英文版微积分考试样题3

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微积分英文版3

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( x3 4cos x sin1) + x ( 3 x2 + 4sin x )
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有限次四则运算的求导法则
(u ± v)′ = u′ ± v′ (uv)′ = u′v + uv′
(Cu)′ = Cu′ ( C为常数 ) u ′ u′v uv′ (v ≠ 0) ( )= 2 v v
2. 曲线的切线斜率
曲线
在 M 点处的切线割线 M N 的极限位置 M T (当时)
y
y = f (x) N
C M
T
切线 MT 的斜率
o α x0
x x
= lim tan
f (x) f (x0 ) 割线 M N 的斜率 tan = x x0
→α
f (x) f (x0 ) k = lim x x0 x→x0
例.
y = x ( x 4cos x sin1) ,
3
3 解: y′ = ( x )′ ( x 4cos x sin1)
( x3 4cos x sin1)′ + x
2 x 1 y′ x=1 = (1 4cos1 sin1) + (3 + 4sin1) 2 7 7 = + sin1 2cos1 2 2
函数的可导性与连续性的关系
定理. 定理证: 设存在 , 因此必有其中故所以函数在点 x 连续 . 在点 x 处可导, 即
x →0
y= x
y
注意: 连续未必可导. 注意函数在点 x 连续未必可导反例: 反例在 x = 0 处连续 , 但不可导.

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。

Denis Auroux -MIT多变量微积分重点试卷汇编

18.02 Practice Exam 2 A
Problem 1. (10 points: 5, 5)
Let f (x, y ) = xy − x4 .
a) Find the gradient of f at P : (1, 1).
b) Give an approximate formula telling how small changes Δx and Δy produce a small change
why not. Problem 3. A ladybug is climbing on a Volkswagen Bug (= VW). In its starting position, the the surface of the VW is represented by the unit semicircle x2 + y 2 = 1, y � 0 in the xy -plane. The road is represented as the x-axis. At time t = 0 the ladybug starts at the front bumper, (1, 0), and walks counterclockwise around the VW at unit speed relative to the VW. At the same time the VW moves to the right at speed 10. a) (15) Find the parametric formula for the trajectory of the ladybug, and ﬁnd its position when it reaches the rear bumper. (At t = 0, the rear bumper is at (−1, 0).) b) (10) Compute the speed of the bug, and ﬁnd where it is largest and smallest. Hint: It is easier to work with the square of the speed. Problem 4. ⎠ 1 2 3 1 � M =� 3 2 2 −1 −1 (a) (5) Compute the determinant of M . b) (10) Find the numbers a and b in the formula for the matrix M −1 . x + 2y + 3z = 0 r = �x, y, z → to 3x + 2y + z = t c) (10) Find the solution � 2x − y − z = 3 d) (5) Compute Problem 5. (a) (5) Let P (t) be a point with position vector � r (t). Express the property that P (t) lies on the plane 4x − 3y − 2z = 6 in vector notation as an equation involving � r and the normal vector to the plane. r d� (b) (5) By diﬀerentiating your answer to (a), show that is perpendicular to the normal vector dt to the plane. r d� . dt as a function of t. � M −1 � ⎠ 1 1 4 1 � a 7 −8 � = 12 b −5 4

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练习题20题

高三英语微积分基础练习题20题1.The rate of change of a function at a certain point is called _____.A.integralB.derivativeC.limitD.continuity答案：B。

本题考查微积分基本概念。

“derivative”是导数的意思，函数在某一点的变化率就是导数。

“integral”是积分；“limit”是极限；“continuity”是连续性。

都不符合题意。

2.The process of finding the area under a curve is known as _____.A.differentiationB.integrationC.limitationD.continuity checking答案：B。

“integration”是积分的意思，找到曲线下的面积的过程就是积分。

“differentiation”是求导；“limitation”是限制；“continuity checking”是检查连续性。

3.If the derivative of a function is positive at a point, then the function is _____ at that point.A.increasingB.decreasingC.constantD.none of the above答案：A。

如果函数在某一点的导数是正的，那么函数在该点是递增的。

“increasing”是递增；“decreasing”是递减；“constant”是常数。

4.The integral of a constant function is _____.A.another constant functionB.a linear functionC.a quadratic functionD.none of the above答案：A。

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)。

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。

英文版微积分考试样题1

9)
Give an appropriate answer. 10) Find the instantaneous rate of change for the function x2 + 7x at x = 2. A) 9 B) 11 C) 4 Determine whether the limit exists. If it exists, find its value. 11) lim f(x) x → nction gives the distances (in feet) traveled in time t (in seconds) by a particle. Find the velocity and acceleration at the given time. 4) s = - 5t3 + 2t2 - 7t + 6, t = 3 4) A) v = - 33 ft/s, a = - 130 ft/s2 C) v = - 130 ft/s, a = - 86 ft/s2 Find the derivative. 5) f(x) = (x3 - 8)2/3 A) f'(x) = x 3 x3 - 8 B) f'(x) = 2x 3 x3 - 8 C) f'(x) = x2 3 x3 - 8 D) f'(x) = 2x2 3 x3 - 8 B) v = - 86 ft/s, a = - 130 ft/s2 D) v = - 130 ft/s, a = -33 ft/s2
8)
Use the properties of limits to help decide whether the limit exists. If the limit exists, find its value. - 3x2 + 3x - 7 9) lim 4x2 + 3 x→∞ A) ∞ B) 3 7 C) 0 D) 3 4

微积分期末考每题五分共二十题总分100分

微積分期末考每題五分共二十題總分100分日期:96.01.181. Suppose the total cost in dollars of manufacturing q units is 5003)(2++=q q q C(a) Use marginal analysis to estimate the cost of manufacturing the 41st unit. (b) Compute the actual cost of manufacturing the 41st unit.2. At a certain factory, the daily output is 3121000,3L K Q =units, where K denotes the firm ’s capital investment measured in units of $1,000 and L denotes the size of the labor force measured in worker-hours. Suppose that the current capital investment is $400,000 and that 1,331 worker-hours of labor are used each day. Use marginal analysis to estimate the effect that an additional capital investment of $1,000 will have on the daily output if the size of the labor force is not changed.3. (a)Find dx dy by implicit differentiation. 06234=-++x y xy x(b)Use implicit differentiation to find the slope of the line that is tangent to the given curve at the specified point.)1,1(;4322=+y x (c) Use implicit differentiation to find 22dxy d if 62322=-y x . 4. The output Q at a certain factory is related to inputs x and y by the equation 32322y xy x Q ++= If the current levels of input are x=10and y=20, use calculus to estimate the change in input y that should be made to offset an increase of 0.5 in input x so that output will be maintained at its current level.5. (a) Find the intervals of increase and decrease for the given function.1493)(23+--=x x x x f(b) Determine the critical numbers of the given function and classify each critical point as a relative maximum, a relative minimum, or neither.1)(2-=x x x F 6.{1}. Determine where the graph of the given function 218)(24+-=x x x f is concave upward and concave downward. Find the coordinates of all inflection points.{2}The first derivative f ’(x) of a certain function f(x) is given.82)('2--=x x x f(a) Find intervals on which f is increasing and decreasing.(b)Find intervals on which the graph of f is concave up and concave down. (c)Find the x coordinates of the relative extrema and inflection points of f .7. (a) Sketch the graph of the given function.229)(x x x f -= (b)Find vertical and horizontal asymptotes of the graph of the given function.435)(22+-=x x x x g 8. Find the absolute maximum and the absolute minimum values (if any) of the given function on the specified interval.(a)44;21232)(23<<-+-+=x x x x x f(b)0;282)(>++=x xx x f 9. An airline determines that when a round-trip ticket between Los Angeles and San Francisco costs p dollars )1600(≤≤p , the daily demand for tickets is201.0256p q -=(a)Find the elasticity of demand. Determine the values of p for which the demand is elastic, inelastic, and of unit elasticity.(b)Interpret the results of part (a) in terms of the behavior of the total revenue as a function of unit price p .(c)What price would you advise the airline to charge for each ticket ?Explain your reasoning.10. A baseball card store can obtain Mel Schlabotnik rookie cards at a cost of $5 per card. The store has been offering the cards at $10 apiece and, at this price, has been selling 25cards per month. The store is planning to lower the price to stimulatesales and estimates that for each 25-cent reduction in the price, 5 more cards will be sold each month. At what price should the cards be sold in order to maximize total monthly profit?11. A manufacturer estimates that if x units of a particular commodity are produced, the total cost will be C(x) dollars, where 33835024)(23++-=x x x x C(a) At what level of production will the marginal cost C ’(x) be minimized?(b) At what level of production will the average cost xx C x A )()(= be minimized? 12. A manufacturer of medical monitoring devices uses 36,000 cases of transistors per year. The ordering cost is $54 per shipment, and the annual cost of storage is $1.20 per case. The transistors are used at a constant rate throughout the year, and each shipment arrives just as the preceding shipment is being used up. How many cases should be ordered in each shipment in order to minimize total cost?。

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)

三年级微积分计算英语阅读理解20题

三年级微积分计算英语阅读理解20题1<背景文章>Calculus is a very interesting subject. It helps us understand how things change. Let's start with the concept of a function. A function is like a magic box. You put something in, and it gives you something out. For example, if you have a function called "add two". When you put in the number 3, it will give you 5. Functions can be represented by equations. Now, let's talk about derivatives. Derivatives tell us how fast a function is changing at a particular point. Imagine you are driving a car. The speedometer shows how fast you are going at a certain moment. That's likea derivative.1. A function is like a ___.A. bookB. magic boxC. pencilD. ball答案：B。

中文解析：文章中明确提到“A function is like a magic box.”，所以答案是魔法盒。

2. When you put 3 into a function called "add two", it gives you ___.A. 4B. 5C. 6D. 7答案：B。

微积分试卷-英文版3

⑵d(arctan x) _ 1/(1+x A2) _____ .1(3) — dx -cotx .sin x(4). (e2x)(n)泰勒展开式（书上有。

）26/3(6) The right propositi on in the follow ing propositi ons is A .A. If lim f (x) exists and lim g(x) does not exist then lim( f(x) g(x)) does notx a x a x aexist.B. If lim f (x), lim g(x)do both not exist then lim( f (x) g(x)) does not exist.x a x a x aC. If lim f (x) exists and lim g(x)does not exist then lim f (x)g(x) does notx a x a x aexist.D. If lim f (x) exists and lim g(x)does not exist thenx a x aman Xf(x) does not exist.g(x)(7)The right propositi on in the followi ng propositi ons isA. If lim f (x) f (a)then f (a) exists.x aB. If lim f (x) f (a) then f (a) does not exist.x aC. If f (a) does not exist then lim f (x) f (a).x aD. 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 __A. lim 叱1x x1 B. lim(1 x)，xC. x dx —x 1 C1 D.a1 x5(10) The right propositi on in the followi ng propositi ons is _ ____ DA. 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].求导为2*x*y(x) (y(x)就是y) 5、Find 严吟 dx . x 2(1 x 2)arcta nx/x A2-arcta nx/(1+x A2)=-(arctanx)/x+ / dx/[x(1+xA2)]=-(arctanx)/x+-xyd ((1+xA2)+ / dx/x=-(arctanx)/x- (1/2) / d(1+xA2)/(1+xA2)+ / dx/x=-(arcta nx)/x-(1/2)l n(1+xA2)+l n|x|+Cexpressi ons is _ C ___“ d bA. (a f(x)dx) f (b) db ad bC. ( f (x)dx) 0 dx aba f(x)dx)b f (x)dx)af (a) f(b) f(a)D. If f (x) has absolute extreme values on [a,b] then f(x) is continuouson [a,b].3、Evaluate 1/24 . Find dy|x °and y (0) if x20 yGt)dte x .隐函数求导。

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。

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 =。

WSET3全套模拟试题

1)Tannat is commonly grown in which South American country?a) Chileb) Uruguayc) Brazild) Argentina2)Which of the following is NOT a system used for training vines?a) Trincadeirab) Guyotc) Gobeletd) Pergola3)What is the main soil type in Bas-Armagnac?a)Chalkb)Limestonec)Clayd)Sand4)Which of the following grapes is commonly used to make Soave?a)Barberab)Verdicchioc)Nebbiolod)Garganega5)Which one of the following wines comes from Australia?a)Tokaji Aszub)Rutherglen Liqueur Muscatc)Carnerosd)Cava6)Where would you find an analyser and a rectifier?a)In a malt whisky distilleryb)In a pot stillc)In a patent stilld)In an Armagnac still7)What is the Western Cape in South Africa?a) A corporate brand producing inexpensive blends and varietal winesb) A viticultural region around Cape Town producing high-quality wines fromthePinotage grapec) A premium quality vineyard on the west coast producing Bordeaux lookalikesd) A catch-all appellation producing varietal wines blended from different regions.8)In red wine production, when are the grapes usually pressed?a)Before fermentationb)During fermentationc)After fermentationd)Not at all9)Asti is produced using a variation of which method of sparkling wine production? a)Traditionalb)Transferc)Tankd)Carbonation10)Which one of the following Chilean regions is renowned for its white wines fromChardonnay and Sauvignon Blanc and its reds from Pinot Noir?a)Casablanca Valleyb)Maipo Valleyc)Rapel Valleyd)Bio-Bio11)Which of these mountains protect the Rheingau?a)Haardtb)Vosgesc)Alpsd)Taunus12)You have been advised to consume NO MORE than the equivalent of 20mm of purealcohol a day. What is the maximum number of small (125mm) glasses of wine at 12% abv you can drink without exceeding these guidelines?a) 1b) 3c) 5d)713)Put the following processes in the correct order for the production of a Scotch Maltwhisky?1.Extraction2.Maturation3.Distillation4.Conversion5.Fermentationa)4, 1, 3, 5, 2b)4, 1, 5, 3, 2c)1, 4, 3, 5, 2d) 1, 5, 3, 2, 4 14)Which one of the following regions is in Western Australia?a)Margaret Riverb)Coonawarrac)Eden Valleyd)Padthaway15)Which one of the following Spanish wine regions produces aromatic white wines withcrisp acidity and aromas of peach and melon from the Verdejo grape variety?a)Navarrab)La Manchac)Ruedad)Tarragona16)Wines from which one of the following appellations would usually sell at a mediumprice?a)Cote-Rotie ACb)Hermitage ACc)Chateau Grillet ACd)Crozes-Hermitage AC17)What is the process of removing yeast deposits from a traditionalmethod sparkling wineknown as?a)Dosageb)Disgorgementc)Autolysisd)Tirage18)Most quality wine is produced by which species of vine?a)Vitis Labruscab)Vitis Viniferac)Vitis Amurensisd)Vitis Botrytis19)What is the main grape variety grown in the Montagne de Reimsa)Pinot Blancb)Pinot Noirc)Pinot Menuierd)Chardonnay20)Which one of the following regions is reputed to produce the most intensely peatyScotch Malt whiskies?a)Islayb)Highlandsc)Lowlandsd)Speyside21)Which one of the following regions has established a reputation for rich, fullbodied,cherry-scented red wines from Pinot Noir?a)Aucklandb)Hawkes Bayc)Gisborned)Martinborough22)What is Oidium?a)An insectb) A pestc) A fungusd) A treatment for vines23)Which one of the following regions is most important for the production of botrytised,sweet wines?a)Weinviertelb)Neusiedlerseec)Egerd)Szekszard24)Which of the following is NOT correctly paired?a)Gavi - Piemonteb)Soave - Venetoc)Rosso Conero - Marched)Frascati - Abruzzi25)What does Erstes Gewachs on a German wine label indicate?a)The wine is enrichedb)The wine is made from late gathered grapesc)The wine is blendedd)The wine is from a top quality vineyard26)What is t he French word for ‘lees stirring’?a)Buttageb)Biologiquec)Battonaged)Barrique27)Which one of the following regions produces large volumes of inexpensive wine?a)Barossa Valleyb)Adelaide Hillsc)Riverinad)Frankland River28)Which one of the following wines is not from Piemonte?a)Vino Nobile di Montepulcianob)Gavid)Dolcetto d’Alba29) A bottle ofwhite wine has small, clear crystals in the bottom. What are theyMOST likely to be?a)Harmless sugar crystalsb)Dangerous particles of glassc)Harmless tartrate crystalsd)Dangerous pesticide residue30)The UK is being considered particularly as a quality source of which sort of wine?a)Ice wineb)Sparkling winec)Fortified wined)Botrytised sweet wine31)Which of the following is classed as a herb liqueur?a)Creme de Mentheb)Creme de Cacaoc)Curacaod)Bison Grass Vodka32)Which of the following statements about Vinho Verde is FALSE?a)The wines should be drunk youngb)It is produced in Northern Portugalc)It may be dry or medium dryd)It is always white33)Which of the following regions is a‘Bordeaux lookalike’ growing the same grapes asBordeaux?a)Cotes du Durasb)Cotes du Luberonc)Cotes de Gascogned)Cotes du Roussillon34)Which one of the following best describes the style of most Fitou AC?a)Spicy, medium-bodied red wineb)Dry, fruity rose winec)Sweet, white, grapy, fortified wined)Dry, high-acid, green-fruit white wine35)What is the minimum maturation period for a VSOP Cognac?a) 2 yearsb) 3 yearsd) 5 years36)The Palomino grape variety is mainly associated with which region?a)The Loire Valleyb)Coonawarrac)Oregond)Jerez37)Which one of the following is an AC for premium priced whiteBurgundy?a)Monrachet ACb)Bourgogne Aligote ACc)Cotes de Beaune Villages ACd)Macon AC38)On a bottle of Tokaji what is the number of Puttonyos a measure ofa)The quality of the wine as assessed by a tasting panelb)The residual sugar content of the winec)The alcohol content of the wined)The acidity of the wine39)Where would you find Tinta Negra Mole?a)Cyprusb)Sicilyc)Sardiniad)Madeira40)Which ONE of the following regions is most important for Cava production?a)Penedesb)Rias Biaxasc)La Manchad)Navarra41)Which one of the following is a black grape variety? a) Tempranillob)Trebbianoc)Viurad)Verdicchio42)The word ‘Meritage’ on a bottle of wine from California indicatesa)The wine is a Bordeaux blendb)The wine has been aged in oak for at least 3 yearsc)The wine is produced from hybrid varietiesd)The wine qualifies for this American QWPSR category43)Which one of the following best describes the climate of Stellenbosch?b)Mediterraneanc)Continentald)Aqueous44)Where is m ost Port is matured?a)Vila Realb)Vila Nova de Gaiac)San Sadurni de Noyad)Sanlucar de Barrameda45)Which of the following environmental conditions is NOT harmful to the wine in abottle?a)Extended warm temperaturesb)Temperature fluctuationsc)Damp and humidityd)Proximity of strong lights46)Which one of the following pairs of words is correctly matched?a)Trocken - Medium dryb)Beerenauslese - Botrytised grapesc)Halbtrocken - Dryd)Dornfelder - Pinot Noir47)How is malolactic fermentation to encouraged take place?a)By keeping the must coolb)By using lots of sulphur dioxidec)By adding more yeastd)By warming the wine48) A Portuguese wine labelled DOC will be equivalent of? a) ACb)VDQSc)Vins de Paysd)Vins de Table49)Which of the following may be used to fine a wine?1.Egg white2.Sulphur dioxide3.Bentonite4.Aglianicoa) 1 and 2b) 2 and 3c) 1 and 3d) 2 and 450)Which coastal Californian region is the furthest north?a)Mendocinob)Montereyc)San Luis Obispod)Santa YnezShort Answer Questions51)a)Identify whether each of the following wines are generally fortified during (D) or after(A) fermentation. Indicate your answer by placing the appropriate letter in the spaceprovided (5 marks)LBV PortOloroso SherryMuscat de RivesaltesRutherglen Liqueur MuscatMalmsey Madeirab)Name the two contrasting styles of Sherry. Describe the production of each. (5 marks)Style 1:Style 2:c)Name 3 grape varieties used in the production of Port (3 marks)d)What effects does the topography of the Douro have on viticulture here? (3 marks)e)Describe the sweetness of Madeira wines generally associated with each of the followingvarieties (4 marks)Malmsey ___________________________________Verdelho _ _________ __________Sercial ____________________________________Bualf) Briefly describe the maturation of wines using the Solera System. Which fortified wineis this method generally associated with? (5 marks)52)a) Draw a line to link each grape variety with an associated region of production. (6marks)b) Briefly describe the climate of three of the regions above, stating each region clearly (6marks)Region: ______________ Region: _________________Region:c) Sauternes is naturally sweet, explain what causes this? (2 marks) GrapeRieslingSemillonChenin BlancSangioveseNebbioloTempranilloRegion Barbaresco Chianti Classico Rioja Sauternes Savennieres Piesportd)What is Eiswein? Describe the style and production methods. (3 marks)e)Suggest a New World alternative for each of the following wines (3 marks)Sancerre ______________________________Pauillac Cru BourgeoisPouilly-Fuissef)The wines of Chianti Classico and Barbaresco are followed by the letters DOCG status.What does this indicate? (3 marks)g)Name 2 other DOCG wines from Italy (2 marks)53)a)For each description of a faulty wine identify the most likely cause from the list provided.Insert the letter attached to the identified fault in the box next to the description. (5 marks)FaultsA.Cork taintB.OxidisedC.Secondary fermentationD.Excess sulphur dioxideE.Volatile acidityFaulty winesA Piesporter Goldtropchen Riesling Spatlesethat is cloudy and slightly fizzy _A South African Chenin Blanc with avinegar aromaA 2004 Chablis with a sherry like aromaAn Australian Shiraz with a wet cardboard aromaA Californian Chardonnay with a burnt matchlike aromab)Select a suitable wine to accompany each of the following dishes from the wine list below?(5 marks)Roast Beef ____________________Pasta with a Tomato based sauce _____________________ _____________Smoked Cheese _ ___________________Delicately flavoured seafood _________________________Fresh fruit tart _________________________Wine List2004 Haut-Medoc Cru Bourgeois2005 Australian Oaked Chardonnay2000 Gevrey-Chambertin1999 Vouvray Moelleux2003 Valpolicella2005 Muscadet Sur-Liec)You are planning a dinner party for12 people. You would like to allowfor each personto drink:1 x 125ml glass of Champagne as a starter1 x 175ml glass of medium bodied white with the first course1 x 250ml glass of full bodied red with the main course1 x 75ml glass of sweet white with the dessert1 x 50ml glass of vintage Port to finish.How many standard 75cl bottles of each do you require? (5 marks)Champagne bottlesWhite bottlesRed bottlesSweet White bottlesPort _ ____ bottlesd)At approximately what temperatures should each of these wines be ideally served? (5marks)Champagne oCWhite oCRed oCSweet White oCPort oCe)The planned wines for your meal allow 9 units of alcohol per person. Howdoes this compare to the recommended daily limits for both Men and Women? What are the implications of this? (5 marks)54)a)Fill in the missing words in the paragraph below. (6 marks)The flowers of nearly all commercial grape varieties are hermaphrodite, that is, they contain both and organs within the sameflower. A occurs when pollen from one grape variety is used tofertilize ovules (eggs) of a different variety of the same species. Some_such as (Pinot Noir X Cinsault) in South Africa are long established in the vineyards of the world. The only way to ensure that one vine hasthe same characteristics as another is to take a and allow it togrow into a new vine.b)For each of the following constituents of the grapewhat are the main elements they canadd to a finished wine? (6 marks)Juice: __________________________________________________ _____Skins: _________________________________________________________Stalks/Pips: ____________________________________________________c)Complete the diagram below to describe the process of alcoholic fermentation (5 marks)Action of ________--------------- --------------------------- > + __ _____________________This process also releases energyd)Give 2 main advantages and 2 main disadvantages for each of the following methods ofharvesting (8 marks).Hand PickingAdvantages: _____________________________________________________ Disadvantages: ___________________________________________________Machine HarvestingAdvantages: _________________ ___________ ___ _________ _ Disadvantages: ___________________________________________________。

三年级微积分应用英语阅读理解20题

三年级微积分应用英语阅读理解20题1<背景文章>Calculus is a very interesting subject. Let me tell you a little bit about it. Calculus is all about change. Functions are very important in calculus.A function is like a magic box. You put something in, and something else comes out. For example, if you have a function that takes a number and doubles it. If you put 3 into this function, you get 6. Isn't that amazing? Functions can be used to describe many things in our world.1. What is calculus all about?A. ColorsB. ChangeC. AnimalsD. Fruits答案：B。

解析：文章中明确提到Calculus is all about change.。

2. What is a function like?A. A bookB. A magic boxC. A pencilD. A chair答案：B。

解析：文中说 A function is like a magic box.。

3. If a function doubles a number and you put 4 into it, what do you get?A. 2B. 4C. 6D. 8答案：D。

解析：因为函数是将数字翻倍，4 翻倍是8。

4. Functions can be used to describe what?A. CarsB. ClothesC. Many things in our worldD. Toys答案：C。

三年级微积分初步英语阅读理解20题

三年级微积分初步英语阅读理解20题1<背景文章>Calculus is a very interesting subject. It helps us understand how things change. Let's start with a simple idea. Imagine you are walking. The distance you walk changes as you move. The rate at which this distance changes is called the change rate. For example, if you walk fast, the change rate is high. If you walk slow, the change rate is low.Now, let's think about a car. The speed of a car is also a kind of change rate. If the car goes fast, the change rate of its position is high. If the car goes slow, the change rate of its position is low.Calculus is all about studying these change rates and how they affect different things.1. What is calculus about?A. ColorsB. Change ratesC. AnimalsD. Fruits答案：B。

解析：文章中明确提到Calculus is all about studying these change rates. 所以答案是B。