最新電大應(yīng)用概率統(tǒng)計試題考試小抄

1、應(yīng)用數(shù)學(xué)一、填空題 （每小題3分，共21分）1已知則2設(shè)且則3已知隨機(jī)變量在0，5內(nèi)服從均勻分布，則4設(shè)袋中有5個黑球、3個白球，現(xiàn)從中隨機(jī)地摸出4個，則其中恰有3個白球的概率為 . 5設(shè)是來自正態(tài)總體的一個樣本，則6有交互作用的正交試驗中，設(shè)與皆為三水平因子，且有交互作用，則的自由度為 .7在minitab菜單下操作，選擇可用來討論 的問題，輸出結(jié)果尾概率為，給定，可做出 的判斷.二、單項選擇題（每小題3分，共15分）1設(shè)為兩隨機(jī)事件，則結(jié)論正確的是（ ） （a）獨(dú)立 （b）互斥 （c） （d）2. 設(shè)與分別為隨機(jī)變量與的分布函數(shù).為使是某一隨機(jī)變量的分布函數(shù)，在下列給定的各組數(shù)值中應(yīng)?。?

2、） （a）（b）（c）（d）3設(shè)和分別來自兩個正態(tài)總體與的樣本，且相互獨(dú)立，與分別是兩個樣本的方差，則服從的統(tǒng)計量為（ ） （a） （b） （c） （d）4. 設(shè)關(guān)于的線性回歸方程為則、的值分別為（ ） （） （a）8.8，-2.4 （b）-2.4，8.8 （c）-1.2，4.4（d）4.4，1.2 5若分布，則服從（ ）分布.（a）（b）（c）（d）四、計算題（共56分）1據(jù)以往資料表明，某一3口之家，患某種傳染病的概率有以下規(guī)律： p孩子得病=0.6 ，p母親得病 | 孩子得病=0.5 ，p父親得病 | 母親及孩子得病=0.4 ，求母親及孩子得病但父親未得病的概率.（8分）2.一學(xué)生接連參

3、加同一課程的兩次考試.第一次及格的概率為0.6，若第一次及格則第二次及格的概率也為0.6；若第一次不及格則第二次及格的概率為0.3.（1）若至少有一次及格則能取得某種資格，求他取得該資格的概率？（2）若已知他第二次已經(jīng)及格，求他第一次及格的概率？（12分） 3假定連續(xù)型隨機(jī)變量的概率密度為，求（1）常數(shù)，數(shù)學(xué)期望，方差；（2）的概率密度函數(shù).（12分）4. 某工廠采用新法處理廢水，對處理后的水測量所含某種有毒物質(zhì)的濃度，得到10個數(shù)據(jù)（單位：mg/l）: 22 , 14 , 17 , 13 , 21 , 16 , 15 , 16 , 19 , 18而以往用老辦法處理廢水后，該種有毒物質(zhì)的平均濃

4、度為19.問新法是否比老法效果好？假設(shè)檢驗水平，有毒物質(zhì)濃度.（12分）（）5. 在某橡膠配方中，考慮三種不同的促進(jìn)劑（a），四種不同份量的氧化鋅（b），每種配方各做一次試驗，測得300%定強(qiáng)如下：定強(qiáng)氧化鋅促進(jìn)劑b1b2b3b4a1 31343539a233363738a335373942試檢驗促進(jìn)劑、氧化鋅對定強(qiáng)有無顯著的影響？（12分）()四. 綜合實驗報告（8分）052應(yīng)用數(shù)學(xué)一、 填空題（每小題2分，共26=12分）1、設(shè)一維連續(xù)型隨機(jī)變量x服從指數(shù)分布且具有方差4，那么x的概率密度 函數(shù)為： 。2、設(shè)一維連續(xù)型隨機(jī)變量x的分布函數(shù)為， 則隨機(jī)變量的概率密度函數(shù)為： 。3、設(shè)總體x服

5、從正態(tài)分布，它的一個容量為100的樣本的均值服從正態(tài)分布 。4、設(shè)是參數(shù)的估計量，若 成立，則稱是的無偏估計量。5、在無交互作用的雙因素試驗的方差分析中，若因素a有三個水平，因素b有四個水平，則誤差平方和sse的自由度 。6、設(shè)關(guān)于隨機(jī)變量y與x的線性回歸方程為，則。 （ ）二、單項選擇題（每小題2分，共26=12分）1、 設(shè)相互獨(dú)立的兩個隨機(jī)變量x、y具有同一分布，且x的分布律為： 則隨機(jī)變量的分布律為（ ） 2、若隨機(jī)變量x的數(shù)學(xué)期望e（x）存在，則（ ） 3、設(shè)x為隨機(jī)變量，下列哪個是x的3階中心矩？（ ） 4、設(shè)兩總體，且未知，從x中抽取一容量為的樣本，從y中抽取一容量為的樣本，對檢驗

6、水平，檢驗假設(shè)： 由樣本計算出來的統(tǒng)計量的觀察值應(yīng)與下列哪個臨界值作比較？（ ）5、在對回歸方程的統(tǒng)計檢驗中，f檢驗法所用的統(tǒng)計量是：（ ） （其中ssr是回歸平方和，sse是剩余平方和，是觀察值的個數(shù)）6、設(shè)總體，從x中抽取一容量為的樣本，樣本均值為，則統(tǒng)計量服從什么分布？（ ） 三、判別題（每小題2分，共26=12分）（請在你認(rèn)為對的小題對應(yīng)的括號內(nèi)打“”，否則打“”）1、設(shè)a、b是兩個隨機(jī)事件，則 （ ）2、設(shè)是服從正態(tài)分布的隨機(jī)變量的分布函數(shù)，則 （ ）3、相關(guān)系數(shù)為零的兩個隨機(jī)變量是相互獨(dú)立的。 （ ）4、如果x、y是兩個相互獨(dú)立的隨機(jī)變量，則 （ ）5、若兩隨機(jī)變量具有雙曲線類型的

7、回歸關(guān)系，則可作適當(dāng)?shù)淖兞看鷵Q轉(zhuǎn)化為線性回歸關(guān)系。（ ） 6、用minitab軟件做有交互作用的雙因素試驗的方差分析時可在菜單中選擇： （ ） 四、計算題（每小題8分，共87=56分）1、 一射手對同一目標(biāo)獨(dú)立進(jìn)行四次射擊，若至少命中一次的概率為，（1） 求該射手的命中率；（2） 求四次射擊中恰好命中二次的概率。2、 如下圖，某人從a點(diǎn)出發(fā)，隨意沿四條路線之一前進(jìn)，當(dāng)他到達(dá)b1，b2，b3，b4 中的任一點(diǎn)時，在前進(jìn)方向的各路線中再隨意選擇一條繼續(xù)行進(jìn)。（1） 求此人能抵達(dá)c點(diǎn)的概率；（2） 若此人抵達(dá)了c點(diǎn)，求他經(jīng)過點(diǎn)b1的概率。b4ab1b2b3c 3、某公共汽車站從早上6時起每隔15分鐘

8、開出一趟班車，假定某人在6點(diǎn)以后到達(dá)車站的時刻是隨機(jī)的，所以有理由認(rèn)為他等候乘車的時間x服從均勻分布，其密度函數(shù)為： ，求（1） 此人等車時間少于5分鐘的概率；此人的平均等車時間e（x）。 4、 設(shè)二維隨機(jī)變量（x，y）的聯(lián)合密度函數(shù)為（1）判斷x與y是否相互獨(dú)立；（2）求概率5、設(shè)某種清漆9個樣本的干燥時間（單位：h）分別為6.0，5.7，5.8，6.5，7.0，6.3，5.6，6.1，5.0，設(shè)干燥時間總體服從正態(tài)分布，求平均干燥 時間的置信度為0.95的置信區(qū)間。 （）6、 某種導(dǎo)線，要求其電阻的標(biāo)準(zhǔn)差不得超過,今在生產(chǎn)的一批導(dǎo)線中取樣品9根，測得，設(shè)總體為正態(tài)分布，問在水平下能否認(rèn)為這

9、批導(dǎo)線的標(biāo)準(zhǔn)差顯著地偏大？ （） 7、 有三臺機(jī)床生產(chǎn)某種產(chǎn)品，觀察各臺機(jī)床五天的產(chǎn)量，由樣本觀察值算出組間平方和，誤差平方和，總離差平方和，試問三臺機(jī)床生產(chǎn)的產(chǎn)品產(chǎn)量間的差異在檢驗水平下是否有統(tǒng)計意義？（）五、綜合實驗（本題8分，開卷，解答另附于數(shù)學(xué)實驗報告中）062應(yīng)用數(shù)學(xué)一、 填空題（每小題2分，共26=12分）1、設(shè)服從01分布的一維離散型隨機(jī)變量x的分布律是：， 若x的方差是，則p=_。2、設(shè)一維連續(xù)型隨機(jī)變量x服從正態(tài)分布，則隨機(jī)變量 的概率密度函數(shù)為_。3、設(shè)二維離散型隨機(jī)變量x、y的聯(lián)合分布律為：則a, b滿足條件：_。4、設(shè)總體x服從正態(tài)分布 ， 是它的一個樣本，則樣本均值的

10、方差是_。5、假設(shè)正態(tài)總體的方差未知，對總體均值 m 作區(qū)間估計?，F(xiàn)抽取了一個容量為n的樣本，以表示樣本均值，s表示樣本均方差，則m 的置信度為1-a 的置信區(qū)間為：_。6、求隨機(jī)變量y與x的線性回歸方程，在計算公式 中，。二、單項選擇題（每小題2分，共26=12分）1、設(shè)a，b是兩個隨機(jī)事件，則必有（ ）2、設(shè)a，b是兩個隨機(jī)事件， 則（ ）3、設(shè)x，y為相互獨(dú)立的兩個隨機(jī)變量，則下列不正確的結(jié)論是（ ）4、設(shè)兩總體未知，從x中抽取一容量為的樣本，從y中抽取一容量為的樣本，作假設(shè)檢驗：所用統(tǒng)計量 服從（ ）5、在對一元線性回歸方程的統(tǒng)計檢驗中，回歸平方和ssr的自由度是：（ ） 6、設(shè)總體，

11、從x中抽取一容量為的樣本，樣本均值為，則統(tǒng)計量服從什么分布？（ ） 三、判別題（每小題2分，共26=12分）（請在你認(rèn)為對的小題對應(yīng)的括號內(nèi)打“”，否則打“”）1、（ ）設(shè)隨機(jī)變量x的概率密度為，隨機(jī)變量y的概率密度為，則二維隨機(jī)變量（x、y）的聯(lián)合概率密度為。 2、（ ）設(shè)是服從標(biāo)準(zhǔn)正態(tài)分布的隨機(jī)變量的分布函數(shù)， x是服從正態(tài)分布的隨機(jī)變量，則有3、（ ）設(shè)二維隨機(jī)變量（x、y）的聯(lián)合概率密度為，隨機(jī)變量 的數(shù)學(xué)期望存在，則4、（ ）設(shè)總體x的分布中的未知參數(shù)的置信度為的置信區(qū)間為 則有。 5、（ ）假設(shè)總體x服從區(qū)間上的均勻分布，從期望考慮，的矩估計是 （是樣本均值）。6、（ ）用mini

12、tab軟件求回歸方程，在菜單中選擇如下命令即可得： 四、計算題（每小題8分，共87=56分）1、某連鎖總店屬下有10家分店，每天每家分店訂貨的概率為p，且每家分店的訂貨行為是相互獨(dú)立的，求（1） 每天訂貨分店的家數(shù)x的分布律；（2） 某天至少有一家分店訂貨的概率。2、現(xiàn)有十個球隊要進(jìn)行乒乓球賽，第一輪是小組循環(huán)賽，要把十支球隊平分成 兩組，上屆冠亞軍作為種子隊分別分在不同的兩組，其余八隊抽簽決定分組， 甲隊抽第一支簽，乙隊抽第二支簽。（1）求：甲隊抽到與上屆冠軍隊在同一組的概率；（2）求：乙隊抽到與上屆冠軍隊在同一組的概率；（3）已知乙隊抽到與上屆冠軍隊在同一組，求：甲隊也是抽到與上屆冠軍隊在

13、同一組的概率。3、已知隨機(jī)變量x服從參數(shù)為的指數(shù)分布，且，求（1）參數(shù)； （2）4、設(shè)一維隨機(jī)變量x的分布函數(shù)為：，求：（1） x的概率密度；（2） 隨機(jī)變量y=2(x+1)的數(shù)學(xué)期望。5、 設(shè)二維隨機(jī)變量（x，y）的聯(lián)合概率密度為 ，求（1）該二維隨機(jī)變量的聯(lián)合分布函數(shù)值；（2）二維隨機(jī)變量（x，y）的函數(shù)z=x+y的分布函數(shù)值fz(1)。6、 用某種儀器間接測量某物體的硬度，重復(fù)測量5次，所得數(shù)據(jù)是175、173、178、174、176，而用別的精確方法測量出的硬度為179(可看作硬度真值)。設(shè)測量硬度服從正態(tài)分布，問在水平a =0.05下，用此種儀器測量硬度所得數(shù)值是否顯著偏低？()7、

14、 某廠生產(chǎn)某種產(chǎn)品使用了3種不同的催化劑（因素a）和4種不同的原料（因素b），各種搭配都做一次試驗測得成品壓強(qiáng)數(shù)據(jù)。由樣本觀察值算出各平方和分別為：ssa=25.17，ssb=69.34，sse=4.16，sst=98.67，試列出方差分析表，據(jù)此檢驗不同催化劑和不同原料在檢驗水平a =0.05下對產(chǎn)品壓強(qiáng)的影響有沒有統(tǒng)計意義？ （）五、綜合實驗（本題8分，開卷，解答另附于數(shù)學(xué)實驗報告中）072 大學(xué)數(shù)學(xué)一、 填空題（每小題2分，本題共12分）1若事件相互獨(dú)立，且，則= ；2設(shè)隨機(jī)變量的分布列為：01234560.10.150.20.30.120.10.03則； 3設(shè)隨機(jī)變量服從參數(shù)為的poi

15、sson分布，且已知，則； 4設(shè) 是來自正態(tài)總體的樣本，則 ； ；5設(shè)是來自總體的一個樣本，則 ；6假設(shè)某種電池的工作時間服從正態(tài)分布，觀察五個電池的工作時間（小時），并求得其樣本均值和標(biāo)準(zhǔn)差分別為：，若檢驗這批樣本是否取自均值為50（小時）的總體，則零假設(shè)為 ，其檢驗統(tǒng)計量為 。 二、單項選擇題（每小題3分，本題共18分）1從數(shù)字1，2，3，4，5中，隨機(jī)抽取3個數(shù)字（允許重復(fù)）組成一個三位數(shù)，其各位數(shù)字之和等于9的概率為（ ）a；b；c；d2如果隨機(jī)變量的密度函數(shù)為，則（ ）a0.875； b； c； d3設(shè)物件的稱重則至少應(yīng)稱多少次？（ ） a16；b15；c4；d204設(shè)隨機(jī)變量x的概

16、率密度函數(shù)為，則常數(shù)c=（ ） a； b5； c2； d5在一個已通過f檢驗的一元線性回歸方程中，若給定的預(yù)測區(qū)間精確表示為（ ）a； b；c；d6樣本容量為時，樣本方差是總體方差的無偏估計量，這是因為（ ） a； b； c； d 三、解下列各題（6小題，共48分）1設(shè)總體，為簡單隨機(jī)樣本，且證明： （6分）2已知連續(xù)型隨機(jī)變量的分布函數(shù)為 試確定常數(shù)； 求； 求的密度函數(shù)（10分） 3若從10件正品、2件次品的一批產(chǎn)品中，無放回地抽取2次，每次取一個，試求第二次取出次品的概率（6分）4設(shè)的密度函數(shù)為 求的數(shù)學(xué)期望和方差； 求與的協(xié)方差和相關(guān)系數(shù)，并討論與是否相關(guān) （8分）5設(shè)二維隨機(jī)變量在區(qū)

17、域上服從均勻分布，其中是由曲線和直線所圍成試求的聯(lián)合分布密度及關(guān)于的邊緣分布密度 與，并判斷是否相互獨(dú)立（10分）6設(shè)隨機(jī)變量服從區(qū)間上的均勻分布，試證明：（為常數(shù)）也服從均勻分布（8分）四、應(yīng)用題：以下是某農(nóng)作物對三種土壤，兩種肥料，每一個處理作四次重復(fù)試驗后所得產(chǎn)量的方差分析表的部分?jǐn)?shù)據(jù)，分別寫出各零假設(shè)，并完成方差分析表，寫出分析結(jié)果 （12分）方差來源平方和自由度均方和值臨界值土壤因素肥料因素2誤差18總和23已知參考臨界值： 五. 綜合實驗報告（10分）請您刪除一下內(nèi)容，o(_)o謝謝！2016年中央電大期末復(fù)習(xí)考試小抄大全，電大期末考試必備小抄，電大考試必過小抄acetylchol

18、ine is a neurotransmitter released from nerve endings (terminals) in both the peripheral and the central nervous systems. it is synthesized within the nerve terminal from choline, taken up from the tissue fluid into the nerve ending by a specialized transport mechanism. the enzyme necessary for this

19、 synthesis is formed in the nerve cell body and passes down the axon to its end, carried in the axoplasmic flow, the slow movement of intracellular substance (cytoplasm). acetylcholine is stored in the nerve terminal, sequestered in small vesicles awaiting release. when a nerve action potential reac

20、hes and invades the nerve terminal, a shower of acetylcholine vesicles is released into the junction (synapse) between the nerve terminal and the effector cell which the nerve activates. this may be another nerve cell or a muscle or gland cell. thus electrical signals are converted to chemical signa

21、ls, allowing messages to be passed between nerve cells or between nerve cells and non-nerve cells. this process is termed chemical neurotransmission and was first demonstrated, for nerves to the heart, by the german pharmacologist loewi in 1921. chemical transmission involving acetylcholine is known

22、 as cholinergic. acetylcholine acts as a transmitter between motor nerves and the fibres of skeletal muscle at all neuromuscular junctions. at this type of synapse, the nerve terminal is closely apposed to the cell membrane of a muscle fibre at the so-called motor end plate. on release, acetylcholin

23、e acts almost instantly, to cause a sequence of chemical and physical events (starting with depolarization of the motor endplate) which cause contraction of the muscle fibre. this is exactly what is required for voluntary muscles in which a rapid response to a command is required. the action of acet

24、ylcholine is terminated rapidly, in around 10 milliseconds; an enzyme (cholinesterase) breaks the transmitter down into choline and an acetate ion. the choline is then available for re-uptake into the nerve terminal. these same principles apply to cholinergic transmission at sites other than neuromu

25、scular junctions, although the structure of the synapses differs. in the autonomic nervous system these include nerve-to-nerve synapses at the relay stations (ganglia) in both the sympathetic and the parasympathetic divisions, and the endings of parasympathetic nerve fibres on non-voluntary (smooth)

26、 muscle, the heart, and glandular cells; in response to activation of this nerve supply, smooth muscle contracts (notably in the gut), the frequency of heart beat is slowed, and glands secrete. acetylcholine is also an important transmitter at many sites in the brain at nerve-to-nerve synapses. to u

27、nderstand how acetylcholine brings about a variety of effects in different cells it is necessary to understand membrane receptors. in post-synaptic membranes (those of the cells on which the nerve fibres terminate) there are many different sorts of receptors and some are receptors for acetylcholine.

28、 these are protein molecules that react specifically with acetylcholine in a reversible fashion. it is the complex of receptor combined with acetylcholine which brings about a biophysical reaction, resulting in the response from the receptive cell. two major types of acetylcholine receptors exist in

29、 the membranes of cells. the type in skeletal muscle is known as nicotinic; in glands, smooth muscle, and the heart they are muscarinic; and there are some of each type in the brain. these terms are used because nicotine mimics the action of acetylcholine at nicotinic receptors, whereas muscarine, a

30、n alkaloid from the mushroom amanita muscaria, mimics the action of acetylcholine at the muscarinic receptors. acetylcholine is the neurotransmitter produced by neurons referred to as cholinergic neurons. in the peripheral nervous system acetylcholine plays a role in skeletal muscle movement, as wel

31、l as in the regulation of smooth muscle and cardiac muscle. in the central nervous system acetylcholine is believed to be involved in learning, memory, and mood. acetylcholine is synthesized from choline and acetyl coenzyme a through the action of the enzyme choline acetyltransferase and becomes pac

32、kaged into membrane-boundvesicles. after the arrival of a nerve signal at the termination of an axon, the vesicles fuse with the cell membrane, causing the release of acetylcholine into thesynaptic cleft. for the nerve signal to continue, acetylcholine must diffuse to another nearby neuron or muscle

33、 cell, where it will bind and activate areceptorprotein. there are two main types of cholinergic receptors, nicotinic and muscarinic. nicotinic receptors are located at synapses between two neurons and at synapses between neurons and skeletal muscle cells. upon activation a nicotinic receptor acts a

34、s a channel for the movement of ions into and out of the neuron, directly resulting indepolarizationof the neuron. muscarinic receptors, located at the synapses of nerves with smooth or cardiac muscle, trigger a chain of chemical events referred to as signal transduction. for a cholinergic neuron to

35、 receive another impulse, acetylcholine must be released from the receptor to which it has bound. this will only happen if the concentration of acetylcholine in the synaptic cleft is very low. low synaptic concentrations of acetylcholine can be maintained via a hydrolysis reaction catalyzed by the e

36、nzyme acetylcholinesterase. this enzyme hydrolyzes acetylcholine into acetic acid and choline. if acetylcholinesterase activity is inhibited, the synaptic concentration of acetylcholine will remain higher than normal. if this inhibition is irreversible, as in the case of exposure to many nerve gases

37、 and some pesticides, sweating, bronchial constriction, convulsions, paralysis, and possibly death can occur. although irreversible inhibition is dangerous, beneficial effects may be derived from transient (reversible) inhibition. drugs that inhibit acetylcholinesterase in a reversible manner have b

38、een shown to improve memory in some people with alzheimers disease. abstract expressionism, movement of abstract painting that emerged in new york city during the mid-1940s and attained singular prominence in american art in the following decade; also called action painting and the new york school.

39、it was the first important school in american painting to declare its independence from european styles and to influence the development of art abroad. arshile gorky first gave impetus to the movement. his paintings, derived at first from the art of picasso, mir, and surrealism, became more personal

40、ly expressive. jackson pollocks turbulent yet elegant abstract paintings, which were created by spattering paint on huge canvases placed on the floor, brought abstract expressionism before a hostile public. willem de koonings first one-man show in 1948 established him as a highly influential artist.

41、 his intensely complicated abstract paintings of the 1940s were followed by images of woman, grotesque versions of buxom womanhood, which were virtually unparalleled in the sustained savagery of their execution. painters such as philip guston and franz kline turned to the abstract late in the 1940s

42、and soon developed strikingly original stylesthe former, lyrical and evocative, the latter, forceful and boldly dramatic. other important artists involved with the movement included hans hofmann, robert motherwell, and mark rothko; among other major abstract expressionists were such painters as clyf

43、ford still, theodoros stamos, adolph gottlieb, helen frankenthaler, lee krasner, and esteban vicente. abstract expressionism presented a broad range of stylistic diversity within its largely, though not exclusively, nonrepresentational framework. for example, the expressive violence and activity in

44、paintings by de kooning or pollock marked the opposite end of the pole from the simple, quiescent images of mark rothko. basic to most abstract expressionist painting were the attention paid to surface qualities, i.e., qualities of brushstroke and texture; the use of huge canvases; the adoption of a

45、n approach to space in which all parts of the canvas played an equally vital role in the total work; the harnessing of accidents that occurred during the process of painting; the glorification of the act of painting itself as a means of visual communication; and the attempt to transfer pure emotion

46、directly onto the canvas. the movement had an inestimable influence on the many varieties of work that followed it, especially in the way its proponents used color and materials. its essential energy transmitted an enduring excitement to the american art scene. science and technology is quite a broa

47、d category, and it covers everything from studying the stars and the planets to studying molecules and viruses. beginning with the greeks and hipparchus, continuing through ptolemy, copernicus and galileo, and today with our work on the international space station, man continues to learn more and mo

48、re about the heavens. from here, we look inward to biochemistry and biology. to truly understand biochemistry, scientists study and see the unseen bystudying the chemistry of biological processes. this science, along with biophysics, aims to bring a better understanding of how bodies work from how w

49、e turn food into energy to how nerve impulses transmit.analytic geometry, branch ofgeometryin which points are represented with respect to a coordinate system, such ascartesian coordinates, and in which the approach to geometric problems is primarily algebraic. its most common application is in the

50、representation of equations involving two or three variables as curves in two or three dimensions or surfaces in three dimensions. for example, the linear equationax+by+c=0 represents a straight line in thexy-plane, and the linear equationax+by+cz+d=0 represents a plane in space, wherea, b, c,anddar

51、e constant numbers (coefficients). in this way a geometric problem can be translated into an algebraic problem and the methods of algebra brought to bear on its solution. conversely, the solution of a problem in algebra, such as finding the roots of an equation or system of equations, can be estimat

52、ed or sometimes given exactly by geometric means, e.g., plotting curves and surfaces and determining points of intersection. in plane analytic geometry a line is frequently described in terms of its slope, which expresses its inclination to the coordinate axes; technically, the slopemof a straight l

53、ine is the (trigonometric) tangent of the angle it makes with thex-axis. if the line is parallel to thex-axis, its slope is zero. two or more lines with equal slopes are parallel to one another. in general, the slope of the line through the points (x1,y1) and (x2,y2) is given bym= (y2-y1) / (x2-x1).

54、 the conic sections are treated in analytic geometry as the curves corresponding to the general quadratic equationax2+bxy+cy2+dx+ey+f=0, wherea, b, fare constants anda, b,andcare not all zero. in solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, , , and , the cosines of the angles the line makes with thex-, y-,andz-axes, respectively; these satisfy the relationship 2+2+2= 1. in the same wa