英文版微积分试卷答案

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高三英语微积分基础单选题60题(答案解析)

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。

微积分英文版第九版课后练习题含答案

微积分英文版第九版课后练习题含答案微积分是数学中重要的一部分，其关注的是一条曲线或曲面的局部性质。

美国著名数学家Stewart所著的微积分英文版第九版是微积分学习的重要教材之一，本文将介绍其课后练习题，并提供答案供大家进行自我学习和测试。

课后练习题微积分英文版第九版的课后练习题共分为两部分，其中Part 1是选择题，Part 2是填空题和证明题，共计约1700道题目。

Part 1中包含了大量的选择题，这些题目主要考察读者对微积分理论的掌握和应用。

大多数题目都要求读者用所学知识推理或计算来获得正确答案。

这些题目中难易程度参差不齐，有一些比较简单，但也有一些比较困难。

Part 2的题目类型较多，包括了填空题、证明题、计算题等。

这些题目内容繁杂、难度较大，需要读者花费很多时间和精力来解答。

这些题目主要是为了检测读者对所学知识的深层次理解和应用能力，考察读者的逻辑思维和推理能力。

答案微积分英文版第九版提供了相应的课后练习答案，可以帮助读者检验自己的答案是否正确，同时也可以帮助读者更好地理解和掌握所学知识。

答案分为Part 1和Part 2两部分，且每部分分别包含了选择题和非选择题的答案。

这些答案详细、准确，提供了完整的解题思路和方法，帮助读者更好地理解题目的解法，并弥补了部分教材中的不足之处。

读者可以通过该教材的官方网站或者其他渠道获得课后练习题的答案。

结论微积分是数学中非常重要的一门学科，对于各个领域的科学研究、技术发展和社会进步都有着举足轻重的作用。

而微积分英文版第九版课后练习题则是培养和检验读者对微积分学习的深刻理解和应用能力的重要途径。

通过对这些题目的研究和答案的掌握，可以帮助读者更好地掌握微积分学科，提升自己的学术能力和科研能力。

AP 微积分 AB 2007 (Form B) 真题与答案

AP® Calculus AB2007 Scoring GuidelinesForm BThe College Board: Connecting Students to College SuccessThe College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 5,000 schools, colleges, universities, and other educational organizations. Each year, the College Board serves seven million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT®, the PSAT/NMSQT®, and the Advanced Placement Program® (AP®). The College Board is committed to the principles of excellence and equity, and that commitment is embodied in all of its programs, services, activities, and concerns.© 2007 The College Board. All rights reserved. College Board, Advanced Placement Program, AP, AP Central, SAT, and the acorn logo are registered trademarks of the College Board. PSAT/NMSQT is a registered trademark of the College Board and National Merit Scholarship Corporation.Permission to use copyrighted College Board materials may be requested online at:/inquiry/cbpermit.html.Visit the College Board on the Web: .AP Central is the official online home for the AP Program: .Question 1Let R be the region bounded by the graph of 22x x y e −= and thehorizontal line and let S be the region bounded by the graph of2,y =22x x y e −= and the horizontal lines and 1y =2,y =as shown above. (a) Find the area of R . (b) Find the area of S .(c) Write, but do not evaluate, an integral expression that gives thevolume of the solid generated when R is rotated about the horizontal line 1.y =222x x e −= when 0.446057,1.553943x =Let and 0.446057P = 1.553943Q =(a) Area of ()2220.5Qx x P14dx −=−=⌠⌡⎪⎩R e3 : ⎪⎨ 1 : integrand1 : limits 1 : answer ⎧(b) when 2221x x e −=0,x =Area of S e Area of R()22201x x dx −=−−⌠⌡− Area of2.06016= 1.546R =OR()()()222220110.219064 1.1078860.219064 1.546Px x x x Qe dx Q P e d −−−+−⋅+−=++=⌠⌠⌡⌡1x⎪⎩3 : ⎪⎨ 1 : integrand 1 : limits 1 : answer ⎧(c) Volume ()()2222121Qx x P e d π−⎛⎞−−⎜⎟⎝⎠⌠⎮⌡x =−3 : {2 : integrand1 : constant and limitsQuestion 2A particle moves along the x -axis so that its velocity v at timeis given by The graph of v is shown abovefor 0t ≥()()2sin .v t t=0t ≤≤ The position of the particle at time t is ()x t and its position at time is 0t =()05x =.(a) Find the acceleration of the particle at time t 3.=(b) Find the total distance traveled by the particle from time 0t =to t 3.==(c) Find the position of the particle at time t 3.(d) For 0t ≤≤ find the time t at which the particleis farthest to the right. Explain your answer.Question 3The wind chill is the temperature, in degrees Fahrenheit ()F ,° a human feels based on the air temperature, in degrees Fahrenheit, and the wind velocity v , in miles per hour ()mph . If the air temperature is 32 then the wind chill is given by and is valid for 56F,°()0.1655.622.1W v v =−0.v ≤≤ (a) Find ()20.W ′ Using correct units, explain the meaning of ()20W ′ in terms of the wind chill.(b) Find the average rate of change of W over the interval 560.v ≤≤ Find the value of v at which theinstantaneous rate of change of W is equal to the average rate of change of W over the interval 560.v ≤≤ (c) Over the time interval hours, the air temperature is a constant 32 At time the windvelocity is mph. If the wind velocity increases at a constant rate of 5 mph per hour, what is the rate of change of the wind chill with respect to time at 0t ≤≤4 F.°0,t =20v =3t = hours? Indicate units of measure.Question 4Let f be a function defined on the closed interval 55x −≤≤ with ()13f=. The graph of ,f ′ the derivative of f , consists of two semicircles and two line segments, as shown above.(a) For −< find all values x at which f has arelative maximum. Justify your answer.5x 5,<5,<(b) For −< find all values x at which the graph of fhas a point of inflection. Justify your answer.5x (c) Find all intervals on which the graph of f is concave upand also has positive slope. Explain your reasoning.(d) Find the absolute minimum value of ()f x over the closed interval 5x 5.−≤≤ Explain your reasoning.Question 5Consider the differential equation 11.2dy x y dx =+−(a) On the axes provided, sketch a slope field for the given differential equationat the nine points indicated.(Note: Use the axes provided in the exam booklet.)(b) Find 2d ydxin terms of x and y . Describe the region in the xy -plane inwhich all solution curves to the differential equation are concave up.(c) Let ()y f x = be a particular solution to the differential equation with theinitial condition ()01f =. Does f have a relative minimum, a relative maximum, or neither at Justify your answer. 0?x =(d) Find the values of the constants m and b , for which y mx b =+ is asolution to the differential equation.y −1Question 6Let f be a twice-differentiable function such that ()2f 5= and ()52f .= Let g be the function given by ()()().g x f f x =(a) Explain why there must be a value c for 25c << such that () 1.f c =−′(b) Show that Use this result to explain why there must be a value k for 2 such that()()2g g =′′5.5k <<()0.g k =′′ (c) Show that if ()0f x =′′ for all x , then the graph of g does not have a point of inflection. (d) Let ()().h x f x x =− Explain why there must be a value r for 25r << such that ()0.h r =。

英文版微积分考试样题3

Problems 37 - 39: Solve the problem. Assume projectile is ideal, launch angle is measured from the horizontal, and launch is over a horizontal surface, unless stated otherwise. 24) An ideal projectile is launched from level ground at a launch angle of 26° and an initial speed of 48 m/sec. How far away from the launch point does the projectile hit the ground? A) ≈ 60 m B) ≈ 230 m C) ≈ 290 m D) ≈ 185 m 25) A projectile is fired with an initial speed of 528 m/sec at an angle of 45°. What is the greatest height reached by the projectile? Round answer to the nearest tenth of a meter. A) 7111.8 m B) 76.2 m C) 69,696.0 m D) 28,447.3 m Find the unit tangent vector of the given curve. 26) r(t) = (5 + 2t7 )i + (4 + 10t7 )j + (8 + 11t7 )k A) T(t) = C) T(t) = 2 2 11 i+ j+ k 15 3 15 2 2 11 i+ j+ k 225 45 225 B) T (t) = 14 14 77 i+ j+ k 15 3 15

高三英语微积分基础单选题20题及答案

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。

高三英语微积分基础单选题40题

高三英语微积分基础单选题40题1. In calculus, the derivative of a constant is _____.A.zeroB.oneC.twoD.three答案解析：A。

在微积分中，常数的导数为零。

选项B、C、D 分别为一、二、三，都不符合常数导数的定义。

2. The integral of x with respect to x is _____.A.xB.x squaredC.x cubedD.x to the fourth power答案解析：B。

对x 积分，结果是x 的平方的一半加上常数C，但这里只考虑积分结果不考虑常数项，所以答案是x 平方。

选项A、C、D 分别为x、x 的立方、x 的四次方，都不是对x 的积分结果。

3. If y = 3x^2 + 2x + 1, then the derivative of y with respect to x is _____.A.6x + 2B.6x - 2C.3x + 2D.3x - 2答案解析：A。

对y = 3x^2 + 2x + 1 求导，3x^2 的导数是6x，2x 的导数是2，1 的导数是0，所以y 的导数是6x + 2。

选项B、C、D 分别为6x - 2、3x + 2、3x - 2，都不符合求导结果。

4. The derivative of sin(x) is _____.A.cos(x)B.-cos(x)C.sin(x)D.-sin(x)答案解析：A。

sin(x)的导数是cos(x)。

选项B、C、D 分别为-cos(x)、sin(x)、-sin(x)，都不是sin(x)的导数。

5. The integral of cos(x) with respect to x is _____.A.sin(x)B.-sin(x)C.cos(x)D.-cos(x)答案解析：A。

对cos(x)积分，结果是sin(x)加上常数C，但这里只考虑积分结果不考虑常数项，所以答案是sin(x)。

英文版-微积分试卷答案

1、 (1) sin 2lim x x x→∞= 0 . (2) d(arctan )x = 1/(1+x^2) . (3) 21d sin x x =⎰ -cotx .(4).2()()x n e = 泰勒展开式（书上有。

） .(5)0x =⎰ 26/3 .2、(6) The right proposition in the following propositions is ____A____.A. If lim ()x a f x →exists and lim ()x a g x →does not exist then lim(()())x af xg x →+does not exist. B. If lim ()x a f x →,lim ()x a g x →do both not exist then lim(()())x af xg x →+does not exist. C. If lim ()x a f x →exists and lim ()x a g x →does not exist then lim ()()x af xg x →does not exist. D. If lim ()x a f x →exists and lim ()x a g x →does not exist then ()lim ()x a f x g x →does not exist. (7) The right proposition in the following propositions is __A______.A. If lim ()()x af x f a →=then ()f a 'exists. B. If lim ()()x af x f a →≠ then ()f a 'does not exist. C. If ()f a 'does not exist then lim ()()x af x f a →≠. D. If ()f a 'does not exist then the cure ()y f x =does not have tangent at (,())a f a .(8) The right statement in the following statements is __D ______. A. sin lim 1x x x→∞= B. 1lim(1)x x x e →∞+= C. 11d 1x x x C ααα+=++⎰ D. 5511d d 11bb a a x y x y =++⎰⎰ (9) For continuous function ()f x , the erroneous expression in the following expressions is __C ____. A.d (()d )()d b a f x x f b b =⎰ B. d (()d )()d b af x x f a a =-⎰ C. d (()d )0d b a f x x x =⎰ D. d (()d )()()d b af x x f b f a x =-⎰ (10) The right proposition in the following propositions is ____D____. A. If ()f x is discontinuous on [,]a b then ()f x is unbounded on [,]a b .B. If ()f x is unbounded on [,]a b then ()f x is discontinuous on [,]a b .C. If ()f x is bounded on [,]a b then ()f x is continuous on [,]a b .D. If ()f x has absolute extreme values on [,]a b then ()f x is continuous on [,]a b .3、Evaluate 2011lim()x x e x x→-- 1/24．Find 0d |x y =and (0)y ''if 20x x x y y t e +=+⎰. 隐函数求导。

TB_chapter2

1. The point ()2,4P lies on the curve x y =. If Q is the point (x , use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value .99.3=x2. The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t ismeasured in seconds. Find the average velocity over the time period [1,3].3.Find the limit.()379lim 25++→x x x4.Find the limit.()51lim 20+-→x x x x5. If ,22)(12++≤≤x x x f for all x find the limit.)(lim 1x f x -→6. Find the limit. 22lim |2|x x x →--7. Evaluate the limit.()xx x 11022lim --→-+8. Use the definition of the derivative to find (2)f '-, where 3()2f x x x =-.9. Find an equation of the tangent line to curve 32y x x =-at the point (2,4).10. Use a graph to find a number N such that 3.031235622<---+x x x whenever N x >.11. If ()g x ()g x '.12. A machinist is required to manufacture a circular metal disk with area 21000cm .a) What radius produces such a disk? b) If the machinist is allowed an error tolerance of 25cm ±in the area of the disk, how close to the idealradius in part (a) must the machinist control the radius?13. Use a graph to find a number δ such that6.0314<-+x whenever .2δ<-x14. For the limit, illustrate the definition by finding values of δ that correspond to .25.0=ε31lim(43)2x x x →+-=15. Determine where f is discontinuous.()20()30333if x f x x if x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩16. For x = 5, determine whether f is continuous from the right, from the left, or neither.17. If a cylindrical tank holds 100,000 gallons of water, which can be drained from the bottom of the tank in an hour,then Torricelli's Law gives the volume of water remaining in the tank after t minutes as2651000,100)(⎪⎭⎫ ⎝⎛-=t t V , 600≤≤tFind the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t ) as a function of t .18.Find the derivative of the function.25314)(x x x f +-=19. If 2313)(tt f += find )(t f '.20. At what point is the function ()|6|f x x =- not differentiable.ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form A1. 0.2501562. 3/m s3. 2634. -∞5. 16. Limit does not exist7. -1/48. 109. 1016y x =-10. 9≥N11.(),3/5-∞12. cm , 0.0445cm13. 81.0≤δ14. 030.0≤δ15. 03at and16. neither17. ⎪⎭⎫ ⎝⎛--=65165200000t y 18. 310-x19. )3(26t t +- 20. 61.The point P (4, 2) lies on the curve .x y = If Q is the point ()x x ,, use your calculator to find the slope of the secant line PQ (correct to six decimal places) for the value of .01.4=x2.The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t is measured in seconds. Find the average velocity over the time period [1,1.5].3.Find the limit, if it exists.44lim |4|x x x →-- 4. Find )(a f '.252)(x x x f -+=5. Find an equation of the tangent line to curve 32y x x =-at the point (2,4).6. Evaluate the function 222)(--=x x x f at the given numbers (correct to six decimal places). Use the results to guess the value of the limit ).(lim 2x f x →7. The graph of f is given. State the numbers at which f is not differentiable.⎪⎭⎫ ⎝⎛→x x x 3cos lim 909. If 66)(12++≤≤x x x f for all x find the limit.)(lim 1x f x -→10.Evaluate the limit.()867lim 25++→x x x11. Evaluate the limit.()x x x 11022lim --→-+12.If an arrow is shot upward on the moon, with a velocity of 70 m/s its height (in meters) after t seconds is given by .99.070)(2t t t H -= With what velocity will the arrow hit the moon?13. The cost (in dollars) of producing x units of a certain commodity is .08.013336,4)(2x x x C ++= Find the average rate of change with respect to x when the production level is changed from 101=x to .103=x14.Let ()20()30333if x f x xif x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩Evaluate each limit, if it exists.0lim ()x f x +→ b.) 0lim ()x f x -→15.If f and g are continuous functions with 3)3(=f and []3)()(3lim 3=-→x g x f x , find ).3(g16.Evaluate the limit. 9lim 9+-→x x17.Find a number δsuch that if |2|x δ-<, then |48|x ε-<, where 0.1ε=.then Torricelli's Law gives the volume of water remaining in the tank after t minutes as2651000,100)(⎪⎭⎫ ⎝⎛-=t t V , 600≤≤tFind the rate at which the water is flowing out of the tank (the instantaneous rate of change of V with respect to t ) as a function of t .19. If ()g x ()g x '.20. For the function f whose graph is shown, state the following.)(lim 4x f x -→ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form B1. 0.2498442.2.625 m/s3. Limit does not exist4. a 101-5.1016y x =- 6.(1.6, 0.7465), (1.8, 0.7257), (1.9, 0.7161), (1.99, 0.7079), (1.999, 0.707195), (2.4, 0.674899), (2.2, 0.690261), (2.1, 0.698482), (2.01, 0.706225), (2.001, 0.707018), Limit = 0.707107 7. 1,0,3-8. 09. 110. 21311. -1/412. -7013. 29.3214. a.) 3 b.) 015. 616. 017. 0.025δ=18. ⎪⎭⎫ ⎝⎛--=65165200000t y 19. (),3/5-∞20. -∞1. A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate inbeats per minute. The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with t = 38 and t = 42.Select the correct answer.a. -89b. 180c. 90d. 100e. 89f. 952. If an arrow is shot upward on the moon with a velocity of 55 m/s, its height in meters after t seconds is given by .04.0552t t h -= Find the average velocity over the interval [1, 1.04].Select the correct answer.a. 54.9194b. 55.0284c. 54.8174d. 54.9184e. 54.90843. The displacement (in feet) of a certain particle moving in a straight line is given by 3/8s t = where t is measuredin seconds. Find the average velocity over the interval [1, 1.8].Select the correct answer.a. 0.865b. 0.654c. 0.765d. 0.756e. 0.745f. 0.7554. For the function f whose graph is shown, find the equations of the vertical asymptotes.Select all that apply.a. x = -7b. x = 9c. x = 5d. x = -3e. x = 10f. x = -25. Find the limit, if it exists55lim |5|x x x →--Select the correct answer.a. 5b. 1-c. 1-d. 0e. limit does not exist6. Find the limit.lim x →-∞Select the correct answer.a. -1/2b. 3c. 3-d. 0e. limit does not exist7. Evaluate the limit.()()62lim 231-+→x x xSelect the correct answer.a. 27b. -45c. -135d. 29e. -1258.If 88)(12++≤≤x x x f for all x , find )(lim 1x f x -→. Select the correct answer.a. 1b. 8c. -1/8d. -1/16e. The limit does not exist9. Evaluate the limit.⎪⎭⎫ ⎝⎛→x x x 5cos lim 90Select the correct answer.a. -5b. 1c. 0d. 5e. The limit does not exist10. Use a graph to find a number δ such that 2.021sin <-x whenever δπ<-6x .Round down the answer to the nearest thousandth.Select the correct answer.a. 218.0≤δb. 368.0≤δc. 401.0≤δd. 251.0≤δe. 425.0≤δ11. A machinist is required to manufacture a circular metal disk with area 1000 cm 2. If the machinist is allowed an error tolerance of ±10 cm 2 in the area of the disk, how close to the ideal radius must the machinist control the radius?Round down the answer to the nearest hundred thousandth.Select the correct answer.a. cm 08898.0≤δb. cm 08908.0≤δc. cm 08999.0≤δd. cm 08913.0≤δe. cm 09913.0≤δ12. Consider the function x e x f /121)(+=. Find the value of -→0)(lim x x f . Select the correct answer.a. 1.5b. -0.1c. 0.1d. 0.9e. 0.513.Choose an equation from the following that expresses the fact that a function f is continuous at the number 6.Select the correct answer.a. 6)(lim =∞→x x fb. )6()(lim 6f x f x =→c. )6()(lim f x f x =∞→d. 0)(lim 6=→x x fe. ∞=→6)(lim x x f14. Determine where f is discontinuous.()20()30333if x f x x if x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩Select the correct answer.a. 03andb. 0onlyc. 3onlyd. 03and -e. 3only -15. Use the definition of the derivative for find (2)f '-, where 3()2f x x x =-.Select the correct answer.a. 4b. 10c. -4d. -10e. none of these16.If ()g x ()g x '.Select the correct answer.a. ()(),00,-∞⋃∞b. [3/5,3/5]-c. [,3/5)-∞d. (),3/5-∞e. ()0,∞17. Find an equation of the tangent line to the curve 353+-=x x y at the point (2, 1).Select the correct answer.a. 138+=x yb. 139--=x yc. 137-=x yd. 137+-=x ye. 157-=x yStewart - Calculus ET 6e Chapter 2 Form C18. The cost (in dollars) of producing x units of a certain commodity is 201.019571,4)(x x x C ++=. Find theinstantaneous rate of change with respect to x when x = 103. (This is called the marginal cost .)Select the correct answer.a. 26.06b. 20.06c. 21.06d. 18.06e. 31.0619. If the tangent line to )(x f y = at (8, 4) passes through the point (5, -32), find )8(f '.Select the correct answer.a. 24)8(='fb. 20)8(='fc. 12)8(-='fd. 12)8(='fe. 32)8(='f20. At what point is the function ()|6|f x x =- not differentiable.Select the correct answer.a. 6b. 6-c. 1d. 1-e. 0ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form C1. c2. d3. f4.a, b, c, f5. e6. a7. c8. a9. c10. a11. a12. e13. b14. a15. b16. d17. c18. c19. d20. a1. The position of a car is given by the values in the following table.Find the average velocity for the time period beginning when t = 2 and lasting 2 seconds.Select the correct answer.a. 35.5b. 47.5c. 39d. 37.5e. 33.52. The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t ismeasured in seconds. Find the average velocity over the time period [1,3].Select the correct answer.a. 3/m sb. 3.5/m sc. 1/m sd. 1.5/m se. none of these3. Find the limit.()71lim 20++→x x x xSelect the correct answer.a. 0b. 71c. 71- d. -∞ e. ∞4. Find the limit.lim x →-∞Select the correct answer.a. -1b. 0c. 1/2d. -∞e. -1/25.The slope of the tangent line to the graph of the exponential function xy 8= at the point (0, 1) is x x x 18lim 0-→. Estimate the slope to three decimal places.Select the correct answer.a. 1.293b. 2c. 2.026d. 1.568e. 2.079f. 2.5566. Find an equation of the tangent line to curve 32y x x =-at the point (2,4).Select the correct answer.a. 1610y x =-b. 108y x =-c. 16y x =-d. 1016y x =+e. none of these7.Find the limit.()10lim tan 1/x x +-→Select the correct answer.a. 0b. ∞c. /2πd. /3πe. π .8. Let |1|1)(2--=x x x F . Find the following limits.),(lim 1x F x +→ )(lim 1x F x -→Select the correct answer.a. both 2b. 2 and 1c. 2 and – 2d. 2 and – 1e. both 19. Use continuity to evaluate the limit.()x x x sin 4sin lim 13+→πSelect the correct answer.a. π13b. - 1c. 0d. ∞e. 110.Let ()20()30333if x f x xif x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩Evaluate the limit, if it exists.0lim ()x f x -→Select the correct answer.e. 3-11. For what value of the constant c is the function f continuous on ()?,∞∞-⎩⎨⎧>-≤+=2527)(2x for cx x for cx x fSelect the correct answer.a. 1=cb. 2=cc. 6=cd. 2-=ce. 7=c12. Find a function g that agrees with f for 25≠x and is continuous on ℜ.xx x f --=255)(Select the correct answer.a. x x g -=51)(b. x x g +=251)(c. x x g +=51)(d. xx g -=51)( e. x x g -=55)(13.Which of the given functions is discontinuous?Select the correct answer.a. 5,5,3121)(<≥⎪⎩⎪⎨⎧-=x x x x fb. 5,5,351)(=≠⎪⎩⎪⎨⎧-=x x x x fc. Both functions are continuous14.Find the limit. 13lim 232-++∞→t t t tSelect the correct answer.a. ∞b. 0c. 3-d. 3e. 215.If ()g x ()g x '.Select the correct answer.a. ()(),00,-∞⋃∞b. [3/5,3/5]-c. [,3/5)-∞d. (),3/5-∞e. ()0,∞16. The cost (in dollars) of producing x units of a certain commodity is .03.013280,4)(2x x x C ++= Find theaverage rate of change with respect to x when the production level is changed from x = 102 to x = 118.Select the correct answer.a. 29.6b. 19.6c. 18.6d. 26.6e. 24.617. Evaluate the limit.|2|lim 2+-→x xSelect the correct answer.a. 2b. 4c. - 2d. 0e. The limit does not exist18. If a ball is thrown into the air with a velocity of 58 ft/s, its height (in feet) after t seconds is given by .11582t t H -=Find the velocity when t = 4.Select the correct answer.a. 27ft/sb. 30ft/sc. 31ft/sd. 25ft/se. 37ft/s19. Is there a number a such that 626lim 223-++++-→x x a ax x x exists? If so, find the value of a and the value of the limit. Select the correct answer.a. a =14, limit equals 1.4b. a =17, limit equals 1.6c. a =28, limit equals 1.4d. a =28, limit equals 1.6e. There is no such number20.If ()g x ()g x '.Select the correct answer.a. ()1/25()352g x x -'=-- b. ()1/21()352g x x '=-- c. ()2()35g x x '=-- d. ()25()352g x x -'=-- e. none of theseANSWER KEYStewart - Calculus ET 6e Chapter 2 Form D1. d2. a3. e4. e5. e6. e7. c8. c9. c10. a11. c12. c13. b14. b15. d16. b17. d18. b19. d20. a1.A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. If P is the point (15, 263) on thegraph of V, fill the table with the slopes of the secant lines PQ where Q is the point on the graph with the corresponding t .Enter your answer to two decimal places.2. The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t ismeasured in seconds. Find the average velocity over the time period [1,1.5].3. If an arrow is shot upward on the moon with a velocity of 57 m/s, its height in meters after t seconds is given by 282.057t t h -=. Find the instantaneous velocity after one second.Select the correct answer.a. 55.46b. 55.35c. 55.25d. 55.36e. 55.374. Given that, 3)(lim 7-=→x f x and 9)(lim 7=→xg x . Evaluate the limit.)()()(2lim 7x f x g x f x -→5. Find an equation of the tangent line to curve 32y x x =-at the point (2,4).Select the correct answer.a. 1610y x =-b. 108y x =-c.16y x =- d. 1016y x =+ e. none of these6. Let ()20()30333if x f x x if x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩Evaluate the limit, if it exists.0lim ()x f x -→Select the correct answer.e. 3-7. For the function f whose graph is shown, find the following.)(lim 7x f x →8.For x = 5, determine whether f is continuous from the right, from the left, or neither.9. Evaluate the limit.()xx x 11077lim --→-+10. Let |1|1)(2--=x x x FFind the following limits.)(lim ),(lim 11x F x F x x -+→→11. Use a graph to find a number δsuch that 3|0.6< whenever |2|x δ-<.Round down the answer to the nearest hundredth.12. Is there a number a such that 6810lim 223-++++-→x x a ax x x exists? If so, find the value of a and the value of the limit.Select the correct answer.a. a =49, limit equals 1.6b. a =13, limit equals 2.2c. a =49, limit equals 2.2d. a =19, limit equals 1.6e. a =49, limit equals 2.713. How close to 2 do we have to take x so that 5x + 3 is within a distance of 0.025 from 13?14. Find a function g that agrees with f for 25≠x and is continuous on .ℜxx x f --=255)( 15. Use the given graph of x x f =)( to find a number δ such that 4.0|2|<-x whenever .|4|δ<-x16.If ()g x ()g x '.17.If ()g x ()g x '.Select the correct answer.a. ()(),00,-∞⋃∞b. [3/5,3/5]-c. [,3/5]-∞d. (),3/5-∞e. ()0,∞18. At what point is the function |6|)(x x f -= not differentiable.19. How close to - 9 do we have to take x so that ()?10000914>+x20.Find the derivative of the function.25314)(x x x f +-=ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form E1. 5, -42.3, 10, -43.6,20, -18.4, 25, -24.1, 30, -17.532. 2.625 m/s3.d 4.-1/2 5. e6. a7.-∞ 8.neither 9. -1/4910. 2, -211. 81.0≤δ12. c13. 005.0|2|<-x14. ()x g +=5115. 44.1≤δ 16. ()1/25()352g x x -'=-- 17. d18. 619. 1.0|9|<+x20. 310-x1. The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t ismeasured in seconds. Find the average velocity over the time period [1,1.5].2. If a ball is thrown into the air with a velocity of 45 ft/s, its height in feet after t seconds is given by 21545t t y -=. Find the instantaneous velocity when 4=t .3. If 5.4)(lim 3=-→x f x , then if )(lim 3x f x → exists, to what value does it converge?Select the correct answer.a. 6.5b. 4.5c. 1d. 2e. 64. For the function f whose graph is shown, find the limit.)(lim 9x f x +-→5. The function has been evaluated at the given numbers (correct to six decimal places). Use the results to guess the value of the limit.112)(--=x x x f________)(lim 1=→x f xSelect the correct answer.a. 1.255039b. 1.911314c. 1.969944d. 1.473889e. 16.Evaluate the limit.()()104lim 251-+→x x x7. Find the limit.lim x →-∞8.Find the limit.()10lim tan 1/x x +-→9.Evaluate the limit and justify each step by indicating the appropriate properties of limits.393198lim 22-++-∞→x x x x x10. Find an equation of the tangent line to the curve 34x y =at the point ()256,4--.11. Find a number δsuch that if |2|x δ-<, then |48|x ε-<, where 0.1ε=.12. Use a graph to find a number δsuch that 1.021sin <-x whenever δπ<-6x .Round down the answer to the nearest thousandth.13. Use the definition of the limit to find values of δ that correspond to 75.0=ε.Round down the answer to the nearest thousandth.()234lim 31=-+→x x x14. Determine where f is discontinuous.()20()30333if x f x x if x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩15. If f and g are continuous functions with 2)2(=f and [],2)()(2lim 2=-→x g x f x find )2(g .16. Find the limit.)(lim 22bx x ax x x +-+∞→17.State the domain.()sin F x =18.Find the derivative of the function using the definition of derivative.22919)(x x x f +-=19. Find a function g that agrees with f for 4≠x and is continuous on ℜ.xx x f --=42)(20. If an arrow is shot upward on the moon, with a velocity of 70 m/s its height (in meters) after t seconds is given by.99.070)(2t t t H -= With what velocity will the arrow hit the moon?ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form F1. 2.625 m/s2. -753. b4. -∞5. e6. 28125-7. -1/28.2π 9. 38 10. 512192+=x y11. 0.025δ=12. 112.0≤δ13. 085.0≤δ14.03at and 15.2 16. 2b a - 17. ),6[∞18.94-x 19.x g +=2120.-701. If 5.4)(lim 2=-→x f x , then if )(lim 2x f x →exists, to what value does it converge?Select the correct answer.a. 2b. 1c. 5d. 4.5e. 1.52. Consider the following function.()111111)(2≥<≤--<⎪⎩⎪⎨⎧--=x x x x x x x fDetermine the values of a for which )(lim x f ax →exists.3. Evaluate the limit and justify each step by indicating the appropriate properties of limits.443398lim 22-++-∞→x x x x x4. Find )(a f '.233)(x x x f -+=5. Guess the value of the limit.3055tan 3lim xx x x -→Select the correct answer.a. 121b. 135c. 134d. 130e. 1256. Given that 8)(lim 7-=→x f x and 10)(lim 7=→x g x .Evaluate the limit.())()(lim 7x g x f x +→7.Evaluate the limit.()()101lim 231-+→x x x8. Evaluate the limit.⎪⎪⎭⎫⎝⎛--→45lim 233x x x9. Find the derivative of the function using the definition of the derivative.2610)(x x x f +-=10.Let |9|81)(2--=x x x FFind the following limits.)(lim ),(lim 99x F x F x x -+→→Select the correct answer.a. 18 and 9b. 18 and - 18c. both 18d. 18 and – 9e. 81 and 911.Use the given graph of x x f =)(to find a number δsuch that 4.0|2|<-x whenever .|4|δ<-x12. Use a graph to find a number δsuch that 5.0|314|<-+x whenever .|2|δ<-x13. For the limit, illustrate the definition by finding values of δthat correspond to .5.0=ε()234lim 31=-+→x x x14. Find the slope of the tangent line to the curve 35x y = at the point (-4, -320).15. At what point is the function |8|)(x x f -= not differentiable.16.Which of the given functions is discontinuous?a. 5,5,3121)(<≥⎪⎩⎪⎨⎧-=x x x x f b. 5,5,351)(=≠⎪⎩⎪⎨⎧-=x x x x fc. Both functions are continuous17.Select the right number for the following limit and prove the statement using the ,δε definition of the limit. 3183lim 23--+→x x x xSelect the correct answer.a. 6b. 8c. 5d. 9e. 1818.Prove the statement using the ,δε definition of the limit.0|2|lim 2=-→x x19.Prove the statement using the ,δε definition of the limit.()241lim 25=--→x x20.Use continuity to evaluate the limit.()x x x sin 3sin lim 17+-→πSelect the correct answer. a. π17- b. ∞ c. -1 d. 0 e. 1ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form G1. d2. ()()()∞--∞-,11,11,3.38 4.a 61- 5.e 6. 27. -728.22/5 9. 112-x10. b11. 44.1≤δ12. 6875.0≤δ13. 056.0≤δ14. 24015. 816. b17. d18. Given 0>ε, we need 0>δsuch that if | x - 2 | δ< then | | x - 2 | - 0 | ε<. But | | x - 2 | | = | x - 2 |. So this is true ifwe pick .εδ=19. Given 0>ε, we need 0>δsuch that if | x - ( - 5 ) | δ< then | ( x 2 - 1 ) - 24 | ε< or upon simplifying we need | x2 – 25| ε<whenever | x + 5 | δ<. Notice that if | x + 5 | < 1 , then- 1 < x + 5 < 1 - 11 < x - 5 < - 9 | x - 5 | < 11. So take =δmin {ε / 11, 1}. Then | x - 5 | < 11 and | x + 5 | ε</ 11, so | ( x 2 - 1 ) - 24 | = | ( x + 5 ) ( x - 5 ) | = | x + 5 | | x - 5 | < (ε / 11 ) ( 11 ) =ε. Therefore, by the definition of a limit, ().241lim 25=--→x x 20.dStewart - Calculus ET 6e Chapter 2 Form H1. The point P (4, 2) lies on the curve x y =. If Qis the point (,x , use your calculator to find the slope of thesecant line PQ (correct to six decimal places) for the value of 99.3=x .Select the correct answer.a. m PQ = 0.250157b. m PQ = 0.250156c. m PQ = - 0.250154d. m PQ = - 0.250156e. m PQ = 0.2501542. The displacement (in meters) of an object moving in a straight line is given by 212/4s t t =++, where t ismeasured in seconds. Find the average velocity over the time period [1,1.5].3. The displacement (in feet) of a certain particle moving in a straight line is given by 83t s =where t is measured in seconds. Find the instantaneous velocity when t = 3.4. If ,5.7)(lim 2=+→x f x then if )(lim 2x f x →exists, to what value does it converge?Select the correct answer. a. 5 b. 8.5 c. 8 d. 11.5 e. 7.55.If f and g are continuous functions with 3)2(=f and [],5)()(3lim 2=-→x g x f x find ).2(g6. The slope of the tangent line to the graph of the exponential function xy 4=at the point (0, 1) is .14lim 0x x x -→ Estimate the slope to three decimal places. Select the correct answer.a. 1.045b. 1.136c. 0.786d. 1.126e. 1.3867. Find an equation of the tangent line to curve 32y x x =-at the point (2,4).Select the correct answer.a. 1610y x =-b. 108y x =-c. 16y x =-d. 1016y x =+e. none of these8. Find the limit.lim x →-∞9. How close to 2 do we have to take x so that 5x + 3 is within a distance of 0.075 from 13?10. Evaluate the limit and justify each step by indicating the appropriate properties of limits.693958lim 2-++-∞→x x x x x11. Find a number δsuch that if |2|x δ-<, then |48|x ε-<, where 0.01ε=.12. Use the given graph of 2)(x x f =to find a number δsuch that 2112<-x whenever δ<-1x .Round down the answer to the nearest hundredth.13.If ()g x ()g x '.14.If ()g x ()g x '.15.Let ()20()30333if x f x x if x x if x ⎧<⎪⎪=-≤<⎨⎪->⎪⎩Evaluate each limit, if it exists.a.) 0lim ()x f x +→b.) 0lim ()x f x -→16.Which of the given functions is discontinuous?Select the correct answer.a. 5,5,3121)(<≥⎪⎩⎪⎨⎧-=x x x x f b. 5,5,351)(=≠⎪⎩⎪⎨⎧-=x x x x fc. Both functions are continuous17.If a ball is thrown into the air with a velocity of 62 ft/s, its height (in feet) after t seconds is given by21662t t H -=.Find the velocity when t = 5.18.Use continuity to evaluate the limit.()x x x sin 6sin lim 8+→πSelect the correct answer.a. ∞b. - 1c. 1d. 0e. π819. Find a function g that agrees with f for 16≠x and is continuous on ℜ.xx x f --=164)( 20. Consider the function .11)(/1x e x f +=Find the value of )(lim 0x f x +→.Select the correct answer.a. -0.8b. -0.5c. 0.3d. 0e. 0.8ANSWER KEYStewart - Calculus ET 6e Chapter 2 Form H1. b2. 2.625 m/s3. 3.3754. e5. 46. e7. e8. -1/29.015.0|2|<-x 10. 38 11. 0.0025δ=12. 22.0≤δ13. ()1/25()352g x x -'=-- 14. (),3/5-∞15. a.) 3 b.) 016. b17. -9818. d19. x g +=4120. d。

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1、 (1) sin 2limx xx→∞= 0 .(2) d(arctan )x = 21d 1+x x(3)21d sin x x =⎰ -cot +C x x(4).2()()x n e = 22n x e .(5)x =⎰26/32、(6) The right proposition in the following propositions is ___A_____.A. If lim ()x af x →exists and lim ()x ag x →does not exist then lim(()())x af xg x →+does not exist.B. If lim ()x af x →,lim ()x ag x →do both not exist then lim(()())x af xg x →+does not exist.C. If lim ()x af x →exists and lim ()x ag x →does not exist then lim ()()x af xg x →does not exist.D. If lim ()x af x →exists and lim ()x ag x →does not exist then ()lim()x af xg x →does not exist. (7) The right proposition in the following propositions is __B______.A. If lim ()()x af x f a →=then ()f a 'exists.B. If lim ()()x af x f a →≠ then ()f a 'does not exist.C. If ()f a 'does not exist then lim ()()x af x f a →≠.D. If ()f a 'does not exist then the cure ()y f x =does not have tangent at (,())a f a .(8) The right statement in the following statements is ___D_____.A. sin lim 1x xx→∞= B. 1lim(1)x x x e →∞+=C.11d 1x x x C ααα+=++⎰ D. 5511d d 11bb a a x y x y =++⎰⎰ (9) For continuous function ()f x , the erroneous expression in the following expressionsis ____D__.A.d (()d )()d b a f x x f b b =⎰B. d (()d )()d ba f x x f a a =-⎰ C. d (()d )0db a f x x x =⎰ D. d (()d )()()d baf x x f b f a x =-⎰(10) The right proposition in the following propositions is __B______.A. If ()f x is discontinuous on [,]a b then ()f x is unbounded on [,]a b .B. If ()f x is unbounded on [,]a b then ()f x is discontinuous on [,]a b .C. If ()f x is bounded on [,]a b then ()f x is continuous on [,]a b .D. If ()f x has absolute extreme values on [,]a b then ()f x is continuous on [,]a b .3、Evaluate 2011lim()x x e x x →-- 201=lim()x x e x x →--01=lim()2x x e x →-01=lim =22x x e →（考点课本4.4节洛比达法则，每年都会有一道求极限的解答题，大多数都是用洛比达法则去求解，所以大家要注意4.4节的内容。

注意洛比达法则的适用范围。

）4．Find 0d |x y =and (0)y ''if 20x x x y y t e +=+⎰.2'()'x x x y y t e +=+⎰（）1'2()'2()1x x y x y x e y x y x e +=⋅+⇒=⋅+-0(20(0)1)0x dy y e dx dx==⋅⋅+-=''(2()1)'2()2'()x xy x y x e y x xy x e =⋅+-=++200-(0)0-01x x y y t e x y e =+⇒=+=⎰0''02(0)20'(0)=3y y y e =+⋅+（）（考察微积分基本定理与微分，书上5.3节）5、 Find 22arctan d (1)xx x x +⎰=22221)arctan d (1)x x x x x x +-+⎰（22arctan arctan =d d (1)x x x x x x -+⎰⎰-12311=-arctan +d arctan +2x x x x x x -⎰22-1221++1=-arctan +d arctan 1+2x x x x x x x x -⎰（） -12211=-arctan +d d arctan 1+2x x x x x x x x --⎰⎰（） -12211=-arctan +In In 1+arctan 22x x x x x ---121=-arctan arctan +C 2x x x - （凑微分求不定积分，积分是微积分的重点及难点，大家一定要掌握透彻。

）6、 Given that 22()1x f x x =+.(1) Find the intervals on which ()f x is increasing or decreasing.22’22212()1x x x x f x x +-⋅=+（）（）2221x x =+（）When ’()00f x x >⇒>’()00f x x <⇒< Therefore, the increasing interval is ()0+∞，, the decreasing interval is ()0-∞，(2) Find the local maximum and minimum values of ()f x’()00f x x =⇒= The function is increasing in interval ()0+∞，, decreasing ininterval ()0-∞，, therefore, the function exist the local minimum value, it is ()0f x =(3)Find the intervals of concavity and the inflection points.'22224222242422181642''()111x x x x x x f x x x x +-+--+⎛⎫=== ⎪+++⎝⎭（）（）（）（）（）4224642''()0133x x f x x x x --+=>⇒><-+（）4224642''()000133x x f x x or x x --+=<⇒-<<<<+（）1''(4f f =Therefore, the concave upward interval are ⎛-∞ ⎝⎭，,⎫+∞⎪⎪⎝⎭, the concave downwardinterval are 0⎛⎫ ⎪ ⎪⎝⎭,0⎛ ⎝⎭and the inflection points are 14⎛⎫ ⎪ ⎪⎝⎭，,14⎫⎪⎪⎝⎭，(4) Find the asymptote lines of the cure ()y f x =2221lim =1111+x x x x→∞=+ T herefore, the liney = 1 is a horizontal asymptote（考点：4.3节，4.5、4.6节。

近几年经常会考一道作图题。

这种题目应该在注意的点主要包括函数的定义域，对称性，增减区间，极值点，凹凸性，拐点，以及渐近线等。

大家参照课本的4.5节进行作图）7、Let R be the region bounded by the curve 1y x=, and the line y x = and 2x =. (a)Evaluate the area of the region R. R=211x dx x -⎰2211=In 2x x -2211=2In21In122⋅--⋅+3=In22- (b)Find the volume of the solid generated by revolving the R about the y -axis .V=212121214)?4)y dy dy y ππ-+-⎰⎰（（12311211443x y y y π⎛⎫ ⎪=-++ ⎪⎝⎭33111142241141423312π⎛⎫=⋅-⋅-⋅+⋅+⋅+-⋅- ⎪⎝⎭83π=（考点：求面积以及体积，课本6.1、6.2节。

这类题目是常考题，较简单。

望同学一定要做相应的题目加以巩固。

）8、 Determine the production level that will maximize the profit for a company with cost and demand functions23()1450360.580.001C x x x x =+-+and ()600.01p x x =-.Solution 2()(600.01)600.01R x x x x x=-=-223()()()600.01(1450360.580.001)P x R x C x x x x x x =-=--+-+320.0010.57241450x x x =-++-'2()0.003 1.142P x x x =-++ Let '()040020P x x or x =⇒==- since x>0, then x=400''()0.006 1.14P x x =-+When x=400 ''()0.006400 1.14 1.260P x =-⋅+=-<32(400)0.0014000.5740024400145035350P =-⋅+⋅+⋅-=Therefore, when the production level is 400 that will maximize the profit 35350（考点：经济函数，课本4.8节。