英文版微积分考试样题2
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2
Factoring the numerator creates: (x -1)(x2 +x+1), thus the
(x – 1) cancels, leaving
limx2 x 1 3 x1
This indicates that the function values are getting very, very, very close to 1 as the x values approach 3.
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Rigorous Study of Limits
Definition of Limit: L is the limit of f(x) as x approaches c if for any value of epsilon (no matter how small) around f(x)=A, there’s a value of sigma around x=c, such that all function values fall into the “target area” of epsilon about L.
08-09(2)Calculus(II)_Exam_Paper(B)参考答案
= −2(m tan m + ln |cos m|) + C (C ∈ R) √ √ = −2 arctan(arcsin 1 − x) arcsin 1 − x + ln √ 2. 交换积分顺序。先对y 积分，再对x积分。 ∫ π ∫ sin x √ 2 Σ = dx cos x 1 + cos2 xdy ∫ = ∫ = =
所以有： x2 z 2 + 2yz y2 = − 2 z + 2xz = − 3
∂y 要求dy ，即求y = y (x, z )的全微分，类似上面求出 ∂x 和 ∂y ∂z （由x， y ， z 的对称性易得） ，所以有：
dy = =
∂y ∂y dx + dz ∂x ∂z −x2 −z 2 dx + dz y 2 + 2yz y 2 + 2yx
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2. 易知过(1, e)点的切线斜率为e，所以切线为y − e = e(x − 1)，即y = ex。 积分区域即为由y = ex 、y = 0、y = ex围成的区域。 ∫ ∫ S = dxdy ∫ 1 ∫ ex dy dx = 0 ex ∫ 1 1 = (ex − ex)dx = e − 1 2 0 3. 沿x轴做切面，得到是一个圆环带，外环是ex ，内环是ex，则每个切 面的面积是π [(ex )2 − (ex)2 ]。再沿x轴累加即得体积。 ∫ V =
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。
微积分练习2
Miscellaneous exercise 81、Each of figures (i) and (ii) shows part of a straight line graph obtained by plotting values of the variables indicated, together with the coordinates of two points on the line. For each case express y as a function of x.2、The variables x and y are related by the equation y = h . The diagram shows the graph of 1 against x. Calculate2x + k ythe values of h and k. The point P lies on the line. Find the value of r.3、Variables x and y arew related by the equation a + b = 2,where a and b are constants. When the graph of 1 against 1 isx y y x drawn, a straight line is obtained. Given that the intercept on the 1 - axis is -0.5 and that the gradient of the line is 0.75 ,ycalculate the value of a and of b.ax where a and b are unknown constants.4、It is expected that the variables x and y are related by the equation y = bGiven sets of experimental values of x and y, determine the variables whose values should be plotted in order to obtain a straight line graph, and explain how the graph may be used to determine the values of a and b.5、The graph of x2y against x is a straight line passing through (-2,1) and (2,7). Find(a) the value of x when y = 10/x2. (b) y in terms of x.6、Variables x and y are related by an equation of the form px2 + qy2 = 2, where p and q are constants. Observed values of the two variables are shown below.x 1 2 3 4 5y 1.83 3.06 4.40 5.77 7.16 By plotting y2 against x2, draw a straight line graph and use it to estimate(a) the value of p and of q, (b) the positive value of x when y = 5.7、The table shows experimental values of two variables x sand y.x 0.5 1.0 1.5 2.0y 14.6 6.8 4.0 2.4It is expected that x and y are related by an equation of the form y = ax + b/x where a and b are constants. Express this equation in a form suitable for drawing a straight line graph.(a)the value of a and of b, (b) the value of y when x = 1.7.8、The variables x and y are known to be connected by the equation y = C a-x . An experiment gave pairs of values of x and y as shown in the table. One of the values of y is subject to an abnormally large error.x 1 2 3 4 5 6 7y 56.20 29.90 25.10 8.91 6.31 3.35 1.78 Plot lg y against x and use the graph to(a) identify the abnormal reading and estimate its correct value,(b) estimate the value of C and of a,(c) estimate the value of x when y = 1.9、Pairs of numerical values (x, y) are collected form an experiment and itis possible that either of the two following equations may be applicable to these data:(a) ax2 +by2 = 1, where a and b are constants. (b) y = cx d, where c and d are constants.10、The table shows experimental values of two variables t and y.t 1 2 3 4 5y 12.2 7.0 4.0 2.3 1.3It is known that t and y are related by an equation of the form y = Ae-bt. By plotting ln y against t, draw a straight line graph for the given data and use it to evaluate A and b.11、It is assumed that x and y obey a law of the form y = ln (ax2 + b). Pairs of values of x and y are recorded in an experiment and tabulated as follows:x 0.2 0.4 0.6 0.8 1.0y 0.322 0.482 0.703 0.948 1.194By means of a straight line graph, verify that the law is valid. Use your graph to estimate(a)the values of a and b, (b) the possible values of x if y = ln 3, (c) the value of y if x = 0.1.12、The variables x and y are connected by the relation y = k(ep)x, where k and p are constants. The graph of ln y against x is given. Find, to 3 significant figures, the values of k and p.13、The period T of oscillation of a pendulum and its length l are related by a law of the form T = al b.Using the following experimental data, estimate(a) a and b,(b) the period of oscillation of a pendulum of length 0.9 m,(c) the length of a pendulum whose period is 1 second.l(metres) 0.4 0.6 0.8 1.0 1.2T(seconds) 1.25 1.55 1.76 2.01 2.1914、It is believed that variables x and y follow a law of the form x + py = qxy. In an experiment, values of y are found for certain values of x. These values are shown in the following table.x 1 2 3 4 5y 0.67 0.91 1.20 1.33 1.43It is suspected that an unusually large error occurs in one of the values of y.By plotting the graph of x/y against x,(a) identify the incorrect value of y and estimate its correct value,(b) estimate the value of x when 5y = 4x,(c) estimate the values of p and q.15、Variables x and y are known to be related by an equation of the form axy -b = a(x2 + bx), where a and b are constants. Observed values of the two variables are shown in the following table:x 1 2 3 4y 4 2.50 2.67 3.25Plot xy - x2 against x and use the graph to estimate(a) the value of y when x = 1.5, (b) the values of a and b.16、The following values of x and y are believed t obey a law of the form x m y n = 200, where m and n are constants.x 1 3 5 7y 14.15 2.72 1.26 0.76Rewrite the given law in a form suitable for drawing a straight line graph. Draw the graph and hence estimate(a) the value of m and of n, (b) the value of x when y =10.。
微积分英文版电子档
Differential CalculusNewton and Leibniz,quite independently of one another,were largely responsible for developing the ideas of integral calculus to the point where hitherto insurmountable problems could be solved by more or less routine methods.The successful accomplishments of these men were primarily due to the fact that they were able to fuse together the integral calculus with the second main branch of calculus,differential calculus.The central idea of differential calculus is the notion of derivative.Like the integral,the derivative originated from a problem in geometry—the problem of finding the tangent line at a point of a curve.Unlile the integral,however,the derivative evolved very late in the history of mathematics.The concept was not formulated until early in the 17th century when the French mathematician Pierre de Fermat,attempted to determine the maxima and minima of certain special functions.Fermat’s idea,basically very simple,can be understood if we refer to a curve and assume that at each of its points this curve has a definite direction that can be described by a tangent line.Fermat noticed that at certain points where the curve has a maximum or minimum,the tangent line must be horizontal.Thus the problem of locating such extreme values is seen to depend on the solution of another problem,that of locating the horizontal tangents.This raises the more general question of determining the direction of the tangent line at an arbitrary point of the curve.It was the attempt to solve this general problem that led Fermat to discover some of the rudimentary ideas underlying the notion of derivative.At first sight there seems to be no connection whatever between the problem of finding the area of a region lying under a curve and the problem of finding the tangent line at a point of a curve.The first person to realize that these two seemingly remote ideas are,in fact, rather intimately related appears to have been Newton’s teacher,IsaacBarrow(1630-1677).However,Newton and Leibniz were the first to understand the real importance of this relation and they exploited it to the fullest,thus inaugurating an unprecedented era in the development of mathematics.Although the derivative was originally formulated to study the problem of tangents,it was soon found that it also provides a way to calculate velocity and,more generally,the rate of change of a function.In the next section we shall consider a special problem involving the calculation of a velocity.The solution of this problem contains all the essential fcatures of the derivative concept and may help to motivate the general definition of derivative which is given below.Suppose a projectile is fired straight up from the ground with initial velocity of 144 fee t persecond.Neglect friction,and assume the projectile is influenced only by gravity so that it moves up and back along a straight line.Let f(t) denote the height in feet that the projectile attains t seconds after firing.If the force of gravity were not acting on it,the projectile would continue to move upward with a constant velocity,traveling a distance of 144 feet every second,and at time t we woule have f(t)=144 t.In actual practice,gravity causes the projectile toslow down until its velocity decreases to zero and then it drops back to earth.Physical experiments suggest that as the projectile is aloft,its height f(t) is given by the formula.The term –16t2 is due to the influence of gravity.Note that f(t)=0 when t=0 and whent=9.This means that the projectile returns to earth after 9 seconds and it is to be understood that formula (1) is valid only for 0<t<9.The problem we wish to consider is this:T o determine the velocity of the projectile at each instant of its motion.Before we can understand this problem,we must decide on what is meant by the velocity at each instant.T o do this,we introduce first the notion of average velocity during a time interval,say from time t to time t+h.This is defined to be the quotient. Change in distance during time interval =f(t+h)-f(t)/h.ength of time intervalThis quotient,called a difference quotient,is a number which may be calculated whenever both t and t+h are in the interval[0,9].The number h may be positive or negative,but not zero.We shall keep t fixed and see what happens to the difference quotient as we take values of h with smaller and smaller absolute value.The limit process by which v(t) is obtained from the difference quotient is written symbolically as follows:The equation is used to define velocity not only for this particular example but,more generally,for any particle moving along a straight line,provided the position function f is such that the differerce quotient tends to a definite limit as h approaches zero.The example describe in the foregoing section points the way to the introduction of the concept of derivative.We begin with a function f defined at least on some open interval(a,b) on the x axis.Then we choose a fixed point in this interval and introduce the differencequotient[f(x+h)-f(x)]/h.Where the number h,which may be positive or negative(but not zero),is such that x+h also lies in(a,b).The numerator of this quotient measures the change in the function when x changes from x to x+h.The quotient itself is referred to as the average rate of change of f in the interval joining x to x+h.Now we let h approach zero and see what happens to this quotient.If the quotient.If the quotient approaches some definite values as a limit(which implies that the limit is the same whether h approaches zero through positive values or through negative values),then this limit is called the derivative of f at x and is denoted by the symbol f’(x) (read as ―f prime of x‖).Thus the formal definition of f’(x) may be stated a s follows Definition of derivative.The derivative f’(x)is defined by the equation。
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)。
山东财经大学大二经济学专业微积分英文试卷及答案2
山东财经大学XX －XX 第一学期期末考试试卷试卷代码：XX 授课课时：48课程名称：微积分I 适用对象：XX 国际学院本科生 试卷命题人：XX 试卷审核人 XX1． (1535=⨯pts) Fill in the blanks:（1）0sin 2lim 1x x kx→=,when k =____________. (2) 1lim(1)2x x x→∞+=____________. (3) If (0)0f =and (0)1f '=, then 0(3)limln(1)h f h h →=+___________. (4) 21()x x d e ==______________. (5) 21=⎰___________. 2． (1535=⨯pts) Choose the best answer for each of the followings:(1) If lim ()x a f x → exists , and lim ()x a g x → does not exist ,then lim[()()]x af xg x →⋅___________. (A) exists (B). may exists or does not exist(C) does not exist (D)the above three statements are false. (2) 2223lim 25x x x ax →∞++=+,when a = ___________. (A) 3 (B) 2(C) 12(D) 13 (3) If 21()1f x x '=+ ,then ()f x =_________ (A) sin x c + (B) cos x c +(C) arctan x c + (D) arccot x c +(4) If ()f x is differentiable , then (ln(2()))f x '=___________.(A) ()()f x f x ' (B) ()2()f x f x ' (C)2()()f x f x ' (D) 2()f x(5) If 20()x F x =⎰, then )(x F '___________.(A) (B)2(C) 23．(7pts) Evaluate 4214lim x x e x x→∞-- 4．(9pts) Evaluate 210lim (cos )x x x +→. 5．(8pts)Find the derivative of the function 211()sin f x x x-=. 6．(10pts) Calculate y '' given that y is defined implicitly as a differentiable function of x by theequation x e y x y =+.7．(8pts) Find A and B given that 1()1x f x Ax B x ≤=+>⎪⎩ is differentiable everywhere. 8．(8pts)Determine the production level that will maximize the profit for a company with cost andprice functions ()1002C x x =+ and ()30p x x =-.9．(12pts) Sketch the graph of 22(1)x y x -=-. 10．(8pts) Suppose that )(x f is continuous on ]1,0[ and differentiable on )1,0(, and0)1()0(==f f . Show that there is at least a number (0,1)c ∈such that()sin ()0f c c f c '-⋅=山东财经大学XX －XX 第一学期 期末考试参考答案与评分标准 试卷代码：XX 授课课时：48课程名称：微积分Ⅰ 适用对象：XX 级国际学院2． (1535=⨯pts) Fill in the blanks:(1)2k = (2) 12e (3) 3 (4) 22e dx(5) 1)2． (1535=⨯pts) Choose the best answer for each of the followings:(1) B (2) C (3) C (4) A (5) B3．(7pts)4421444lim lim 3216lim 222x x x x xx e x e x xe →∞→∞→∞---===∞分分分 4．(9pts)2200ln(cos )1lim 01(sin )cos lim 212lim(cos )333x x x x x x x x x x e ee +→++→→⋅--===分分分. 5．(8pts)122111()2sin ()612sin 8f x x x x x x x --'=+-=分分.6．(10pts) 2214161()(1)(1)8(1)()(1)(1)10(1)x x x xx x x x xx x x x x xx e y e y y e y y e e y e y e e y e y e e y e y e e y e e ''+=+-'=-'+-+-⋅''=-'+-+-⋅=-分分分分 7．(8pts)11111lim()1lim (1)24lim 5()2lim lim 7118x x x x x Ax B A B f A B Ax B Ax A A x x A B +--++→→→→→+=+==+==+--==--==分分分分分 8．(8pts). 22()()302()()()281002()2280228()202R x xp x x x P x R x C x x x P x x x P x ==-=-=-+-'=-+==''=-<分分分分9． (12pts) 222211223334422lim0lim 0(1)(1)22lim lim 2(1)(1)11144()1(1)042122(6)066x x x x x x x x x x x x x y x x x x xy x x y x x x y x +-→∞→-∞→→--==----=-∞=-∞---'=-=-+=--'==-''=-=''==分分.10．(8pts) Setting auxiliary function cos ()()x F x e f x -=⋅, …… 2’ Thus, by the known conditions we have that )(x F is continuous on [0,1]and differentiable on (0,1), and (0)(1)0F F ==. …… 5’Hence, there exists one number at least (0,1)c ∈for which 0)(='c F by Rolle ’s theorem,cos ()sin ()0c f c c f c '⋅-⋅=. That is ()sin ()0f c c f c '-⋅= …… 8’。
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。
英文版-微积分试卷答案汇编
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。
微积分 高等数学 英文练习题以及答案(1)
lim
n
sin n 0. n
(n 1, 2, ) 。
n
n
(4) xn sin
n ； 2
1 证：因为 sin x 1，所以 sin n n1 有界，而 lim 0 ，根 n n
据题目已知的结论有：
则 xn 1,
(n 1, 2, ) ，lim xn 1 ， 但是 lim xn 不存在。
3
Exercise 1-2 Limits of Sequences 1. Observe the behavior of the general term of each of the following sequences as n increases infinitely, determine the sequences that have limits and write out the limit if it exists: (1) xn
(7) xn cos
1 ； n
极限存在，为 1。
n 4 n 4 n 4 4 1 2 0.1 2 n n n( n 4 n) 2n
2 2
只要 n 20或n 5 即可。
2
xn a n .
(8) xn ln
1 . n
所以取 N 5 ， 则当 n N ， xn a 即 成立。
Xi’an Jiaotong-Liverpool University
Subgroup
Name
ID No.
1
Preliminaries: Functions 1. Find the natural domains of the following functions: (1)
微积分 高等数学 英文练习题以及答案(2)
Exercise 2-1 Concept of Derivative1. The motion of an object along the s -axis follows the law2s t t =+(m). Find:(1) the average speed of the object during the time interval from the 1st second to the 2nd second; 解：2(2)(1)(22)(11)421s s v -==+-+=- (m/sec)(2) the Instantaneous velocity of the object at the 2nd second.解：()()12v t s t t '==+； 所以(2)(2)5v s '== (m/sec)2. Compare the following limits with the definition of the derivative and then point out the relation between A and 0()f x '. (Assume that ()0x f ' exists) (1) ()()A xx f x x f x =∆-∆-→∆000lim ；解：()()()0000lim x fx x fx A f x x∆→-∆-'=-=--∆(2) ()A x f n x f n n =⎥⎦⎤⎢⎣⎡-⎪⎭⎫ ⎝⎛+∞→001lim解：()()0001lim1n f x f x n A f x n→∞⎡⎤⎛⎫+- ⎪⎢⎥⎝⎭⎣⎦'==(3) ()()00limh fx h fx h A h→+--=.(Hint:()()()()()()000000f x h f x h f x h f x f x h f x hhh+--+---=-)解：()()()()00000lim →+---⎛⎫=- ⎪⎝⎭h f x h f x f x h f x A h h ()()()()()000000lim 2→+---⎛⎫'=+= ⎪-⎝⎭h f x hf x f x h f x f x h h3. Find 0()lim sin 2→x f x x, if (0)0f = and (0)2f '=.解：00()()(0)lim lim sin 20sin 2→→-=⋅-xx f x f x f xx x x00()(0)lim lim 0sin 21(0)12→→-=⋅-'=⋅=x x f x fxx x f4. Find derivatives of the following functions by using the derivative formula of power functions:(1)y =解：32==y x ，所以311223322-'==y x x(2)y =； 解：13-==y x，所以141331133---'=-=-y xx(3)3y x =解：1163355+===y x x x，所以1125161655'==y x x5. Take two points with the abscissa 11=x and 33=x on the parabola 2x y =and draw a secant line through the two points. Find the point on the parabola at which the tangent line to the curve is parallel to this secant line.解：当11=x 时，11y =；当23x =时，29y =。
微积分英文版第八版
微积分英文版第八版1、He has made a lot of films, but ____ good ones. [单选题] *A. anyB. someC. few(正确答案)D. many2、If you do the same thing for a long time, you'll be tired of it. [单选题] *A. 试图B. 努力C. 厌倦(正确答案)D. 熟练3、23．Hurry up! The train ________ in two minutes. [单选题] * A．will go(正确答案)B．goC．goesD．went4、( ) You had your birthday party the other day,_________ [单选题] *A. hadn't you?B. had you?C. did you?D. didn't you?(正确答案)5、If you want to be successful one day, you have to seize every _______ to realize your dream. [单选题] *A. changeB. chance(正确答案)C. chairD. check6、98．There is a post office ______ the fruit shop and the hospital. [单选题] *A．atB．withC．between(正确答案)D．among7、If you get _______, you can have some bread on the table. [单选题] *A. happyB. hungry(正确答案)C. worriedD. sad8、--It is Sunday tomorrow, I have no idea what to do.--What about _______? [单选题] *A. play computer gamesB. go fishingC. climbing the mountain(正确答案)D. see a film9、My brother is _______ actor. He works very hard. [单选题] *A. aB. an(正确答案)C. theD. one10、He always did well at school _____ having to do part-time jobs every now and then. [单选题] *A despite ofB. in spite of(正确答案)C. regardless ofD in case of11、11．No one ________ on the island(岛). [单选题] *A．liveB．lives(正确答案)C．livingD．are living12、The manager gave one of the salesgirls an accusing look for her（）attitude towards customers. [单选题] *A. impartialB. mildC. hostile(正确答案)D. opposing13、12．Who will ________ the Palace Museum after Shan Jixiang retires? [单选题] *A．in chargeB．in charge ofC．be in charge of (正确答案)D．be in the charge of14、I repeated my question several times. [单选题] *A. 到达B. 惊奇C. 重复(正确答案)D. 返回15、My camera is lost. I am ______ it everywhere.（）[单选题] *A. looking atB. looking for(正确答案)C. looking overD. looking after16、Tony is a quiet student, _______ he is active in class. [单选题] *A. soB. andC. but(正确答案)D. or17、—Who came to your office today, Ms. Brown?—Sally came in. She hurt ______ in P. E. class. （）[单选题] *A. sheB. herC. hersD. herself(正确答案)18、_____ whether robots will one day have vision as good as human vision. [单选题] *A. What is not yet knownB. It is not yet known(正确答案)C. As is not yet knownD. This is not yet known19、Last week they _______ in climbing the Yuelu Mountain. [单选题] *A. succeeded(正确答案)B. succeedC. successD. successful20、( )He gave us____ on how to keep fit. [单选题] *A. some advicesB. some advice(正确答案)C. an adviceD. a advice21、The storybook is very ______. I’m very ______ in reading it. （）[单选题] *A. interesting; interested(正确答案)B. interested; interestingC. interested; interestedD. interesting; interesting22、The blue shirt looks _______ better on you than the red one. [单选题] *A. quiteB. moreC. much(正确答案)D. most23、10．Mum, let me help you with your housework, so you ________ do it yourself. [单选题] * A．don’t need to(正确答案)B．need toC．don’t needD．need24、We should _______ a hotel before we travel. [单选题] *A. book(正确答案)B. liveC. stayD. have25、—Where are you going, Tom? —To Bill's workshop. The engine of my car needs _____. [单选题] *A. repairing(正确答案)B. repairedC. repairD. to repair26、The boy lost his（）and fell down on the ground when he was running after his brother. [单选题] *A. balance(正确答案)B. chanceC. placeD. memory27、The car is _______. It needs washing. [单选题] *A. cleanB. dirty(正确答案)C. oldD. new28、If you don’t feel well, you’d better ask a ______ for help. [单选题] *A. policemanB. driverC. pilotD. doctor(正确答案)29、Guilin is _______ its beautiful scenery. [单选题] *A. famous for(正确答案)B. interested inC. fond ofD. careful with30、—Where ______ you ______ for your last winter holiday?—Paris. We had a great time. （）[单选题] *A. did; go(正确答案)B. do; goC. are; goingD. can; go。
高三英语微积分基础单选题20题
高三英语微积分基础单选题20题1. In the function f(x) = 3x^2 + 5x - 2, the derivative of f(x) at x = 1 is:A. 11B. 8C. 10D. 12答案：A。
本题考查导数的基本运算。
首先对函数f(x) 求导得到f'(x) = 6x + 5，将x = 1 代入f'(x)，得到f'(1) = 6×1 + 5 = 11。
选项B 计算错误；选项C 和D 也是计算错误。
2. If the integral of f(x) from 0 to 2 is 10, and the integral of f(x) from0 to 1 is 4, then the integral of f(x) from 1 to 2 is:A. 6B. 8C. 14D. 16答案：A。
根据定积分的性质，从a 到b 的积分等于从a 到c 的积分加上从 c 到 b 的积分。
所以从1 到 2 的积分为从0 到 2 的积分减去从0 到1 的积分，即10 - 4 = 6。
选项B、C、D 计算错误。
3. The slope of the tangent line to the curve y = x^3 at the point (1, 1) is:A. 1B. 3C. 2D. 4答案：B。
对y = x^3 求导得y' = 3x^2，将x = 1 代入得斜率为3×1^2 = 3。
选项A、C、D 计算错误。
4. The area under the curve y = 2x + 1 from x = 1 to x = 3 is:A. 10B. 12C. 8D. 14答案：A。
先求出定积分，∫(2x + 1)dx = x^2 + x，代入上限3 和下限1，得到(3^2 + 3) - (1^2 + 1) = 12 - 2 = 10。
九年级微积分入门英语阅读理解25题
九年级微积分入门英语阅读理解25题1<背景文章>Calculus is a fundamental branch of mathematics that has had a profound impact on various fields of study. It consists of two main concepts: derivatives and integrals.The derivative can be thought of as the rate of change of a function. For example, if we have a function that describes the position of an object over time, the derivative of that function will give us the velocity of the object. Geometrically, the derivative represents the slope of the tangent line to a curve at a given point. The concept of the derivative was developed independently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century. Newton was mainly interested in using calculus to solve problems in physics, such as the motion of celestial bodies. Leibniz, on the other hand, developed a more formal notation for calculus that is still widely used today.Integrals, on the other hand, are related to the concept of accumulation. If we have a function that represents a rate of change, for example, the rate at which water is flowing into a tank, the integral of that function over a certain time interval will give us the total amount of water that has accumulated in the tank during that time. Integrals can also be used to findthe area under a curve.In mathematics, calculus is used in many areas such as solving differential equations, which are equations that involve derivatives. These equations are used to model a wide variety of phenomena, from the spread of diseases to the behavior of electrical circuits. In physics, calculus is essential for understanding concepts like motion, force, and energy. In engineering, it is used for designing structures, analyzing control systems, and optimizing processes. In economics, calculus can be used to analyze functions such as cost, revenue, and profit.1. <问题1>What does the derivative of a function represent?A. The total amount of a quantity.B. The rate of change of a function and the slope of the tangent line to a curve at a given point.C. The area under a curve.D. The maximum value of a function.答案：B。
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∫
A) x2 + 3x +
3 ln 10x - 1 + C 10
B) x2 - 3x -
3 ln 10x - 1 + C 10
C) x2 - 3x + ln 10x - 1 + C
பைடு நூலகம்
D) 2x + 3 - ln 10x - 1 + C
Find the value of the derivative of the inverse of the function. 5) Given that f(x) = x3 - 4x + 5, - 1 ≤ x ≤ 1, find (f- 1)'(5) A) 1 8 B) 1 4 C) 1 13 D) 1 2
12) y = 8 4x2 + 9x A) 88x + 9 · ln 8 C) (8x + 9) 8 4x2 + 9x · log8 x dy by using logarithmic differentiation. dx 13) y = (cos x)x A) ln x(cos x)x- 1 C) (cos x)x (ln cos x - x tan x) B) ln cos x - x tan x D) (cos x)x (ln cos x + x cot x) B) (8x + 9) 8 4x2 + 9x · ln 8 D) (8x + 9) 8 4x2 + 9x
29)
∫ 3 cos4 6x dx
A) C) 1 3 cos3 6x sin 6x + sin 12x + C 8 32 1 1 3 cos3 6x sin 6x + x + sin 6x + C 4 8 32 B) D) 1 1 3 cos2 6x sin 6x + x + sin 12x + C 8 8 16 9 1 1 x + sin 12x + sin 24x + C 8 8 64
30)
∫ 7 sec4 x dx
A) C) 7 tan3 x + C 3 B) 7(sec x + tan x)5+ C D) 7 tan x + 7 tan3 x + C 3 7 tan3 x + C 3
31)
∫
sin 6x cos 4x dx A) C) 1 1 cos 10x sin 10 x + C 20 20 1 1 cos 10x - cos 2x + C 20 4 B) D) 1 1 sin 2x + sin 10x+ C 4 20 1 1 sin 2x sin 10x+ C 4 20
Evaluate the integral by completing the square. dx 22) x2 + 6x + 18
∫
A) 2x + 6 ln x2 + 6x + 18 + C C) 1 x+3 tan- 1 3 3 + C
x+3 B) 3 tan- 1 3 D) 1 -1 x + 3 sin 3 3
+ C + C
Evaluate the integral by reducing (dividing) the improper fraction and using a substitution if necessary. 4t3 + t2 + 64t 23) dt t2 + 16
∫
A) 4t - ln t2 + 16 + C C) 2t2 + t + tan-1 t +C 16
Find the derivative of y with respect to x, t, or θ, as appropriate. 6) y = e7 - 9x A) - 9 ln (7 - 9x) Find the derivative. 7) g(x) = e3x cos(2x) A) e3x cos(2x) - 2e3x sin(2x) C) 3e3x cos(2x) + e3x sin(2x) Evaluate the integral. 8) B) - 3e3x sin(2x) D) 3e3x cos(2x) - 2e3x sin(2x) B) -9e7 - 9x C) e- 9 D) 7e7 - 9x
10)
∫
e1/x dx 4x2 A) e1/x +C 4 B) e-1/x +C 4 C) e1/x +C 4 D) - 4 e1/x + C
Find
dy . dx 11) y = log7(x2 - 6x - 2) A) 2x - 6 (x2 - 6x - 2) ln 7 B) 2x - 6 (x2 - 6x - 2) C) 1 (x2 - 6x - 2) ln 7 D) (2x - 6) ln 7 (x2 - 6x - 2)
MATH 2414 - Sample MC problems for the Final Exam Find the derivative of y with respect to x, t, or θ, as appropriate. 1-x 1) y = ln Use log properties to split it up, differentiate, and find a common denominator. (x + 2)4 A) (x + 2)4 1-x B) 3x - 6 (x + 2)5 C) ln 5x - 6 (x + 2)5 D) 3x - 6 (x + 2)(1 - x)
Find the derivative. 2) y = ln (ln 8x) 1 A) 8x Evaluate the integral. cos x dx 3) 1 + 2 sin x
B)
1 ln 8x
C)
1 x ln 8x
D)
1 x
∫
A) 2 sin x + C C) ln 1 + 2 sin x + C
27)
∫
π/2 x sin 3x dx A) - 9.0000 B) -0.1111 C) 0.1111 D) 0.5236
0
28)
∫ 6 cos3 3x dx
A) 2 sin 3x C) 2 sin 3x 2 cos3 3x + C 3 2 sin3 3x + C 3 B) 2 sin 3x + 2 sin3 3x + C 3 D) 6 sin 3x - 2 sin3 3x + C
25)
∫ x3 ln 3x dx
A) 1 4 1 5 x ln 3x x +C 4 20 1 4 x +C 4 B) D) 1 4 1 4 x ln 3x x +C 4 16 1 4 1 4 x ln 3x + x +C 4 16 C) ln 3x -
26)
∫ arccos x dx
A) x arccosx + C) x arccosx 1 - x2 + C 1 - x2 + C B) x arccosx - 2 1 - x2 + C D) x arccosx 1 1 - x2 +C
B) 2t2 + t - 4tan- 1
t +C 4
D) 2t2 + t - ln t2 + 16 + C
Evaluate the integral. 24)
∫ e2x x2 dx
A) C) x2 2x x 2x 1 2x e - e + e +C 2 2 4 x2 2x x 2x e - e +C 2 2 B) D) x2 2x 1 e - xe2x + e2x + C 2 4 x2 2x x 2x 1 2x e - e + e +C 2 4 4
5x B) 2sec- 1 x +C ln 5 D) 2sin- 1 x - 5 x + C
Find Dx y. 18) y = tanh2 9x A) 18 tanh 9x sech2 9x C) 18 tanh2 9x sech 9x Find the derivative of y. 19) y = ln (sinh 2x) A) 2 coth 2x B) 1 sinh 2x C) coth 2x D) 2 csch 2x B) - 18 tanh 9x csch2 9x D) 2 tanh 9x sech2 9x
x2 - 4 x - sec- 1 2 2
34)
∫
dx 2 (x + 25)3/2 A) x 5 25 + x2 x 25 25 - x2 +C 25 - x2 +C x B) x 25 25 + x2 5 x 25 - x2 +C
C)
+
D)
+C
Evaluate the integral by partial fractions. dx 35) 2 x x2 - 25
B) 2 ln 1 + 2 sin x + C D) 1 ln 1 + 2 sin x + C 2
Evaluate the integral by reducing (dividing) the improper fraction and using a substitution if necessary. 20x2 + 28x 4) dx 10x - 1