第二章地图的数学基础习题及参考答案.

2.-第二章课后习题及答案D3. (Q4) What information is used by a processrunning on one host to identify a process running on another host?Answer: The IP address of the destination host and the port number of the destination socket.4. (Q6) Referring to Figure 2.4, we see thatnone of the application listed in Figure2.4 requires both no data loss and timing.Can you conceive of an application that requires no data loss and that is also highly time-sensitive?Answer: There are no good example of an application that requires no data loss and timing. If you know of one, send an e-mail to the authors5. (Q9) Why do HTTP, FTP, SMTP, and POP3 run on top of TCP rather than on UDP?Answer: The applications associated with those protocols require that all application data be received in the correct order and without gaps. TCP provides this service whereas UDP does not.6. (Q11) What is meant by a handshaking protocol?Answer: A protocol uses handshaking if the two communicating entities first exchange control packets before sending data to each other. SMTP uses handshaking at the application layer whereas HTTP does not.7. (Q13) Telnet into a Web server and senda multiline request message. Include inthe request message the If-modified-since:header line to force a response message with the 304 Not Modified status code.Answer: Issued the following command (in Windows command prompt) followed by the HTTP GET message to the “” web server:> telnet 80Since the index.html page in this web server was not modified since Fri, 18 May 2007 09:23:34 GMT, the following output was displayed when the above commands were issued on Sat, 19 May 2007. Note that the first 4 lines are the GET message and header lines input by the user and the next 4 lines (starting from HTTP/1.1 304 Not Modified) is the response from the web server.8. (Q14) Consider an e-commerce site thatwants to keep a purchase record for each of its customers. Describe how this can be done with cookies.Answer: When the user first visits the site, the site returns a cookie number.This cookie number is stored on the user’s host and is managed by the browser.During each subsequent visit (and purchase), the browser sends the cookie number back to the site. Thus the site knows when this user (more precisely, this browser) is visiting the site.9. (Q15) Suppose Alice, with a Web-basede-mail account (such as Hotmail or gmail), sends a message to Bob, who accesses his mail from his mail server using POP3.Discuss how the message gets from Alice’s host to Bob’s host. Be sure to list theseries of application-layer protocols that are used to move the message between the two hosts.Answer: Message is sent from Alice’s host to her mail server over HTTP. Alice’s mail server then sends the message to Bob’s mail server over SMTP. Bob then transfers the message from his mail server to his host over POP3.10. (Q10) Recall that TCP can be enhancedwith SSL to provide process-to-process security services, including encryption.Does SSL operate at the transport layer or the application layer? If the application developer wants TCP to be enhanced with SSL, what does the developer have to do?Answer: SSL operates at the application layer. The SSL socket takes unencrypteddata from the application layer, encrypts it and then passes it to the TCP socket. If the application developer wants TCP to be enhanced with SSL, she has to include the SSL code in the application.11. (Q16) Print out the header of an e-mailmessage you have recently received. How many Received: header lines are there?Analyze each of the header lines in the message.Answer: from 65.54.246.203 (EHLO ) Received：(65.54.246.203) bywith SMTP; Sat, 19 May 200716:53:51 -0700from ([65.55.135.106]) by Received: with MicrosoftSMTPSVC(6.0.3790.2668); Sat, 19 May 2007 16:52:42 -0700Received: from mail pickup service by with MicrosoftSMTPSVC; Sat,19 May 200716:52:41 -0700Message-ID:<BAY130-F26D9E35BF59E0D18A819AFB9310@p hx.gbl>Received: from 65.55.135.123 byby130fd.bay130.hotmail.msn.com with HTTP; Sat, 19 May 200723:52:36 GMTFrom: "prithula dhungel"<prithuladhungel@>To: prithula@Bcc:Subject: Test mailDate: Sat, 19 May 2007 23:52:36 +0000Mime-Version:1.0Content-Type: Text/html; format=flowedReturn-Path: prithuladhungel@Figure: A sample mail message headerReceived: This header field indicates the sequence in which the SMTP servers send and receive the mail message including the respective timestamps.In this example there are 4 “Received:”header lines. This means the mail message passed through 5 different SMTP servers before being delivered to the receiver’s mail box. The last (forth) “Received:” heade r indicates the mail message flow from the SMTP server of the sender to the second SMTP server in the chain of servers. The sender’s SMTP server is at address 65.55.135.123 and the second SMTP server in the chain is . The thir d “Received:” header indicates the mail message flow from the second SMTPserver in the chain to the third server, and so on.Finally, the first “Received:” header indicates the flow of the mail message from the forth SMTP server to the last SMTP server (i.e. the receiver’s mail server) in the chain.Message-id: The message has been given this numberBAY130-F26D9E35BF59E0D18A819AFB9310@ph x.gbl(by. Message-id is a unique string assigned by the mail system when the message is first created.From: This indicates the email address of the sender of the mail. In the given example, the sender is prithuladhungel@To: This field indicates the email address of the receiver of the mail. In the example, the receiver is prithula@Subject: This gives the subject of the mail (if any specified by the sender). In the example, the subject specified by the sender is “Test mail”Date: The date and time when the mail was sent by the sender. In the example, the sender sent the mail on 19th May 2007, at time 23:52:36 GMT.Mime-version: MIME version used for the mail. In the example, it is 1.0.Content-type: The type of content in the body of the mail message. In the example, it is “text/html”.Return-Path: This specifies the emailaddress to which the mail will be sent if thereceiver of this mail wants to reply to the sender. This is also used by the sender’s mail server for bouncing back undeliverable mail messages of mailer-daemonerror messages. In the example, the return path is“prithuladhungel@”.12. (Q18) Is it possible for anorganization’s Web serve r and mail server to have exactly the same alias fora hostname (for example, )? Whatwould be the type for the RR that contains the hostname of the mail server?Answer: Yes an organization’s mail server and Web server can have the same alias for a host name. The MX record is used to map the mail server’s host name to its IP address.13. (Q19) Why is it said that FTP sends control information “out-of-band”?Answer: FTP uses two parallel TCP connections, one connection for sending control information (such as a request to transfer a file) and another connection for actually transferring the file. Because the control information is not sent over the same connection that the file is sent over, FTP sends control information out of band.14. (P6) Consider an HTTP client that wantsto retrieve a Web document at a given URL.The IP address of the HTTP server is initially unknown. What transport and application-layer protocols besides HTTP are needed in this scenario?Answer:Application layer protocols: DNS and HTTPTransport layer protocols: UDP for DNS;TCP for HTTP15. (P9) Consider Figure2.12, for whichthere is an institutional network connected to the Internet. Suppose that the average object size is 900,000 bits and that the average request rate from the institution’s browsers to the origin servers is 10 requests per second.Also suppose that the amount of time it takes from when the router on the Internet side of the access link forwards an HTTP request until it receives the response is two seconds on average (see Section 2.2.5). Model the total average response times as the sum of the averageaccess delay (that is, the delay from Internet router to institution router) and the average Internet delay. For the average access delay, use △/(1-△β), where △ is the average time required to send an object over the access link and β is the arrival rate of objects to the access link.a. Find the total average response time.b. Now suppose a cache is installed in theinstitutional LAN. Suppose the hit rate is 0.6. Find the total response time.Answer:a.T he time to transmit an object of sizeL over a link or rate R is L/R. The average time is the average size of the object divided by R:Δ= (900,000 bits)/(1,500,000 bits/sec) = 0.6 secThe traffic intensity on the link is (1.5requests/sec)(0.6 sec/request) = 0.9. Thus, the average access delay is (0.6 sec)/(1 - 0.9) = 6 seconds. The total average response time is therefore 6 sec + 2 sec = 8 sec.b.T he traffic intensity on the access linkis reduced by 40% since the 40% of the requests are satisfied within the institutional network. Thus the average access delay is (0.6 sec)/[1 –(0.6)(0.9)] = 1.2 seconds. The responsetime is approximately zero if the request is satisfied by the cache (which happens with probability 0.4); the average response time is 1.2 sec + 2 sec = 3.2 sec for cache misses (which happens 60% of the time). So the average response time is (0.4)(0 sec) +(0.6)(3.2 sec) = 1.92 seconds. Thus theaverage response time is reduced from 8 sec to 1.92 sec.16. (P12) What is the difference betweenMAIL FROM: in SMTP and From: in the mail message itself?Answer: The MAIL FROM: in SMTP is a message from the SMTP client that identifies the sender of the mail message to the SMTP server. The From: on the mail message itself is NOT an SMTP message, but rather is just a line in the body of the mail message.17. (P16) Consider distributing a file of F= 5 Gbits to N peers. The server has an upload rate of u= 20 Mbps, and each peers=1 Mbps and an has a download rate of diupload rate of u. For N = 10, 100, and 1,000 and u = 100 Kbps, 250 Kbps, and 500 Kbps, prepare a chart giving the minimumdistribution time for each of the combinations of N and u for both client-server distribution and P2P distribution.Answer: For calculating the minimum distribution time for client-server distribution, we use the following formula:D cs = max {NF/u s , F/d min }Similarly, for calculating the minimum distribution time for P2P distribution, we use the following formula:D P2P = max{F/u s ,F/d min ,NF/( u s +∑u i n i=1 )}Where,F = 5 Gbits = 5 * 1024 Mbitsu s = 20 Mbpsd min = d i = 1 MbpsClient Server:N10 100 1000200 Kbps 10240 51200 512000u 600 Kbps 10240 51200 512000 1Mbps 10240 51200 512000Peer to Peer:N10 100 1000200 Kbps 10240 25904.3 47559.33 U 600 Kbps 10240 13029.6 16899.641 Mbps 10240 10240 10240。

第二章1. (Q2) For a communication session between a pair of processes, which process is the client and which is the server?Answer: The process which initiates the communication is the client; the process that waits to be contacted is the server..2. (Q3) What is the difference between network architecture and application architecture?Answer: Network architecture refers to the organization of the communication process into layers (e.g., the five-layer Internet architecture). Application architecture, on the other hand, is designed by an application developer and dictates the broad structure of the application(e.g., client-server or P2P)3. (Q4) What information is used by a process running on one host to identify a process running on another host?Answer: The IP address of the destination host and the port number of the destination socket.4. (Q6) Referring to Figure 2.4, we see that none of the application listed in Figure 2.4 requires both no data loss and timing. Can you conceive of an application that requires no data loss and that is also highly time-sensitive?Answer: There are no good example of an application that requires no data loss and timing.If you know of one, send an e-mail to the authors5. (Q9) Why do HTTP, FTP, SMTP, and POP3 run on top of TCP rather than on UDP?Answer: The applications associated with those protocols require that all application data be received in the correct order and without gaps. TCP provides this service whereas UDP does not.6. (Q11) What is meant by a handshaking protocol?Answer: A protocol uses handshaking if the two communicating entities first exchange control packets before sending data to each other. SMTP uses handshaking at the application layer whereas HTTP does not.7. (Q13) Telnet into a Web server and send a multiline request message. Include in the request message the If-modified-since: header line to force a response message with the 304 Not Modified status code.Answer: Issued the following command (in Windows command prompt) followed by the HTTP GET message to the “” web server:> telnet 80Since the index.html page in this web server was not modified since Fri, 18 May 2007 09:23:34 GMT, the following output was displayed when the above commands were issued on Sat, 19 May 2007. Note that the first 4 lines are the GET message and header lines input by the user and the next 4 lines (starting from HTTP/1.1 304 Not Modified) is the response from the web server.8. (Q14) Consider an e-commerce site that wants to keep a purchase record for each of its customers. Describe how this can be done with cookies.Answer: When the user first visits the site, the site returns a cookie number. This cookie number is stored on the user’s host and is managed by the browser. During each subsequent visit (and purchase), the browser sends the cookie number back to the site. Thus the site knows when this user (more precisely, this browser) is visiting the site.9. (Q15) Suppose Alice, with a Web-based e-mail account (such as Hotmail or gmail), sends a message to Bob, who accesses his mail from his mail server using POP3. Discuss how the message gets from Alice’s host to Bob’s host. Be sure to list the series of application-layer protocols that are used to move the message between the two hosts.Answer: Message is sent from Alice’s host to her mail server over HTTP. Alice’s mail server then sends the message to Bob’s mail server over SMTP. Bob then transfers the message from his mail server to his host over POP3.10. (Q10) Recall that TCP can be enhanced with SSL to provide process-to-process securityservices, including encryption. Does SSL operate at the transport layer or the application layer? If the application developer wants TCP to be enhanced with SSL, what does the developer have to do?Answer: SSL operates at the application layer. The SSL socket takes unencrypted data from the application layer, encrypts it and then passes it to the TCP socket. If the application developer wants TCP to be enhanced with SSL, she has to include the SSL code in the application.11. (Q16) Print out the header of an e-mail message you have recently received. How manyReceived: header lines are there? Analyze each of the header lines in the message.Answer: from 65.54.246.203 (EHLO )Received：(65.54.246.203) by with SMTP; Sat, 19 May 2007 16:53:51 -0700from ([65.55.135.106]) by Received: with Microsoft SMTPSVC(6.0.3790.2668); Sat, 19 May 2007 16:52:42 -0700 Received: from mail pickup service by with Microsoft SMTPSVC; Sat,19 May 2007 16:52:41 -0700Message-ID: <BAY130-F26D9E35BF59E0D18A819AFB9310@phx.gbl>Received: from 65.55.135.123 by with HTTP; Sat, 19 May 2007 23:52:36 GMTFrom: "prithula dhungel" <prithuladhungel@>To: prithula@Bcc:Subject: Test mailDate: Sat, 19 May 2007 23:52:36 +0000Mime-Version:1.0Content-Type: Text/html; format=flowedReturn-Path: prithuladhungel@Figure: A sample mail message headerReceived: This header field indicates the sequence in which the SMTP servers send and receive the mail message including the respective timestamps.In this example there are 4 “Received:” header lines. This means the mail message passed through 5 different SMTP servers before being delivered to the receiver’s mail box. The last (forth) “Received:” header indicates the mail message flow from the SMTP server of the sender to the second SMTP server in the chain of servers. The sender’s SMTP server is at address 65.55.135.123 and the second SMTP server in the chain is . The third “Received:” header indicates the mail message flow from the second SMTP server in the chain to the third server, and so on.Finally, the first “Received:” header indicates the flow of the mail message from the forth SMTP server to the last SMTP server (i.e. the receiver’s mail server) in the chain.Message-id: The message has been given this numberBAY130-F26D9E35BF59E0D18A819AFB9310@phx.gbl(by .Message-id is a unique string assigned by the mail system when the message is first created.From: This indicates the email address of the sender of the mail. In the givenexample, the sender is prithuladhungel@the following formula:D cs = max {NF/u s , F/d min }Similarly, for calculating the minimum distribution time for P2P distribution, we use the following formula:D P2P = max{F/u s ,F/d min ,NF/( u s +∑u i n i=1 )} Where,F = 5 Gbits = 5 * 1024 Mbits u s = 20 Mbps d min = d i = 1 MbpsClient Server:N 10 100 1000 200 Kbps10240 51200 512000 u 600 Kbps10240 51200 512000 1Mbps10240 51200 512000Peer to Peer:N 10 100 1000 200 Kbps10240 25904.3 47559.33 U 600 Kbps10240 13029.6 16899.64 1 Mbps10240 10240 10240。

2.-第二章课后习题及答案第二章1. (Q2) For a communication session betweena pair of processes, which process is theclient and which is the server?Answer: The process which initiates the communication is the client; the process that waits to be contacted is the server. .2. (Q3) What is the difference between network architecture and application architecture?Answer: Network architecture refers to the organization of the communication process into layers (e.g., the five-layer Internet architecture). Application architecture, on the other hand, is designed by an application developer and dictates the broad structure of the application (e.g., client-server or P2P)3. (Q4) What information is used by a processrunning on one host to identify a process running on another host?Answer: The IP address of the destination host and the port number of the destination socket.4. (Q6) Referring to Figure 2.4, we see thatnone of the application listed in Figure2.4 requires both no data loss and timing.Can you conceive of an application that requires no data loss and that is also highly time-sensitive?Answer: There are no good example of an application that requires no data loss and timing. If you know of one, send an e-mail to the authors5. (Q9) Why do HTTP, FTP, SMTP, and POP3 run on top of TCP rather than on UDP?Answer: The applications associated with those protocols require that all application data be received in the correct order and without gaps. TCP provides this service whereas UDP does not.6. (Q11) What is meant by a handshaking protocol?Answer: A protocol uses handshaking if the two communicating entities first exchange control packets before sending data to each other. SMTP uses handshaking at the application layer whereas HTTP does not.7. (Q13) Telnet into a Web server and senda multiline request message. Include inthe request message the If-modified-since:header line to force a response message with the 304 Not Modified status code.Answer: Issued the following command (in Windows command prompt) followed by the HTTP GET message to the “” web server:> telnet 80Since the index.html page in this web server was not modified since Fri, 18 May 2007 09:23:34 GMT, the following output was displayed when the above commands were issued on Sat, 19 May 2007. Note that the first 4 lines are the GET message and header lines input by the user and the next 4 lines (starting from HTTP/1.1 304 Not Modified) is the response from the web server.8. (Q14) Consider an e-commerce site thatwants to keep a purchase record for each of its customers. Describe how this can be done with cookies.Answer: When the user first visits the site, the site returns a cookie number.This cookie number is stored on the user’s host and is managed by the browser.During each subsequent visit (and purchase), the browser sends the cookie number back to the site. Thus the site knows when this user (more precisely, this browser) is visiting the site.9. (Q15) Suppose Alice, with a Web-basede-mail account (such as Hotmail or gmail), sends a message to Bob, who accesses his mail from his mail server using POP3.Discuss how the message gets from Alice’s host to B ob’s host. Be sure to list theseries of application-layer protocols that are used to move the message between the two hosts.Answer: Message is sent from Alice’s host to her mail server over HTTP. Alice’s mail server then sends the message to Bob’s mail server over SMTP. Bob then transfers the message from his mail server to his host over POP3.10. (Q10) Recall that TCP can be enhancedwith SSL to provide process-to-process security services, including encryption.Does SSL operate at the transport layer or the application layer? If the application developer wants TCP to be enhanced with SSL, what does the developer have to do?Answer: SSL operates at the application layer. The SSL socket takes unencrypteddata from the application layer, encrypts it and then passes it to the TCP socket. If the application developer wants TCP to be enhanced with SSL, she has to include the SSL code in the application.11. (Q16) Print out the header of an e-mailmessage you have recently received. How many Received: header lines are there?Analyze each of the header lines in the message.Answer: from 65.54.246.203 (EHLO ) Received：(65.54.246.203) bywith SMTP; Sat, 19 May 200716:53:51 -0700from ([65.55.135.106]) by Received: with MicrosoftSMTPSVC(6.0.3790.2668); Sat, 19 May 2007 16:52:42 -0700Received: from mail pickup service by with MicrosoftSMTPSVC; Sat,19 May 200716:52:41 -0700Message-ID:<BAY130-F26D9E35BF59E0D18A819AFB9310@p hx.gbl>Received: from 65.55.135.123 byby130fd.bay130.hotmail.msn.com with HTTP; Sat, 19 May 200723:52:36 GMTFrom: "prithula dhungel"<prithuladhungel@>To: prithula@Bcc:Subject: Test mailDate: Sat, 19 May 2007 23:52:36 +0000Mime-Version:1.0Content-Type: Text/html; format=flowedReturn-Path: prithuladhungel@Figure: A sample mail message headerReceived: This header field indicates the sequence in which the SMTP servers send and receive the mail message including the respective timestamps.In this example there are 4 “Received:”header lines. This means the mail message passed through 5 different SMTP servers before being delivered to the receiver’s mail box. The last (forth) “Received:” header indicates the mail message flow from the SMTP server of the sender to the second SMTP server in the chain of servers. The sender’s SMTP server is at address 65.55.135.123 and the second SMTP server in the chain is . The third “Received:” header indicates the mail message flow from the second SMTPserver in the chain to the third server, and so on.Finally, the first “Received:” header indicates the flow of the mail message from the forth SMTP server to the last SMTP server (i.e. the receiver’s mail server) in the chain.Message-id: The message has been given this numberBAY130-F26D9E35BF59E0D18A819AFB9310@ph x.gbl(by. Message-id is a unique string assigned by the mail system when the message is first created.From: This indicates the email address of the sender of the mail. In the given example, the sender is prithuladhungel@To: This field indicates the email address of the receiver of the mail. In the example, the receiver is prithula@Subject: This gives the subject of the mail (if any specified by the sender). In the example, the subject specified by the sender is “Test mail”Date: The date and time when the mail was sent by the sender. In the example, the sender sent the mail on 19th May 2007, at time 23:52:36 GMT.Mime-version: MIME version used for the mail. In the example, it is 1.0.Content-type: The type of content in the body of the mail message. In the example, it is “text/html”.Return-Path: This specifies the emailaddress to which the mail will be sent if thereceiver of this mail wants to reply to the sender. This is also used by the sender’s mail server for bouncing back undeliverable mail messages of mailer-daemonerror messages. In the example, the return path is“prithuladhungel@”.12. (Q18) Is it possible for anorganization’s Web server and mail server to have exactly the same alias fora hostname (for example, )? Whatwould be the type for the RR that contains the hostname of the mail server?Answer: Yes an organization’s mail server and Web server can have the same alias for a host name. The MX record is used to map the mail server’s host name to its IP address.13. (Q19) Why is it said that FTP sends control information “out-of-band”?Answer: FTP uses two parallel TCP connections, one connection for sending control information (such as a request to transfer a file) and another connection for actually transferring the file. Because the control information is not sent over the same connection that the file is sent over, FTP sends control information out of band.14. (P6) Consider an HTTP client that wantsto retrieve a Web document at a given URL.The IP address of the HTTP server is initially unknown. What transport and application-layer protocols besides HTTP are needed in this scenario?Answer:Application layer protocols: DNS and HTTPTransport layer protocols: UDP for DNS;TCP for HTTP15. (P9) Consider Figure2.12, for whichthere is an institutional network connected to the Internet. Suppose that the average object size is 900,000 bits and that the average request rate from the institution’s browsers to the origin servers is 10 requests per second.Also suppose that the amount of time it takes from when the router on the Internet side of the access link forwards an HTTP request until it receives the response is two seconds on average (see Section 2.2.5). Model the total average response times as the sum of the averageaccess delay (that is, the delay from Internet router to institution router) and the average Internet delay. For the average access delay, use △/(1-△β), where △ is the average time required to send an object over the access link and β is the arrival rate of objects to the access link.a. Find the total average response time.b. Now suppose a cache is installed in theinstitutional LAN. Suppose the hit rate is 0.6. Find the total response time.Answer:a.T he time to transmit an object of sizeL over a link or rate R is L/R. The average time is the average size of the object divided by R:Δ= (900,000 bits)/(1,500,000 bits/sec) = 0.6 secThe traffic intensity on the link is (1.5requests/sec)(0.6 sec/request) = 0.9. Thus, the average access delay is (0.6 sec)/(1 - 0.9) = 6 seconds. The total average response time is therefore 6 sec + 2 sec = 8 sec.b.T he traffic intensity on the access linkis reduced by 40% since the 40% of the requests are satisfied within the institutional network. Thus the average access delay is (0.6 sec)/[1 –(0.6)(0.9)] = 1.2 seconds. The responsetime is approximately zero if the request is satisfied by the cache (which happens with probability 0.4); the average response time is 1.2 sec + 2 sec = 3.2 sec for cache misses (which happens 60% of the time). So the average response time is (0.4)(0 sec) +(0.6)(3.2 sec) = 1.92 seconds. Thus theaverage response time is reduced from 8 sec to 1.92 sec.16. (P12) What is the difference betweenMAIL FROM: in SMTP and From: in the mail message itself?Answer: The MAIL FROM: in SMTP is a message from the SMTP client that identifies the sender of the mail message to the SMTP server. The From: on the mail message itself is NOT an SMTP message, but rather is just a line in the body of the mail message.17. (P16) Consider distributing a file of F= 5 Gbits to N peers. The server has an upload rate of u= 20 Mbps, and each peers=1 Mbps and an has a download rate of diupload rate of u. For N = 10, 100, and 1,000 and u = 100 Kbps, 250 Kbps, and 500 Kbps, prepare a chart giving the minimumdistribution time for each of the combinations of N and u for both client-server distribution and P2P distribution.Answer: For calculating the minimum distribution time for client-server distribution, we use the following formula:D cs = max {NF/u s , F/d min }Similarly, for calculating the minimum distribution time for P2P distribution, we use the following formula:D P2P = max{F/u s ,F/d min ,NF/( u s +∑u i n i=1 )}Where,F = 5 Gbits = 5 * 1024 Mbitsu s = 20 Mbpsd min = d i = 1 MbpsClient Server:N10 100 1000200 Kbps 10240 51200 512000u 600 Kbps 10240 51200 512000 1Mbps 10240 51200 512000Peer to Peer:N10 100 1000200 Kbps 10240 25904.3 47559.33 U 600 Kbps 10240 13029.6 16899.641 Mbps 10240 10240 10240。

高中数学（必修一）第二章 基本不等式练习题（含答案解析）学校:___________姓名：___________班级：_____________一、解答题 1．已知a b ，比较2a ab +与23ab b -的大小，并证明.2．设a ，b 为正实数，求证：()()()2233338a b a b a b a b +++≥．3．求函数1(3)3y x x x =+>-的最小值.4．（1）把49写成两个正数的积，当这两个正数各取何值时，它们的和最小？（2）把12写成两个正数的和，当这两个正数各取何值时，它们的积最大？5．已知圆C 的圆心在坐标原点，且过点(M . (1)求圆C 的方程；(2)已知点P 是圆C 上的动点，试求点P 到直线40x y +-=的距离的最小值；(3)若直线l 与圆C 相切，且l 与,x y 轴的正半轴分别相交于,A B 两点，求ABC 的面积最小时直线l 的方程.6．已知a ，b R +∈，求证：()114a b a b ⎛⎫++≥ ⎪⎝⎭．7．函数π()2sin()10,||2f x x ωϕωϕ⎛⎫=++>< ⎪⎝⎭图像过点π,13⎛⎫ ⎪⎝⎭，且相邻对称轴间的距离为π2．(1)求,ωϕ的值；(2)已知ABC 的内角A ，B ，C 所对的边分别为a ，b ，c ．若32A f ⎛⎫= ⎪⎝⎭，且2a =，求ABC 面积的最大值．8．小张于年初支出50万元购买一辆大货车，第一年因缴纳各种费用需支出6万元，从第二年起，每年都比上一年增加支出万元，假定该车每年的运输收入均为25万元.小张在该车运输累计收入超过总支出后，考虑将大货车作为二手车出售，若该车在第x 年年底出售，其销售收入为25x -万元（国家规定大货车的报废年限为10年）.（1）大货车运输到第几年年底，该车运输累计收入超过总支出？（2）在第几年年底将大货车出售，能使小张获得的年平均利润最大？ （利润=累积收入+销售收入-总支出）9．高一（3）班的小北为我校设计的冬季运动会会徽《冬日雪花》获得一等奖．他的设计灵感来自三个全等的矩形的折叠拼凑，现要批量生产．其中会徽的六个直角（如图2阴影部分）要利用镀金工艺上色．已知一块矩形材料如图1所示，矩形 ABCD 的周长为4cm ，其中长边 AD 为 x cm ，将BCD △沿BD 向ABD △折叠，BC 折过去后交AD 于点E ．(1)用 x 表示图1中BAE 的面积；(2)已知镀金工艺是2元/2cm ，试求一个会徽的镀金部分所需的最大费用．10．已知ABC 的内角A ，B ，C 的对边分别为a ， b ，c ，A 为锐角，cos cos 3cos b A a B c A +=． (1)求cos A ；(2)若2a =，求ABC 面积的最大值．11．已知(2,5)x ∈-，求(2)(5)y x x =+-的最大值，以及y 取得最大值时x 的值．12．下列结论是否成立？若成立，试说明理由；若不成立，试举出反例．(1)若0ab >，则a b +≥(2)若0ab >2≥；(3)若0ab <，则2b aa b+≤-．13．已知a ，b ，c 均为正实数.(1)求证：a b c ++≥(2)若1a b +=，求证：11119a b ⎛⎫⎛⎫++≥ ⎪⎪⎝⎭⎝⎭.14．已知x ＞2，求函数4()2f x x x =+-的最小值．15．已知抛物线C ：()220y px p =>的焦点为F ，直线l 过F 且与抛物线C 交于A ，B 两点，线段AB 的中点为M ，当3AB p =时，点M 的横坐标为2. (1)求抛物线C 的方程；(2)若直线l 与抛物线C 的准线交于点D ，点D 关于x 轴的对称点为E ，当DME 的面积取最小值时，求直线l 的方程.16．如图，动物园要以墙体为背面，用钢筋网围成四间具有相同面积的矩形虎笼．(1)现有可围36m 长钢筋网的材料，每间虎笼的长、宽各设计为多少时，可使每间虎笼的面积最大？(2)若每间虎笼的面积为220m ，则每间虎笼的长、宽各设计为多少时，可使围成四间虎笼的钢筋网总长最小？17．已知 5<4x ，求函数14145y x x =-+- 的最大值．参考答案：1．见解析【解析】利用作差法比较大小. 【详解】解：223a ab ab b +>-，证明如下：()2222232()a ab ab b a ab b a b +--=-+=-.a b ≠2()0a b ∴-> 223a ab ab b ∴+>-【点睛】本题考查作差法比较两式的大小关系，属于基础题. 2．证明见解析【分析】利用基本不等式计算可得；【详解】解：因为a ，b 为正实数，所以a b +≥222a b ab +≥，332a b +≥=当a b =时取等号，所以()()()223333228a b a b a b ab a b +++≥⨯=，即()()()2233338a b a b a b a b +++≥，当且仅当a b =时取等号；3．5【分析】式子化为1333x x +-+-，再利用基本不等式即可求解. 【详解】因为3x >， 所以30x ->，所以133353y x x =+-+≥=-， 当且仅当133x x -=-即4x =时取等号，此时取得最小值5.4．（1）当7x y ==时，x y +取得最小值14；（2）当6x y ==时，xy 取得最大值36【解析】（1）设0x >，0y >，49xy =，然后利用基本不等式求得x y +的最小值，根据基本不等式等号成立的条件，求得,x y 的值.（2）设0x >，0y >，12x y +=，然后利用基本不等式求得x y ⋅的最大值，根据基本不等式等号成立的条件，求得,x y 的值.【详解】（1）设0x >，0y >，49xy =，由均值不等式，得214x y xy +=， 当且仅当x y =时，取等号．由,49,x y xy =⎧⎨=⎩得7x y ==，即当7x y ==时，x y +取得最小值14．（2）设0x >，0y >，12x y +=，由均值不等式，得22123622x y x y +⎛⎫⎛⎫⋅== ⎪ ⎪⎝⎭⎝⎭．当且仅当x y =时，取等号．由,12,x y x y =⎧⎨+=⎩得6x y ==．即当6x y ==时，xy 取得最大值36.【点睛】本小题主要考查利用基本不等式求最值，属于基础题. 5．(1)224x y +=(2)2(3)0x y +-【分析】（1）利用两点间距离公式可求得半径r ，由此可得圆C 方程； （2）利用点到直线距离公式可求得圆心到直线距离d ，可知最小值为d r -；（3）设():10,0x yl a b a b+=>>，由圆心到直线距离等于半径，结合基本不等式可知当a b ==ABC面积取得最小值，由此可得直线l 方程. （1）由题意知：圆心()0,0C ，半径2r CM ===，∴圆C 的方程为：224x y +=.（2）圆心到直线40x y +-=的距离d r ==，∴点P 到直线40x y +-=的距离最小值为2d r -=.（3）设直线():10,0x yl a b a b+=>>，即0bx ay ab ， 则圆心到直线l 距离2d ==，ab ∴=≥a b ==，解得：8ab ≥， ∴当a b ==ABC 面积取得最小值142ab =，则直线1l =，即0x y +-=. 6．见解析【分析】()11a b a b ⎛⎫++ ⎪⎝⎭展开并运用基本不等式即可得证.【详解】()11224b a a b a b a b ⎛⎫++=++≥+= ⎪⎝⎭，当且仅当b a a b =即a b =时等号成立.【点睛】本题考查基本不等式的应用，属于基础题. 7．(1)2ω=，π3ϕ=；(2)2+【分析】（1）由题干条件得到最小正周期，进而求出2ω=，待定系数法求出π3ϕ=；（2）先由32A f ⎛⎫= ⎪⎝⎭求出π6A =，利用余弦定理，基本不等式求出8bc ≤+. (1)由题意得：()f x 的最小正周期πT =，由于0>ω，故2ππω=，解得：2ω=，又2π32sin()11ϕ++=，所以2ππ,3k k Z ϕ+=∈，即2ππ,3k k Z ϕ=-∈，又π||2ϕ<，所以2πππ,32k k Z <∈-，解得：1766k <<，k Z ∈，故1k =，此时π3ϕ=，综上：2ω=，π3ϕ=； (2)2sin()33π12A f A ⎛⎫= ⎪⎝++=⎭，所以sin()1π3A +=，因为()0,πA ∈，所以ππ4π,333A ⎛⎫+∈ ⎪⎝⎭，则ππ32A +=，解得：π6A =，又2a =，所以由余弦定理得：224cos 2b c A bc +-==，则224b c +=，由基本不等式得：222b c bc +≥，即42bc ≥，解得：8bc ≤+b c =时等号成立，故ABC 面积最大值为1sin 22bc A ≤8．（1）第三年；（2）第5年.【解析】（1）求出第x 年年底，该车运输累计收入与总支出的差，令其大于0，即可得到结论； （2）利用利润＝累计收入+销售收入﹣总支出，可得平均利润，利用基本不等式，可得结论． 【详解】（1）设大货车运输到第x 年年底，该车运输累计收入与总支出的差为y 万元， 则y ＝25x ﹣[6x +x （x ﹣1）]﹣50＝﹣x 2+20x ﹣50（0＜x ≤10，x ∈N ）由﹣x 2+20x ﹣50＞0，可得10﹣＜x ＜，∈2＜10﹣＜3，故从第3年，该车运输累计收入超过总支出； （2）∈利润＝累计收入+销售收入﹣总支出，∈二手车出售后， 小张的年平均利润为(25)y x y x +-=＝19﹣（x +25x）≤19﹣10＝9，当且仅当x ＝5时，等号成立， ∈小张应当在第5年年底将大货车出售，能使小张获得的年平均利润最大． 【点睛】思路点睛：首先构建函数的模型一元二次函数，再解一元二次不等式，再利用基本不等式求最值.9．(1)()223cm 12S x x x ⎡⎤⎛⎫=-+<< ⎪⎢⎥⎝⎭⎣⎦；(2)当 AD cm 时，一个会徽的镀金部分所需的最大费用为(36-元.【分析】（1）设ED a =cm ，根据条件可得222x x a x-+=，然后利用面积公式即得；（2）利用基本不等式即得.（1）因为AD x =cm ，所以()2AB x =-cm ， 设 ED a = cm ，则()AE x a =-cm ，因为AEB C ED '∠=∠，EAB DC E '∠=∠，AB DC '=， 所以Rt Rt BAE DC E '≌△△，所以BE ED a ==cm ， 在Rt BAE △中，由勾股定理得222BA AE BE +=， 即()()2222x x a a -+-=， 解得222x x a x-+=，所以22x AE x a x-=-=， 所以BAE 的面积()()22112232223cm 1222x x x S AB AE x x x x x x --+-⎡⎤⎛⎫=⋅=-⋅==-+<< ⎪⎢⎥⎝⎭⎣⎦． 所以BAE 的面积()223cm 12S x x x ⎡⎤⎛⎫=-+<< ⎪⎢⎥⎝⎭⎣⎦；（2）设一个会徽的镀金费用为y 元，则(26212312336BAE y Sx x ⎡⎤⎛⎫=⋅⋅=⨯-+≤⨯-=- ⎪⎢⎥⎝⎭⎣⎦当且仅当2xx=，12x <<，即x所以当AD cm 时，一个会徽的镀金部分所需的最大费用为(36-元． 10．(1)1cos 3A =；【分析】（1）由正弦定理、两角和的正弦公式求cos A 的值；（2）由同角三角函数间的基本关系求sin A 的值，根据余弦定理和基本不等式求bc 的最大值，最后根据三角形的面积公式求ABC 面积的最大值即可． (1)因为cos cos 3cos b A a B c A +=，由正弦定理得sin cos cos sin 3sin cos B A B A C A +=， 所以()sin 3sin cos A B C A +=，所以sin 3sin cos C C A =． 在ABC 中，sin 0C ≠， 所以1cos 3A =；(2)由（1）知1cos 3A =，由22sin cos 1A A +=，A 为锐角，得sin A =由余弦定理可知222123b c a bc +-=，因为2a =， 所以2233122b c bc +-=， 所以22212336bc b c bc +=+≥，所以3bc ≤，当且仅当b c ==所以1sin 2ABC S bc A =△所以ABC 11．当32x =时，y 取得最大值494【解析】根据基本不等式，求得y 的最大值，根据基本不等式等号成立的条件，求得此时x 的值.【详解】∈(2,5)x ∈-，∈20,50x x +>->，∈22549(2)(5)24x x y x x ++-⎛⎫=+-=⎪⎝⎭． 当且仅当25x x +=-，即32x =时，取等号．即当32x =时，y 取得最大值494．【点睛】本小题主要考查利用基本不等式求最值，属于基础题. 12．(1)不成立，理由见解析； (2)成立，理由见解析； (3)成立，理由见解析；【分析】取特殊值判断（1），由均值不等式判断（2）（3）. （1）取1,2a b =-=-满足0ab >，此时a b +≥不成立； （2）0ab >，0,0a bb a∴>>，2，当a b =时等号成立. （3）0ab <，0,0b aa b∴<<，2b a b a a b a b ⎡⎤⎛⎫⎛⎫∴+=--+-≤-=- ⎪ ⎪⎢⎥⎝⎭⎝⎭⎣⎦，当a b =-时等号成立. 13．(1)证明见解析 (2)证明见解析【分析】（1）利用基本不等式证明即可；（2）由112111⎛⎫⎛⎫++=+ ⎪⎪⎝⎭⎝⎭a b ab 利用基本不等式求最值即可.（1）因为a ，b ，c 都是正数，所以 ()()()(1122++=+++++≥⎡⎤⎣⎦a b c a b b c a c=，当且仅当a b c ==时，等号成立，所以a b c ++≥ （2）211111122211111119142a b a b a b ab ab ab ab a b +⎛⎫⎛⎫++=+++=++=+≥+=+= ⎪⎪⎝⎭⎝⎭+⎛⎫⎪⎝⎭， 当且仅当12a b ==时等号成立. ∈11119a b ⎛⎫⎛⎫++≥ ⎪⎪⎝⎭⎝⎭. 14．6【解析】利用基本不等式可求函数的最小值.【详解】解：∈2x >，∈20x ->，故44()222622f x x x x x =+=-++≥=--， 当且仅当4x =时等号成立，故()f x 的最小值为6.15．(1)24y x =(2)1x y =±+【分析】（1）设()()1122,,,A x y B x y ，根据焦点弦的性质得到12||AB x x p =++，从而求出p ，即可得解； （2）设:1l x ty =+，联立直线与抛物线，消元、利用韦达定理得到M y ，从而得到M x ，则()1||12DEM M S DE x =⋅+最后利用基本不等式求出最小值，即可得解； （1）解：设()()1122,,,A x y B x y ，由题知12||43AB x x p p p =++=+=时，2p =，故抛物线方程为24y x =；（2）解：设:1l x ty =+，联立抛物线方程得2440y ty --=，∈1222M y y y t +==，2121M M x ty t =+=+，而21,D t ⎛⎫-- ⎪⎝⎭，21,E t ⎛⎫- ⎪⎝⎭， 所以()()21141||1224||822||||DEM M S DE x t t t t ⎛⎫=⋅+=⋅⋅+=+≥ ⎪⎝⎭， 当且仅当||1t =时等号成立，故直线l 的方程为1x y =±+．16．(1)长为9m 2，宽为18m 5(2)长为5m ，宽为4m【分析】（1）设每间老虎笼的长为m x ，宽为m y ，则每间老虎笼的面积为S xy =，可得出4536x y +=，利用基本不等式可求得S 的最大值，利用等号成立的条件求出x 、y 的值，即可得出结论；（2）设每间老虎笼的长为m x ，宽为m y ，则20xy =，利用基本不等式可求得钢筋网总长45x y +的最小值，利用等号成立的条件求出x 、y 的值，即可得出结论.（1）解：设每间老虎笼的长为m x ，宽为m y ，则每间老虎笼的面积为S xy =，由已知可得4536x y +=，由基本不等式可得()2211458145m 202025x y S xy x y +⎛⎫==⋅⋅≤⨯= ⎪⎝⎭， 当且仅当454536x y x y =⎧⎨+=⎩，即当92185x y ⎧=⎪⎪⎨⎪=⎪⎩时，等号成立， 因此，每间虎笼的长为9m 2，宽为18m 5时，可使得每间虎笼的面积最大. （2）解：设每间老虎笼的长为m x ，宽为m y ，则20xy =，钢筋网总长为()4540m x y +≥=，当且仅当4520x y xy =⎧⎨=⎩，即当54x y =⎧⎨=⎩时，等号成立， 因此，每间虎笼的长为5m ，宽为4m 时，可使围成四间虎笼的钢筋网总长最小. 17．2 【分析】将14145y x x =-+-变形为[()1]54454y x x=--++-，利用基本不等式即可求得答案. 【详解】根据题意，函数()114545444554y x x x x ⎡⎤=-++=--++⎢⎥--⎣⎦ ， 又由54x <，则540x ->，则()154254x x -+≥-， 当且仅当15454x x -=-时，即1x =时取等号， 则1[(54)]424254y x x=--++≤-+=-, 故函数14145y x x =-+-的最大值为2.。