托马斯微积分-Thomas` CALCULUS 课后习题答案cap11b
托马斯微积分ThomasCALCULUS课后习题答案附录
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托马斯微积分第13版第七章答案
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CHAPTER 7 TRANSCENDENTAL FUNCTIONS
7.1 INVERSE FUNCTIONS AND THEIR DERIVATIVES 1. Yes one-to-one, the graph passes the horizontal line test. 2. Not one-to-one, the graph fails the horizontal line test. 3. Not one-to-one since (for example) the horizontal line y 2 intersects the graph twice. 4. Not one-to-one, the graph fails the horizontal line test. 5. Yes one-to-one, the graph passes the horizontal line test. 6. Yes one-to-one, the graph passes the horizontal line test. 7. Not one-to one since the horizontal line y 3 intersects the graph an infinite number of times. 8. Yes one-to-one, the graph passes the horizontal line test. 9. Yes one-to-one, the graph passes the horizontal line test. 10. Not one-to one since (for example) the horizontal line y 1 intersects the graph twice. 11. Domain: 0 x 1, Range: 0 y 12. Domain: x 1, Range: y 0
托马斯微积分
Figure 2.43: The balloon in Example 3.
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 7
Figure 2.31: sin (x°) oscillates only /180 times as often as sin x oscillates. Its maximum slope is /180. (Example 9)
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 15
Figure 2.51: The position of the curve y = (a h – 1) /h, a > 0, varies continuously with a.
Chapter 2 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved. Chapter 2ET, Slide 16
微积分课后习题六答案
微积分课后习题六答案微积分课后习题六答案微积分是一门重要的数学学科，它研究的是函数的变化和极限。
在学习微积分的过程中，课后习题是巩固知识和提高能力的重要途径。
本文将为大家提供微积分课后习题六的答案，希望能帮助大家更好地理解和掌握微积分知识。
1. 求函数f(x) = x^2在区间[0,1]上的定积分。
解：根据定积分的定义，我们可以将区间[0,1]等分成n个小区间，每个小区间的长度为Δx = (1-0)/n = 1/n。
然后，我们在每个小区间中选择一个代表点xi，计算函数在该点的函数值f(xi)，并将其乘以小区间的长度Δx。
最后，将所有小区间的函数值乘以对应的长度Δx后相加，即可得到定积分的近似值。
当n趋向于无穷大时，这个近似值将趋向于定积分的真实值。
即：∫[0,1] x^2 dx = lim(n→∞) ∑[i=1,n] f(xi)Δx= lim(n→∞) ∑[i=1,n] (xi)^2 * (1/n)= lim(n→∞) (1/n) * ∑[i=1,n] (xi)^2由于区间[0,1]上的任意小区间长度都是相等的，所以我们可以将其简化为：∫[0,1] x^2 dx = lim(n→∞) (1/n) * ∑[i=1,n] (i/n)^2= lim(n→∞) (1/n) * ∑[i=1,n] i^2/n^2= lim(n→∞) (1/n^3) * ∑[i=1,n] i^2根据数学公式∑[i=1,n] i^2 = n(n+1)(2n+1)/6，代入上式，得到：∫[0,1] x^2 dx = lim(n→∞) (1/n^3) * [n(n+1)(2n+1)/6]= lim(n→∞) (2n^3 + 3n^2 + n)/(6n^3)= 1/3所以，函数f(x) = x^2在区间[0,1]上的定积分为1/3。
2. 求函数f(x) = e^x在区间[0,2]上的定积分。
解：与上题类似，我们可以将区间[0,2]等分成n个小区间，每个小区间的长度为Δx = (2-0)/n = 2/n。
托马斯微积分 几何与多元微积分B上 - 副本
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参考答案
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1).理解序列、子序列、有界序列的概念， 以及递归法定义序列。 2). 会判别序列的敛散性, 会求序列极限。
8.3 无穷级数 8.4 非负项级数 8.5交错级数、绝对收敛和条件收敛
1).理解常数项级数的概念、了解收敛级数的性质。 2). 对正项级数、交错级数会判断其敛散性。 3). 对任意项级数会判断绝对收敛与条件收敛。
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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。
大学_微积分(刘书田著)课后习题答案下载
微积分（刘书田著）课后习题答案下载微积分（刘书田著）课后答案下载人类对自然的认识永远不会止步，微积分这门学科在现代也一直在发展着。
以下列举了几个例子，足以说明人类认识微积分的水平在不断深化。
在黎曼将柯西的积分含义扩展之后，勒贝格又引进了测度的概念，进一步将黎曼积分的含义扩展。
例如著名的狄利克雷函数在黎曼积分下不可积，而在勒贝格积分下便可积。
[6]前苏联前苏联著名数学大师舍盖索伯列夫为了确定偏微分方程解的存在性和唯一性，建立了广义函数和广义导数的概念。
这一概念的`引入不仅赋予微分方程的解以新的含义，更重要的是，它使得泛函分析等数学工具得以应用到微分方程理论中，从而开辟了微分方程理论的新天地。
美国美籍华裔数学大师陈省身所研究的微分几何领域，便是利用微积分的理论来研究几何，这门学科对人类认识时间和空间的性质发挥着巨大的作用，并且这门学科至今仍然很活跃。
前不久由俄罗斯数学家佩雷尔曼完成的庞加莱猜想便属于这一领域。
中国中国的数学爱好者发现了积乘和微商，使微积分的内容进一步拓展。
微积分（刘书田著）：基本内容数学分析研究函数，从量的方面研究事物运动变化是微积分的基本方法。
这种方法叫做数学分析。
从广义上说，数学分析包括微积分、函数论等许多分支学科，但是现在一般已习惯于把数学分析和微积分等同起来，数学分析成了微积分的同义词，一提数学分析就知道是指微积分。
微积分微积分的基本概念和内容包括微分学和积分学。
微分学的主要内容包括：极限理论、导数、微分等。
积分学的主要内容包括：定积分、不定积分等。
微积分（刘书田著）：现代发展点击此处下载微积分（刘书田著）课后答案。
微积分课后题答案
微 积 分 课 后 习 题 答 案习 题 一 （A ）1．解下列不等式，并用区间表示不等式的解集：（1）74<-x ； （2）321<-≤x ；（3）)0(><-εεa x ； （4）)0,(0><-δδa x ax ；（5）062>--x x ；（6）022≤-+x x ．解：1)由题意去掉绝对值符号可得：747<-<-x ，可解得j .113．x <<-即)11,3(-. 2）由题意去掉绝对值符号可得123-≤-<-x 或321<-≤x ，可解得11≤<-x ,53<≤x .即]5,3[)1，1(⋃-3）由题意去掉绝对值符号可得εε<-<-x ,解得εε+<<-a x a .即)a , (εε+-a ；4）由题意去掉绝对值符号可得δδ<-<-0x ax ,解得ax x ax δδ+<<-00，即ax a x δδ+-00 , () 5）由题意原不等式可化为0)2)(3(>+-x x ,3>x 或2-<x 即）（3， 2） ， (∞+⋃--∞. 6）由题意原不等式可化为0)1)(2(≤-+x x ,解得12≤≤-x .既1] , 2[-.２．判断下列各对函数是否相同，说明理由： （1）x y =与x y lg 10=； （2）xy 2cos 1+=与x cos 2；（3）)sin (arcsin x y =与x y =；（4）)arctan (tan x y =与x y =；（5）)1lg(2-=x y 与)1lg()1lg(-++=x x y ； （6）xxy +-=11lg 与)1lg()1lg(x x x +--=．解：1)不同，因前者的定义域为) , (∞+-∞，后者的定义域为) , 0(∞+； 2)不同,因为当))(2 , )212((ππ23k k x k ++∈+∞-∞- 时，02cos 1 >+x ,而0cos 2<x ；3）不同，因为只有在]2, 2[ππ-上成立； 4）相同；5)不同，因前者的定义域为) , (11) , (∞+⋃--∞)，后者的定义域为) , 1(∞+； 6）相同3．求下列函数的定义域（用区间表示）： （1）1)4lg(--=x x y ； （2）45lg 2x x y -=；（3）xx y +-=11； （4）)5lg(312x x x y -+-+-=； （5）342+-=x x y ；（6）xy xlg 1131--=；（7）xy x-+=1 lg arccos 21； （8）6712arccos2---=x x x y ．解：1)原函数若想有意义必须满足01>-x 和04>-x 可解得 ⎩⎨⎧<<-<41 1x x ,即)4 , 1()1 , (⋃--∞.2)原函数若想有意义必须满足0452>-x x ,可解得 50<<x ,即)5 , 0(.3)原函数若想有意义必须满足011≥+-xx,可解得 11≤<-x ,即)1 , 1(-. 4)原函数若想有意义必须满足⎪⎩⎪⎨⎧>-≠-≥-050302x x x ,可解得 ⎩⎨⎧<<<≤5332x x ,即) 5 , 3 (] 3 , 2 [⋃,3]. 5)原函数若想有意义必须满足⎪⎩⎪⎨⎧≥--≥+-0)1)(3(0342x x x x ,可解得 ⎩⎨⎧≥-≤31x x ,即(][) , 3 1 , ∞+⋃-∞.6)原函数若想有意义必须满足⎪⎩⎪⎨⎧≠-≠>0lg 100x x x ,可解得⎩⎨⎧><<10100x x ,即) , 10()10 , 0(∞+⋃. 7)原函数若想有意义必须满足01012≤≤-x 可解得21010--≤<x 即]101 , 0()0 , 101[22--⋃- 8)原函数若想有意义必须满足062>--x x ,1712≤-x 可解得) 4 , 3 (] 2 , 3 [⋃--.4．求下列分段函数的定义域及指定的函数值，并画出它们的图形： （1）⎪⎩⎪⎨⎧<≤-<-=43,13,922x x x x y ，求)3( , )0(y y ；（2）⎪⎪⎩⎪⎪⎨⎧∞<<+-≤≤-<=x x x x x x y 1, 1210,30,1，求)5( , )0( , )3(y y y -．解：1）原函数定义域为：)4 , 4(-3)0(==y 8)3(==y .图略2)原函数定义域为：) , (∞+-∞31)3(-=-y 3)0(-==y 9)5(-=y y(5)＝-9.图略5．利用x y sin =的图形，画出下列函数的图形：(1)1sin +=x y ； （2）x y sin 2=； （3）⎪⎭⎫⎝⎛+=6sin πx y ．解：x y sin =的图形如下(1)1sin +=x y 的图形是将x y sin =的图形沿沿y 轴向上平移1个单位(2)x y sin 2=是将x y sin =的值域扩大2倍。