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a r X i v :h e p -l a t /0409150v 1 27 S e p 20043-d lattice SU(3)free energy to four loops∗MIT-CTP-3541;UPRF-2004-18F.Di Renzo a ,A.Mantovi a ,V.Miccio a ,Y.Schroder b ,C.Torrero a aDipartimento di Fisica,Universit`a degli Studi di Parma,and INFN,Gruppo Collegato di Parma,ItaliabCenter for Theoretical Physics,MIT,Cambridge,MA,USAWe report on the perturbative computation of the 3d lattice Yang-Mills free energy to four loops by means of Numerical Stochastic Perturbation Theory.The known first and second orders have been correctly reproduced;the third and fourth order coefficients are new results and the known logarithmic IR divergence in the fourth order has been correctly identified.Progress is being made in switching to the gluon mass IR regularization and the related inclusion of the Faddeev-Popov determinant.1.IntroductionThe main goal of finite temperature QCD is to characterize the deconfinement transition bet-ween the low temperature regime,ruled by con-finement,and the high temperature regime ruled by asymptotic freedom.In this respect the free energy density,i.e.the pressure ,is a sort of ‘theoretical laboratory’in which to study such a transition,and,eventually,a potential candidate observable in heavy ion collisions.Evidently,the large T limit of the pressure is that of an ideal gas of non-interacting particles,p ∝T 4;unfortu-nately,lattice simulations across the whole tem-perature range are not feasible and our efforts and understanding are confined to the opposite ends of the temperature range:namely,the low temperature regime (up to some 4÷5times the transition temperature T c ∼200MeV)which is accessible to actual computational resources,and the high temperature regime at which the pres-sure exhibits a purely perturbative behaviour.Dimensional reduction [1]has been applied [2]to fill the gap:4d QCD is matched to 3d SU (3)Yang-Mills (YM)coupled to a Higgs field in the adjoint representation;such a theory can then be matched to 3d SU (3)YM,which captures the ultrasoft degrees of freedom;both these reduc-tions have been successfully performed in the∗Presentedby Andrea Mantovischeme.Finally,3d YM has to be treated non-perturbatively,the only effective means being the lattice.Indeed lattice computations can be con-sistently incorporated into the dimensional reduc-tion strategy:the theory is superrenormalizable and all divergences can be computed perturba-tively;this allows a clean matching of the schemes in the continuum.It is well known that comput-ing at high orders in LPT is quite hard a task;in this respect the approach of Numerical Stochastic Perturbation Theory (NSPT)[3]is a very efficient one.Let f ≡−V ln Z be the free energy associated with the Wilson action S W =β0 P (1−ΠP ),β0=2N c /(a 4−d g 20)being the dimensionless bare lattice inverse coupling in d dimensions.To com-pute the free energy one can revert to the com-putation of the plaquette,being1−ΠP =2a d∂β01β0+c 2(N c ,d )β30+c 4(N c ,d )616g 2(3)12−a−1c3648g60+O(a).so that thefirst four coefficients uniquely deter-mine the matching to the continuum;c1and c2 being long known,c3and c4were the aims of the computation.The presence has already been established[4]of a logarithmic IR divergence at fourth loop order;it was our aim to recover the scheme independent coefficient of such a loga-rithm:the tilde on c4reminds us that the IR divergence has to be isolated and subtracted by means of a definite IR regulator,which we choose to be the finite lattice size L inherent to any lat-tice simulation.putational setupWe have inserted the expansion Uµ(n)=1+ 8i=1β−i3subtracted c4.4.PerspectivesA definite IR regulator enables one to isolate afinite part out of the4loops contribution to the plaquette;we have been employing the finite volume inherent to any computer simulation.A mass is a commonly used regulator in continuum computations;the coupling itself(dimensionful in 3d YM)is quite a natural IR regulator for non-perturbative lattice simulations.Despite the fact that the coefficient of the logarithmic divergence is universal it is clear that definite coefficients re-late different regularizations.We are interested in the perturbative matching between the lattice and a continuum perturbative scheme:in both approaches the same gluon mass IR regulator can be employed so that the same mismatch shows up with respect to data coming from nonpertur-bative computer simulations with the coupling as IR regulator.Employing massive gluons is the na-tural extension of our computational programme. Inserting a mass into the gluon propagator amounts to switching to the Lie algebra for the definition of the new lattice action and this we perform at a definite step of the computation.It is well known that the Lie algebra YM action is made out of a gaugefixing term and the ghost action resulting from the FP determinant.Our NSPT strategy entails the construction of the Eu-ler scheme solution U′=exp(−ǫF)U to Eq.(4); the quantity F belongs to the Lie algebra and our algorithm entails going back and forth from the group to the algebra:at that stage we include the new contribution to the action.The strategy for a determinant is well established[6,7]and is employed by the Parma group in the unquenched NSPT approach to lattice QCD[8];the determi-nant is simulated without introducing fermion or ghostfields and the inversions of the nonlocal ma-trix is performed perturbatively via an efficient FFT by a back and forth procedure from Fourier space:the zeroth order of any propagator is di-agonal in momentum space and the construction of successive orders does not require any matrix inversion.The implementation of such a strategy is in progress;signals of the stabilization of gauge dependent quantities as well as of the invariance of gauge invariant quantities make us confident about the reliability of the whole procedure. REFERENCES1. E.Braaten and A.Nieto,Phys.Rev.D53,3421(1996);2.K.Kajantie,ine,K.Rummukainen andY.Schroder,Phys.Rev.Lett.86,10(2001);3. F.Di Renzo,E.Onofri,G.Marchesini andP.Marenzoni,Four loop result in SU(3)lat-tice gauge theory by a stochastic method:lat-tice correction to the condensate,Nucl.Phys.B426(1994)675,[hep-lat9405019].4.K.Kajantie,ine,K.Rummukainen andY.Schroder,Phys.Rev.sss D67,105008 (2003);5. F.Di Renzo, A.Mantovi,V.Miccio andY.Schroder,J.High Energy Phys.05(2004) 006;6.G.G.Batrouni,G.R.Katz,A.S.Kronfeld,G.P.Lepage,B.Svetitski and K.G.Wilson,Phys.Rev.D32(1985)2736.7.G.Burgio,F.Di Renzo,G.Marchesini,E.Onofri,M.Pepe,L.Scorzato,Nucl.Phys.B (Proc.Suppl.)73(1999)853.8. F.Di Renzo,L.Scorzato,Nucl.Phys.B(Proc.Suppl.)94(2001)567.。
高三英语微积分基础单选题60题(答案解析)1.The derivative of a constant is_____.A.0B.1C.the constant itselfD.undefined答案：A。
解析：任何常数的导数都是0。
选项B，1 不是常数的导数。
选项C，常数本身不是常数的导数。
选项D，常数的导数不是未定义。
2.The integral of a constant times a function is equal to_____.A.the constant times the integral of the functionB.the integral of the function plus the constantC.the function times the constantD.the constant divided by the integral of the function答案：A。
解析：常数乘以函数的积分等于常数乘以函数的积分。
选项B，是函数积分加常数不是常数乘以函数积分的结果。
选项C，函数乘以常数不是积分的结果。
选项D，常数除以函数积分错误。
3.The derivative of a sum of two functions is_____.A.the sum of the derivatives of the two functionsB.the product of the derivatives of the two functionsC.the quotient of the derivatives of the two functionsD.the negative of the sum of the derivatives of the two functions答案：A。
解析：两个函数之和的导数等于两个函数导数之和。
选项B，不是乘积。
选项C，不是商。
初一英语数学方程求解步骤单选题40道1. Solve the equation: 2x + 5 = 11. What is the first step?A. Add 5 to both sidesB. Subtract 5 from both sidesC. Divide both sides by 2D. Multiply both sides by 2答案：B。
本题考查一元一次方程求解的第一步。
原方程为2x + 5 = 11，第一步应是两边同时减去5，得到2x = 6。
选项A 是错误的，加5 会使方程变得更复杂。
选项C 是第二步，得到2x = 6 之后两边同时除以2 求出x。
选项D 也是错误的操作。
2. To solve the equation 3x - 7 = 8, what should you do first?A. Add 7 to both sidesB. Subtract 7 from both sidesC. Multiply both sides by 3D. Divide both sides by 3答案：A。
此方程3x - 7 = 8，求解第一步应是两边同时加7，得到3x = 15。
选项B 会使方程错误。
选项C 和D 都不是第一步操作。
3. When solving the equation 4x + 9 = 21, which is the correct first step?A. Subtract 9 from both sidesB. Add 9 to both sidesC. Divide both sides by 4D. Multiply both sides by 4答案：A。
对于方程4x + 9 = 21，首先应两边同时减去9，得到4x = 12。
选项B 操作错误。
选项C 是第二步。
选项D 不是正确的第一步。
4. Solve: 5x - 12 = 18. What is the initial step?A. Add 12 to both sidesB. Subtract 12 from both sidesC. Multiply both sides by 5D. Divide both sides by 5答案：A。
I. Multiple-choice Questions (40 points, 1 point each)1. The number of vertices in a tree with n edges is:A. n+1B. nC. n-1D. n-2Answer: C2. The binary operation defined on the set of natural numbers is given by ab = a+b+1. Which of the following is not a binary operation?A. 1B. 2C. 3D. 4Answer: B3. The truth value of the statement "p ∨ q" is:A. TrueB. FalseC. UnknownD. Cannot be determinedAnswer: A4. The function f: R → R defined by f(x) = 2x + 3 is:A. One-to-oneB. OntoC. Both one-to-one and ontoD. Neither one-to-one nor ontoAnswer: A5. The number of distinct permutations of the letters in the word "MATH" is:A. 3!B. 4!C. 5!D. 6!Answer: BII. Fill in the blanks (20 points, 2 points each)6. A _______ is a finite set of symbols.Answer: alphabet7. The binary operation + on the set of integers is commutative if a + b = b + a for all _______ a and b.Answer: integers8. A _______ is a subset of a set that contains all the elements of the set.Answer: superset9. The binary operation on the set of natural numbers is associative if(a b) c = a (b c) for all _______ a, b, and c.Answer: natural numbers10. The _______ of a graph is the number of edges in the graph.Answer: number of edgesIII. Short Answer Questions (30 points, 3 points each)11. Prove that the set of even numbers is not a subset of the set of odd numbers.Proof: Let A be the set of even numbers and B be the set of odd numbers. We can show that A is not a subset of B by providing a counterexample. For example, 2 is an even number and belongs to A, but it does notbelong to B because it is not an odd number. Therefore, A is not asubset of B.12. Determine whether the function f: R → R defined by f(x) = x^2 isone-to-one or onto.Solution: To determine if the function is one-to-one, we need to checkif f(a) = f(b) implies a = b for all a and b in R. If we take a = 2 andb = -2, we have f(2) = 2^2 = 4 and f(-2) = (-2)^2 = 4. Since f(2) = f(-2) but 2 ≠ -2, the function is not one-to-one. To determine if thefunction is onto, we need to check if for every y in R, there exists anx in R such that f(x) = y. Since the square of any real number is non-negative, the function is onto.13. Find the sum of the first 10 terms of the arithmetic sequence with a first term of 3 and a common difference of 2.Solution: The formula for the sum of the first n terms of an arithmetic sequence is given by Sn = n/2 (2a + (n-1)d), where a is the first term and d is the common difference. Substituting a = 3, d = 2, and n = 10, we have Sn = 10/2 (23 + (10-1)2) = 5 (6 + 18) = 5 24 = 120.IV. Long Answer Questions (10 points)14. (5 points) Prove that the product of two even numbers is an even number.Proof: Let a and b be even numbers. Then, there exist integers m and n such that a = 2m and b = 2n. The product of a and b is given by ab =(2m)(2n) = 4mn. Since 4mn is divisible by 2, the product of two even numbers is an even number.15. (5 points) Solve the following system of linear equations using Cramer's rule:2x + 3y = 74x - y = 1Solution: The determinant of the coefficient matrix is given by D = |23| = 21 - 34 = -10. The determinant of the matrix obtained by replacing the first column with the constants is given by D1 = |7 3| = 71 - 34 = -5. The determinant of the matrix obtained by replacing the second column with the constants is given by D2 = |2 1| = 21 - 31 = -1. Therefore, x = D1/D = -5/-10 = 1/2 and y = D2/D = -1/-10 = 1/10.。
2024年浙江省宁波市英语初三上学期期中复习试卷及答案指导一、听力部分（本大题有20小题，每小题1分，共20分）1、What does the girl want to do?a) Play basketball.b) Watch TV.c) Go shopping.Recording: (Boy) “Hey, do you want to play some basketball after school?” (Girl) “Not really, I was thinking about going shopping instead.”Answer: c) Go shopping.Explanation: The girl responds to the boy’s suggestion by saying she’d rather go shopping than play basketball, making ‘Go shopping’ the correct answer.2、Where are the speakers?a) At a restaurant.b) In a library.c) At a movie theater.Recording: (Woman) “Excuse me, could you please keep it down? We’re trying to study over here.” (Man) “Oh, sorry, we didn’t realize it was so quiet aroundhere.”Answer: b) In a library.Explanation: The woman asks the man to keep his voice down because people are studying, indicating they are in a quiet place such as a library, not a restaurant or movie theater where noise levels are typically higher.3、What is the weather like in the city where the woman is visiting?A. Sunny.B. Rainy.C. Cloudy.D. Windy.Answer: BExplanation: In the conversation, the woman mentions that it’s been raining all day in the city she is visiting, so the correct answer is “Rainy.”4、Why does the man think it’s important to learn a second language?A. To improve his grades.B. To travel abroad easily.C. To understand different cultures.D. To find a good job.Answer: CExplanation: The man in the conversation states that learning a second language helps him understand and appreciate different cultures, making option C the correct answer.5、What does the boy want to drink?A. Coffee.B. Tea.Audio: (The audio clip would play here.)Answer: B. Tea.Explanation: In the conversation, the boy says, “I’m not a fan of caffeine;I think I’ll have some tea instead.” This clearly indicates that he prefers tea over coffee because he doesn’t like caffeine.6、Where does the dialogue take place?A. At a restaurant.B. At a clothing store.Audio: (The audio clip would play here.)Answer: A. At a restaurant.Explanation: The dialogue includes phrases such as “Can I get the check, please?” and “I’ll have the steak,” which suggest that the conversation is happening in a restaurant setting rather than a clothing store.7、Question: What is the weather like today according to the conversation?A) It’s sunny.B) It’s cloudy.C) It’s raining.D) It’s windy.Answer: BExplanation: The conversation mentions that it’s a bit cloudy today, so the correct answer is B) It’s cl oudy.8、Question: How does the man feel about the movie he just watched?A) He enjoyed it very much.B) He thinks it’s boring.C) He didn’t like it at all.D) He needs to see it again.Answer: AExplanation: The man says, “It was an amazing movie! I really enjoyed it,” which indicates that he enjoyed it very much, so the correct answer is A) He enjoyed it very much.9、(Listen to a dialogue about a school trip planning)Question: What activity does the boy suggest adding to the school trip itinerary?A. Visiting a museumB. Going to the beachC. Watching a play at the local theaterAnswer: C. Watching a play at the local theaterExplanation: In the dialogue, the boy mentions that he thinks it would be enriching for everyone to experience the culture of the destination by watchinga local performance at the theater, which isn’t part of the initial plan.10、(Follow-up question based on the same dialogue)Question: How does the girl respond to the boy’s suggestion?A. She agrees but suggests doing it another day.B. She disagrees because she thinks it will be boring.C. She likes the idea and proposes they combine it with visiting a museum.Answer: A. She agrees but suggests doing it another day.Explanation: The girl responds positively to the idea but mentions that the current schedule is already tight, so she proposes postponing the theater visit to another less busy day during the trip.11.You are listening to a conversation between a student and a teacher ina classroom.Student: “Mr.Smith, can you explain how the quadratic formula works again?”Teacher: “Certainly, the quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0. It is derived from the completing the square method. The formula is x = (-b ± √(b^2 - 4ac)) / (2a). Do you understand that?”Question: What does the teacher say about the quadratic formula?A) It’s a method to find the roots of any equation.B) It’s a formula to solve quadratic equations.C) It’s a technique for completing the square.D) It’s a shortcut for fin ding the sum of roots.Answer: B) It’s a formula to solve quadratic equations.Explanation: The teacher explicitly states that the quadratic formula isused to solve quadratic equations.12.You are listening to a news report about a new environmental initiative.News Anchor: “In a bid to combat climate change, the city government has announced the introduction of a new recycling program. This program will require all residents to separate their waste into recyclable and non-recyclable materials. The goal is to reduce the amount of waste sent to landfills by 50% within the next five years. How do you think this initiative will impact the environment?”Question: What is the main goal of the new recycling program?A) To encourage people to use more plastic products.B) To increase the number of parks in the city.C) To reduce the amount of waste in landfills.D) To provide jobs for recycling workers.Answer: C) To reduce the amount of waste in landfills.Explanation: The news report clearly states that the aim of the recycling program is to reduce waste sent to landfills by 50%.13.W: Have you finished your science homework yet?M: No, I haven’t. I’m going to the library to get some books on phy sics.Question: Where is the boy going?A) To the classroomB) To the libraryC) To the science labD) To the teacher’s officeAnswer: B) To the libraryExplanation: The boy says he is going to the library to get some books on physics, which indicates that he is going to the library for his homework.14.W: How was your weekend, Tom?M: It was quite busy. I had to help my father with the gardening. But on Sunday, we went to the beach and had a great time.Question: What did the boy do on Sunday?A) He went to the beachB) He helped his father with gardeningC) He studied for examsD) He went to the libraryAnswer: A) He went to the beachExplanation: The boy mentions that on Sunday, he went to the beach and had a great time, which is the correct answer. The other activities were mentioned but not on Sunday.15.You are listening to a conversation between two friends, Alice and Bob.A: Hey Bob, how was your weekend?B: It was pretty good, Alice. I went hiking with my friends on Sunday. We climbed up a mountain and had a great view at the top.A: That sounds amazing! Do you often go hiking?B: Yes, I love outdoor activities. We usually go hiking once a month.Question: How often does Bob go hiking?Answer: B) Once a month.Explanation: Bob mentions that he and his friends usually go hiking oncea month, so the correct answer is B) Once a month.16.You are listening to a news report about a new study.News Anchor: According to a recent study, eating a healthy diet can help improve cognitive function in older adults. Researchers found that individuals who consumed a balanced diet rich in fruits, vegetables, and whole grains showed better memory and concentration.Question: What did the study find about a healthy diet?Answer: A) It improves memory and concentration in older adults.Explanation: The news report directly states that the study found that a healthy diet, rich in fruits, vegetables, and whole grains, can help improve memory and concentration in older adults. Therefore, the correct answer is A) It improves memory and concentration in older adults.17.What are the speakers mainly discussing?A. The weather forecast for the next weekB. The importance of wearing a coat in cold weatherC. Their favorite winter activitiesD. The need for an extra blanket in the dormitoryAnswer: CExplanation: The speakers are discussing their favorite winter activities,which is indicated by the words “fun” and “exciting” in the conversation.18.Why does the man decide not to go for a walk in the park?A. He is worried about his health.B. The weather is too cold.C. He doesn’t have enough time.D. He is not interested in walking.Answer: BExplanation: The man mentions that it is freezing outside, which implies that the weather is too cold for him to go for a walk in the park.19.You hear a conversation between two students discussing their favorite subject.A)Math is their favorite subject because it’s challenging.B)English is their favorite subject because it’s interesting.C)Science is their favorite subject because it’s engaging.Answer: BExplanation: The conversation indicates that the students find English engaging and enjoyable, making it their favorite subject.20.You hear a weather forecast for a city in China.A)The weather will be sunny and warm with a high of 28°C.B)The weather will be cloudy with a chance of rain and a high of 22°C.C)The weather will be windy with a low of 10°C and a high of 15°C.Answer: BExplanation: The weather forecast mentions that it will be cloudy with a chance of rain and ahigher temperature during the day, which corresponds to option B.二、阅读理解（30分）Reading Passage:The following passage is about the importance of exercise for students.Exercise is not only beneficial for our physical health, but it also has a significant impact on our mental well-being. In recent years, more and more studies have shown that regular exercise can improve students’ academic performance, reduce stress, and increase their overall happiness.One study conducted by the University of California, Berkeley, found that students who engaged in moderate exercise for at least 30 minutes a day showed better cognitive function and memory retention compared to those who did not exercise. The study also suggested that physical activity can help students focus better and reduce the symptoms of ADHD.Moreover, exercise is a great way to relieve stress. A study published in the Journal of Physical Activity and Health revealed that students who exercised regularly had lower levels of cortisol, the stress hormone, in their bloodstream. This, in turn, helped them to handle stress more effectively and improve their emotional well-being.In addition to its mental health benefits, exercise also contributes to better academic performance. A survey conducted by the American Council on Exercise showed that students who participated in sports or physical activitiesscored higher on standardized tests than their non-athletic peers. This is because exercise improves blood circulation, which in turn enhances the delivery of oxygen and nutrients to the brain, leading to better cognitive function.With all these benefits, it is essential for students to incorporate regular exercise into their daily routines. However, many students struggle to find the time to exercise. To help them, schools can offer physical education classes, organize extracurricular sports activities, and encourage students to participate in community sports programs.Questions:1.According to the passage, what is one of the benefits of exercise for students’ academic performance?A) Improved physical healthB) Enhanced cognitive function and memory retentionC) Lower stress levelsD) Better emotional well-being2.What is the main idea of the second paragraph?A) Exercise is a great way to relieve stress.B) Regular exercise can improve students’ academic performance.C) Students with ADHD can benefit from exercise.D) Physical activity can help students focus better.3.Which of the following statements is NOT supported by the passage?A) Exercise can help students with ADHD.B) Regular exercise can improve students’ academic performance.C) Students who do not exercise have lower levels of cortisol.D) Physical activity enhances the delivery of oxygen and nutrients to the brain.Answers:1.B) Enhanced cognitive function and memory retention2.A) Exercise is a great way to relieve stress.3.C) Students who do not exercise have lower levels of cortisol.三、完型填空（15分）Section III: Reading ComprehensionDirections: There are 10 blanks in the following passage. For each blank, there are four choices marked A, B, C, and D. Choose the one that best fits into the passage. Then mark the corresponding letter on Answer Sheet 2 with a single line through the centre.The following is a story about a young girl named Lily who discovered the beauty of nature.One sunny morning, Lily decided to take a walk in the nearby forest. She had always been fascinated by nature and loved to explore the wonders of the world around her. As she walked deeper into the forest, she noticed a small, clear stream that flowed gently over the rocks.1.The story begins by describing a__________morning.A. cloudyB. sunnyC. rainyD. windyLily followed the stream and came across a beautiful flower that she had never seen before. It had petals of various colors, and its scent was so sweet that it made her stop in her tracks.2.Lily felt__________when she saw the flower.A. boredB. excitedC. scaredD. hungryContinuing her journey, Lily came upon a small, serene lake. The water was so clear that she could see the fish swimming below. She sat by the lake and took in the peacefulness of the scene.3.The lake was described as __________.A. dirtyB. sereneC. loudD. darkAs the afternoon sun began to set, Lily realized she had walked quite far. She needed to find her way back home. However, she was not worried, as she had learned the basics of navigation from her father.4.Lily was not worried about finding her way back because she __________.A. had a mapB. had a phoneC. knew the wayD. was too tiredAs she walked back, Lily reflected on her adventure. She realized that nature had a way of teaching her valuable lessons about life and the world around her.5.Lily’s adventure taught her about __________.A. the importance of technologyB. the beauty of natureC. the dangers of the forestD. the need for exerciseAnswer Key:1.B2.B3.B4.C5.B四、语法填空题（本大题有10小题，每小题1分，共10分）1、The students are going to the library_____________we_____________need some new books.A. because / areB. since / doC. if / willD. when / willAnswer: BExplanation: The correct answer is “since / do” because “since” is used to introduce a reason or cause, and “do” is used to agree with the present need expressed in the second clause. The sentence means “The students are going to the library because we need some new books.”2、In the_____________days, people used to travel by horseback, which was_____________than traveling by car.A. past, much slowerB. old, fasterC. present, as fastD. future, much fasterAnswer: AExplanation: The correct answer is “past, much slower” because “past” refers to a time that is in the past, and “much slower” correctly contrasts the speed of travel by horseback in the past with the speed of travel by car, which is generally faster.3、In the___________of the park, there is a beautiful lake that reflects the___________of the sky.A. middle, colorsB. center, blueC. middle, colors of the skyD. center, blue skyAnswer: CExplanation: The correct phrase to describe the location within the park is “in the middle of the park.” Additionally, the sentence is referring to the colors of the sky reflected in the lake, so “colors of the sky” is the correct choice.4、She___________her homework yesterday, but now she has finished it.A. didn’t doB. ha sn’t doneC. doesn’t doD. didn’t have doneAnswer: AExplanation: The sentence describes an action that happened in the past, which is a completed action. The correct past simple negative form to use is “didn’t do,” indicating that she did not do her home work yesterday. The other options are incorrect because they suggest actions in the present perfect or future perfect, which do not match the context of the sentence.5.The teacher,____________was very strict with us, retired last month.A. whoseB. whoC. thatD. whichAnswer: BExplanation: The correct answer is “who” because it is used as a relative pronoun to refer to “The teacher” in the non-restrictive relative clause “who was very strict with us.”6.It was not until yesterday that he____________the truth of the matter.A. realizedB. had realizedC. was realizingD. have realizedAnswer: AExplanation: The correct answer is “realized” because the sentence is using the structure “It was not until + past time + that + past simple,” indicating that the realization of the truth occurred in the past.7.In the 7 morning, I 8 up early and decided to go for a run in the park.A. last / gotB. first / gotC. next / gotD. early / gotAnswer: B. first / gotExplanation: The correct answer is “first” because it indicates the first morning in a sequence, which is the context given. “Got up” is the correct phrasal verb to use here, meaning to wake up.8.She 9 her homework last night and now she feels 10 to take a break.A. finished / tiredB. finishes / tiredC. finishing / tiredD. finished / tiredlyAnswer: A. finished / tiredExplanation: The correct answer is “finished” because the sentence is referring to an action that was completed in the past. “Tired” is the correct adjective to describe how she f eels after finishing her homework. “Finishes” would be incorrect because it is in the present tense, not past tense. “Finishing” would also be incorrect as it is the present participle, not the past simple form. “Tiredly” is an adverb and does not fit the context of describing her feeling after finishing her homework.9.I______(be) reading a book when you______(call) me yesterday.A. was, calledB. were, calledC. was, were calledD. were, was calledAnswer: AExplanation: In the first part of the sentence, “I” is the subject, and “was” is the past continuous tense of “be,” indicating that the action was in progress at a specific past time. In the second part, “you” is the subject, and “called” is the past simple tense of “call,” indicating that the action w as completed at a specific past time. Therefore, the correct answer is “was, called.”10.If I______(have) more time, I______(do) my homework yesterday.A. had, would have doneB. have, would doC. had, would doD. have, would have doneAnswer: AExplanation: This sentence is a conditional sentence with a past perfect tense in the first clause and a past perfect tense in the second clause. The first clause uses “If I had more time” to express a hypothetical situation in the past, and the second clause uses “I would have done” to express the result of that hypothetical situation. Therefore, the correct answer is “had, would have done.”五、简答题（本大题有5小题，每小题2分，共10分）1、What are the main differences between a novel and a play in terms of narrative structure and audience engagement?答案：1.Narrative Structure:a.A novel typically follows a linear narrative, with a clear beginning,middle, and end, allowing readers to follow the story sequentially.b.A play, on the other hand, uses a non-linear structure, often with scenesand acts that can be experienced out of chronological order, depending on the performance.2.Audience Engagement:a.In a novel, the audience engages with the story through reading andimagination, creating their own mental images of the characters andsettings.b.In a play, the audience engages through live performance, experiencingthe story through the actors’ performan ces and the physical setting of the stage.解析：This question requires an understanding of the basic differences between a novel and a play. The answer outlines the distinct narrative structures and the ways in which audiences engage with each form of literature. It highlights the linear nature of novels versus the more flexible and non-linear structure of plays, as well as the different methods of audience engagement, such as imagination in novels and live performance in plays.3、What are the main differences between the past simple tense and the present perfect tense in English grammar?答案：1.The past simple tense is used to describe actions or states that happened at a specific time in the past, without any reference to the present.2.The present perfect tense is used to describe actions or states that have happened at an unspecified time in the past and have a present result.解析：This question tests the understanding of the past simple and present perfecttenses in English. The past simple tense is typically used for past events without a clear connection to the present, while the present perfect tense is used to show a connection between the past and the present, often emphasizing the result or outcome of the action. For example, “I visited the museum yest erday” (past simple) vs.“I have visited the museum before” (present perfect).4、How do you form the future continuous tense in English?答案：1.The future continuous tense is formed by combining the word “will” with the base form of the verb plus “ing.”2.It is used to express actions or states that will be in progress at a specific time in the future.解析：This question focuses on the formation of the future continuous tense, which is used to describe actions that will be ongoing at a particular time in the future. It is important to note that the future continuous tense is not as common as the simple present tense for future actions. For example, “I will be reading a book at 6 PM tomorrow” or “They will be watching a movie tonight.” This tense emphasizes the ongoing nature of the action at the specified future time.5、After reading the passage about environmental protection, list three ways in which individuals can contribute to reducing pollution in their daily lives. Then, explain why you think one of these methods is the most effective.Answer:•Reduce, Reuse, and Recycle (the 3Rs)•Use public transportation, carpool, bike, or walk instead of driving alone •Conserve water and energy at homeExplanation for the most effective method:I believe that using public transportation, carpooling, biking, or walking instead of driving alone is the most effective way to reduce pollution. This is because the transportation sector is a major source of greenhouse gas emissions, contributing significantly to air pollution and climate change. By choosing more sustainable transport options, we can drastically cut down on the number of vehicles on the road, leading to less carbon dioxide and other pollutants being released into the atmosphere. Additionally, it helps to reduce traffic congestion, which not only lowers emissions but also improves the overall quality of life in urban areas by decreasing noise and travel time.解析：本题考查学生对环保知识的理解以及他们将这些知识应用到日常生活中的能力。
2.12.比:ratio 比例:proportion 利率:interest rate 速率:speed 除:divide 除法:division 商:quotient 同类量:like quantity 项:term 线段:line segment 角:angle 长度:length 宽:width高度:height 维数:dimension 单位:unit 分数:fraction 百分数:percentage3.(1)一条线段和一个角的比没有意义,他们不是相同类型的量.(2)比较式通过说明一个量是另一个量的多少倍做出的,并且这两个量必须依据相同的单位.(5)为了解一个方程,我们必须移项,直到未知项独自处在方程的一边,这样就可以使它等于另一边的某量.4.(1)Measuring the length of a desk, is actually comparing the length of the desk to that of a ruler.(3)Ratio is different from the measurement, it has no units. The ratio of the length and the width of the same book does not vary when the measurement unit changes.(5)60 percent of students in a school are female students, which mean that 60 students out of every 100 students are female students.2.22.初等几何:elementary geometry 三角学:trigonometry 余弦定理:Law of cosines 勾股定理/毕达哥拉斯定理:Gou-Gu theorem/Pythagoras theorem 角:angle 锐角:acute angle 直角:right angle 同终边的角:conterminal angles 仰角:angle of elevation 俯角:angle of depression 全等:congruence 夹角:included angle 三角形:triangle 三角函数:trigonometric function直角边:leg 斜边:hypotenuse 对边:opposite side 临边:adjacent side 始边:initial side 解三角形:solve a triangle 互相依赖:mutually dependent 表示成:be denoted as 定义为:be defined as3.(1)Trigonometric function of the acute angle shows the mutually dependent relations between each sides and acute angle of the right triangle.(3)If two sides and the included angle of an oblique triangle areknown, then the unknown sides and angles can be found by using the law of cosines.(5)Knowing the length of two sides and the measure of the included angle can determine the shape and size of the triangle. In other words, the two triangles made by these data are congruent.4.(1)如果一个角的顶点在一个笛卡尔坐标系的原点并且它的始边沿着x轴正方向，这个角被称为处于标准位置.(3)仰角和俯角是以一条以水平线为参考位置来测量的,如果正被观测的物体在观测者的上方,那么由水平线和视线所形成的角叫做仰角.如果正被观测的物体在观测者的下方,那么由水平线和视线所形成的的角叫做俯角.(5)如果我们知道一个三角形的两条边的长度和对着其中一条边的角度,我们如何解这个三角形呢?这个问题有一点困难来回答,因为所给的信息可能确定两个三角形,一个三角形或者一个也确定不了.2.32.素数:prime 合数:composite 质因数:prime factor/prime divisor 公倍数:common multiple 正素因子: positive prime divisor 除法算式:division equation 最大公因数:greatest common divisor(G.C.D) 最小公倍数: lowest common multiple(L.C.M) 整除:divide by 整除性:divisibility 过程:process 证明:proof 分类:classification 剩余:remainder辗转相除法:Euclidean algorithm 有限集:finite set 无限的:infinitely 可数的countable 终止:terminate 与矛盾:contrary to3.(1)We need to study by which integers an integer is divisible, that is , what factor it has. Specially, it is sometime required that an integer is expressed as the product of its prime factors.(3)The number 1 is neither a prime nor a composite number;A composite number in addition to being divisible by 1 and itself, can also be divisible by some prime number.(5)The number of the primes bounded above by any given finite integer N can be found by using the method of the sieve Eratosthenes.4.(1)数论中一个重要的问题是哥德巴赫猜想,它是关于偶数作为两个奇素数和的表示.(3)一个数,形如2p-1的素数被称为梅森素数.求出5个这样的数.(5)任意给定的整数m和素数p,p的仅有的正因子是p和1,因此仅有的可能的p和m的正公因子是p和1.因此,我们有结论:如果p是一个素数,m是任意整数,那么p整除m,要么(p,m)=1.2.42.集:set 子集:subset 真子集:proper subset 全集:universe 补集:complement 抽象集:abstract set 并集:union 交集:intersection 元素:element/member 组成:comprise/constitute包含:contain 术语:terminology 概念:concept 上有界:bounded above 上界:upper bound 最小的上界:least upper bound 完备性公理:completeness axiom3.(1)Set theory has become one of the common theoretical foundation and the important tools in many branches of mathematics.(3)Set S itself is the improper subset of S; if set T is a subset of S but not S, then T is called a proper subset of S.(5)The subset T of set S can often be denoted by {x}, that is, T consists of those elements x for which P(x) holds.(7)This example makes the following question become clear, that is, why may two straight lines in the space neither intersect nor parallel.4.(1)设N是所有自然数的集合,如果S是所有偶数的集合,那么它在N中的补集是所有奇数的集合.(3)一个非空集合S称为由上界的,如果存在一个数c具有属性:x<=c对于所有S中的x.这样一个数字c被称为S的上界.(5)从任意两个对象x和y，我们可以形成序列（x，y），它被称为一个有序对，除非x=y，否则它当然不同于（y，x）.如果S和T是任意集合，我们用S*T表示所有有序对（x,y),其中x术语S，y属于T.在R.笛卡尔展示了如何通过实轴和它自己的笛卡尔积来描述平面的点之后，集合S*T被称为S和T的笛卡尔积.2.52.竖直线:vertical line 水平线:horizontal line 数对:pairs of numbers 有序对:ordered pairs 纵坐标:ordinate 横坐标:abscissas 一一对应:one-to-one 对应点:corresponding points圆锥曲线:conic sections 非空图形:non vacuous graph 直立圆锥:right circular cone 定值角:constant angle 母线:generating line 双曲线:hyperbola 抛物线:parabola 椭圆:ellipse退化的:degenerate 非退化的:nondegenerate任意的:arbitrarily 相容的:consistent 在几何上:geometrically 二次方程:quadratic equation 判别式:discriminant 行列式:determinant3.(1)In the planar rectangular coordinate system, one can set up aone-to-one correspondence between points and ordered pairs of numbers and also a one-to-one correspondence between conic sections and quadratic equation.(3)The symbol can be used to denote the set of ordered pairs（x，y）such that the ordinate is equal to the cube of the abscissa.（5）According to the values of the discriminate，the non-degenerate graph of Equation （iii） maybe known to be a parabola， a hyperbolaor an ellipse.4.（1）在例1，我们既用了图形，也用了代数的代入法解一个方程组（其中一个方程式二次的，另一个是线性的）。
高三英语微积分基础单选题20题及答案1. In calculus, the derivative of a constant is _____.A. zeroB. oneC. itselfD. undefined答案：A。
常数的导数是零。
选项B“one”错误，常数的导数不是1。
选项C“itself”错误，常数的导数不是它本身。
选项D“undefined”错误，常数的导数是确定的，为零。
2. The process of finding the derivative is called _____.A. integrationB. differentiationC. summationD. multiplication答案：B。
求导数的过程叫做微分。
选项A“integration”是积分。
选项C“summation”是求和。
选项D“multiplication”是乘法。
3. If y = x², then the derivative of y with respect to x is _____.A. 2xB. x²C. 2x²D. x/2答案：A。
y = x²的导数是2x。
选项B“x²”错误，不是它本身。
选项C“2x²”错误，系数错误。
选项D“x/2”错误，计算错误。
4. The integral of a constant times a function is equal to the constant times the integral of the function. This is known as _____.A. the power ruleB. the product ruleC. the chain ruleD. the constant multiple rule答案：D。
常数乘以函数的积分等于常数乘以函数的积分，这被称为常数倍数法则。
gre考试数学真题试卷GRE考试数学真题试卷一、选择题（每题1分，共20分）1. If the function f(x) = 3x^2 - 2x + 1, what is the value of f(1)?A. 0B. 2C. 3D. 4E. 52. What is the derivative of the function g(x) = 4x^3 - x^2 + 7?A. 12x^2 - 2xB. 12x^2 + 2xC. 12x^3 - 2xD. 12x^3 + 2xE. 12x^3 - 2x^23. The area under the curve of y = x^2 from x = 0 to x = 2 is:A. 2B. 4C. 8D. 10E. 124. If a and b are the roots of the quadratic equation x^2 +5x + 6 = 0, what is the value of a + b?A. -3B. -2C. -1D. 0E. 15. The slope of the line passing through the points (2, 3) and (4, 7) is:A. 1B. 2C. 3D. 4E. 56. What is the value of sin(30°)?A. 1/2B. √2/2C. √3/2D. 2/√3E. 1/√27. The integral of the function h(x) = 3x + 2 is:A. x^3 + 2x + CB. x^3 + 2x^2 + CC. x^2 + 2x + CD. 3x^2 + 2x + CE. 3x^2 + 2x^3 + C8. The equation of a circle with center (3, 4) and radius 5 is:A. (x - 3)^2 + (y - 4)^2 = 25B. (x - 3)^2 + (y - 4)^2 = 1C. (x - 3)^2 + (y - 4)^2 = 100D. (x - 3)^2 + (y - 4)^2 = 625E. (x - 3)^2 + (y - 4)^2 = 09. The volume of a sphere with radius 4 is:A. 256πB. 512πC. 1024πD. 2048πE. 4096π10. If the sequence 2, 6, 18, 54, ... is a geometric sequence, what is the common ratio?A. 2B. 3C. 4D. 5E. 6二、填空题（每题2分，共20分）11. If the sum of the first n terms of an arithmetic sequence is given by S_n = n^2, then the 5th term of the sequence is__________.12. The equation of the line perpendicular to y = 2x - 1 and passing through the point (1, 3) is __________.13. The value of the definite integral ∫(0 to 1) x^2 dx is__________.14. If the function f(x) = sin(x) + cos(x), then f''(x) is__________.15. The area of a triangle with vertices at (0,0), (3,0), and (0,4) is __________.16. The limit of the function (1 + 1/n)^n as n approaches infinity is __________.17. The value of e^(iπ) is __________.18. The standard deviation of the data set {2, 4, 6, 8, 10} is __________.19. If a fair coin is tossed 5 times, the probability of getting exactly 3 heads is __________.20. The value of the binomial coefficient C(n, k) when。
