SKKU Fall, 2014

Calculus II

                            고급 미분적분학2

             Prof. : Sang-Gu LEE (이상구 교수)

       http://matrix.skku.ac.kr/Cal-Book/

http://matrix.skku.ac.kr/Calculus-Story/index.htm

http://matrix.skku.ac.kr/Cal-Book1/Ch1/ 

http://matrix.skku.ac.kr/Cal-Book1/Ch2/ 

...

http://matrix.skku.ac.kr/Cal-Book1/Ch15/

     Final PBL Report   (종합- 강좌의 기록)

미적분학 Lab: http://matrix.skku.ac.kr/Lab-Book/Sage-Lab-Manual-1.htm

선형대수 Lab: http://matrix.skku.ac.kr/Lab-Book/Sage-Lab-Manual-2.htm

Appendix: http://matrix.skku.ac.kr/Cal-Book/Appnd/index.htm

 강의, 녹화, 학생 문제풀이, 녹화, QnA, 공지사항, 참고자료, 읽을거리 등

Chapter 9. Infinite Sequences and Infinite Series

9.1 Sequences and Series   http://youtu.be/rz8ZS4Y_cvc 

문제풀이 by 문지호  http://youtu.be/Qo0MArZG2EA

문제풀이 by 이원준  http://youtu.be/O6y1v5fJA0k  

9.2 Tests for convergence of series with positive terms

http://youtu.be/mxwSv6ApZ2g 

문제풀이 by 김범윤  http://youtu.be/1flKAnlv9LA

9.3 Alternating Series and Absolute Convergence  http://youtu.be/NtSitFNv9Mk

문제풀이 by 계성곤  http://youtu.be/e_5D0dzrqwc  

9.4 Power Series  http://youtu.be/426kkrMArgs

문제풀이 by 배성준  http://youtu.be/R3AcB12z2kk

9.5 Taylor, Maclaurin, and Binomial Series   http://youtu.be/3zSPSvYHJQI

문제풀이 by 우시명  http://youtu.be/NSFrYRYZ6Qc

Chapter 10. Parametric Equations and Polar Coordinates

10.1 Parametric Equations    http://youtu.be/hQGCZk1tpuA

문제풀이 by 문지호  http://youtu.be/uz1DkKVeD2k

문제풀이 by 임효정  http://youtu.be/Ybs68e0iMZI  

10.2 Calculus with Parametric Curves   http://youtu.be/QFMSbGKhoX4

문제풀이 by 장찬영  http://youtu.be/yF5oZOQVnCE

10.3 Polar Coordinates    http://youtu.be/lKPJeAGw0ZA

문제풀이 by 계성곤  http://youtu.be/smAmDRK-tWY

문제풀이 by 황인철  http://youtu.be/4hoVKvk8dq0

10.4 Areas and Lengths in Polar Coordinates

http://youtu.be/qHEl7KHOAfE 

문제풀이 by 곽주현  http://youtu.be/LRmasW9uqYY

10.5 Conic Section

http://matrix.skku.ac.kr/2014-Album/Quadratic-form/index.htm 

문제풀이 by 변희성  http://youtu.be/ONItxvlsnb8

문제풀이 by 이한울  http://youtu.be/CZ9SHMtqVy4  

Chapter 11. Vectors and the Geometry of Space

11.1 Three-Dimensional Coordinate Systems

문제풀이 by 김태현  http://youtu.be/_s_2T1VVob8

11.2 Vectors

문제풀이 by 오교혁  http://youtu.be/BFgh6irMqsc

11.3 The Dot Product

11.4 The Vector or Cross Product

11.5 Equations of Lines and Planes

문제풀이 by 구본우  http://youtu.be/lxuGE_Erthg

11.6 Cylinders and Quadric Surfaces  

Chapter 12. Vector Valued Functions

12.1 Vector-Valued Functions and Space Curves

http://youtu.be/0pvywjBjsQw 

문제풀이 by 최양현  http://youtu.be/jvMI6OzdR_I

12.2 Calculus of Vector Functions

문제풀이 by 김동윤  http://youtu.be/VS5rPyOjP2I

12.3 Arc Length and Curvature

*12.4 Motion Along A Space Curve: Velocity and Acceleration  

Chapter 13. Partial Derivatives

13.1 Multivariate Functions

문제풀이 by 구본우  http://youtu.be/As_0AYApHlM

13.2 Limits and Continuity of Multivariate Functions

13.3 Partial Derivatives    http://youtu.be/LR89Ct3cEDY  

문제풀이 by 김동윤  http://youtu.be/rSYLp1mSMXY

13.4 Differentiability and Total Differentials

문제풀이 by 김범윤  http://youtu.be/qDmCWBiXbIA

13.5 The Chain Rule   http://youtu.be/r3dGYL1vkEU

문제풀이 by 김유경  http://youtu.be/vzN5By6qzvM

13.6 Directional Derivatives and Gradient     http://youtu.be/o8L_ShRANjo

문제풀이 by 김태현  http://youtu.be/2_7TOUuzJoE

13.7 Tangent Plane and Differentiability     http://youtu.be/uOf-5YHKGI4

문제풀이 by 서용태  http://youtu.be/GDkE8OqUvsk

13.8 Extrema of Multivariate Functions   http://youtu.be/oDZUkOEszOQ

문제풀이 by 오교혁  http://youtu.be/FWmk_MasIjE

13.9 Lagrange Multiplier    

문제풀이 by 이원준  http://youtu.be/YMGdQWBzyrI

Chapter 14. Multiple Integrals

14.1 Double Integrals     http://youtu.be/jZ2pAmPZYOE

문제풀이 by 이인행  http://youtu.be/w8g9fgcEP4A

14.2 Double Integrals in Polar Coordinates    http://youtu.be/olQgihl5aZg

문제풀이 by 이지석  http://youtu.be/jpsObxtZ50A

14.3 Surface Area     http://youtu.be/p9R0TTLfBzk

14.4 Cylindrical Coordinates and Spherical Coordinates

http://youtu.be/q3FKd2UxV_I 

문제풀이 by 최양현  http://youtu.be/F9u6pMubVRs

14.5 Triple Integrals    http://youtu.be/r1tzH9Ibbqk

문제풀이 by 이인행  http://youtu.be/C-uPM3km96k 

14.6 Triple Integrals in Cylindrical and Spherical Coordinates

http://youtu.be/xd0U4_C2ePY 

14.7 Change of Variables in Multiple Integrals  http://youtu.be/INn-bkgXYNg

Chapter 15. Vector Calculus

15.1 Vector Differentiation  http://youtu.be/q0aVmUCXgTI

문제풀이 by 김동윤  http://youtu.be/iSUME4Q1WPM

15.2 Line Integrals   http://youtu.be/wHINlpNXYaU

문제풀이 by 김범윤  http://youtu.be/ZdRjCfJeHM8

15.3 Independence of the Path   http://youtu.be/jGGOL3QDj1Y

문제풀이 by 김유경  http://youtu.be/TreCe8ESEiU

15.4 Green’s Theorem in Plane  http://youtu.be/WxdTbaSb_ZI

문제풀이 by 서용태  http://youtu.be/wLTHYaANwtI

15.5 Curl and Divergence  http://youtu.be/IswmJUCTeNA

문제풀이 by 오교혁  http://youtu.be/j7F3xVNdHvA

15.6 Surface and Area   http://youtu.be/xX6tNVpegbs

15.7 Surface Integrals  http://youtu.be/nrzIrM4doLo

문제풀이 by 이원준  http://youtu.be/s_MRgW2By38

15.8 Stokes’ Theorem   http://youtu.be/t4skc_PzJvg

15.9 Divergence Theorem  http://youtu.be/3BmcFr81kuQ

문제풀이 by 최주영  http://youtu.be/vGMLoGWF1Is

[실습실]     http://matrix.skku.ac.kr/Cal-Book/

Part I  Single Variable Calculus 

http://matrix.skku.ac.kr/Cal-Book/part1/part1.html

Part II  Multivariate Calculus 

http://matrix.skku.ac.kr/Cal-Book/part2/part2.html 

[Grapher]

http://matrix.skku.ac.kr/cal-lab/sage-grapher-integral2.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-inverse.html

http://matrix.skku.ac.kr/cal-lab/cal-Newton-method.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-derivatives.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-integral.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-integral2.html

http://matrix.skku.ac.kr/cal-lab/SKKU-Cell-Matrix-Calculator.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-butterfly.html
http://matrix.skku.ac.kr/cal-lab/sage-grapher-cochleoid.html
http://matrix.skku.ac.kr/cal-lab/sage-grapher-dewdrop.html      
http://matrix.skku.ac.kr/cal-lab/sage-grapher-epicycloid.html  
http://matrix.skku.ac.kr/cal-lab/sage-grapher-flower.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-pinwheel.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Gear-Curve.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Bicorn.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Cartesian-Oval.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Double-Folium.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Figure-Eight-Curve.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Folium.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Involute-of-a-Circle.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Kappa-Curve.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Lame-Curves.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Lituus.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Nephroid.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Pearls-of-Sluze.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Serpentine.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Tricuspoid.html  

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Fermat-Spiral.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Freeth-Nephroid.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Durer-Shell-Curves.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Newton-Diverging-Parabolas.html

http://matrix.skku.ac.kr/cal-lab/sage-grapher-Talbot-Curve.html

Calculus with Sage p.568 Problem 9.1 #5

Solved by 이인행, Revised by 김태현, Finalized by SGLee and Final OK by SGLee

5. Determine whether the sequence converges or diverges. If it converges, find the limit.

[Another solution]

We try to use the squeeze theorem.

For all natural number n2, we know and

       0= = 0.

So we can easily find that

[CAS] http://sage.skku.edu/   또는 https://sagecell.sagemath.org/  

var('x, i, n')

p1 = plot((x^2 - 2*x + 3)/(2*x^3 + 2), (x, 1, 50), rgbcolor=(1,0,0))

p2 = list_plot([(i^2 - 2*i + 3)/(2*i^3 + 2) for i in range(0, 51, 1)], rgbcolor=(0,0,1))

show(p1+p2)

Answer : The series is convergent and the limit is 0.            ■

Calculus with Sage p.575 Problem 9.1 #7

Solved by 최양현 Revised by 김건호 Finalized by 이인행 Final OK by SGLee

7. Determine whether the sequence converges or diverges. If it converges, find the limit.

Sol)

http://matrix.skku.ac.kr/cal-lab/cal-10-1-7.html 

[CAS] http://sage.skku.edu/   또는 https://sagecell.sagemath.org/ 

var('x, i, n')

p1=plot((sin(x))^2/(3^x), (x, 1, 20), rgbcolor=(1, 0, 0))

p2=list_plot([(i,(sin(i))^2/(3^i)) for i in range(1, 21, 1)], rgbcolor=(0, 0, 1))

show(p1+p2)

Answer : = 0

Note : 아래 Sage graph는 이 0으로 수렴함을 한눈에 보여준다.     ■

Calculus with Sage p.571 Problem 9.1 #10

Solved by 김범윤 Revised & Finalized by 이인행 Final OK by SGLee

10. Determine whether the sequence converge or diverges. If it converges, find the limit

Sol)    

        By L'Hopital's Rule,

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n')

a(n) = (3/n)^(2/n)

limit(a(n), n=+oo)

Ans : 1

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, a, n')

p1=plot((3/x)^(2/x), (x,1,20), rgbcolor=(1,0,0))

p2=list_plot([(a,(3/a)^(2/a)) for a in range(1,21,1)], rgbcolor=(0,0,1))

show(p1+p2)

Note : 위의 Sage graph는 이 1로 수렴함을 보여준다.                          ■

 Solved by 오교혁 Revised by 최양현 Finalized by 최양현 Final Ok by SG LEE

20. Determine whether the following series is convergent or divergent. Find the sum if it is convergent.

Sol)

But, is not convergent and is convergent.

Thus, is divergent

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n) =  ((3/n)-(1/(3)^n)) sum(a(n), n, 1, +oo)

sum(0(n-3^(n+1))*3^(-n)/n, n, 1, +infinity)

Answer: divergent infinity.                                                     ■

Calculus with Sage p.574 Problem 9.1 #21

Solved by 최양현 Revised by 김건호 Finalized by 이인행 Final OK by SGLee

21. Determine whether the series converges or diverges. If it converges, find the sum.

Sol)

Answer : =

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n,s')

a(n) = (1/7)^n

b(n) = (3/7)^n

c(n) = (5/7)^n

s = sum(a(n), n, 1, +oo) + sum(b(n), n, 1, +oo) + sum(c(n), n, 1, +oo)

show (s)

Answer :

Note : 한번에 3항을 더하려고 하면 가 되어 위와 같이 단순화하여 더하는 것이 현명하다.

                                                                                ■ 

Calculus with Sage p.575 Problem 9.1 #22

Solved by 최양현 Revised & Finalized by 이인행 Final OK by SGLee

22. Determine whether the following series is convergent or divergent. Find the sum if it is convergent.

Sol)    

        Let = .

        By L'Hopital's Rule,

         =

        Since , the series is divergent.                 ■

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n)=(sqrt(n))/(log(n+1)) lim(a(n), n= +oo)

+Infinity

Note : 위 Sage 명령어는  이 발산함을 확인한 것이다.

Calculus with Sage p.575 Problem 9.1 #23

Solved by 김유경 Revised by 이인행 Revised by 이원준 Final OK by SGLee

23. Determine whether the series is convergent or divergent. Find the sum if it is convergent.

Sol)     =   ()

        therefore, is convergent.

         can be obtained by sage.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n) =  (sin(2))^n sum(a(n), n, 1, +oo).n()

Answer : 10.0250455417417 - 3.37373543539345e-15*I

Note : 이 와 같이 표현될 수 있고, 그 값이 유한하므로 은 수렴한다.

                                                                                ■　

Calculus with Sage p559 Exercise 9.1 #24

Solved by 이원준 Revised and Finalized by 이한울 Final OK by SGLee

(Old) Determine whether the following series is convergent or divergent. Find the sum if it is convergent.

Sol) Let .

    We have .

    So the series is divergent.

Solved with Sage) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell No.1
Code	var('k n x') sum(ln(1+1/k), k,1,n)
Answer	log(n + 1)

Cell No.2
Code	plot(sum(ln(1+1/k), k,1,x))
Answer	Graph 1

Cell No.3
Code	limit(sum(ln(1+1/k), k,1,n),n=+oo)
Answer	+Infinity

■

Calculus with Sage p.570 Problem 9.1 #26

Solved by 이인행 Final OK by SGLee

26. Express the number as a ratio of integers.

Sol)

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n) =  83/(10^(2*n)) s = sum(a(n), n, 1, +oo) show(s) R = RealField(200) R(s)

Answer : = 0.83838383838383838383838383838383838383838383838383838383838 ■

Calculus with Sage p561 Exercise 9.2 #02

Solved by 김범윤 Revised by 이원준 finalized by 오교혁 Final OK by SGLee

Determine whether the series is convergent or divergent using the Integral Test.

Sol) 

Let .

We have Since (by L’ Hopital’s rule)

So, this series is convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Calculus with Sage p 562 Exercise 9.2 #05

Solved by 오교혁 Revised by 최양현  Finalized by 김범윤 Final OK by SGLee

Determine whether the series is convergent or divergent using the Integral Test.

Sol) The function is continuous, positive on ,

    and also decreasing since when , so we can use the Integral Test and .

.

Let

So,

Hence by the Integral Test, converges.

[Sage]

var('n') f=1/(n*ln(n)^2) integral(f, n, 2, +oo)

Answer

1/log(2)

So, the series is convergent.             ■

Calculus with Sage p 561 Exercise 9.2 #03

solved by 서용태 revised by 오교혁 Finalized by 김범윤 Final OK by SGLee

Determine whether the series is convergent or divergent using the Integral Test.

Sol)

 because when .

Let .

 is continuous on . when .

So, is decreasing for .

Now use Integral Test.

Hence, is convergent.

By the Comparison Test, is also convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') f=e^(1/n^2)/n^2 integral ( f, n, 1, oo)

Answer

1/2+e*gamma(-1/2, 1)

.

So, the series is convergent.

cf) gamma(a,z)=        ■

Solved by 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 563 Exercise 9.2 #07

Find the values of  for which the series is convergent.

Sol )

We have already shown (in Exercise 4) that when , the given series is divergent. Let us assume .

 is continuous, positive on ,and also decreasing since

, If . So, if we can use the Integral Test and .

For Integral Test,

Thus the series converges for .         ■

Solved 오교혁 Revised by 최양현 Finalized by 김범윤 Final OK by SGLee

Calculus with Sage p 563 Exercise 9.2 #09

Test for convergence or divergence of the series.

Sol)

So, is convergent by comparison with (The p-series. )

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') f=(1)/(n^2) integral(f,n,1,oo)

1

Answer 

 is convergent. So, is convergent.       ■

Solved by 오교혁 Revised by 최양현 Finalized by 김범윤 Final OK by SGLee

Calculus with Sage p 563 Exercise 9.2 #10

Test for convergence or divergence of the series.

Sol)

Use the limit comparison test with

Since convergent, convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') u(n)=(5^n+2^n)/(8^n+3^n) limit(u(n+1)/u(n), n=+oo)

Answer

5/8

By ratio test, is convergent.         ■

Calculus with Sage p.575 Problem 9.2 #11

Solved by 김범윤 Revised & Finalized by 이인행 Final OK by SGLee

11. Test for convergence or divergence of the series.

Sol)    

Since, the given series is absolutely convergent by the Ratio Test and therefore the series is convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n)=2^n*factorial(n)/n^n limit(a(n+1)/a(n), n=+oo)

Answer : 2*e^(-1)

bool(2*e^(-1)<1)

Answer : True

Note : Sage 명령어로 ratio가 수렴하는 것과 수렴값이 1보다 작은 것을 확인하였다. ■

Solved by 김범윤 Revised by 오교혁 Finalized by 김범윤

Calculus with Sage p 564 Exercise 9.2 #12

Test for convergence or divergence of the series.

Sol 1)

Using Integral Test.

by L’ Hopital’s rule,

by L’ Hopital’s rule,

So, this series is convergent.

Sol 2)

Let ,.

.

.

By ratio test, this series is convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') u(n)=n^2/3^n limit(u(n+1)/u(n), n=+oo)

Solved by 김유경 Revised by 김동윤 Revised by 오교혁 Finalized by 김유경 Final OK by SGLee

Section 9.2 Exercises 13

Test for convergence or divergence of the series.

Sol) Let and . =.

The limit comparison test 

  is convergent. ( is convergent by p-series test. )   ■

Solved by 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 565 Exercise 9.2 #17

Find the radius of convergence of .

Sol)

Let .

Then,

Therefore, the series converges for all and diverges for .

Thus the radius of convergence is .      ■

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n, x') a(n) = (e^n)/(sqrt(4^n+1)) rho(n)=(abs(a(n)))^(1/n) limit(rho(n), n=oo)

      e^(-1/2^log(4) + 1)                                       

solved by 서용태 finalized by 김유경 Final OK by SGLee

3. Find a formula for the general term of the sequence, assuming that the pattern of thee first few terms continues.

             {}

Sol)

 , , , ⋯

∴

Sage를 이용하여서 를 그리고 의 포인트 마다 점을 찍어 보고 의 극한 값을 구해보았다.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x i n') p1 = plot((1/(x^2+1)), (x, 1, 20)) p2 = list_plot([1/(i^2+1) for i in range(0, 20, 1)], rgbcolor=(1,0,0)) show(p1+p2) limit(1/(n^2+1), n=+oo)

Solved by 이원준 Finalized by 이원준 Final OK by SGLee

HONOR CALCULUS Ch 9.3 Page 596 Exercise 9

Test whether the series is absolutely convergent, conditionally convergent, or divergent.

Sol) 

Let

So this series is divergent by ratio test.            ■

Solved by 오교혁 Revised by 최양현 Finalized  by 김유경 Final OK by SGLee

14. Determine whether the sequence converges of diverges. If it converges, find the limit.

Sol)

By`L prime Hopital prime s`Rule.

Hence,`this`series`is`convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n) = (ln(n)^2)/(n) lim(a(n), n=+oo

0

So is convergent to 0

Sage를 이용하여서 을  그려 보았습니다.

var('n') a(n) = (ln(n)^2)/(n) plot(a(n),(n,1,10000))

그래프에서 n이 무한이 커지면 a(n)이 0으로 수렴한 다는 것을 알 수 있다.          ■

Solved by 오교혁 Revised by 최양현 Finalized by 김유경 Final OK by SGLee

15. Determine whether the sequence converges or diverges. If it converges, find the limit.

Sol)

Since ,

Therefore,

Hence, this series is convergent.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') a(n) = (2^n+2/(e^n) lim(a(n), n=+oo)

    0

So is convergent to 0

Sage를 이용하여서 를 그려보면 다음과 같다.

var('n') a(n) = (2^n+2)/(e^n) plot(a(n),(n,1,100))

n이 커짐에 따라서 a(n)이 0으로 접근하는 것을 볼 수 있다.       ■

Solved by 오교혁 Revised by 구본우 Finalized by 김범윤 Final OK by SGLee

Calculus with Sage p 568 Exercise 9.3 #11

Test whether the series is absolutely convergent, conditionally convergent, or divergent

Sol 1) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n') f=cos(n*pi/3)/(factorial(n)) sum( f, n, 1, +oo)

    e-1

so is convergent.       ■

Sol 2)

Absolutely convergent test
for all n.

Let .

Using ratio test,

 is absolutely convergent.

Hence, is also absolutely convergent.    ■

Solved by 김유경 Revised by 김동윤 Finalized by 서용태 Final OK by SGLee

Section 9.3  13.

13) Test whether the series is absolutely convergent, conditionally convergent, or divergent.

Series :

Sol)

Let .

Use the ratio test to determine whether a is absolutely convergent.

=  = =1

Therefore, the ratio test is inconclusive, so I use the alternating series test.

Let

Then,   , so . Also,

 is conditionally convergent, by the alternating series test. ■

Calculus with Sage  9.4 (Power Series)

Solved by 김태현 Revised by 순샤오웨이 Finalized by 이인행 Final OK by SGLee

1 Determine the radius of convergence and interval of convergence of the following series.

9.4 #3 (Old). Determine the radius of convergence and interval of convergence of the following series.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('n')

u(n)=1/factorial(2*n)

rho=limit(abs(u(n+1)/u(n)), n=+oo)

rho

Solved by 최양현

Finalized by 이원준

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Solved by 김유경 Revised by 김동윤 Finalized by 서용태 Final OK by SGLee

Section 9.4

13) Determine the interval of convergence of a power series representation for the function f(x)

Series :

Sol)

  (because power series)

The interval of convergence is (-1,1) because this series converges when .

Solved by 김유경 Revised by 김동윤 Finalized by 서용태 Final Ok by SGLee

Section 9.4

18) Evaluate the indefinite intergral as a power series and fine the radius of convergence.

Function : 

Solution :

  ( by power series)

 is convergent when .

Therefore, the interval of convergence is (-1,1).

Solved by 최주영 Revised by 김태현 Final OK by S. G. LEE

9.5 Exercises (Taylor, Maclaurin, and Binomial Series)

6.  Obtain the Taylor series for f(x) about a, where f(x) = , a=2

Sol)

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x')

f(x)=exp(-2*x)

g(x)=f.taylor(x,2,4)

p1=plot(f,(x,0,4))

p2=plot(g,(x,0,4),color="red",linestyle='--')

show(p1+p2)

print g

x |--> 2/3*(x - 2)^4*e^(-4) - 4/3*(x - 2)^3*e^(-4) + 2*(x - 2)^2*e^(-4) - 2*(x - 2)*e^(-4) + e^(-4)                                 ■

Solved by 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 577 Exercise 9.5 #07

Obtain the Taylor series for about , where

,

Sol)

If ,

So,

   ■

Solved by 이원준 Finalized by 이원준 Final OK by SGLee

HONOR CALCULUS Ch 9.5 Page 632 Exercise 10

Find the Maclaurin series for the given function.

 (Use ).

Sol) 

First we know Maclaurin series of

(from page 618 Example3  ).

So we have .

That is,

■

Calculus with Sage Problem 9.5 #13

Solved by 최주영 Revised by 구본우 Finalized by 이인행 and 서용태 Final OK by SGLee

13. Evaluate the indefinite integral as an infinite series.

Sol)

         as Maclaurin series.

        so,

 (C is constant, )

Also, we can get the result by Sage.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x') f=sin(x)/x g=f.taylor(x,0,10) integral(g,x)

Ans : -1/439084800*x^11 + 1/3265920*x^9 - 1/35280*x^7 + 1/600*x^5 - 1/18*x^3 + x  ■

solved by 오교혁 revised and finalized by 김태현 Final OK by S. G. LEE

9.5 #15 If and , describe the set of all points such that .

is the is the

so the .

so the is the sphere which sphere's center is

sol) 

we can rewrite as

so we can find the equation of the sphere

and I'll show the graph by letting () to easily visualize.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y,z')

implicit_plot3d(9==(x-3)^2+(y-3)^2+(z-3)^2, (x, 0, 6), (y, 0, 6), (z, 0, 6), opacity=0.5)

Solved by 김유경 Revised by 김동윤 Finalized by 서용태 Final OK by SGLee

Section 9.5 Exercise

18) Obtain the binomial series and radius of convergence of the function.

Sol)

 is convergent when

Therefore the radius of convergence of this function is 1.

Solved by 이원준 Finalized by 이원준 Final OK by SGLee

HONOR CALCULUS Ch 9.5 Page 636 Exercise 22

Evaluate using the binomial series where .

Sol)

And Maclaurin series of ;

So we can evaluate by comparison term of with .

  => 

  So we have .

■

NOTE : By using Maclaurin series we can evaluate (for ) easily.

Presented by 김태현

Chapter. 11.1 Exercises.

[1-2] Draw the surface in

Chapter. 11.1 1.

<Sol> [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x,y,z') implicit_plot3d(2*x-3*y+z==1, (x, -5, 5), (y, -5, 5), (z, -5, 5))
Answer

Chapter. 11.1 2.

<Solve>

Code	var('x,y,z') implicit_plot3d(x^2-y^2==3, (x, -5, 5), (y, -5, 5), (z, -5, 5))
Answer

Chapter. 11.1 3. Find the lengths of the sides of the triangle with vertices A(1,3,-2), B(3,1,-3) and C(2,-1,-1) is ABC a right triangle? Is it an isosceles trangle?

<Solve>

By hand :

so we can find this triangle isosceles triangle

By Sage :

Code	A=(1,3,-2) B=(3,1,-3) C=(2,-1,-1) show(point3d([A,B,C])+line([A,B])+line([B,C])+line([C,A])) AB=sqrt((1-3)^2+(3-1)^2+(-2-(-3))^2); BC=sqrt((3-2)^2+(1-(-1))^2+(-3-(-1))^2); CA=sqrt((2-1)^2+(-1-3)^2+(-1-(-2))^2); print AB print BC print CA
Answer	3 3 3*sqrt(2)

Chapter. 11.1 4. Find the distance from (3,-4,5) to each of the following

 (a) The x-axis    (b) The y-axis     (c) The z-axis

 (d) The xy-plane  (e) The yz-plane   (f) The xz-plane

<Sol> By hand

 (a) (b) (c) 5 (d) 5 (e) 3 (f) 4

By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	x = (3,-4,5) a=sqrt(x[0]^2) #yz평면과의 거리 b=sqrt(x[1]^2) #zx평면과의 거리 c=sqrt(x[2]^2) #xz평면과의 거리 d=sqrt(x[1]^2 +x[2]^2) #x축과의 거리 e=sqrt(x[0]^2 +x[2]^2) #Y축과의 거리 f=sqrt(x[0]^2 +x[1]^2) #Z축과의 거리 a,b,c,d,e,f
Answer	3, 4, 5, sqrt(41), sqrt(34), 5

Chapter. 11.1 5. Find an equation of the sphere with center (2,-4,1) and radius 3. What is the intersection of this sphere with the yz-plane?

<Sol> By hand : An equation of sphere : , and intersection of this sphere and yz-plane can be obtained by replacing x=o

By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x, y, z') s1=implicit_plot3d((x-2)^2+(y+4)^2+(z-1)^2==9, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s2=implicit_plot3d(x==0, (x,-10,10), (y,-10,10), (z,-10,10), color='green', opacity=0.5) s1+s2
Answer

Chapter. 11.1 6. Find an equation of the sphere that passes through the point (-1, 4, 3) and has center (5, 1, -1)

<Sol> By hand : distance between (-1, 4, 3) and (5, 1, -1) is the radious of the sphere

distance

equation of the sphere

[7-8] Show that the equation represents a sphere, and find its center and radius

Chapter. 11.1 7.

<Sol> By hand : we can find equation of sphere :

center : (4,-3,6) radius

Chapter. 11.1 8.

<Sol> By hand : equation of the sphere：

center : (2,-3,0) radius

Chapter. 11.1 9. (a) Prove that the midpoint of the line segment from

 to is

(b) Find the lengths of the medians of the triangle with vertices A(1,2,3), B(0,3,-1 and V(5,1,-2)

<Sol> By hand : midpoint of the line AB

lengths of the of the medians of the triangle with vertices

[10-16] Determine the Region of represented by the equation or inequality

Chapter. 11.1 10.

<Sol> By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x, y, z') s1=implicit_plot3d(x==8, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s1
Answer

Chapter. 11.1 11.

<Sol> By Sage

Code	var('x, y, z') s1=implicit_plot3d(z==2, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s2=implicit_plot3d(z==3, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s3=implicit_plot3d(z==4, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s1+s2+s3
Answer

Chapter. 11.1 12.

<Sol> By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code

var('x, y, z') s1=implicit_plot3d(y==0, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s2=implicit_plot3d(y==1, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s3=implicit_plot3d(y==2, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s4=implicit_plot3d(y==3, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s5=implicit_plot3d(y==4, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s6=implicit_plot3d(y==5, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s7=implicit_plot3d(y==6, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s8=implicit_plot3d(y==7, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s1+s2+s3+s4+s5+s6+s7+s8

Answer

Chapter. 11.1 13.

<Sol> By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x, y, z') s1=implicit_plot3d(x^2+y^2==3, (x,-3,3), (y,-3,3), (z,-2,0.5), color='red', opacity=0.3) s2=implicit_plot3d(z==-1, (x,-3,3), (y,-3,3), (z,-2,0.5), color='green', opacity=0.5) s3=implicit_plot3d(x^2+y^2==3, (x,-3,3), (y,-3,3), (z,-0.99,-1.01), color='blue') s1+s2+s3
Answer

Chapter. 11.1 14.

<Sol> By Sage

Code	var('x, y, z') s1=implicit_plot3d(x^2+y^2==9, (x,-5,5), (y,-5,5), (z,-10,10), color='blue', opacity=0.3) s1
Answer

Chapter. 11.1 15.

<Sol> outside surface of sphere

By Sage

Code	var('x, y, z') s1=implicit_plot3d(x^2+y^2+z^2==2*x, (x,-2,2), (y,-2,2), (z,-2,2), color='blue', opacity=0.3) s1
Answer

Chapter. 11.1 16.

<Sol> inside surface of cylinder

By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x, y, z') s1=implicit_plot3d(x^2+z^2==9-2*z, (x,-5,5), (y,-5,5), (z,-5,5), color='blue', opacity=0.3) s1
Answer

Chapter. 11.1 17. The half-space consisting of all points to the left of the yz-plane

<Sol>

By Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code

var('x, y, z') s1=implicit_plot3d(x==0, (x,-10,10), (y,-10,10), (z,-10,10), color='blue', opacity=0.2) s2=implicit_plot3d(x==1, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s3=implicit_plot3d(x==2, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s4=implicit_plot3d(x==3, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s5=implicit_plot3d(x==4, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s6=implicit_plot3d(x==5, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s7=implicit_plot3d(x==6, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s8=implicit_plot3d(x==7, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s9=implicit_plot3d(x==8, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s0=implicit_plot3d(x==9, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.9) s1+s2+s3+s4+s5+s6+s7+s8+s9+s0

Answer

Chapter. 11.1 18. The solid rectangular box in the first octant bounded by the planes x=1, y=3 and z=2.

Sol) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Code	var('x, y, z') s1=implicit_plot3d(x==0, (x,-10,10), (y,-10,10), (z,-10,10), color='blue', opacity=0.3) s2=implicit_plot3d(y==3, (x,-10,10), (y,-10,10), (z,-10,10), color='red', opacity=0.3) s3=implicit_plot3d(z==2, (x,-10,10), (y,-10,10), (z,-10,10), color='green', opacity=0.3) s1+s2+s3
Answer

Calculus with Sage p 643 Exercise 11.1.  8

Solved by 김동윤 Revised & Finalized by 김유경 Final OK by SGLee

Show that the equation represents a sphere and find its center and radius.

Sol)

Ans) Center : Radius :

Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') p1=implicit_plot3d(4*x^2+4*y^2+4*z^2-20*x+24*y==5, (x,-10, 10),(y,-10, 10), (z,-10, 10), opacity=0.2, color="red") show(p1)

Calculus with Sage 11.1.  #13

Solved by 김동윤 Revised & Finalized by 김유경 Final OK by SGLee

Determine the region of represented by the equation of inequality.

Sol)

So the region consists of those points whose distance from the point (1,1,0) is greater than

This is the set of all point outside the sphere with radius and center (1,1,0)

Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') s1=implicit_plot3d(x^2+y^2+z^2==2*x+2*y, (x,-1,3), (y,-1,3), (z,-2,2), color='red', opacity=0.3) s1

빨간 구가 나타내는 부분 바깥 부분이 이 나타내는 부분이다.    ■

Solved by 김범윤 revised finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 643 Exercise 11.2 #01

Determine .

(a) , .

(b) , .

Sol)

(a) ,

(b) ,                ■

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector([2, 5]) b=vector([-3, 1]) a-b

(-5, -4)

a=vector([5, -3]); b=vector([10, 7]); a-b

(5, 10)                                               

Solved by 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 643 Exercise 11.2 #02

Determine ,

, .

Sol)

Let .

Then,

So, .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

v1=vector([-1,2,3]) v2=vector([-2,1,-2]) v2-v1

Answer

(-1, -1, -5)

■

Solved by 오교혁 Revised by 최양현  Finalized by 최양현 Final OK by SG LEE

11-2 #3 Find the sum of the given vectors.

Sol)

   <-1,4>+<3,-1>=<2,3>

Sage)

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector([-1, 4]); b=vector([3, -1]); a+b

The solution of this sum of vector is [2, 3]                            ■

Solved by 김범윤 finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 643 Exercise 11.2 #04

Find the sum of the given vectors.

,

Sol)

If, ,

.

So, answer is                   ■

[Sage] [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

v1=vector([0, 1, -4]); v2=vector([0, 2, 0]); v1+v2

Solution is (0, 3, -4).

Solved 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 643 Exercise 11.2 #05

Compute .

, .

sol)

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector([5,3]) b=vector([-3,2]) print a.norm() print a+b print 2*a-3*b print (a-b).norm()

Answer

sqrt(34) (2, 5) (19, 0) sqrt(65)

■

Solved by 오교혁 Revised by 최양현 Finalized by 김범윤 Final OK by SGLee

Calculus with Sage p 644 Exercise 11.2 #06

Compute and .

The first vector a is

the second vector b is

Sol)

 is the length of vector.

So, .

Sum of vector is the sum of each vector components. So is .

 is the subtract of each vector components. so is .

is the length of a vector

 is .

So, is

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector([5, -1, 3]) b=vector([-1, 3, -2]) print a.norm() print a.norm() print a+b print 2*a-3*b print (a-b).norm()

Answer

sqrt(35) (4, 2, 1) (13, -11, 12) sqrt(77)

   ■

Solved by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 644 Exercise 11.2 #07

Compute and .

.

Sol)

, ,

So, ,

[Sage] [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector([1,-3,1]) b=vector([0,1,3]) print a.norm() print a+b print 2*a-3*b print (a-b).norm()

Answer

sqrt(11) (1, -2, 4) (2, -9, -7) sqrt(21)

■

Solved by 최양현 Revised by 김범윤 Finalized by 오교혁 Final OK by SGLee

Calculus with Sage p 644 Exercise 11.2 #08

Compute .

,

Sol)

Let and .

Then

[Sage] [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

http://matrix.skku.ac.kr/cal-lab/cal-11-2-8.html  

a=vector([3, -4, 0]); b=vector([1, -1, 1]); print a.norm() print a+b print 2*a-3*b print (a-b).norm()

Answer

5 (4, -5, 1) (3, -5, -3) sqrt(14)

■ 

Sage와 손으로 푼것과의 답이 일치한다.

Solved by 오교혁 Revised by 최양현 Finalized by 김범윤 Final OK by SGLee

Calculus with Sage p 644 Exercise 11.2 #09

Determine a unit vector that has the same direction as .

Sol)

Length of is

So, is unit vector.

[Sage] [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

v1= vector([7,-3,1]); v2= v1.norm(); v3= v1/v2; print v3;

Answer

(7/59*sqrt(59), -3/59*sqrt(59), 1/59*sqrt(59))

    ■

Solved by 김범윤 Revised by 오교혁 Finalized by 최양현 Final OK by SGLee

Calculus with Sage p 645 Exercise 11.2 #11

A clothesline is tied between two poles, 6m apart. The line is quite tight and has negligible sag. When a wet shirt with a mass of 0.8kg is hung at the middle of the line, the midpoint is pulled down 6cm. Find the tension in each half of the clothesline.

Sol)

Let and represent the tension vectors in each side of the clothe line as shown in the figure.

Let , .

,

and , and

So,

in ,

So,

,

Sage) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

T1=vector([50, 1]) T2=vector([-50, 1]) print T1+T2 print 0.8*9.8/2*T1 print 0.8*9.8/2*T2

Answer:

(0,2) (196.000, 3.9200) (-196.000, 3.9200)

(0,2) mean Sum of tensions and  (196.000, 3.9200) mean tension T1 vector

and (-196.000, 3.9200) mean tension T2 vector.       ■

solved by 김동윤 revised & finalized by 김유경 Final OK by SGLee

11.3.8 Compute the angle between the vectors

Sol)

(a) =<5,-2>, =<3,3>

Hence the angle between and is

(b) = <3, -1, 2>, =<1, 2, 3>

Hence the angle between and is

Sage [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

def anglebetween(a,b):

    return arccos(a.dot_product(b)/(a.norm()*b.norm()))

a=vector([5,-2])

b=vector([3, 3])

c=vector([3,-1,2])

d=vector([1,2,3])

print "The angle between a and b is", anglebetween(a,b).n(), "radians"

print "The angle between a and b is", anglebetween(a,b).n()*180/pi.n(), "degrees"

print "The angle between a and b is", anglebetween(c,d).n(), "radians"

print "The angle between a and b is", anglebetween(c,d).n()*180/pi.n(), "degrees"

결과

The angle between a and b is 1.16590454050981 radians

The angle between a and b is 66.8014094863518 degrees

The angle between a and b is 1.04719755119660 radians

The angle between a and b is 60.0000000000000 degrees

print arccos(3/(58)^0.5).n()

print arccos(0.5).n()Sage

결과1.16590454050981

1.04719755119660

=> 손으로 푼것과 sage를 이용한 것의 답이 일치한다.                 ■

solve by 김유경 revised by 이지석 finalized by 서용태 Final OK by SGLee

Section 11.4

3) Find the cross product and verify that it is orthogonal to both a and b.

    =<3,1,-1> =<-2,1,1>

Sol)

[Sage] [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

a=vector(QQ, [3, 1,-1]) b=vector(QQ, [-2,1,1])  c=a.cross_product(b)   show(c)

Answer : (2,−1,5)            ■

Solved by 김태현 Final OK by S. G. LEE

11.4 #10 변형

Find the area of the parallelogram with vertices A(0,1), B(2,4), C(4,1) and D(2,-2)

Sol)

1 ) Using hand writing.

,

Area of the parallelogram : 12

2 ) Using a Sage tools. [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

A=vector((0,1)) C=vector((2,1)) B=vector((1,4)) D=vector((1,-2)) show(point([A,B,C,D])+line([A,B])+line([B,C])+line([C,D])+line([D,A]))

Area of the parallelogram : 12

Solved by 김태현 Final OK by S. G. LEE

11.4 #12 변형

Find the vector perpendicular to the plane through the points P,Q and R. P(1,0,0), Q(3,1,1), R(4,-1,-4).

Sol)

1) Using hand writing.

,

 

Answer : n=<-5,5,-5>.

2) Using a Sage tools. [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

P=vector([1,0,0]); Q=vector([3,1,-1]); R=vector([4,-1,-4]); PQ=Q-P PR=R-P PQ.cross_product(PR)

Answer : n=<-5,5,-5>.

Solved by 김태현 Final OK by S. G. LEE

11.4 #15 Find the area of triangle PQR, P(2,0,-1), Q(3,1,6), R(-1,3,4).

Sol) 

1) Using hand writing.

=(1,1,7) =(-3,3,5) =22  ∴=11

 

Answer : = 11.

2) Using a Sage tools. [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

P=vector([2,0,-1]); Q=vector([3,1,6]); R=vector([-1,3,4]); PQ=Q-P; PR=R-P; CP=PQ.cross_product(PR); 1/2*CP.norm()

Answer:  = 11.

Exercise 23] if a=, , and , then find .

Solved by 김태현 Revised and Finalized by 이한울 Finalized by Sang-gu Lee

Sol)  Using Sage) http://math1.skku.ac.kr/home/pub/1339/

1) Using hand writing.

Answer : b = (1,0,0) or (0,0,-1).

2) Using a Sage tools. [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell NO.	1
Code	var('x y z') a=vector((1,0,1)) b=vector((x,y,z)) c=a.cross_product(b) c
Answer	(-y, x - z, y)

Cell NO.	2
Code	eq=(c[0]==0, c[2]==0) solve(eq,y)
Answer	[[y == 0]]

Cell NO.	3
Code	print b.norm()
Answer	sqrt(abs(x)^2 + abs(y)^2 + abs(z)^2)

Cell NO.	4
Code	eq=(x-z==1, sqrt(x^2+z^2)==1) solve(eq,x,z)
Answer	[[x == 0, z == -1], [x == 1, z == 0]]

Answer : b = (1,0,0) or (0,0,-1).

Calculus with Sage p.669 Problem 11.5 #12

Solved by 구본우 Revised by 김건호 Final OK by SG Lee

12. Determine whether the lines L1 and L2 are parallel, skew, or intersecting. If they intersect, find the point of intersection.

Sol)

By this, two lines are not parallel.

And if they had a point of intersect A (a, b, c),

The two lines do not have point(s) of intersection

Answer: Two lines are not parallel and there's no point of intersection, so they are skewed lines.

Using Sage we can easily show that they are skewed lines:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('s, t')

r_1=vector([1,4,-1])

d_1=vector([2,3,-1])

r_2=vector([-1,0,1])

d_2=vector([1,-2,-4])

Ls=r_1+s*d_1

Lt=r_2+t*d_2

print "Clearly the two lines are not parallel and does not have intersection(s)"

parametric_plot3d(Ls,(s,-4,4),color='red',thickness=3)+parametric_plot3d(Lt,(t,-3,3))         

Calculus with Sage Problem 11.5 #15

Solved by 최주영 Finalized by 이지석 Refinalized by 이인행 Final OK by SGLee

(new) 15. Find an equation of plane through the point (3, 7, -2) and with normal vector  4i +2j -3k.

Sol) 

The vector is a normal vector to the plane

and is a point of the plane. Then

or is the equation of the plane.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(4*x+2*y-3*z==32,(x,-30,60),(y,-21,40),(z,-21,40),color='black',opacity=0.6) show(P1)

                                                                          ■

Calculus with Sage p.687 Problem 11.5 #14
Solved by 최주영 Finalized by 이지석 Refinalized by 이인행 Final OK by SGLee

(new) 14. Find an equation of the plane through the point (2, -3, 4) and perpendicular to the vector (3, 3, 7).

Sol) 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(3*x+3*y+7*z==25,(x,-14,14),(y,-14,14),(z,-14,14),color='black',opacity=0.6) show(P1)

                                ■                                      

Calculus with Sage p688 Problem 11.5 #16

Solved by 최주영 Revised by 이지석 Finalized by 이인행 Final OK by SGLee

(new)16. Which of the following four planes are parallel?

 :       :

 :   :

Sol) 

         = : normal vector of the plane

         = : normal vector of the plane

∴ and   : same plane

Let :    :

 and are paralell because their normal vector = <2,-3,4> and

 = <-4,6,-8> are parallel.

(Two normal vectors and are parallel (=0 or ) if and only if ×=0)

∴ and are parallel.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(2*x-3*y+4*z == 7,(x,-7,7),(y,-7,7),(z,-7,7),color='black',opacity=0.9) P2=implicit_plot3d(-4*x+6*y-8*z==-3,(x,-7,7),(y,-7,7),(z,-7,7),color='yellow',opacity=0.9) show(P1+P2)

                         ■

Calculus with Sage p.669 Problem 11.5 #17

Solved by 김건호 Finalized by 김태현 Final OK by SG Lee

17. Which of the following four lines are parallel?

Sol) 

We can solve each equations for t to get direction numbers:

the direction vector of the four lines are given below:

Answer: Since , indicating the two lines are parallel.

Using Sage it is also possible to visualize that and is parallel.¹⁾

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('s, t, u, v')

r_1=vector([1,0,3])

d_1=vector([2,1,-4])

r_2=vector([1,0,13])

d_2=vector([4,2,-1])

r_3=vector([1,0,1])

d_3=vector([1,2,-2])

r_4=vector([1,1,4])

d_4=vector([4,2,-8])

Ls=r_1+s*d_1

Lt=r_2+t*d_2

Lu=r_3+u*d_3

Lv=r_4+v*d_4 

print "Clearly only the red and green lines are intersecting"

parametric_plot3d(Ls,(s,-4,4),color='red',thickness=3)+parametric_plot3d(Lt,(t-3,3))+parametric_plot3d(Lu,(u,-3,3),color='black')+parametric_plot3d(Lv,(v-3,3),color='green',thickness=3)

11.5 #17

Solved by 김태현 Final OK by S. G. LEE

Find an equation for the surface obtained by rotating the parabola about the z-axis

Sol)

영어 표현력이 부족하여 이 풀이는 한글로 적겠습니다.

z-x평면 위의 포물선 을 z축을 회전축으로 돌렷을때 z=c평면 위의 회젼체의 한 점 한 점 (a,b,c)와 z축 위의 점 (0.0.c)사이의 거리는 일정하므로

이라는 방정식을 얻을 수 있다.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y,z')

implicit_plot3d(z==x^2+y^2, (x, -5, 5), (y, -5, 5), (z, -5, 5), opacity=0.5)

Calculus with Sage p.668 Problem 11.5 #18

Solved by 서용태 Revised & Finalized by 이인행 Final OK by SGLee

18. Find an equation of the plane through the given point with normal vector which is the direction of the line with the given parametric equations.

                        ; , ,

Sol)

The vector is a normal vector to plane, and is a point of the plane.

Therefore, .

∴

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Calculus with Sage p.669 Problem 11.5 #20

Solved by 구본우 Revised by 김건호 Finalized by 이지석 Final OK by SGLee

20. Find the distance from the point to the given plane.

(3, 1, 5);

Sol)

The distance from the point to the plane is given by the equation:

Applying and to above equation:

Using Sage, it is possible to obtain the same result:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

n=vector(QQ, [2, -1, 3])

d= -4

p=vector(QQ, [3, 1, 5])

dis=abs(n.dot_product(p)+d)/n.norm()

dis 

    8/7*sqrt(14)

Answer: the distance from the point to the plane is   ■

Calculus with Sage p.688 Problem 11.5 #20

Solved by 최주영 Revised by 이지석 Finalized by 이인행 Final OK by SGLee

(new) 20. Find the distance from the point to the given plane.

                    ;

Sol)

 The distance =

Using Sage, it is possible to obtain the same result:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

n=vector(QQ, [7, -3, 2])

d= -2

p=vector(QQ, [2, 2, 4])

dis=abs(n.dot_product(p)+d)/n.norm()

dis

Answer: the distance from the point to the plane is   7/31*sqrt(62)           ■

Calculus with Sage p.669 Problem 11.5 #21

Solved by 구본우 Revised by 김건호 Final OK by SG Lee

21. Find the distance from the point to the given plane.

(1, 4, -2);

Sol)

The distance from the point to the plane is given by the equation:

Applying and to above equation:

Using Sage, it is possible to obtain the same result:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

n=vector(QQ, [4, -4, 4])

d= -1

p=vector(QQ, [1, 4, -2])

dis=abs(n.dot_product(p)+d)/a.norm()

dis 

   7/4*sqrt(3)

Answer: the distance from the point to the plane is .■

Calculus with Sage p.668 Problem 11.5 #22

Solved by 구본우

Revised by 이지석

Finalized by 이인행

Final OK by SGLee

22. Find the distance between the given parallel planes

                 : , :

Sol) 

There is one point on

The distance from the point to the plane is given by the equation:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(2*x+ y - z == 2,(x,-7,7),(y,-7,7),(z,-7,7),color='blue',opacity=0.3) P2=implicit_plot3d(2*x+ y - z == -3,(x,-7,7),(y,-7,7),(z,-7,7),color='red',opacity=0.3) show(P1+P2)

11.5 #5 변형

solved by 김태현

Final OK by S. G. LEE

Sol)  [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y,z')

implicit_plot3d(y^2+4*z^2-x^2==0, (x, -5, 5), (y, -5, 5), (z, -5, 5), opacity=0.5)

11.5 #5 변형

solved by 김태현

Final OK by S. G. LEE

xyz=3

sol) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y,z')

implicit_plot3d(x*y*z==3, (x, -5, 5), (y, -5, 5), (z, -5, 5), opacity=0.5)

Solved by 김태현

Final OK by S. G. LEE

11.5 #20 변형

 Find the distance from the point to the given plane

sol.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

n=vector(QQ, [2,-1,2])

d=-4

p=vector(QQ, [3,2,5])

dis=abs(n.dot_product(p)+d)/n.norm()

10/3

Calculus with Sage p.688 Problem 11.5 #23

Solved by 최주영

Revised by 이지석

Finalized by 이인행

Final OK by SGLee

23. Find the distance between the given parallel planes.

                         : , :

Sol) 

There is one point on

The distance from the point to the plane is given by the equation:

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(3*x+2*y-4*z== 4,(x,-7,7),(y,-7,7),(z,-7,7),color='blue',opacity=0.3) P2=implicit_plot3d(6*x+4*y-8*z == 2,(x,-7,7),(y,-7,7),(z,-7,7),color='red',opacity=0.3) show(P1+P2)

Calculus with Sage p.669 Problem 11.5 #25

Solved by 김태현

Revised by 이인행

Finalized by 이인행

Final OK by SGLee

25. Prove that the distance between the parallel planes and is

Proof) Let be any point on the plane ,

then the distance between the parallel planes is equivalent to the distance from the point to the plane .

The distance from the point to the plane is given by the equation:

Since the point is a point on the plane , it satisfies .

So

■

Note : 를 만족하는 점 에 대하여 평면 까지의 최단거리를 구하는 것이 위 증명보다 더 좋다.

Calculus with Sage p.669 Problem 11.5 #26

Solved by 구본우

Revised by 이지석

Finalized by 이인행

Final OK by SGLee

26. Plot the two planes and . Find the line of intersection of two planes and hence plot this.

Sol) 

        , then .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z') P1=implicit_plot3d(x+y+z== 1,(x,-7,7),(y,-7,7),(z,-7,7),color='blue',opacity=0.3) P2=implicit_plot3d(2*x-y+z == 2,(x,-7,7),(y,-7,7),(z,-7,7),color='red',opacity=0.3) show(P1+P2)

Solved by 김범윤

Revised by 김범윤

Finalized by 김범윤

Final OK by SGLee

Calculus with Sage p 790 Exercise 12.1 #1

Find the domain of the vector functions.

Sol)

So,

 ■

Solved by 김범윤

Revised by 이인행

Finalized by 김범윤

Final OK by SGLee

Calculus with Sage p 790 Exercise 12.1 #4

Find the limit.

Sol)

,

Sage

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') a=limit((t-2)/(t^2-4),t=2) b=limit(sqrt(t+1),t=2) c=limit((sin(t))/((t-1)*cos(t)),t=2) print a,b,c

Answer

1/4 sqrt(3) sin(2)/cos(2)

So, the answer is .          ■

Calculus with Sage Exercise 12.1 #6

Solved by 이원준

Finalized by 이원준

Final OK by SGLee

Sketch curves with the given vector equations. Indicate with an arrow the direction in which increases.

.

Sol) Using Sage. [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

■

Solved by 김범윤

Revised by 이인행

Finalized by 김범윤

Final OK by SGlee

Calculus with Sage p 791 Exercise 12.1 #7

Sketch curves with the given vector equations. Indicate with an arrow the direction in which increases.

Sol)

Sage

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') r=vector([t^2,t,t^3]) f=parametric_plot3d(r,(t,0,4),thickness=2) t1=1 t2=2 t3=3 dt=0.5 a1=arrow3d(r(t=t1),r(t=t1+dt),thickness=5) a2=arrow3d(r(t=t2),r(t=t2+dt),thickness=5) a3=arrow3d(r(t=t3),r(t=t3+dt),thickness=5) f+a1+a2+a3

Answer

positive octant, plane, plane, plane.

■

Solved by 오교혁

Revised by 최양현

        Finalized by 최양현

Final OK SG LEE

12.1 # 8 Sketch curves with the given vector equations. Indicate with an arrow the direction in which  increases.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Calculus with Sage Exercise 12.1 #9

Solved by 이원준

Finalized by 이원준

Final OK by SGLee

Sketch curves with the given vector equations. Indicate with an arrow the direction in which increases.

Sol) Using Sage.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

■

Calculus with Sage p 793 Exercise 12.1 #10

Solved by 김범윤

Revised & Finalized by 이인행

Refinalized by 이인행

Final OK by SGLee

10. Find vector equations and parametric equations for the line segment to .

                                , .

Sol)

        Let,

        Vector equation

          ()

        Parametric equation

         , 

                                                                        ■

Calculus with Sage Exercise 12.1 #11

Solved by 이원준

Finalized by 이원준

Final OK by SGLee

Find vector equations and parametric equations for the line segment to .

,

Sol) 

Vector equations :

Parametric equations :

■

Solved by 김범윤

Revised by 이인행

Finalized by 김범윤

Fianl OK by SGLee

Calculus with Sage p 794 Exercise 12.1 #13

Show that the curve with parametric equations , , is the curve of intersection of the surfaces and .

Sol)

 , period : . So, .

Sage

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z, t'); p1=plot3d(x^2/4, (x, -2, 2), (y, -2, 2), color='blue', opacity=0.5); p2=implicit_plot3d(x^2+y^2==4, (x, -2, 2), (y, -2, 2), (z, 0, 1), color='green', opacity=0.5); p3=parametric_plot3d((2*sin(t), 2*cos(t), (sin(t))^2), (t, 0, 2*pi), color='red',thickness=5); show(p1+p2+p3);

Answer

and //, and , ,

■

Note : 면 두 개의 그래프를 그리고 그 교선이 문제에서 주어진 곡면과 일치하는지를 보는 코드를 짜보았다. 확인을 쉽게 하기 위하여 투명도를 0.5로 주었다.

Solved by 김범윤

Revised by 이인행

Finalized by 김범윤

Final OK by SGLee

Calculus with Sage p 795 Exercise 12.1 #16

Find vector functions that represent the curves of intersection of the two surfaces.

The cylinder and the surface .

Sol)

Parametric equation is

Then

So, is the vector equation of the answer.

Sage

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z, t'); p1=plot3d(2*x*y, (x, -sqrt(7), sqrt(7)), (y, -sqrt(7), sqrt(7)), color='green', opacity=0.5); p2=implicit_plot3d(x^2+y^2==7, (x, -sqrt(7), sqrt(7)), (y, -sqrt(7), sqrt(7)), (z, -14, 14), color='blue', opacity=0.5); p3=parametric_plot3d((sqrt(7)*cos(t), sqrt(7)*sin(t), 7*sin(2*t)), (t, 0, 2*pi), color='red',thickness=4); show(p1+p2+p3);

Answer

, //, ,

■

Note : 문제에서 주어진 두 면을 세이지를 통해 표현 한 뒤, 그 위에 문제에서 구한 답을 표현해서 답()이 맞는지 확인해 보았다.

확인을 쉽게 하기 위하여 투명도를 0.5로 주었다.

Solved by 김범윤

Revised by 이인행

Finalized by 김범윤

Final OK by SGLee

Calculus with Sage p 796 Exercise 12.1 #19

If two objects travel through space along two different curves, it is important to know whether they will collide (Will a missile hit its moving target? Will two aircraft collide?) The curves might intersect, but we need to know whether the objects are in the same position at the same time. Suppose the trajectories of two particles are given by the vector function.

, for .

Do the particles collide?

Sol)

                If the particles collide,

                ,

                , ,

                but, there is no , satisfying those equations.

                So, the particles do not collide.

Sage

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') a=parametric_plot3d((t^2, 5*t-9, t^2),(0, 5),color='red',thickness=2) b=parametric_plot3d((3*t-2, t^2, 4*t-5),(0, 5),color='blue',thickness=2) a+b

Answer

No collision.■

Note : 벡터함수로 표현된 두 곡선을 Sage를 이용하여 그려본 뒤 교점이 없음을 확인함.

Calculus with Sage Exercise 12.2 #12

Solved by 이원준

Finalized by 이원준

Final Ok by SGLee

Find the unit tangent vector at the point with the given value of the parameter .

, .

Sol)

So , .

Therefore, .

■

12.3 EXERCISES (Arc Length and Curvature)

1-5. Find the length of the curve.

Calculus with Sage p830 Exercise 12.3 #1

Solved by 이한울

Old) Find the length of the curve.

     , .

Sol)  .

     .

     .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell NO.	1
Code	var('t') r(t)=(4*t,3*cos(t),3*sin(t)) parametric_plot3d(r(t),(t,-5,5))
Answer

Cell NO.	2
Code	dr=diff(r(t),t) s(t)=sqrt((dr[0]^2+dr[1]^2+dr[2]^2).simplify_trig()); length=integral(s(t),t,-5,5) length
Answer	50

■

 2. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-2.html 

  [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') r(t)=(2*sin(t)-t*cos(t), 2*t, 2*cos(t)+t*sin(t)) dr=diff(r(t), t); s(t)=sqrt((dr[0]^2+dr[1]^2+dr[2]^2).simplify_trig()); length=integral(s(t), t, 0, sqrt(5)) print length

  1/2*sqrt(5)*sqrt(10) + 5/2*arcsinh(1)

3. , .

 .

     .

     .

 4. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-4.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') x(t)=t y(t)=2 z(t)=ln(t) dx(t)=diff(x(t), t) dy(t)=diff(y(t), t) dz(t)=diff(z(t), t) s(t)=sqrt(dx(t)^2+dy(t)^2+dz(t)^2)

length=integral(s(t), t, 1, sqrt(3)) print length

  -sqrt(2) - 1/2*log(sqrt(2) - 1) + 1/2*log(sqrt(2) + 1) - 1/2*log(3) + 2

5. , .

 ,

      ,

      .

6-8. Reparametrize the curve with respect to arc length measured from the point in the direction of increasing .

Calculus with Sage p831 Exercise 12.3 #6

Solved by 이한울

Old) Reparametrize the curve with respect to arc length measured from the point in the direction of increasing .

.

Sol) Since , .

    Thus , this implies .

    Substituting in , we have

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell NO.	1
Code	var('t') r(t)=(2-5*t,4*t,-(1+3*t)) parametric_plot3d(r(t),(t,0,5))
Answer

Cell NO.	2
Code	dr=diff(r(t),t) s(t)=sqrt((dr[0]^2+dr[1]^2+dr[2]^2).simplify_trig()) S=integral(s(t),t,0,t) S
Answer	5*sqrt(2)*t

■

7.

 Since , .

      Thus   .

      Substituting in , we have .

 8. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-8.html 

  [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') x(t)=3*t*sin(t) y(t)=3*t*cos(t) z(t)=--2*sqrt(2)*t^(3/2) dx(t)=diff(x(t), t) dy(t)=diff(y(t), t) dz(t)=diff(z(t), t) s(t)=sqrt(dx(t)^2+dy(t)^2+dz(t)^2) length=integral(s(t), t, 1, sqrt(3)) print length

Answer :  3*sqrt(3)

9-10. Find the unit tangent , unit normal vectors and the curvature .

Calculus with Sage p832 Exercise 12.3 #9

Solved by 이한울

Old) Find the unit tangent , unit normal vectors and the curvature .

     .

Sol) , .

    Hence .

    , .

    Hence .

    .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell NO.	1
Code	var('t') r(t)=(5*cos(t),5*sin(t),4*t) dr=diff(r(t),t) s(t)=sqrt((dr[0]^2+dr[1]^2+dr[2]^2).simplify_trig()); show(s(t))
Answer	√41

Cell NO.	2
Code	T(t)=(dr[0]/s(t),dr[1]/s(t),dr[2]/s(t)) show(T(t))
Answer	(−541√41sin(t),541√41cos(t),441√41)

Cell NO.	3
Code	dT=diff(T(t),t) d(t)=sqrt((dT[0]^2+dT[1]^2+dT[2]^2).simplify_trig()); show(d(t))
Answer	5√141

Cell NO.	4
Code	N(t)=(dT[0]/d(t),dT[1]/d(t),dT[2]/d(t))  show(N(t))
Answer	(−√141√41cos(t),−√141√41sin(t),0)

Cell NO.	5
Code	k=d(t)/s(t) k.simplify_full()
Answer	5/41

■

11-13. Use Theorem 2 to find the curvature. (7>>>2)

11. .

 ,

      ,

      .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

t=var('t') 

r=vector((3*t,3+t,3-t^2)) 

rprime=diff(r,t)

rrprime=diff(rprime,t)

curvature = abs(rprime.cross_product(rrprime))/abs(rprime)^3

k=curvature.full_simplify()

show(k)

Answer :

 12. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-12.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

r(t)=(3+4*t^(3/2), 6*t,3/2*t^2)

dr=diff(r(t), t)

ddr=diff(r(t), t, 2)

ANSWER=(dr.cross_product(ddr)).norm()/(dr.norm())^3

print ANSWER

Answer : sqrt(abs(-9*sqrt(t))^2 + abs(-18/sqrt(t))^2 + 324)/(abs(3*t)^2 + abs(6*sqrt(t))^2 + 36)^(3/2)

13. .

 ,

      .

      .

14. Find the curvature of at the point .

 ,

      ,

           ,

      ,

                  ,

      , .

15. Find the curvature of at the point .

 ,

      ,

      ,

      .

16-18. Use Formula 6 to find the curvature.

Calculus with Sage p834 Exercise 12.3 #16

Solved by 이한울

Old) Use Formula 6 to find the curvature.

     .

Sol)

    .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

Cell NO.	1
Code	var('x') y=x^2+1 dy=diff(y,x) ddy=diff(dy,x) k=abs(ddy)/(1+dy^2)^(3/2) k
Answer	2/(4*x^2 + 1)^(3/2)

■

17. .

 ,

      .

18. .

 ,

      .

 19-20. Find the vectors , and at the given point, and plot at some point.

19. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-19.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

r(t)=(t+1, 2*t, t^2)

dr=diff(r(t), t)

T=dr/dr.norm()

dT=diff(T, t)

N=dT/dT.norm()

B=T.cross_product(N)

print T.subs(t=1)

print N.subs(t=1)

print B.subs(t=1)

  (1/3, 2/3, 2/3)

  (-2/15*sqrt(5), -4/15*sqrt(5), 1/3*sqrt(5))

  (2/5*sqrt(5), -1/5*sqrt(5), 0)

t=var('t') 

r=vector((t+1, 2*t, t^2))

T=diff(r,t)/norm(diff(r,t))

N=diff(T,t)/norm(diff(T,t))

B=T.cross_product(N)

x=1

C=parametric_plot3d(r,(t,0,pi),thickness=2,color='goldenrod')

A=arrow3d(r(t=x), r(t=x)+T(t=x),thickness=1.1)+arrow3d(r(t=x), r(t=x)+N(t=x),color='red',thickness=1.1)+arrow3d(r(t=x), r(t=x)+B(t=x),color='green',thickness=1.1)+point3d((cos(x),sin(x),x),pointsize=20)

C+A

20. , .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-20.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

r(t)=(exp(t)*cos(t), exp(t)*sin(t), sqrt(2)*exp(t))

dr=diff(r(t), t)

T=dr/dr.norm()

dT=diff(T, t)

N=dT/dT.norm()

B=T.cross_product(N)

print T.subs(t=0)

print N.subs(t=0)

print B.subs(t=0)

  (1/2, 1/2, 1/2*sqrt(2))

  (-sqrt(1/2), sqrt(1/2), 0)

  (-1/2*sqrt(1/2)*sqrt(2), -1/2*sqrt(1/2)*sqrt(2), sqrt(1/2))

t=var('t') 

r=vector([exp(t)*cos(t), exp(t)*sin(t), sqrt(2)*exp(t)])

T=diff(r,t)/norm(diff(r,t))

N=diff(T,t)/norm(diff(T,t))

B=T.cross_product(N)

x=0

C=parametric_plot3d(r,(t,-pi/2,pi/2),thickness=2,color='goldenrod')

A=arrow3d(r(t=x), r(t=x)+T(t=x),thickness=1.1)+arrow3d(r(t=x), r(t=x)+N(t=x), color='red',
  thickness=1.1) + arrow3d(r(t=x),  r(t=x) + B(t=x), color='green', thickness=1.1) +
  point3d((cos(x), sin(x), x), pointsize=20)

C+A

21-22. Find equations of the normal plane and osculating plane of the curve at the given point, and plot the graphs.

21. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-21.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z, t')

r(t)=(2*sin(t), 5*t, 2*cos(t))

dr=diff(r(t), t)

T=dr/dr.norm()

dT=diff(T, t)

N=dT/dT.norm()

B=T.cross_product(N)

N=N.subs(t=pi)

expand(N[0]*(x-0)+N[1]*(y-5*pi)+N[2]*(z+2)==0)

  sqrt(1/29)*sqrt(29)*z + 2*sqrt(1/29)*sqrt(29) == 0

B=B.subs(t=pi)

expand(B[0]*(x-0)+B[1]*(y-5*pi)+B[2]*(z+2)==0)

  -10*pi*sqrt(1/29) + 5*sqrt(1/29)*x + 2*sqrt(1/29)*y == 0

t=var('t') 

r=vector([2*sin(t), 5*t, 2*cos(t)])

T=diff(r,t)/norm(diff(r,t))

N=diff(T,t)/norm(diff(T,t))

B=T.cross_product(N)

x=pi

C=parametric_plot3d(r,(t,0,2*pi),thickness=2,color='goldenrod')

A=arrow3d(r(t=x), r(t=x) + T(t=x), thickness=1.1) + arrow3d(r(t=x),  r(t=x) + N(t=x),
  color='red', thickness=1.1) + arrow3d(r(t=x),  r(t=x) + B(t=x), color='green',
  thickness=1.1) + point3d((cos(x), sin(x), x), pointsize=20)

C+A

22. .

http://matrix.skku.ac.kr/cal-lab/cal-13-3-22.html 

var('x, y, z, t')

r(t)=(t^2, 2/3*t^3, t)

dr=diff(r(t), t)

T=dr/dr.norm()

dT=diff(T, t)

N=dT/dT.norm()

B=T.cross_product(N)

N=N.subs(t=1)

expand(N[0]*(x-1)+N[1]*(y-2/3)+N[2]*(z-1)==0)

  -1/3*x + 2/3*y - 2/3*z + 5/9 == 0

B=B.subs(t=1) expand(B[0]*(x-1)+B[1]*(y-2/3)+B[2]*(z-1)==0)

 

  -2/3*x + 1/3*y + 2/3*z - 2/9 == 0

t=var('t')  r=vector([t^2, 2/3*t^3, t]) T=diff(r,t)/norm(diff(r,t)) N=diff(T,t)/norm(diff(T,t)) B=T.cross_product(N) x=1 C=parametric_plot3d(r,(t,0,pi/2),thickness=2,color='goldenrod') A=arrow3d(r(t=x), r(t=x) + T(t=x), thickness=1.1) + arrow3d(r(t=x), r(t=x) + N(t=x),   color='red', thickness=1.1) + arrow3d(r(t=x),  r(t=x) + B(t=x), color='green',   thickness=1.1) + point3d((cos(x), sin(x), x), pointsize=20) C+A

23. At what point on the curve is the tangent plane parallel to the plane

    ?

http://matrix.skku.ac.kr/cal-lab/cal-13-3-23.html 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x, y, z, t')

r(t)=(t^4,  3*t, t^2)

dr=diff(r(t), t)

T=dr/dr.norm()

T=T.subs(t=1)

expand(T[0]*(x-r(1)[0])+T[1]*(y-r(1)[1])+T[2]*(z-r(1)[2])==0)

  4/29*sqrt(29)*x + 3/29*sqrt(29)*y + 2/29*sqrt(29)*z - 15/29*sqrt(29) == 0.

24. The curvature at a point of a curve is defined as , where is the angle of inclination of the tangent line at , as shown in the figure. Thus, the curvature is the absolute value of the rate of change of with respect to arc length. It can be regarded as a measure of the rate of change of direction of the curve at and will be studied in greater detail in Chapter 12.

    (a)For a parametric curve , derive the formula

        .

       where the dots indicate derivatives with respect to ; that is, . [Hint: Use and Equation to find . Then use the Chain Rule to find .]

    (b)For a curve as the parametric curve , , show that the formula in part (a) becomes

       .

(a) Here ,

   , and

   . Thus .

(b) Here .

   By applying the result of (a) to this case, we obtain that 

   .

25. (a) Show that the curvature at each point of a straight line is .

    (b) Show that the curvature at each point of a circle of radius is .

(a) For a straight line, we parametrize . Since is a straight line,

    is a constant, and hence is zero. By #30-(b), we have .

(b) . Then,

    and . By problem 30 (a), we have .

Calculus with Sage 12.4 #1,    Solved by 이인행

1. Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for specified value of

Sol) [CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

r(t)=(t^3+1, t)

v=diff(r(t), t)

a=diff(v, t)

s=v.norm()

print v, a, s

v2=v.subs(t=2)

a2=a.subs(t=2)

print v2, a2

p1=parametric_plot(r(t), (t, 0, 3))

p2=line([r(2), r(2)+v2], color='red')

p3=line([r(2), r(2)+a2], color='green')

show(p1+p2+p3)

Note : 위치벡터함수를 미분하면 속도벡터함수, 속도벡터함수를 미분하면 가속도벡터함수인 관계를 이용하였다.

       Sage에서는 diff 와 parametric_plot 이 핵심적으로 이용되었다.

       .norm() 이란 각성분의 제곱합의 제곱근 값을 의미한다. .subs(t=a)는 t의 함수에 a를 대입한 값을 의미한다. line([점의 좌표, 점의 좌표])을 이용하여 선분을 그릴 수 있다.

Calculus with Sage p.848 Problem 12.4 #6

Solved by 구본우, Revised by 이인행. Finalized by 이인행

Final OK by SGLee

6. Find the velocity, acceleration and speed of a particle with the given position function.

Sol)

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

r(t)= (2*t^2 + 1, t^3, 2*t^2 - 1)

v= diff(r(t), t)

a= diff(v(t), t)

s= v.norm()

p1 = parametric_plot3d(r(t), (t, 0, 20), color='blue', thickness = 2)

show (p1)

r(t); v; a; s

(2*t^2 + 1, t^3, 2*t^2 - 1)

(4*t, 3*t^2, 4*t)

(4, 6*t, 4)

sqrt(abs(3*t^2)^2 + 2*abs(4*t)^2)

                                                                                                                                    ■

Note : 위치벡터함수를 미분하면 속도벡터함수, 속도벡터함수를 미분하면 가속도벡터함수인 관계를 이용하였다. Sage에서 diff를 이용하여 다른 어려운 함수가 주어져도 풀 수 있습니다. 속도의 크기를 구할 떄에는 .norm()을 이용하면 간단히 구할 수 있습니다.

알게된 점 : 일일이 각 성분을 정의하고 diff 하는 것보다 r(t)를 벡터로 정의하고 한번에 diff 하는 것이 편하다. 선 그릴때, thickness 를 조정해주는 것이 보기 좋다.

Calculus with Sage 12.4 #11

Solved by 구본우

Revised by 이인행

Finalized by 이인행

Final OK by SGLee

11. Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position

Sol)

        .

        Since , we have .

        .

        Since , we have .

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

a(t)= (0, 0, 2)

v= integral(a(t), t)

v0= vector([1, -1, 0])

V= v+v0

r= integral(V, t)

r0= vector([0, 0, 0])

R= r+r0

p1= parametric_plot3d(R, (t, 0, 4), color='blue', thickness=2)

show(p1)

a(t); V; R

Note : 위치벡터를 미분하면 속도벡터, 속도벡터를 미분하면 가속도벡터인 관계를 알고, 초기값들을 이용하여 가속도 벡터를 적분해서 위치벡터를 구하였다. Sage에서 integral 을 이용하여 손쉽게 적분할 수 있다. 다른 함수가 주어져도 Sage를 이용하면 간단히 구할 수 있습니다.

알게된 점 : 초기값을 더하려고 하는데 Sage에서 그냥 v0=(1, -1, 0)으로 선언하고 더했더니 더해지지 않았습니다. r0를 vector([ , , ])로 선언하고 더하거나 빼야 된다는 것을 알았습니다.

Solved by 서용태 Revised by SGLee Finalized by 김범윤 Refinalized by 이인행

Calculus with Sage p 851 Exercise 12.4 #13

13. Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.

                            .

Sol )

             .

        So, .

        .

        So,

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t')

a(t)= (2, 2, 6*t)

v= integral(a(t), t)

v0= vector([0, 0, 0])

V= v+v0

r= integral(V, t)

r0= vector([1, 0, 1])

R= r+r0

p1= parametric_plot3d(R, (t, 0, 4), color='blue', thickness=2)

show(p1)

a(t); V; R

(2, 2, 6*t)

(2*t, 2*t, 3*t^2)

(t^2 + 1, t^2, t^3 + 1)

Note : 책에서 와 의 적분 과정에서 항의 계산 실수가 있어서 바로잡았다.

Calculus with Sage 12.4 # 15

Solved by 구본우 Revised by SGLee Finalized by 최양현 Refinalized by 이인행

15, The position function of a particle is given by . When is the speed a minimum?

Sol :

         When , the speed is minimum.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') r(t)=(t^2, t, t^2-4*t) v=diff(r(t), t) s=v.norm() s0=diff(s, t) solve(s0==0, t)

[t==1]

plot (s, (t, 0, 2))

Note : 교재의 코드에 오류가 있어 수정하였다. Sage 코드에서 위치벡터함수를 구하고자 하는 다른 함수로 바꾸면 원하는 답을 구할 수 있습니다.

Calculus with Sage 12.4 #16

Solved by 이인행

16. Find the tangential and normal components of the acceleration vector.

                                .

Sol) 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('t') r(t)=(t-t^3, t^2, 0) dr=diff(r(t), t) ddr=diff(r(t), t, 2) T=dr.dot_product(ddr)/dr.norm() print T N=(dr.cross_product(ddr)).norm() / dr.norm() print N

  2*(3*(3*t^2 - 1)*t + 2*t)/sqrt(abs(-3*t^2 + 1)^2 + abs(2*t)^2)

  sqrt(abs(6*t^2 + 2)^2)/sqrt(abs(-3*t^2 + 1)^2 + abs(2*t)^2)           ■

Note : 가속도벡터함수는 속도벡터함수를 미분함으로써 구할 수 있다. Sage에서 .dot_product와 .cross_product를 이용하여

짜놓은 코드를 이용하여 다른 위치벡터함수를 안다면 가속도벡터함수의 접선방향과 법선방향의 성분값을 간단히 구할 수 있습니다.

CH 13. Partial Derivatives

13장1절 

http://matrix.skku.ac.kr/Cal-Book/part2/CS-Sec-13-1-Sol.html  

 http://youtu.be/As_0AYApHlM

13.1-1 구본우 최주영

Solved by 구본우 revised by 최주영

13.1  1-1. If , find  ..

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y'); f(x,y)=x^3-3*x*y+y^3; f(3,3);

1-2. If , find .

var('x,y'); f(x,y)=x^3-3*x*y+y^3; f(1/x,3/y);

1-3. If , find ..

var('x,y'); f(x,y)=x^3-3*x*y+y^3; print (f(x+k,y)-f(x,y))/k;

.

13.1 #5 & 6 - 이지석 최주영

Solved by 이지석 Revised by 최주영

5. Find the range of the given function.

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/

var('x,y,z'); implicit_plot3d(1+exp(x*y)==z,(x,-2.5,2.5),(y,-2.5,2.5),(z,0,3),opacity=0.3)