB. cC. chen a daiD. dai正确答案：B。

3.下面C++代码最后执行后输出是( )。

A. 1B. 2C. 3D. 4正确答案：D。

4.下面C++代码执行后的结果是( )。

A.B.C.D.正确答案：C。

5.下面C++代码执行后输出是（）。

A. 1,2,3B. 1C. 2D. 3正确答案：C。

6.如果变量x的地址是0x6ffe14,下面C++代码执行以后输出的是（）。

A. 0x6ffe11B. 0x6ffe14C. 0x6ffe18D. 0x6ffe15正确答案：C。

7.在C++中,执行下面代码后，输出的是（）。

A. 400B. 200C. 20D. 100正确答案：A。

8.下列C++语句执行以后结果是true的是（）。

A. 3&&falseB. 5&&2C. 101&&000D. 4&true正确答案：B。

9.在如下的C++代码中实现了对字符串中出现的26个字母的个数统计，横线处应填入是（）。

A. alpha[i]=alpha[i-1]+1;B. alpha[i]=alpha[i]+1;C. alpha[i+1]=alpha[i]+1;D. alpha[i-1]=alpha[i]+1;正确答案：A。

Economics 2020a / HBS 4010 / HKS API-111FALL 2010Solutions to Practice Problems for Lectures 1 to 41.1. Quantity Discounts and the Budget Constraint(a) The only distinction between the budget line with a quantity discount and an ordinary budget line is the change in price for purchases of x beyond Q* units. This change in price induces a change in slope of the budget line.Therefore, the budget line is piecewise linear with(1)slope = -p / q = -1 if 0 < x < Q*.(2)slope = -p’ / q = -1/4 if 0 > Q*.We now need to identify the y-intercept and point of change of slope to fully describe the budget constraint.y-interceptIf the consumer purchases only good y, she can purchase W / q = 32 / 4 = 8 units. So, one boundary point for the budget constraint is point A = (0, 8).Change of slopeThe slope of the budget constraint changes at x = Q*.At x = Q*, the consumer spends 4Q* on good x and has wealth W – 4Q* to spend on good y – corresponding to a total of (W – 4Q*) / q = 8 – Q* units of y given that W = 32 and q = 4.That is, the point on the budget line where the price of x (and thus the slope of the budget line) changes is B = (Q*, 8 – Q*).Piecewise Linear Budget ConstraintThe equation for the first (of two) linear segment of the budget constraint is y= y-intercept + slope * x,which in this case yields the equationy = 8 – x.We can solve for the second linear segment of the budget constraint by using the equation y = constant + slope * x.We know that the slope of this second segment of the budget constraint is -1/4 and the and that it starts at the point B = (Q*, 8-Q*). This gives the equation8-Q* =constant – ¼ * Q*OR constant = 8 – (3/4) Q*Thus, the second component of the budget constraint is given byy = 8 – (3/4) Q* - (1/4) x.With this as background, we can now substitute the appropriate values of Q* to find the budget constraints in (a1), (a2), (a3).(a1) Component 1: Straight line y = 8 – x between A = (0, 8) and B = (5, 3).Component 2: Straight line y = (17 – x ) / 4 between B = (5, 3) and C = (17, 0).(a2) Component 1: Straight line y = 8 – x between A = (0, 8) and B’ = (16/3, 3).Component 2: Straight line y = 4 – x /4 between B’ = (16/3, 8/3) and C = (16, 0).(a3) Component 1: Straight line y = 8 – x between A = (0, 8) and B’’ = (6, 2).Component 2: Straight line y = (14 - x ) / 4 between B’’ = (6, 2) and C = (14, 0).The graph below illustrates the budget set with Q* = 6. The boundary of the budget set is in bold. The dotted line indicates the convex combination of the intercepts of the budget set (0, 8) and (14, 0), showing that the budget set is not convex.Problem 1.1: Budget Set if Q* = 6246810121416Good 1G o o d 2(b) The quantity discount affects the budget constraint if the point (Q*, 0) lies within the budget constraint – i.e. if the consumer can afford to purchase enough units of good x to reach the quantity discount: p Q* < W, or Q* < w / p .(c) The budget set is not convex for any positive value of Q* that influences the budget constraint – i.e. Q* < W / p .It is straightforward to see this in a graph. The line between any point on the budget constraint with x < Q* and any other point on the budget constraint with x > Q* is entirely outside the budget set. The quantity discount expands the budget constraint, and thereby expands the set of points included in convex combinations of the budget constraints – including many points that remain outside the budget set.To make the same point algebraically, suppose that Q* = 6 and consider the straight line between the x-intercept (32 – 3Q*, 0) = (14, 0) and the y-intercept (0, W / q) = (0, 8) of the budget set. (It is also possible to make this same argument without assuming a specific value of Q*, but then the notation may make the exposition rather unwieldy.)A convex combination of these two points takes the form [14 α, 8(1- α)]. For α = ½, this convex combination produces the bundle [7, 4], which has associated value 41 (10 units at original prices 4 per unit and one unit of x purchased at quantity discount), and is not affordable with budget 32.Intuitively, each convex combination of points on the boundary of the budget set would be in the budget set based on the average prices of those goods. At (14, 0), the average price of x is 32 / 14, while at (0, 8) the average price of y is 4. Thus, the bundle (7, 4) would be affordable at average price 32/14 for x and average price 4 for y. But with the quantity discount, the average price of x declines with the number of units purchased – exactly the opposite of what would be required for convex combinations to remain in the budget set.1.2. Discounts on Initial Units and the Budget Constraint(a) As in Problem 1.1, The only distinction between the budget line with a quantity discount and an ordinary budget line is the change in price for purchases of x beyond Q* units. This change in price induces a change in slope of the budget line.Therefore, the budget line is piecewise linear withslope = -p / q = -1/2 if 0 < x < Q*.slope = -p’ / q = -1 if 0 > Q*.We now need to identify the y-intercept and point of change of slope to fully describe the budget constraint.y-interceptIf the consumer purchases only good y, she can purchase W / q = 24 / 4 = 6 units. So, one boundary point for the budget constraint is point A = (0, 6).Change of slopeThe slope of the budget constraint changes at x = Q*.At x = Q*, the consumer spends 2Q* on good x and has wealth W – 2Q* to spend on good y – corresponding to a total of (W – 2Q*) / q = 8 – Q* / 2 units of y given thatW = 24 and q = 4.That is, the point on the budget line where the price of x (and thus the slope of the budget line) changes is B = (Q*, 6 – Q* / 2).Piecewise Linear Budget ConstraintThe equation for the first (of two) linear segment of the budget constraint is y= y-intercept + slope * x,which in this case yields the equationy = 6 – x / 2.We can solve for the second linear segment of the budget constraint by using the equation y = constant + slope * x.Using slope = -1 and the values (Q*, 6- Q*/2), we can solve for the constant and identify the constant on the second piece of the budget constraint: y = c onstant + slope * x:6 - Q* / 2=constant – Q*6 + Q* / 2 = constant,Thus, the second component of the budget constraint is given byQ* / 2 – x.y = 6+With this as background, we can now substitute the appropriate values of Q* to find the budget constraints in (a1), (a2), (a3).(a1)Component 1: Straight line y = 6 – x / 2 between A = (0, 6) and B = (2, 5).Component 2: Straight line y = 7 - x between B = (2, 5) and C = (7, 0).(a2)Component 1: Straight line y = 6 – x / 2 between A = (0, 6) and B’ = (4, 4).Component 2: Straight line y = 8 –x between B’ = (4, 4) and C = (8, 0).(a3)Component 1: Straight line y = 6 – x / 2 between A = (0, 6) and B’’ = (6, 3).Component 2: Straight line y = 9 - x between B’’ = (6, 3) and C = (9, 0).Problem 1-2: Budget Set with Q* = 42468G o o d 2The graph below illustrates the budget set with Q* = 6. By contrast to the graph for Problem 1.1 above, it is apparent from inspection that this budget set is convex.(b) The budget set is convex for every positive value of Q*,It is straightforward to see this in a graph. The line between any point on the budget constraint with x < Q* and any other point on the budget constraint with x > Q* is always inside the budget set. By comparison to a pricing rule that maintains the same (lower initial) price for all units of good y , the price increase for later units of y causes the budget constraint to contract in a way that ensures that all convex combinations of points on the boundary of the budget set are still contained in the budget set.1.3.Revealed Preference and Transitivity (Problem 1 on Problem Set 1)Solution will be included with Problem Set 1 Solutions1.4.(Not) The Subsidy Principle (Problem 2 on Problem Set 1)Solution will be included with Problem Set 1 Solutions1.5 Revealed Preference and Compensated Demand(a) Suppose that the new bundle chosen by the consumer is (x*, y*). The Law of Compensated Demand says that (x* - 5)(8 – 10) + (y* - 5)(12 – 10) < 0 for the first case where prices change to p1 = 8 and p2 = 12 and wealth stays at 100. This simplifies to the equation 2 (y* - 5) – 2(x* - 5) < 0, or x* > y*. The original bundle (5, 5) is still available under these conditions. So, the Law of Compensated Demand rules out all bundles with x* < 5.Similarly, The Law of Compensated Demand says that(x* - 5)(16 – 10) + (y* - 5)(24 – 10) < 0for the second case where prices change to p1 = 16 and p2 = 24 and wealth increases to 200. This simplifies to the equation 6x* + 14y* - 100 < 0. The budget constraint is given by 16 x* + 24 y* = 200 since the consumer spends all money on goods. Rearranging terms to isolate y*, we find y* = 25 / 3 – 2x* / 3. Substituting back into the inequality, we find6 x* - 14 (2/3) x* + 14 * 25 / 3 – 100 < 0,which simplifies to x* > 5. Once again, the Law of Compensated Demand rules out all bundles with x* < 5. In fact, these two cases correspond to the budget line, since prices and wealth simply double from the first case to the second.(b) Let (x*, y*, z*) be the bundle chosen under the new conditions. By the Law of Compensated Demand,(x* - 5)(8 – 10) + (y* - 10) (10 – 10) + (z* - 10) (12 – 10) < 0,which simplifies to x* > z*.But it is possible for demand for any individual good to increase or decrease. For example, the bundle (2, 13.4, 0) satisfies this inequality with demand for good x falling despite its drop in price, while demand for good 2 increases and demand for good z falls. Similarly, the bundle (11, 0, 31/6) satisfies this inequality with demand for goods x and z increasing while demand for good 2 falls.Note that (2, 13.4, 0) and (11, 0, 31/6) were not available at the original prices, so WARP does not rule out the possibility of choosing either of them over (5, 5, 5), the bundle selected at the original prices.While we cannot conclude that demand for any one particular good must increase or fall, we can rule out some combinations of changes. If demand for good z rises, then demand for good x must rise even more, while if demand for good x falls, then demand for good 3 must fall even more. It seems natural that demand for good x would rise and demand for good z would fall, since the price of good x declined while the price of good z increased, but this need not be the case.OPTIONAL PROBLEM FROM MWG4. MWG 2.D.4:You should be able to identify the non-convexity of this budget set immediately: if you were to draw a line connecting the points (24,0) and (a , M), the points on this line would lie outside the budget set. What this means in practical terms is that the individual can choose to have 24 hours of leisure (no work) and no consumption, or a hours of leisure and M consumption, but if the individual were to choose an amount of leisure exactly ½ of the way between a and 24, they would get less than ½ M in consumption. [The same is true if you pick any other fraction.] This makes it unlikely that individuals will choose (leisure, consumption) bundles that place them on the non-convex portion of a budget set. Try drawing some indifference curves and you will see that depending on how steep or flat these curves are, people will tend to locate either at (24,0) or (a ,M). Moreover, two people with very similar indifference curves might end up choosing very different bundles because of this non-convexity.To demonstrate this non-convexity mathematically, the strategy is to find the slope of the line connecting (24,0) and (a ,M), and then show that at the point on this line whereleisure=16 (or alternatively, work=8 hours), you get a consumption level above the point on the budget set (16,b ). Thus, the first helpful step is to find the values for a and b .The point (16,b ) is attained with 8 hours of labor at a wage rate of s per hour, so b =8s. The value of a indicates how much leisure you will have left, if you work enough hours to consume M. You can consume M by working 8 hours at wage rate s and (16-a ) hours at wage rate s’:'81616'8')16(8s s M a a s s M s a s M −−=−=−−+=If we work 8 hours (leisure=16), where would this put us on the line between (24,0) and (a ,M)? If we calculate the slope of the line with respect to work hours instead of withrespect to leisure hours, we get rise/run = (M-0)/(24-a ). Working 8 hours would thus give consumption of: '888s s M M −+ And then we need only show that this quantity is larger than 8s. There are many ways to do this; the most elegant is to notice that if we changed s’ to s in the above expression: s s M M s s M M 88888==−+ Since we know from the problem that s’>s, s s M M s s M M s s M M 88888'888==−+>−+ This demonstrates that the line is outside the budget set. NOTE: A full mathematical proof is ideal, but not necessary. Instead, a descriptive answer showing a clear understanding of the mathematical meaning of convexity as well as its practical interpretation in this problem, along with an appropriate graphical illustration, would be sufficient. 2.1 Transformations of Utility Functions (a) When x = y , u(x , y ) = xy = x 2. The desired transformation function f solves the equation f (u (x ,y )) = x when x = y OR f (x 2) = x , which can be solved by inspection: f (z ) = SQRT(z ). Then v (x , y ) = f (u (x , y )) = SQRT(xy ) = x 1/2 y 1/2Next compute the Marginal Rate of Substitution for u and for v .∂u / ∂x = y∂u / ∂y = x ,so(∂u / ∂y) / (∂u / ∂x) = x / y .∂v / ∂x = 1/2 x -1/2 y 1/2 ∂v / ∂y = 1/2 x 1/2 y -1/2so (∂v / ∂y) / (∂v / ∂x) = x / y.(b) When x = y, u(x, y) = 2x + y2= 2x + x2.The desired transformation function solves the equationf(u(x, y)) = x when x = yOR f(2x + x2) = x.This indicates that f(x) is the inverse function to g(x) = 2x + x2To invert this function, solve the equationx2z=2x+z + 1 = 2x + x2 + 1z + 1 = (x+1)2SQRT (z + 1) = x + 1SQRT(z + 1) – 1 = xSo if g(x) = 2x + x2, then g -1(x) = SQRT(x+1) – 1 – i.e. f(x) = SQRT(x+1) – 1. Then, v(x, y) = f(u(x,y)) = SQRT(2x + y2 – 1) – 1.Next compute the Marginal Rate of Substitution for u and for v.∂u / ∂x = 2 ∂u / ∂y = 2y,so (∂u / ∂y) / (∂u / ∂x)= y..∂v / ∂x = (2x + y2 – 1) -1/2∂v / ∂y = y (2x + y2 – 1) -1/2so (∂v / ∂y) / (∂v / ∂x)= y.2.2 Convex Preferences and Quasiconcave Utility FunctionsConsider the utility functions u1(x, y) = xy, u2(x, y) = x + y,u3(x, y) = x + SQRT(y).(a) As shown in the graph, these three utility functions produce indifference curves with qualitatively distinct properties. For indifference curves corresponding to u1, convex combinations of bundles on an indifference curve lie above the indifference curve. For indifference curves corresponding to u2, convex combinations of bundles on an indifference curve lie exactly on the indifference curve. For indifference curves corresponding to u3, convex combinations of bundles on an indifference curve lie below the indifference curve. That is, u1 appears to be strictly quasiconcave, u2 appears to be weakly quasiconcave and u3 does not appear to be quasiconcave.Problem 2.1 Graph102030(b) Taking partial derivatives:∂u 1 / ∂x = y ∂u 1 / ∂y = x;∂u 2 / ∂x = 1 ∂u 2 / ∂y = 1;∂u 3 / ∂x = 1∂u 3 / ∂y = 2y.MRS 1 = (∂u 1 / ∂x) / (∂u 1 / ∂y)= y / x.MRS 2 = (∂u 1 / ∂x) / (∂u 1 / ∂y)= 1.MRS 1 = (∂u 1 / ∂x) / (∂u 1 / ∂y)= 1 / 2y.Along an indifference curve as x increases and y decreases, MRS 1 is decreasing, MRS 2 is constant, and MRS 3 is increasing. This indicates that u 1 is strictly quasiconcave, u 2 is weakly quasiconcave, and u 3 is not quasiconcave.2.3 Skiing & Violins(a)Graph for Problem 2.32468101214161820(b) As shown in the graph, a convex combination of two points on the same indifference curve lies below the indifference curve, indicating that with these indifference curves, the consumer strictly prefers the weighted average of utilities of two different bundles to the convex combination of those bundles. Thus, the preferences represented by these utility functions are not convex.(c) The lowest indifference curve is tangent to the budget line at B = (10, 10). The second indifference curve intersects with the budget line at A = (0, 15) and at (20, 5). The third indifference curve intersects with the budget line only at the boundary pointC = (30, 0). Comparing these points and indifference curves, the optimal bundle on the budget line is at C = (30, 0). All other bundles on the budget line lie below the last indifference curve and therefore C = (30, 0) is strictly preferred to all other options on the given budget line.(d) The nonconvexity of preferences induces a choice of bundle that specializes in consumption of a single good. In essence, once we knew that the preferences were nonconvex as shown (with indifference curves that are mirror images of those for convex preferences) then it was almost certain that the optimal bundle would be at one of the ends of the budget line at A = (0, 15) or C = (30, 0).2.4 Piecewise Linear Indifference Curves and Optimal Bundles(a) First equate each bundle to a different bundle with x = y and then use transitivity to compare the bundles. The consumer is indifferent between bundle (x 1, y 1) and bundle (k 1, k 1) where k 1 = (x 1 + 2 y 1) / 3. The consumer is indifferent between bundle (x 2, y 2) and bundle (k 2, k 2) where k 2 = (2 x 2 + y 2) / 3.The consumer’s preference between the bundles can be identified from the comparison between k 1 and k 2. If k 1 > k 2, i.e. (x 1 + 2 y 1) / 3 > (2 x 2 + y 2) / 3, then the consumer strictly prefers (x 1, y 1) to (x 2, y 2). Similarly, if k 2 > k 1, i.e. (x 2 + 2 y 2) / 3 > (2 x 1 + y 1) / 3, then the consumer strictly prefers (x 2, y 2) to (x 1, y 1). Finally, if k 1 = k 2, i.e. (x 1 + 2 y 1) / 3 > (2 x 2 + y 2) / 3, then the consumer is indifferent between (x 1, y 1) and (x 2, y 2).(b) On a graph, show this consumer’s indifference curves through (4, 4) and (8, 8).Graph for Problem 2.4(b)24681012(c) As shown in the graph below, the bundle (4, 4) lies on the budget line and is the optimal choice.Graph for Problem 2.4(c)12345678optimal choice. Graph for Problem 2.4(d)24681012With equal prices, as in (c), the consumer will naturally choose a bundle with equal amounts of each good. However, if the price of one good is much larger than the other (as in (d), when p > 2q) then the degree of the consumer’s preference for equal consumption is outweighed by the difference in prices and the consumer will specialize consumption to just one good.2.5 Perfect ComplementsSolution will be included with Problem Set 1 Solutions3.1 Cobb-Douglas Utility(a) Start by writing the Lagrangian:L(x, y) = xα y1-α– λ (px + qy - W)Taking partial derivatives gives the first-order conditions∂L / ∂x = α xα-1 y1-α– λp =α (x / y) α-1– λp = 0; (1)∂L / ∂y = (1-α) xα y –α – λq =(1-α) (x / y)α – λq =0; (2) ∂L / ∂λ = W – px – qy = 0. (3) Rearrange and solve for x and y however you like. For example, you could start by solving (1) and (2) for λ, then using those two equalities to solve for y as a function of x.λ= α (x / y)α-1 / pq;λ= (1-α) (x / y)α/Setting these equal,α (x / y)α-1 / p = (1-α) (x / y)α / q;OR α (y / x) / p = (1-α) / q;OR y = (1-α) px / αq. (4) Finally, we can solve for the optimal bundle by substituting (4) into (3) – which is simply the budget constraint.First rewrite (3) asx = (W – qy) / p (5) then substitute (5) into (4):y = (1- α) (W – qy) / αqα) W / αq - (1- α) y / αy =(1-ORα) W / q - (1- α) yOR αy =(1-α) W / q. (6) ORy =(1-This is the formula for the choice of y in the optimal bundle. Substituting (6) into (5) yields the formula the choice of x in the optimal bundle.α) W) / p(1-x =(W-OR x = αW/ p. (7)(b) Use the same FOCs as above to solve for the Lagrange multiplier:q;λ= (1-α) (x / y)α/From (6) and (7),x / y = αq / (1-α) p,implying λ= (1-α) (αq / (1-α) p)α/ q,which can be simplified slightly toλ= (α / p)α[(1-α) / q)1-α.(c) Returning to the results from (a), the optimal choice of x is linear in W, αand 1 / p,and similarly the optimal choice of y is linear in W, 1-αand 1 / q. One property of these results is that total expenditure on x is equal to px*(p, W) = αW, while total expenditureon y is equal to qx*(p, W) = (1-α)W. That is, the parameter αdetermines the proportionof wealth spent on each good, then the prices p and q, translate that proportional expenditure into particular quantities of x and y.(d) Optimal BundleWith transformed utility function v(x, y) = α ln x + (1-α) ln y,L(x, y) = α ln x + (1-α) ln y – λ (px + qy - W)Taking partial derivatives gives the first-order conditions∂L / ∂x = α / x– λp = 0; (1’)∂L / ∂y = (1-α) / y – λq=0; (2’) ∂L / ∂λ = W – px – qy = 0. (3’) Solving (1) and (2) for λ,λ= α / px;λ= (1-α) / qy;Setting these equal,α / px = (1-α) / qyOR y= (1-α) px / αq,which is identical to (4). Thus, the optimal bundle is unchanged by the transformation of the utility function .Lagrange MultiplierNow solving λ = (1-α) / qy for λgivesλ= (1-α) / q[(1- α) W / q]ORλ= 1 / W .Using the interpretation of λas the “shadow price of the budget constraint”, the number of utility units achieved per additional unit of money varies with the form of the utility function. Therefore, since the transformation of the utility function changed the definition of one unit of utility, the Lagrange multiplier must also change as a result of this transformation.3.2. Skiing & Violins revisited(a) This is a separable utility function with associated first-order condition (MRS form)(∂u / ∂x) / p = (∂u / ∂y) / qOR 10 / p= 2y / q.With p = 10 and q = 20, the first-order condition is given by10 / 10 = 2y / 20 OR y = 10.(b) From 2.3, we know that the utility function is not quasiconcave – so that preferences are not convex and in fact, the first-order condition identifies a global minimum (i.e. minimum utility available on the budget constraint) rather than a maximum.(c) When the first-order conditions identify a minimum rather than a maximum, then the optimal bundle must be at one of the endpoints of the budget line, in this case (30, 0) and (0, 15). Computing u(30, 0) = 300 and u(0, 15) = 225, we conclude that the optimal bundle is (30, 0), since this gives higher utility in the comparison of the endpoints.3.3 Comparative Statics with One Good (Problem 3 on Problem Set 2)Solution will be included with Problem Set 2 Solutions3.4 Comparative Statics and the Marginal Rate of Substitution (Problem 1 on Problem Set 2)Solution will be included with Problem Set 2 Solutions3.5 Utility Maximization with Quantity Discount (Problem 2 on Problem Set 2)Solution will be included with Problem Set 2 Solutions3.6. Utility Maximization with Discounts on Initial Units(a) The budget constraint is piecewise linear, with slope -1/2 between the pointA = (0, 6) and pointB = (2, 5) (the point where the quantity discount applies). The budget constraint is “kinked” at point B, with slope changing to -1 betweenB = (2, 5) andC = (7, 0).The natural method for solving this problem is to find the optimal bundle on each separate linear portion of the budget line, and then compare these two bundles(a1) Between A = (0, 6) and B = (2, 5)On this part of the budget line, p = 2, q = 4 and the first-order condition is given by (∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x / 2.The equation for the line between A and B is y = 6 – x / 2. Substituting the first-order condition into the budget line equation gives the solution x = 6, corresponding to a point beyond the end of the line between A and B. This indicates that the first-order conditions for a maximum cannot be satisfied on the line between A and B and instead that the optimal bundle on this line must be at either A or B. Since A yields utility 0, the optimal bundle on this line must be at B.TECHNICAL COMMENT: The first-order condition identifies the optimal bundle for a budget line from (0, 6) to (12, 0) – the budget line that applies with no increase in the price of x for x > Q*. Under these conditions, the optimal bundle would be (6, 3), producing utility 18. When the first-order condition is not satisfied on the line between A and B, the utility function must be strictly increasing (or strictly decreasing) on that line. In this case, the utility function starts at 0 at point A and is strictly increasing in x untilreaching point B where x = 2. Therefore, the maximum value of the utility function on the line between A and B occurs at the boundary, point B.(a2). Between B = (2, 5) and C = (7, 0),On this part of the budget line, p = 4, q = 4 and the first-order condition is given by (∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x.The equation for the line between B and C is y = 7 – x. Substituting the first-order condition into the budget line equation gives the condition 2x = 7, which yields the solution x = 3.5, corresponding to the bundle (3.5. 3.5). This is the unique point on the line between B and C that satisfies the first-order condition.(a3) Comparing the candidate solutions.We have two candidate bundles for the consumer’s optimum. Bundle (2, 5) yields utility 10 and is the optimal choice on the budget line between A and B. By comparison, bundle (3.5, 3.5) yields utility 49/4 and is the optimal choice on the budget line between B and C. Since 49/4 > 10, bundle (7/2, 7/2) is the global optimum.(b) The solution follows the same method as in (a), but with slightly different budget equations throughout the analysis. As before, the budget constraint is piecewise linear, with slope -1/2 between the point A = (0, 6) and point B’ = (4, 4) (the point where the quantity discount applies) and separate slope -1 between B’ = (4, 4) and C = (8, 0).(b1) Between A = (0, 6) and B’ = (4, 4),On this part of the budget line, p = 2, q = 4 and the first-order condition is given by (∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x / 2.As in (a1), the first-order condition identifies a bundle beyond the end of the line between A and B’. so the optimal bundle on this part of the budget line is at a boundary point, in this case at B’ = (4, 4).(b2) Between B’ = (4, 4) and C = (8, 0),On this part of the budget line, p = 4, q = 4 and the first-order condition is given by (∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x.The equation for the line between B’ and C is y = 8 – x. Substituting the first-order condition into the budget line equation gives 2x = 8, which yields the solutionx = 4, corresponding to the bundle (4. 4). This is the unique point on the line between B’ and C’ that satisfies the first-order condition.(b3) Comparing the candidate solutions.We’ve identified the same bundle (4, 4) as the optimal bundle on each separate component of the budget constraint. Therefore, the bundle, which lies exactly at the boundary between the two parts of the budget constraint, must be the global optimum.(c) The solution follows the same method as above, but with again slightly different budget equations throughout the analysis. As before, the budget constraint is piecewise linear, with slope -1/2 between the point A = (0, 6) and point B’’ = (6, 3) (the point where the quantity discount applies) and separate slope -1 between B’’ = (6, 3) and C = (9, 0).(c1) Between A = (0, 6) and B’ = (6, 3)On this part of the budget line, p = 2, q = 4 and the first-order condition is given by(∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x / 2.identifying a bundle (6, 3) that lies at the boundary of the line between A and B’’.(c2) Between B’’ = (6, 3) and C’’ = (9, 0),On this part of the budget line, p = 4, q = 4 and the first-order condition is given by(∂u / ∂x) / (∂u / ∂y)) = p / q OR y = x.The equation for the line between B’’ and C’’ is y = 9 – x. Substituting the first-order condition into the budget line equation gives 2x = 9, which corresponds to a bundle with x = 4.5 that does not lie on the line between B’’ and C’’. The optimal bundle on this line must be at one of the boundaries and in this case is at point B’’ = (6, 3).(c3) Comparing the candidate solutions.We’ve identified the same bundle (6, 3) as the optimal bundle on each separate component of the budget constraint. Therefore, the bundle, which lies exactly at the boundary between the two parts of the budget constraint, must be the global optimum. (d) For purchases of good x beyond Q* units, the prices of the two goods are equal, so with the given utility function, the consumer would then want to set the quantities of these two goods equal. However, if Q* is relatively large by comparison to W, the consumer has reason to consume more of x than y (setting x = 2y if possible) in order to take advantage of the reduced price for x on initial purchases.In (a) and (b), when Q* is low, the consumer simply sets x = y and the value of Q* only affects the total number of units of each good that the consumer can choose.。