SKKU Spring, 2014

   Honor Calculus 1     

                                    (고급미분적분학1)

                               담당교수 Prof : LEE, Sang-Gu

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) 

Name(학생의 이름) : ***

Major(전공) : ***

SN (학번) : *** etc

e-mail (이-메일): ***

   < 강의, 녹화, 학생 문제풀이, 녹화, QnA 1140건, 공지사항 80건, 참고자료, 읽을거리 등 >

[개인성찰노트] 자기평가, 동료평가, PBL에 대한 학생 반응!

1. 이번 학습과정에서 배운 내용은 무엇입니까?

  이번 학습에서는 중간고사 범위에 비해 새로운 내용들이 많이 있었습니다. 크게 적분, 수열, 매개변수 방정식, 극좌표로 나눌 수 있습니다. 수열의 경우 수렴판정법에 대해 자세히 배웠고 taylor, maclaurin, binomial 수열에 대해 새로 배웠습니다. 극좌표도 고등학교 때에는 하지 않은 내용으로 데카르트 좌표계를 벗어난 새로운 좌표계를 배웠습니다.

2. 학습과정에서 어떤 방법을 통해 학습했는지 구체적으로 적어주세요. 

  이번 학습과정에서도 중간 때와 마찬가지로 sage tool을 이용하였고 qna에 문제를 올리고 수정받는 방식으로 학습했습니다. 이번에는 특히 sage grapher를 이용하여 매개변수 방정식의 변화를 애니메이션 형태로 볼 수 있었습니다.

3. 이번 학습과정에서 내용이나 방법면에서 인상 깊었던 점은 무엇입니까? 

   위에서 말한 바와 같이 sage grapher를 처음 사용해본 것이 인상 깊었습니다. 단순히 그림을 그리는 것에서 끝나는 것이 아니라 변화과정까지 시각적으로 볼 수 있어 직관적인 이해가 되었습니다.

4. 이번 학습과정에서 다른 과목의 수강이나, 학교 공부 외, 취업 후 등에 적용할 만 점은?

   이번 학습과정에서 배운 내용 중 특히 수열 부분에서 모든 함수를 다항함수로 바꿀 수 있게 되어서 아무리 복잡한 식이 나오더라도 계산기로 계산할 수 있게 되었는데 나중에 만약 취직을 하게 되면 쓰게 되지 않을까 싶습니다.

* 기말고사 서술형 문제 예 : 문제 풀어 올리고 질문과 답을 하면서 공부한 학생은 별도로 시험공부를 할 필요가 없을 것입니다.

 < 기말고사 서술형 문제 예 >

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

우리는 생활하면서 ‘주어진 곡선의 일부분의 길이나 그 곡선 위의 어떤 점에서의 접선은 어떻게 구하나?’ 또 ‘주어진 곡면의 일부분 표면적은 어떻게 구하나?’와 같은 질문에 접하게 된다. 일반적으로 표현하면, 시간에 따라 두 변량(Variable quantity)이 변하는 비율은 어떻게 계산할 것인가? 하는 문제와 만나게 되는 것이다. 이 모든 것들을 계산할 수 있게 하는 방법이 수많은 과학자들을 거쳐 뉴우톤과 라이프니츠에 의하여 체계적인 방법으로 정리되어 소개되었으며2), 이와 관련된 다양한 연구들은 인류 역사 발전에 절대적인 영향을 끼쳤다. 그 발전 과정을 보면 ...

1. 라이프니츠가 생각한 <미분의 개념> 은 무엇인가?

2. Archimedes를 비롯한 수학자들은 원의 경우에 적용한 유사한 방법을 확장하여 타원형이나 더 불규칙한 도형의 면적에 적용하였다. 이 방법의 아이디어는 무엇인가?

3. 미분을 배우는 이유는 무엇일까?

4. 컴퓨터 단층 촬영 장치인 CT에는 적분의 어떤 아이디어가 이용된 것인가?

5. 19세기에 Bernard Bolzano, Augustin-Louis Cauchy, Karl Weierstrass 등을 거치면서, 극한을 엄밀한 논리(argument)를 이용하여 완전한 형식화 방법은 무엇인가?

6. 미분과 적분을 이어주는 정리는 무엇인지 말하고, 그 이유에 대하여 아는 대로 서술하시오~~

7. 행성들의 운동을 나타내는 Kepler의 법칙을 설명하기 위하여 미적분학을 개발한 사람은 누구인가?

8. 근사와 Taylor급수가 왜 중요한지 아는 대로 서술해 보시오.

9.  differential  dx  는 어떤 식으로 이용되나?

10. *** 을 구하기 위하여 거치는 과정을 단계별로 서술하시오.

  등 등 우리가 배운 내용에서 기말고사에  출제할 수 있는  문제를 생각 중이랍니다.

 QnA 에 위의 질문에 대한  답을 하면서, 연습을 해 두기를 권한답니다.

    (여러분의 활동 중 같이 공유할 좋은 질문과 답이 시험 문제가 될 가능성이 가장 높을 것입니다.)

따라서 질문과 답을 하면서 공부한 학생은 별도로 시험공부를 할 필요가 없을 것입니다.

 행운을 빌어요!

Good luck on your (서술형, Story telling) Midterm Exam and PBL~

 담당교수 드림   From your Prof.

◆ Your QnA Records

제목: 0의0승은 어떻게 처리해야하나요?

  그래프 그려보니 그냥 1로 취급하는 것 같던데 그런 식으로 봐도 되는 건가요?

제목: dir='minus'은 무슨 의미인가요?

  dir='minus' 어떤 명령어죠? 그리고 x^x에서 음수에서는 그래프가 그려지면 안되는 거 아닌가요?

제목: Smart learning environment

 smart learning 영상을 보기 전에 솔직히 책이 사라진 교실이란 상당히 집중이 안되있을 거란 생각이 많이 들었다 . 하지만 영상속의 모습은 자발적이었으며 집중도가 높았다.  특히 실험실에서 이제 책이라는 것이 주어진 지식을 얻는 것이 아닌 자신만의 새로운 책을 만들어가는 과정을 보고 깊게 감명 받았다.

  스마트러닝은 한 과목에 국한되어있지 않고 실험, 체육, 음악 외 다양한 분야에서 활용될 수 있는 새로운 교육의 패러다임을 제시하고 있다.

http://www.youtube.com/watch?v=Ll1P_57qluo

제목: Calculus-History-p

  발견이란 것이 항상 관측을 바탕으로 하지만, 관측을 한다고만 해서 발견이 이뤄지지않는다는 것을  케플러의 사례를 통해 알게되었습니다.  Tycho Brahe는 평생을 육안으로 천문관측을 한 사람으로, 육안으로 볼 수 있는 극한까지 이른 최고의 관측 천문학자였지만, 그의 데이터를 바탕으로 일정한 법칙을 알아낸 사람은 결국 Kepler였습니다. Tyco Brahe 역시 같은 데이터를 가지고 있었지만 발견하지 못했고 Kepler는 그 데이터를 분석함으로서 발견을 이뤄냈습니다. 이와같은 일화를 통해서 수학에서의 분석의 중요함을 느꼈습니다.

제목: 미적분학을 배우는 이유에 대한 생각

  미적분학에 핵심은 변화에 대한 관찰 그리고 그것을 통한 어느 시점에서의 상황의 예측정도라고 생각합니다. 미적분학에 대한 일상생활관련 문제를 보더라도 대포를 쏘아올린 것이 포물선을 그려 얼마정도 후에 어느 위치에 있을 거라던지, 물을 붓는데 이것의 높이변화와 부피변화를 구하라던지와 같은 문제들입니다. 이러한 일상생활관련 문제들이 단지 문제를 위한 문제는 아닐거라고 생각합니다. 그리고 또한 위 같은 문제들이 묻는것은 대체로 미래의 어느 때입니다. 따라서 위 같은 사실을 종합적으로 고려해볼때 우리가 미적분학을 배우는 이유는 현실을 관찰하고, 그 관찰한 것을 이용해서 '미래를 예측하기 위함' 이라고 생각합니다.

제목: sage에서 dir의 의미: limit(f(x), x=a, dir='plus') f(x)의 x=a+ 에서의 극한값

  이번에 세이지를 통해서 극한값을 구하는 중에 명령어 dir의 의미와 극한값을 구하는 법을 알게 되었습니다.

limit(f(x), x=a)            f(x)의 x=a 에서의 극한값

limit(f(x), x=a, dir='plus')       f(x)의 x=a+ 에서의 극한값

limit(f(x), x=a, dir='minus')      f(x)의 x=a- 에서의 극한값

'dir'=direction 방향을 나타내는데 쓰입니다.

http://matrix.skku.ac.kr/sage   에서 lim을 검색하면 극한계산에 대해서 나옵니다.

제목 : Arc length 구하는 아이디어

　　곡선의 길이를 구하는 것은 그 곡선을 미세하게 분해해서 보는 것으로 시작한다. 이때 곡선을 매우 작게 쪼게면 직선과 같다. 이때 그 직선을 직각삼각형의 빗변으로 생각하면, x의 미소변화량과 y의 미소변화량이 각각 밑변과 높이가 되고, 따라서 피타고라스의 정리를 이용하여 구할 수 있다((x^2+y^2)^1/2).

제목 : Center of mass를 배우는 이유 - 내용요약

질량중심을 배우는 이유는 질량중심이라는 것이 특정한 상황에서 구하기 쉬울 수 있고, 그 질량중심을 안다면 계산이 쉬워지기 때문이다.

 In physics, the center of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero.

Calculations in mechanics are simplified when formulated with respect to the center of mass.

In the case of a single rigid body, the center of mass is fixed in relation to the body, and if the body has uniform density, it will be located at the centroid.

The center of mass may be located outside the physical body.

The center of mass is the mean location of all the mass in a system.

( 질량중심을 잡음으로써 계산이 간단해진다. 하나의 강체(간단히 변형이 일어나지않는 물체)에서 질량중심은 고정적이고 질량의 밀도가 같다면 질량중심은 중심점에 위치하게된다. 질량점은 물체밖에 있을수도 있고, 시스템에서 모든 질량의 실질적 위치가 된다.)

2. SKKU Calculus  - Record  (기록)

[동영상 강의]

1.1 History of Calculus            http://youtu.be/ODfMaHgIhAc

How to manage our class Review  http://youtu.be/XWEQFlv4jKc

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

Chapter 1. Functions     

http://youtu.be/cl8GqIWIRD0

문제풀이 by 곽주현  http://youtu.be/BNKUzSohiD8

문제풀이 by 장찬영 http://youtu.be/x0E0ZMxZ3Og

문제풀이 by 임효정  http://youtu.be/vx7GCWY68Zw

Chapter 2. Limits and Continuity

2.1 Limits of functions    http://youtu.be/VBCeAllP1M0

문제풀이 by 장재철-이훈정, http://youtu.be/LZSmRPAAXME

문제풀이 by 황인철 http://youtu.be/hj8d-j_DGf4

2.2 Continuity             http://youtu.be/zGxx3PUCTnM

문제풀이 by 이훈정 http://youtu.be/azrkT1RP4-c

Chapter 3. Theory of Differentiation

3.1 Definition of Derivatives, Differentiation  http://youtu.be/A-vDsF9ulTs

문제풀이 by 김태현  http://youtu.be/7wTBWuk2CzU

3.2 Derivatives of Polynomials, Exponential Functions, Trigonometric Functions, The product rule   http://youtu.be/XXMnCESesfQ

문제풀이 by 조건우  http://youtu.be/Ei5KGW9vZhE

3.3 The Chain Rule and Inverse Functions  http://youtu.be/HfScHEsPfKI

문제풀이 by 유휘의 http://youtu.be/aSKm12922FE

3.4 Approximation and Related Rates   http://youtu.be/ViRwEJ0Wfkw

문제풀이 by 김종민  http://youtu.be/JmBOv6_D6qA

Chapter 4. Applications of Differentiation

4.1 Extreme values of a function   http://youtu.be/mXVU8OqIHJY

문제풀이 by 김태영  http://youtu.be/_V4MryNEzWY

4.2 The Shape of a Graph    http://youtu.be/cZrAF_77On4

문제풀이 by 김태영  http://youtu.be/SVOWADHlzV8

4.3 The Limit of Indeterminate Forms and L’Hospital’s Rule 

http://youtu.be/vp-gck5-gKE 

문제풀이 by 신종희  http://youtu.be/gR2luDDPsMY

4.4 Optimization Problems   http://youtu.be/k0NtkmZFnh8

문제풀이 by 이승철  http://youtu.be/AELEV2ElaeQ

4.5 Newton’s Method  http://youtu.be/VxCfl2JzMYU

문제풀이 by 이승철  http://youtu.be/fdBHQ46g9RE

Chapter 5. Integrals

5.1 Areas and Distances   http://youtu.be/mT_oxlD6RSA

문제풀이 by 남택현  http://youtu.be/Y_nCn76RPmY

5.2 The Definite Integral   http://youtu.be/GIm3Oz58Ti8

문제풀이 by 남택현  http://youtu.be/iUsf1h_hTAE

5.3 The Fundamental Theorem of Calculus   http://youtu.be/Zf1HT2H2fbA

문제풀이 by 정승찬 &Kim  http://youtu.be/Pa4Z38KkDVY

5.4 Indefinite Integrals and the Net Change Theorem 

http://youtu.be/E6I3EDzAVuU 

5.5 The Substitution Rule   http://youtu.be/h7tmvmNOliU

문제풀이 by 이한울 http://youtu.be/0TMbpCPO4Uc

5.6 The Logarithm Defined as an Integral  http://youtu.be/kD0Z9PqetsA

문제풀이 by 이한울 http://youtu.be/ymDImdIQ90c

   미적분학 with Sage Midterm Exam    http://youtu.be/QAEI7A2DMMM

Chapter 6. Applications of Integration

6.1 Areas between Curves   http://youtu.be/o53phm5cqJE

6.2 Volumes   http://youtu.be/4-ChOAFbJAs

문제풀이 by 김종민  http://youtu.be/Fd4Mguf2dbU

6.3 Volumes by Cylindrical Shells   http://youtu.be/qM1izf8qeX8

문제풀이 by 신영찬  http://youtu.be/gNaKkA0UNHg      

6.4 Work   http://youtu.be/u3ZaJWhKy6k

문제풀이 by 김건호  http://youtu.be/SmIo2yaxNsY

6.5 Average Value of a Function   http://youtu.be/zmEeGmwQTB0

문제풀이 by 신종희  http://youtu.be/BVahd-DJoe8 

Chapter 7. Techniques of Integration

7.1 Integration by Parts   http://youtu.be/WX-6C9tCneE

문제풀이 by 이인행  http://youtu.be/jKCAGJ4HqvQ

7.2 Trigonometric Integrals   http://youtu.be/sIR0zNGQbus

문제풀이 by 김태현  http://youtu.be/ytETYf1wLbs

7.3 Trigonometric Substitution   http://youtu.be/avTqiEUi8u8

문제풀이 by 이훈정  http://youtu.be/utTQHIabTyI 

7.4 Integration of Rational Functions by the Method of Partial Fractions

http://youtu.be/KLTHp_7G4cI 

문제풀이 by 장재철  http://youtu.be/SkNW_bax0YI

7.5 Guidelines for Integration   http://youtu.be/Fgn8U4We60o

문제풀이 by 김대환 http://youtu.be/-N9Fe_Arp2c

7.6 Integration Using Tables    http://youtu.be/tn9jLkgTMp8

문제풀이 by 조건우  http://youtu.be/EnEQ9ZS3B_k

7.7 Approximate Integration     http://youtu.be/hg2pw1n1cZI

7.8 Improper Integrals      http://youtu.be/rquxbYrC0Yc

문제풀이 by 이송섭  http://youtu.be/C3kb4c9nLXM

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

Chapter 8. Further Applications of Integration

8.1 Arc Length   http://youtu.be/7OVqI20z_Bw

문제풀이 by 남택현  http://youtu.be/A8N-mDD0ja8

8.2 Area of a Surface of Revolution   http://youtu.be/Eq4i2A8eKxA

문제풀이 by 정승찬  http://youtu.be/yZFJDJgTJfw

8.3 Applications of Integral Calculus    http://youtu.be/1ZAJeP16pAQ

8.4 Differential equations      http://youtu.be/uHfOjz8I4-s

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

3. Linear Algebra (English)

[Linear Algebra Syllabus (선형대수학 수업계획서)]

http://matrix.skku.ac.kr/2017-Album/LA-syllabus.htm  

Linear Algebra, First Class, Syllabus-Review  https://youtu.be/43nhECDzfiE 

(English Textbook)  http://goo.gl/t3JcNP  

[Lectures Recorded]

http://matrix.skku.ac.kr/2017-Album/2017-Spring-Lectures.htm

http://matrix.skku.ac.kr/2015-LA-FL/Linear-Algebra-Flipped-Class-SKKU.htm 

Linear Algebra Lecture Note (English) http://matrix.skku.ac.kr/LA/   

Linear Algebra Lecture Note (Korean) http://matrix.skku.ac.kr/LA-K/ 

Linear Algebra Simulations:  http://matrix.skku.ac.kr/LinearAlgebra.htm

Chapter 1. Vectors 

*1.1 Vectors in n-space and *1.2 Inner product and Orthogonality

https://youtu.be/f6eKIuLE-Ko

1.3  Vector Equations of lines and planes  https://youtu.be/MR1md8R1T_g

Chapter 2. Linear system of equations

2.1 Linear System of Equations, https://youtu.be/JbfSo5G6JR0

    Review  https://youtu.be/nhYG5uuGHqU

2.2 Gaussian and Gauss-Jordan elimination, https://youtu.be/ySncbrZTdMk

2.2–2.3 Exercise https://youtu.be/ySncbrZTdMk  https://youtu.be/khbfoZBFfvA

2.4  Exercises, https://youtu.be/khbfoZBFfvA

Chapter 3. Matrix and Matrix Algebra

3.1 Matrix operation https://youtu.be/rDt3EOGl9lg

3.2 Inverse matrix https://youtu.be/o2iT6ZT5WIU

3.3 Elementary matrix  https://youtu.be/DubGO81dTAI

3.4 (part 1) Subspace  https://youtu.be/gNm7yzk8ess

3.4 (part 2) Linear independence https://youtu.be/LswYaDbj4ds 

3.5 Solution set of a linear system and matrix https://youtu.be/7zZDGgPGE4s

3.6 Special matrices and Sec 3.8  Exs/Sol https://youtu.be/gve7cYW3W9I

*3.7 LU-decomposition  http://youtu.be/lKJPnLCiAVU

  Student Review : Ch3-Ch2-Ch1   https://youtu.be/m4ZEHknJMZY

Chapter 4. Determinant

4.1 Definition and Properties of the Determinants  https://youtu.be/ltxi0hCUILg

4.2 Cofactor Expansion/ Appl of Determinants https://youtu.be/Yn5qu_062sA

4.3 Cramer's Rule https://youtu.be/rAmfnERqfU8

*4.4 Application of Determinant  https://youtu.be/APsZ33BBOVs

4.5 Eigenvalues/Eigenvectors &4.6 Excercise https://youtu.be/s1OI74nr660 

Chapter 5. Matrix Model 

5.1 Lights out Game http://youtu.be/_bS33Ifa29s

5.2 Power Method  http://youtu.be/CLxjkZuNJXw

5.3 Linear Model (Google) http://youtu.be/WNUoXLh8i_E

    Project: http://youtu.be/coNq48CW6Pg 

   - Ch 5 Matrix Model 학생 발표 https://youtu.be/4u9LtmX7lvk 

Chapter 6. Linear Transformations

6.1 Matrix as a Function (Transformation)  https://youtu.be/Es4BfHnIq7g

6.2 Geometric Meaning of LT (part 1) https://youtu.be/V6m0PKQm6es

    (part 2) https://youtu.be/qeRmhJQIphI  

6.3 Kernel and Range  https://youtu.be/7OfNTNl6IjI

6.4 Composition of LT and Invertibility   https://youtu.be/Im7uaogKySw  

*6.5 Computer Graphics with Sage  https://youtu.be/45zSkGN7inw

6.6 Exercises  https://youtu.be/pJgIaHpIhsM

Chapter 6. QnA Review https://youtu.be/snQsn2J_tuA  

  LA Midterm PBL 1 Presentation https://youtu.be/WG-HFdER5Ro  

  LA PBL and Ch6 and Ch4 Student Review https://youtu.be/xq4YyRtzTKg

      http://matrix.skku.ac.kr/2016-album/LA-Sol-Ch-1-2-3-4-6/index.html 

      http://matrix.skku.ac.kr/LA-Lab/Solution/ 

      http://matrix.skku.ac.kr/2016-album/LA-Main-Theorems/index.html 

 Sample Midterm Exam: http://matrix.skku.ac.kr/LA/2016-S-LA-Midterm-Final-Solution.pdf

LA Midterm Exam 

 http://matrix.skku.ac.kr/2017-Album/2017-S-LA-Midterm-Exam-Final-3.pdf

 LA Midterm Exam Sol 

 http://matrix.skku.ac.kr/2017-Album/2017-S-LA-Midterm-Exam-Solution-Final3.pdf

 LA Midterm Exam Sol 

 http://matrix.skku.ac.kr/2017-Album/LA-Midterm-Exam-Solution.htm  (image)

 2017 Spring LA Midterm Exam Review https://youtu.be/DOhahD4Nb44

Chapter 7. Dimension and Subspaces 

7.1 and 7.2 (Review)  Bases and dimensions, Basic spaces  

https://youtu.be/45J08qGSzmk

7.3,7.4, 7.5, Rank-Nullity theorem, Rank theorem, Projection theorem 

             https://youtu.be/WytZezfNAiI

*7.6 Least square solution  https://youtu.be/GwHh5lh5wEs

 Grade/QnA/ Review: https://youtu.be/eEAHVs351u8  

7.7 Gram-Schmidt orthonormalization process  https://youtu.be/Px6Gaks9fXQ

* 7.8 QR-Decomposition; Householder

   https://youtu.be/gQ7gxTx5f9k 

7.9 Coordinate vectors  https://youtu.be/VR9FoZDQmAo

Chapter 8. Diagonalization

8.1 Matrix Representation of LT  https://youtu.be/LpIR47W_stw

8.2 Similarity and Diagonalization  https://youtu.be/wqrLcfSeL8Q

8.3 Diagonalization with orthogonal matrix  https://youtu.be/5Sg-Edczw_g  

8.4 Quadratic forms and Sec *8.5 Appl.  https://youtu.be/mjAr3ddevE8  

8.6 SVD and Pseudo-Inverse  https://youtu.be/KU5l-XWDJuo   

8.7 Complex eigenvalues and eigenvectors  https://youtu.be/-l7uTfYHjFU

8.8 Hermitian, Unitary, Normal Matrices  https://youtu.be/NRTmmmC-L9k

*8.9 Linear system of differential equations 

http://www.hanbit.co.kr/EM/sage/1_chap6.html 

https://youtu.be/9IeskMZ_hn4

Chapter 9. General Vector Spaces 

9.1 Axioms of Vector Space, https://youtu.be/RnKjspG65AM

9.2 Inner product spaces; *Fourier Series, https://youtu.be/J0s8AkP4E38

9.3 Isomorphism, https://youtu.be/WiZZtF0c1hY

Chapter 10. Jordan Canonical Form

10.1 Finding the Jordan Canonical Form with a Dot Diagram

https://youtu.be/8fwPPOg8LW0   https://youtu.be/1E3wXN1oZyc

*10.2 Jordan Canonical Form and Generalized Eigenvectors, 

https://youtu.be/YrRnCByzxNM    https://youtu.be/yJ7n0icjtNA

https://youtu.be/lK4_Kp6P_N4

10.3 Jordan Canonical Form and CAS, https://youtu.be/YrRnCByzxNM  

http://youtu.be/LxY6RcNTEE0

 학생 문제 풀이, https://youtu.be/y4173MpjoxE   http://youtu.be/9-G3Fd2xOW0 Chapter 10 http://www.youtube.com/watch?v=adWzUKKmO2k

4. Linear Algebra (Korean)

[선형대수학 Korean Lectures – 우리말 강의 (동영상)]

PBL - Flipped Learning   http://youtu.be/Mxp1e2Zzg-A  

Lecture 1 Introduction     http://youtu.be/w7IzR4nGa3Q  

(Korean)  http://matrix.skku.ac.kr/2015-Album/Big-Book-LinearAlgebra-Eng-2015.pdf

(디지털 교과서)  http://matrix.skku.ac.kr/LA-K/

Chapter 1. Vectors 

1.1 벡터 and 1.2 내적  http://youtu.be/aeLVQoPQMpE  

1.3 벡터방정식         http://youtu.be/4UGACWyWOgA  

Chapter 2. Linear system of equations

2.1 선형연립방정식  http://youtu.be/CiLn1F2pmvY 

2.2 Gauss-Jordan 소거법  http://youtu.be/jnC66zvqHJI 

Chapter 3. Matrix and Matrix Algebra

3.1 행렬연산  http://youtu.be/DmtMvQR7cwA 

3.2, 3.3 역행렬과 기본행렬  http://youtu.be/GCKM2VlU7bw  

3.4 부분공간  http://youtu.be/HFq_-8B47xM 

3.5 해공간  3.6 특수행렬   http://youtu.be/daIxHJBHL_g  

Chapter 4. Determinant

4.1 행렬식  http://youtu.be/DM-q2ZuQtI0 

4.2 여인자 전개와 역행렬  http://youtu.be/XPCD0ZYoH5I 

4.3 크래머의 법칙 4.4. Appl, 4.5 고유값, 고유벡터

    http://youtu.be/OImrmmWXuvU 

Chapter 6. Linear Transformations

6.1 선형변환  http://youtu.be/YF6-ENHfI6E 

6.2 선형변환의 기하학적 의미  http://youtu.be/cgySDj-OVlM 

6.3 핵과 치역  http://youtu.be/9YciT9Bb2B0

6.4 선형변한의 합성과 역행렬  http://youtu.be/EOlq4LouGao 

  LA Midterm Exam  http://youtu.be/R3F3VNGH8Oo   

Chapter 7. Dimension and Subspaces 

7.1 기저와 차원  http://youtu.be/or9c97J3Uk0 

7.2 주요 부분공간들  https://youtu.be/BC9qeR0JWis 

7.3 Rank Nullity Theorem  http://youtu.be/ez7_JYRGsb4 

7.4 계수정리  http://youtu.be/P4cmhZ3X7LY 

7.5 정사영정리  http://youtu.be/GlcA4l8SmlM 

7.6* 최소제곱해 https://youtu.be/BC9qeR0JWis

7.7 Gram-Schmidt의 정규직교화과정  http://youtu.be/gt4-EuXvx1Y  

7.8* QR-분해, Householder transformations   https://youtu.be/crMXPi2lgGs

7.9 좌표벡터  http://youtu.be/M4peLF7Xur0

Chapter 8. Diagonalization

8.1 선형변환의 행렬표현    http://youtu.be/gn5ve1tXD7k 

                           http://youtu.be/jfMcPoso6g4  

8.2 닮음과 행렬의 대각화  http://youtu.be/xirjNZ40kRk 

8.3 직교대각화             http://youtu.be/jimlkBGAZfQ  

8.4 이차형식  http://youtu.be/vWzHWEhAd-k 

8.5* Appl of Quadratic Function  http://youtu.be/cOW9qT64e0g  

8.6 Singular Value Decomposition  https://youtu.be/ejCge6Zjf1M 

8.7 and 8.8 복소고유값, 복소고유벡터, 정규행렬  http://youtu.be/8_uNVj_OIAk  

Chapter 9. General Vector Spaces 

9.1 and 9-2 일반벡터공간, 내적공간  http://youtu.be/m9ru-F7EvNg  

9.3 동형사상   http://youtu.be/frOcceYb2fc 

Chapter 10. Jordan Canonical Form

10.1 Jordan 표준형  http://youtu.be/NBLZPcWRHYI 

10.3  Jordan Canonical Form with Sage  http://youtu.be/LxY6RcNTEE0 

  (15 주차)  복습과 프로젝트 발표

 Math, Art and 3D Printing  http://youtu.be/olTfft1cuGw  

 PBL 보고서 by 김병찬 &우시명  http://youtu.be/hUDuQ8e8HsU  

             by 손홍철  http://youtu.be/woyS_EYWiDs  

             by  박민  http://youtu.be/E-5m65-8Ea8 

             by 전승준  http://youtu.be/JHT6aTQhr-A 

             by 김태용, 이학현, 이종화  http://youtu.be/JFVM4KRr2nc  

  (16주차) 기말고사  

  http://matrix.skku.ac.kr/2015-Album/CLA-Final-Sample-Exam.pdf 

[실습실]

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

http://matrix.skku.ac.kr/2018-album/LA-Sec-1-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-1-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-1-3-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-2-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-2-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-3-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-4-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-5-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-3-6-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-4-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-4-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-4-5-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-6-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-6-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-6-3-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-6-4-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-3-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-4-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-7-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-7-9-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-8-1-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-8-2-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-8-3-lab.html 

http://matrix.skku.ac.kr/2018-album/LA-Sec-8-8-lab.html 

 http://matrix.skku.ac.kr/2018-album/LA-Sec-9-1-lab.html 

 http://matrix.skku.ac.kr/2018-album/LA-Sec-9-3-lab.html

Problems in Chapter 1

Solved by 김민수 Revised by 배성준 Finalized by 김민수

Refinalized and Final OK by SGLee

Page 13 Exercise 1.2 No.7 (New)

Q : Graph the function. Specify the intervals where the function is increasing and where it is decreasing.

.

Sol)

Graph From sage

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

Answer: 

(i) Symmetric with respect to the -axis.

(ii) Increasing on and .

(iii) Decreasing on and .                                         ■

Solved by 우시명  Revised by ShaoweiSun  Finalized by 우시명

Final OK by SGLee

Calculus with Sage p.13 Chapter 1 Problem 1.2-10

Graph the following function. What symmetry, if any, do the graphs have?

Specify the intervals where the function is increasing and where it is decreasing.   

Sol) 

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

1. It is not symmetric.

            (See the graph to make sure it is not symmetric.)

2. [Find Decreasing Interval and Increasing Interval]

Use derivative to find the point where the graph starts to decrease or increase.

Find derivative : [CAS]

Derivative :

Solve the equation for :

Answer :

(, ) : Decreasing Interval

(, ) : Increasing Interval                                       ■

Solved by 계성곤

Revised by 배성준

Finalized by 계성곤

Page 21 Exercise 1.3 (New) NO.6

Draw the graph of given function.

Sol) 

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

plot(exp(-1/sqrt(x^2+3)), x, -10, 10, color='blue')

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

plot(exp(-1/sqrt(x^2+3)), x, -1000, 1000, color='blue')

The graph's asymptote is ■

Solved by 계성곤

Revised by 배성준

Finalized by 계성곤

Refinalized by 이송섭

Final OK by SGLee

Page 30 Exercise 1.4 (New) NO.12

Graph the function.

Sol) 

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

var('x')

plot((x^3-6)^(1/3), x, 2, 10, color='blue')

                             ■

Problems in Chapter 2

Solved by 이송섭

Resolved by 배성준

Finalized by 이송섭

Refinalized and Final OK by SGLee

Page 47 Exercise 2.1 (Old) No.3

Find the following limit.

Sol)

Since  and,                  

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

limit(sin(x)/abs(cos(x)), x=pi/2)

+Infinity

Answer:                                              ■

Solved by 변희성

 Revised by 배성준

Refinalized and Final OK by SGLee

page 47 Exercise 2.1 4 (New)

Find the following limit or explain why the limit does not exist.

Sol)

If ,

.

If ,

=>     

=>   The limit does not exist.                                                 ■

Solved by 이송섭

Finalized by 배성준

Refinalized by 이송섭

Refinalized and Final OK by SGLee

P.49 Chaptor2.1 exercise14 (old)

Let .

Find all positive integer such that .

Sol)

Then the limit should satisfy .

1)   when ,

2)   When ,

3)   When and is even,

4)   When and is odd,

Thus, only (3) satisfies the condition, that is .

Answer:                                         ■

Solved by 이송섭

Revised by 우시명

Revesed by 배성준

Finalized by 이송섭

Refinalized by 이송섭

Final OK by SGLee

p.50 Chapter2.1 Exercise No.19(old)

19. Use the argument to prove that

Proof)

Let .  [Find > 0]

Take

If ,

Then .

 whenever

We haver proved .                                               ■

[side cal]

Solved by 배성준

Finalized by 이송섭

Final OK by SGLee

Page 50 Exercise 2.1 (New) No.19

Use the argument to prove that .

Proof)

Let .  [Find > 0]

Take

If ,

Then .

 whenever

We haver proved .                                               ■

[side cal]

Epsilon Delta Proof

Solved by 배성준

Revised by 이송섭

Finalized by 문지호

Refinalized by 배성준

Final OK by SGLee

Page 50 Exercise 2.1 (New) No.20

17. Use the argument to prove that

if and .

Proof)

Let ,  [Find  > 0]

Take

If ,

then

    whenever .      █

(side cal)

Epsilon Delta Proof

Solved by 이송섭

Revised by 이송섭

Finalized by 이송섭

Final OK by SGLee

[1] 연속함수인 경우(아주 간단한 예)

Calculus with Sage P.51 Chapter 2.1-24 (Old).

Using the argument prove

Proof)

Let ,  [Find  > 0]

Take .

If ,

then .

 whenever .                            ■

(side cal)

.  

Final OK by SGLee

[2]  연속함수인 경우(예)

 Show .

Proof. ∀ >0 [Find ]

   Let  =

If |-| < ,

then 

    ( = )

 =

    ■              

<side cal.>

               □

Final OK by SGLee

[3]  연속함수인 경우(예)

Show.

Proof. ∀ >0 [Find ]

   Let =

If     ,

then

      ( = )

      =

       =

       =

         = ■

 <side cal.>

                              □

Epsilon Delta Proof

ReFinalized and Final OK by SGLee

[4] 연속함수인 경우 (min 기법 예)

Show .

Proof. ∀ >0 [Find ]

  Let {}

 If ,

 then =

                     ■

<side cal.>

         □

Epsilon Delta Proof

[4] 연속함수인 경우 (min 기법 예)

Show

Proof. ∀ε>0,  Find δ

  choose {}

 If

= 

　Show

Proof. ∀ε>0,  Find δ

   Let =min{}

ReFinalized and Final OK by SGLee

[4] 연속함수인 경우 (min 기법 예) 　

Show .

Proof. ∀ >0 [Find ]

   Let

If

then =

             ■           

<side cal.>

Take .

If , then .

=

                □

ReFinalized and Final OK by SGLee

[7] 불연속인 점에서의 경우 limit (예) 

Show .

Proof. ∀ >0  [Find ]

 Let .

  If  ,

  then 

       =

          ■

<side cal.>

   by (*)

 and

  □   

https://sagecell.sagemath.org/ (*)

Final OK by SGLee

[8] 발산의 경우 limit (예) 

Show .

Proof. ∀ Large number ,[Find ]

   Let   ()

If  ,

then   ■

<side cal.>

Since ,

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

Solved by 이송섭

Finalized by 김민수

Final OK by SGLee

Page 63 Exercise 2.2 No.18 (New)

1. Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.

Proof)

Let

Then .

Since is a real-valued continuous function on the interval and , there is a root such that by the intermediate value theorem.                             █

[Sage] 그림을 그려보면 1과 2 사이에서 근이 존재함을 확인 할 수 있다.   

True

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

Solved by 계성곤

Finalized by 이송섭

Final OK by SGLee

Page 63 Exercise 2.2 (New) NO.18

Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.

Proof)

Let .

The function is continuous on the domain .

We can easily check and .

By the intermediate value theorem, there exists s.t. .

   has at least one real root.                       ■

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

var('x')

f(x)=sqrt(3*x^2-x^3)-1.5

P=plot(f,0,3)

show(P, aspect_ratio=1)

bool(f(1)<0)

True

bool(f(2)>0)

True            

Solved by 배성준 Revised by 계성곤 Revised by 배성준 Finalized by 배성준

Final OK by SGLee

Page P64 Exercise 2.2 (New) No.20

Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.

, (0,1)  

Proof)                  

Let =.

 is continuous on the domain .

We can easily check and .

By the intermediate value theorem, there exists s.t. .

This implies that there is a root of the given equation in the specified interval.  ■ 

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

var('x')

f(x)=(ln(2*x^2))^2/(exp(x^2)+3/x)

g(x)=x^6

p1=plot(f(x)-g(x), 0, 1, exclude=[0], ymax=1, ymin=-1)

show(p1)

True

True     

Solved by 문지호

Fixed by 이송섭

Finalize by 문지호

Final OK by SGLee

Page 64 exercise 23 (old)

Show that the following equation has at least one real root.

Proof)

Define a function .

The function is continuous on the domain .

We can easily check > 0 and < 0.

By the intermediate value theorem, there exists s.t. .

  has at least one real root.    ■

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

sage: P = plot(e^x, x, 0, 1, linestyle="--", color='red')

sage: Q = plot(4*sin(x),x,0, 1)

sage: show(P+Q)

[설명]  과 가 [0, 1] 사이의 한점에서 만나는 것을 두 가지 방법으로 쉽게 확인 할 수 있었다. Solved by 김민수

Solved by 계성곤

Page 64 Exercise 2.2 (New) NO.23

Show that the following equation has at least one real root.

Sol) 

we may draw both and in one graph to find intersections. It shows the function has one real root in 0<x<1.

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

p1=plot(ln(x)+3, x, 0, 9, color='blue')

p2=plot(4*cos(x), x,0, 9, color='red')

show(p1+p2, ymax=5, ymin=-4)

The following Sage commands give the value of the intersection in the interval 0<x<1.

find_root(ln(x)+3==4*cos(x), 0, 1)

Answer : 0.8024194649325627■

Problems in Chapter 3

Solved by 우시명

Revised by Shaowe Sun

Finalized by 우시명

Final OK by SGLee

Calculus with Sage p.70 Chapter 3.1 Problem 5 (New)

Differentiate the following function, if it exists.

Sol) Use sage to find derivative.

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

var('t')

f(t)=(4*t-3)^3*(3*t-2)^2*(2*t-1)

diff(f(t),t).show()

Answer :

              ■        

Solved by 배성준

Revised by Shaowe Sun

Finalized by 배성준

Final OK by SGLee

Page 70  Chapter 3.1 Exercise No 7. (New)

Differentiate the following function using Definition, if it exists.

Sol)

                                               ■

Solved by 우시명

Finalized by 이송섭

Final OK by SGLee

Calculus with Sage p.70 Chapter 3.1 Problem 7 (New)

Differentiate the following function, if it exists.

 Sol) Use sage and find derivative.

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

var('t')

f(t)=t^(t-1)/t+1

diff(f(t),t)

((t - 1)/t + log(t))*t^(t - 1)/t - t^(t - 1)/t^2

Answer :                                ■

Solved by 김요섭

Finalize by 문지호

Refinalized and Final OK by SGLee

Chapter 3.1 page 70 exercise 8 (old)

Is the function

differentiable at ?

Sol)

The function is differentiable at if and only if exists.

=>

   The function is differentiable at     ■

Solved by 계성곤

Finalized by 계성곤

Refinalized by 이송섭

Final OK by SGLee

Page 83 Exercise 3.2 (New) NO.7

The normal line to a curve at a point is the line that passes through and is perpendicular to the tangent line to at . Find an equation of the normal line to the curve at the point (0, 1)

Sol) 

Slope of normal line: -1

Since normal line L pass through point (0,1)

Answer: Normal line is                                     ■

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

f(x)=1+e^x*sin(x)

df(x)=diff(f(x),x)

y(x)=-1/df(0)*x+1

y(x)

Answer : -x+1

p1=plot(f(x),x,-5,5, color='blue');

p2=plot(y(x),x,-5,5, color='red');

show(p1+p2,ymax=10,ymin=-5)

Solved by 계성곤

Revised by 김요섭

Finalized by 계성곤

Page 84 Exercise 3.2 (New) NO.9

Let . Find the values of and that make differentiable everywhere.

Sol) 

Note that is differentiable everywhere except =2. For to be differentiable at 2,

, so . And also have to be continuous at 2. , so ,

Therefore, and .■

Solved by 계성곤

Finalized by 이송섭

Final OK by SGLee

Page 85 Exercise 3.2 (New) NO.14

Find derivatives of the following function.

Sol) 

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

f(x)=(tan(x))^2*csc(x)/exp(x)

df(x)=diff(f(x),x)

print df(x)

Answer :

-e^(-x)*tan(x)^2*csc(x)*cot(x) + 2*(tan(x)^2 + 1)*e^(-x)*tan(x)*csc(x) - e^(-x)*tan(x)^2*csc(x)                                                                                    ■  

Solved by 우시명

Finalized by 이송섭

Final OK by SGLee

Calculus with Sage p.85 Chapter3  Problem3.2-15

Find derivatives of the following functions.(New)

Sol) 

Use Sage to find its derivative.

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

-(log(x)*sin(x) - cos(x)/x)*sin(x)/x^(-cos(x)) + cos(x)/x^(-cos(x))

Answer:          ■

Solved by 우시명

Finalized by 문지호

Final OK by SGLee

Calculus with Sage p.108 Chapter 3.3 Problem 12 (New)

Find the th derivative of . 

Sol) 

Find etc. And predict .

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

var('x')

f(x)=x^4+2/(2-x)

print "f'(x)=", diff(f(x), x).factor()

print "f''(x)=", diff(f(x), x, 2).factor()

print "f^(3)(x)=", diff(f(x), x, 3).factor()

print "f^(4)(x)=", diff(f(x), x, 4).factor()

print "f^(5)(x)=", diff(f(x), x, 5).factor()

print "f^(6)(x)=", diff(f(x), x, 6).factor()

print "f^(7)(x)=", diff(f(x), x, 7).factor()

print "f^(8)(x)=", diff(f(x), x, 8).factor()

print "f^(9)(x)=", diff(f(x), x, 9).factor()

f'(x)= 2*(2*x^5 - 8*x^4 + 8*x^3 + 1)/(x - 2)^2

f''(x)= 4*(3*x^5 - 18*x^4 + 36*x^3 - 24*x^2 - 1)/(x - 2)^3

f^(3)(x)= 12*(2*x^5 - 16*x^4 + 48*x^3 - 64*x^2 + 32*x + 1)/(x - 2)^4

f^(4)(x)= 24*(x^5 - 10*x^4 + 40*x^3 - 80*x^2 + 80*x - 34)/(x - 2)^5

f^(5)(x)= 240/(x - 2)^6

f^(6)(x)= -1440/(x - 2)^7

f^(7)(x)= 10080/(x - 2)^8

f^(8)(x)= -80640/(x - 2)^9

f^(9)(x)= 725760/(x - 2)^10

Answer :                                      ■

Solved by 변희성

Finalized by 계성곤

Page 109 Exercise 3.3 (New) NO.17

In problems below, find

Sol) 

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

var('x')

f(x)=ln((x-1)^(1/3)/(x^2-1)^(1/2))

diff(f(x),x)

Answer : -1/3*sqrt(x^2 - 1)*(3*(x - 1)^(1/3)*x/(x^2 - 1)^(3/2) - 1/((x - 1)^(2/3)*sqrt(x^2 - 1)))/(x – 1)^(1/3)■

Solved by 계성곤

Page 109 Exercise 3.3 (New) NO.19

In problem, find .

Sol) 

var('x')

diff((csc(x))^(sqrt(x+3)))

Answer :

-1/2*(2*sqrt(x + 3)*cot(x) - log(csc(x))/sqrt(x + 3))*csc(x)^sqrt(x + 3)

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

                                                                               ■

Solved by 김요섭

Finalize by 문지호

Refinalize by 계성곤

Final OK by SGLee

Chapter 3.3 Page 109 exercise 21 (new)

Find if .

Sol)  

Answer:

                                  ■

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

var('x');

df(x)=diff(x^(ln(x)),x);

ddf(x)=diff(df(x), x);

df(x), ddf(x)

show(df(x))

show(ddf(x))

Answer

p.s. Chain rule was used each time when differentiation is done.

Solved by 계성곤

Page 109 Exercise 3.3 (New) NO.21

Find if

Sol) 

var('x')

df(x)=diff((arcsec(x))^(arctan(x)),x);

df(x)

Answer : (log(arcsec(x))/(x^2 + 1) + arctan(x)/(sqrt(-1/x^2+

1)*x^2*arcsec(x)))*arcsec(x)^arctan(x)

ddf(x)=diff(df(x),x)

ddf(x)

Answer : (log(arcsec(x))/(x^2 + 1) + arctan(x)/(sqrt(-1/x^2 +

1)*x^2*arcsec(x)))^2*arcsec(x)^arctan(x) - (2*x*log(arcsec(x))/(x^2 +

1)^2 + 2*arctan(x)/(sqrt(-1/x^2 + 1)*x^3*arcsec(x)) - 2/(sqrt(-1/x^2 +

1)*(x^2 + 1)*x^2*arcsec(x)) + arctan(x)/((-1/x^2 + 1)*x^4*arcsec(x)^2) +

arctan(x)/((-1/x^2 + 1)^(3/2)*x^5*arcsec(x)))*arcsec(x)^arctan(x)

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

Solved by 우시명

Finalized by 계성곤

Page 109 Exercise 3.3 (New) NO.22

Find  and  of the following functions.

,

Sol) 

To find , derive about and derive about . Using foregoing two, find

var('t, a, b')

x(t)=a*sec(t)

y(t)=b*tan(t)

dx(t)=diff(x(t), t)

dy(t)=diff(y(t), t)

dydx=dy(t)/dx(t)

print "dy/dx=", dydx

Answer : dy/dx= (tan(t)^2 + 1)*b/(a*tan(t)*sec(t))

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

To find , derive two times and follow forgoing process.

var('t, a, b')

x(t)=a*sec(t)

y(t)=b*tan(t)

dx(t)=diff(x(t), t)

dy(t)=diff(y(t), t)

dydx=dy(t)/dx(t)

print "d2y/dx2=", (diff(dydx, t)/dx(t))

Answer : d2y/dx2= ((tan(t)^2 + 1)*b/(a*sec(t)) - (tan(t)^2 + 1)^2*b/(a*tan(t)^2*sec(t)))/(a*tan(t)*sec(t))

https://sagecell.sagemath.org/   ■

Solved by 변희성

Finalized by 배성준

Refinalized by 이송섭

Final OK by SGLee

P.110 Chapter 3.3 Exercises problem 30(New)

Given , find y'' at the point (1,2).

Sol)

Answer:                                            ■

Solved by 우시명

Finalized by 문지호

Refinalized and Final OK by SGLee

Calculus with Sage p.119 Chapter 3.4  Problem 1 (New)

Use differential to approximate .

Sol) 

Let .

Then at .

Thus,

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

var('x, dx');

f(x)=(x+36)^(1/2);

dy(x)=diff(f(x),x)*dx;

(f(0)+dy(x=0, dx=7)).n()

Answer : 6.58333333333333                                            ■

Solved by 계성곤

Finalized by 이송섭

Refinalized and Final OK by SGLee

Page 119 Exercise 3.4 (New) NO.2

Use differential to approximate the following quantity.

Sol) 

Let .

Then

Thus,

Set and at

Answer:                                        ■

Solved by 변희성

Finalized by 계성곤

Refinalized by 이송섭

Final OK by SGLee

Page 121 Exercise 3.4 (New) NO.12

Water is being pumped at a rate of 10 liters per minute into a tank shaped like a globe. The tank has a radius 10 meters. How fast is the water level rising when the depth of the water is 15 meters?

Sol) 

The tank is shaped like a globe. So in 15 meters, the surface is above the hemisphere.

Answer:    (meter/minute)                                 ■

Problems in Chapter 4

Solved by 우시명 Revised by 변희성 Finalized by 문지호

Revised by 배성준 Finalized by 문지호

Final OK by SGLee

Calculus with Sage p.132 Chapter 4.1  Problem 6 (New)

Find all critical numbers of the given function.

Sol)

Therefore critical numbers of are .

[CAS]  Draw the graph by using Sage.

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

Answer: critical numbers of are .     █

Solved by 문지호

 Finalized by 배성준

Final OK by SGLee

Page 132. Ch4.1 Exercise No 8. (New)

2. Find all critical numbers of the given function.

Sol) 

     =

=> when ()

Critical numbers : ()

[CAS]  Draw the graph by using Sage.

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

Answer: ()                          █

Solved by 이수헌

Revised by Sun Shaowei

Finalized by 문지호

Final OK by SGLee

Calculus with Sage p.134 Chapter 4.1 Problem 16(New)

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

Solved by 이수헌

Revised by Sun Shaowei

Finalized by 문지호

Final OK by SGLee

Calculus with Sage p.134 Chapter 4.1 Problem 18 (New)

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

Solved by 우시명

revised by 변희성

Finalized by 배성준

Refinalized and Final OK by SGLee

Calculus with Sage p.135 Chapter 4.1  Problem 20 (NEW)

Prove the inequality.

Sol) 

[CAS] Draw the graph of

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

In this graph, when .

                                                ■

Solved by 우시명

revised by 변희성

Fianlized by 배성준

Refinalized and Final OK by SGLee

Calculus with Sage p.135 Chapter 4.1  Problem 23 (New)

Prove the inequality using the Mean Value Theorem.

Proof) 

Let .

By the Mean Value Theorem there exist in such that

Since for all , .           ■

[CAS]  Draw the graph of .

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

var('x')

f=sin(x)-sin(x+1)

plot(f,-100,100,color='red')

Solved by 우시명

revised by 변희성

Finalized by 배성준

Refinalized and Final OK by SGLee

Calculus with Sage p.135 Chapter 4.1  Problem 20 (New)

Prove the inequality.

Proof) 

[CAS] Draw the graph of

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

  when

                                                     ■

Solved by 우시명

revised by 변희성

Finalized by 배성준

Refinalized and Final OK by SGLee

Calculus with Sage p.145 Chapter 4.2  Problem 4 (New)

Find the local maximum and minimum values of . In addition, find the intervals on which is increasing and decreasing, and the intervals of concavity and the inflection points, sketch a graph of .

Sol) 

[CAS] Draw the graph, and find the point or interval.

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

Answer: 

(a) Local maximum : , Local minimum: Not exist;

(b) Increase Interval: (, 0); Decreasing Interval: (0, )       

(c) 

in ,

So, there is no inflection point on the interval .

The graph is concave downward(위로 볼록) on (,)                   ■

Solved by 이수헌

Revised by 우시명

Finalized by Sun Shaowei

Final OK by SGLee

Calculus with Sage p.146 Chapter 4.2 Problem 10 (New)

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

Answer: 

Solved by 이수헌

Revised by 우시명

Finalized by Sun Shaowei

Final OK by SGLee

Calculus with Sage p.146 Chapter 4.2 Problem 11 (New)

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

Solved by 이수헌

Revised by  문지호

Finalized by 문지호

Refinalized and Final OK by SGLee

 Calculus with Sage p.147 Chapter 4.2 Problem 12 (New)

Sol)

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

,

,

Answer: Vertical asymptotes : ,  Horizontal asymptote : .            ■

Solved by 우시명

revised by 변희성

Finalized by 배성준

Final OK by SGLee

 Calculus with Sage p.147 Chapter 4.2 Problem 12 (New)

Find the vertical and horizontal asymptotes of .  

,

Sol) 

Draw the graph

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

Answer: vertical asymptote : No, horizontal asymptote : y=0.    ■

Solved by 문지호

Revised by Saowei Sun

Finalized by 배성준

Refinalized and Final OK by SGLee

Page 149. Ch4.2 Exercise No 24. (Old)

Prove the Concavity Test.

Concavity Test

Let be a function whose second derivative extists on an open interval .

(i) If for all in ,

          then the graph of is concave upward (위로 오목) on .

                                             (증가율이 점점 커진다는 의미)

(ii) If for all in ,

         then the graph of is concave downward (아래로 오목) on .

Proof) 

(i) BWOC (By the Way Of Contradiction)

   Suppose is not concave upward on (when for all in ,).

 => There are SOME such that and .   

By the mean value theorem, for some .

  at . It occurs a contradiction.

   : concave upward on .

(ii) Similarly.                                                              █

Solved by 문지호

revised by 변희성

Finalized by 계성곤

Final OK by SGLee

Page 149. Chapter 4.2 Exercise 25 (New)

Use CAS to find and when

Sol)

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

var('x')

f=(4*x^sqrt(2)-1)*sin(x)

df=diff(f,x)

ddf=diff(df,x)

show(df)

show(ddf)

Answer:

                  ■

Solved by 문지호

Revised by 계성곤

Finalized by 배성준

Final OK by SGLee

Page 155. Chapter 4.3 Exercise 14 (Old)

Find .

Sol) 

Apply L'Hospital's rule for a form of type .

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

var('x')

f=(x^3-2*x^2+2*x-1)/(x^3-x^2)

P1=plot(f, -0.5,1.5, ymax=2)

P2=plot(1,-0.5,1.5, linestyle='--', color='red')

show(P1+P2)

limit(f,x=1)

Answer:  Limit is 1                                                      ■

Solved by 우시명

revised by 변희성

Finalized by 배성준

Final OK by SGLee

Calculus with Sage p.155 Chapter 4.3  Problem 16 (New)

Find the following limit.

Sol) 

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

1

Answer : 1                                                ■

Solved by 문지호

Finalized by 이송섭

Final OK by SGLee

Page 156. Ch4.3 Exercise No 26. (New)

Find , .

Sol-1)

=     (by L'Hospital's rule) 

                                                              █

Sol-2)

=> (since is a continuous function, )

Answer :                                                █

Solved by 문지호

Revised by 계성곤

Finalized by 계성곤

Final OK by SGLee

Page 157. Chapter 4.3 Exercise 29 (Old)

 Let be a continuous function with and .

Find .

Sol 1)

Sol 2) Apply L'Hospital's theorem, since and

Answer:                            ■

Solved by 문지호

Finalized by 문지호

Final OK by SGLee

p.170 Chapter 4.5 Exercise No.13 (new)

Compute , the third approximation to the root of the given equation using Newton's method with the specified initial approximation .

   ,

Sol) 

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

Answer:                   █

Problems in Chapter 5

Solved by 배성준

Revised by 김요섭

Finalized by 문지호

Final OK by SGLee

Page 177 Exercise 5.1 (New) No.1

 Find the area under the curve from 0 to 5.

Sol)

Thus the length of each sub-interval is and the th sub-interval is given by . now we apply the right end formula to find required area.

 Side Calculus)

Answer:                                    ■

    http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 문지호

ReSolved by 김요섭

Revised by 이송섭

Fianllized by 우시명

Final OK by SGLee

p.177 Chapter 5.1 exercise No.2(new)

2. (New) Find the area of the region under the graph of from 0 to 2.

Sol-1) 

                                  ■

Sol-2) 

[CAS]

https://sagecell.sagemath.org/         

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

Answer:                           ■

Solved by 배성준

Revised by 김요섭

Finalized by 문지호

Final OK by SGLee

Page 177 Exercise 5.1 (New) No.3

Find the area under the curve from to , where .

Sol)

 when

=>.

Answer:                                               █

Solved by 배성준

finalized by 김요섭

Final OK by SGLee

Page 188 Exercise 5.2 (New) No.1

Find the Riemann sum by using the Midpoint Rule with the given value of to approximate the integral.

,

sol)

Let . With the interval width is

 and midpoints are for . So the Riemann sum is

                                                         ■

 http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 변희성

Revised by 우시명

Finalized by 배성준

Final OK by SGLee

Calculus with Sage p.188 Chapter 5.2  Problem 1 (New)

Find the Riemann sum by using the Midpoint Rule with the given value of n to approximate the integral.

Sol)

Let .  Also, , and midpoints are 4.5, 7.5, 10.5.

So, the Riemann sum is

Answer:                                           ■

                                                                         http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 계성곤

Revised by 배성준

Finalized by 김민수

Final OK by SGLee

Page 188 Exercise 5.2 (New) NO.4

8.  Find the Riemann sum by using the Midpoint Rule with the given value of to approximate the integral.

,

Sol) 

Let . With the interval width is and midpoints are

 ()

So the Riemann sum is

       =

Answer :                        ■

    http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 김요섭

Revised by 이송섭

Finallized by 우시명

Refinalized by 배성준

Final OK by SGLee

p.188 Chapter5.2 No.5 (new)

5. (New) Express the limit as a definite integral on the given interval.

Sol)

 then, .

Answer:                       ■

 http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 배성준

Revised by 김요섭

Finalized by 계성곤

Final OK by SGLee

Page 188 Exercise 5.2 (New) No.6

Express the limit as a definite integral in the given interval.

, [1, 6]

Sol)

          ■

 http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 계성곤

Finalized by 배성준

Final OK by SGLee

Page 189 Exercise 5.2 (New) NO.8

Express  the limit as a definite integral on the given interval.

, [2, 15]

Sol) 

=

=

Answer :                                            ■

 http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 문지호

Revised by 김요섭

Finalized by 배성준

Final OK by SGLee

p.189 Chapter 5.2 Exercise No.12 (Old)

Determine whether the statement is true or false. If it is true, explain why. If it is false, give a counter example.

If is differentiable on , then

Sol) 

Theorem 1 : If function is continuous on , then is integrable on .

Since is integrable on , is continuous on by Theorem 1.

And by property 2(page 185), is true.              

Answer:  True                             █

Solved by 김요섭

Revised by 이송섭

Finalized by 우시명

Refinalized by 이송섭

Final OK by SGLee

p.189 Chapter5.2 exercise No.16

Show that  if and are continuous and and , then                                   .

Sol1)

Let and

 and are continuous, so and exist.

So                                            ■

Solved by 변희성

Revised by 우시명

Finalized by 배성준

Final OK by SGLee

Calculus with Sage p.189 Chapter 5.2  Problem 19 (New)

Evaluate the intergral.  (You should mention which method you use.)

Sol) 

                          ■

[CAS] Draw the graph.

https://sagecell.sagemath.org/  

var('x')

f(x)=3*x+5

p1=plot(f(x), (x,-10,-5))+plot(f(x), (x,-5,-5/3), fill=true)+plot(f(x), (x,-5/3,10), fill=true)

p2=parametric_plot((-5,x),(x,-15,40), linestyle='--')

p3=parametric_plot((10,x),(x,-15,40), linestyle='--')

show(p1+p2+p3)

Solved by 문지호

Finalized by 문지호

Final OK by SGLee

p. 190 Chapter 5.2 Exercise 20 (New)

Evaluate the integral. (You should mention which method you use)

Sol) 

[FTC] Find the anti-derivative of the integrand.

                                            ■

[CAS]   https://sagecell.sagemath.org/

var('x,t');

f(x)=(e-1)*x^2-3

p1=plot(f(x),(x,-1,1))+plot(f(x),(x,1,e),fill="axis")+plot(f(x),(x,e,4))

p2=parametric_plot((1,t),(t,0,45),linestyle='--')

p3=parametric_plot((e,t),(t,0,45),linestyle='--')

show(p1+p2+p3) 

print integral(f(x),x,1,e)

-10/3*e - 1/3*e^3 + 1/3*e^4 + 10/3 

Answer:                         ■

Solved by 변희성

Revised by 우시명

Finalized by 문지호

Final OK by SGLee

Calculus with Sage p.190 Chapter 5.2  Problem 21

 Evaluate the intergral. (You should mention which method you use.)

Sol) 

[CAS] Draw the graph.

https://sagecell.sagemath.org/ 

var('x');

f(x)= -x^2-x+6

print integral(f(x),x,-3,6)

plot(f(x),(x,-5,0))+plot(f(x),(x,-3,6),fill=true)+plot(f(x),(x,6,9))

Answer:                             ■

Solved by 배성준

Revised by 김요섭

Finalized by 계성곤

Refinalized and Final OK by SGLee

Page 191 Exercise 5.2 (New) No.25

Evaluate the integral by interpreting it as a sum of the areas.

Sol)

Let

=>  

=>

Since , let and .

=>   and

=>

Answer:                                                        ■

Solved by 계성곤

Finalized by 김민수

Refinalized by 계성곤

Refinalized by 우시명

Final OK by SGLee

Page 191 Exercise 5.2 (New) NO.26

Evaluate the integral by interpreting it as a sum of the areas.

Sol) 

Let

⇔

    and

∴

Answer :                                                        ■

Solved by 계성곤

Finalized by 김민수

Final OK by SGLee

Page 191 Exercise 5.2 (New) NO.28

 Prove that

Proof) 

By using the end point rule,

        =

        =

        =

∴

Answer :                                  ■

 http://matrix.skku.ac.kr/cal-lab/Area-Sum.html : Riemann Sum을 이용하여 적분의 근사값을 구하는 과정 시각화 를 이용하여 확인할 수 있다.

Solved by 김요섭

Revised by 이송섭

Finallized by 우시명

Refinalized by 배성준

Revised by 문지호

Re Finalize by 문지호

Final OK by SGLee

p.192 Chapter5.2 Exercise No.32(new).

Verify the inequality

Proof) 

Let , and define the -th derivative function of .

=> for 

=>    ( )

Therefore  .

We have proved that    .                      ■

[CAS] http://sage.skku.edu/  또는 https://sagecell.sagemath.org/  이용해서 보일 수 있다.

Solved by 김요섭

Revised by 김민수

Finalized by 계성곤

Refinallized by 우시명

Final OK by SGLEE

Page 199 Exercise 5.3 (New) NO.6

6. (New) Find the derivative of the function.

=

Sol)

Answer :                    ■

Solved by 계성곤

Finalized by 김민수

Final OK by SGLee

Page 199 Exercise 5.3 (New) NO.7

Find the derivative of the function.

Sol) 

Answer :                                                             ■

      Solved by 배성준

finalized by 김요섭

Final OK by SGLee

Page 199 Exercise 5.3 (New) No.8

Calculate the integral using Part 2 of the FTC.

Sol)

Answer:                                     ■

Solved by 이송섭

Revised by 문지호

Finalized by 계성곤

Final OK by SGLee

p.200 Chapter 5.3 exercise No.13 (new)

 Let and Find .

Sol)

Let . Then,

Answer:                                  █

Solved by 문지호

Finalized by 배성준

Final OK by SGLee

p. 201 Chapter 5.3 Exercise 18 (New)

When    where , find .

Sol) 

Since , .

Let .

=> 

=> 

=> 

Answer:                                             ■

Solved by 김요섭

Revised by 계성곤

Finalized by 문지호

Final OK by SGLee

Page 207 Exercise 5.4 (New) NO.2

Verify by differentiation that the formula is correct.

Proof-1)

Let . Then .

. 

                                     █

Proof-2)

                                                          █ 

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/ 이용해서도 구할 수 있다.

Solved by 김요섭

Revised by 계성곤

Finalized by 문지호

Final OK by SG LEE

Page 207 Exercise 5.4 (New) NO.2

Verify by differentiation that the formula is correct.

Sol)

Let

⟹

∴   ( is an integral constant.)

Answer :       ■

Solved by 배성준

finalized by 김요섭

Refinalized and Final OK by SGLee

Page 207 Exercise 5.4 (New) No.3

Verify by differentiation that the formula is correct.

Proof)

Let (). Then .

Since ,

When , ; when , .

=>

Therefore          ■

Solved by 변희성

Revised by 문지호

Finalized by 계성곤

Re fianlized by 배성준

Final OK by SGLee

p.207 Chapter 5.4 exercise No.3(new)

Verify by differentiation that the formula is correct.

Proof-2)

Let . Then .

. 

Proof-1)

Answer)         █

Solved by 변희성

Revised by 문지호

Finalized by 계성곤

Refinalized 배성준

Final OK by SGLee

p.207 Chapter 5.4 exercise No.4 (new)

Verify by differentiation that the formula is correct.

Proof)

Let . Then .

.

.

                                                  █

[CAS] http://sage.skku.edu/ 또는 https://sagecell.sagemath.org/ 이용해서도 구할 수 있다.

Solved by 문지호

Finalized by 배성준

Final OK by SGLee

p. 208 Chapter 5.4 Exercise 20 (Old)

Evaluate the integral.

Sol) 

Substitute to

Woframalpha :

Answer:           ■

Solved by 문지호

Finalized by 이송섭
Final OK by SGLee

p. 217 Chapter 5.5 Exercise 9 (New)

Find the indefinite integral.

Sol) 

Let then .

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

Answer:                          ■

Solved by 문지호

Finalized by 배성준

Final OK by SGLee

p. 218 Chapter 5.5 Exercise 20 (New)

Evaluate the definite integral, if it exists.

Sol) 

Let then .

Since this is even function,

.

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

Answer:                                                   ■

Solved by 김요섭

Revised by 배성준

Finalized by 문지호

Final OK by SGLee

p.228 Chapter 5.6 Exercise No.5 (Old)

Show that by using an integral.

Proof) 

                                             █

Solved by 김요섭

Revised by 배성준

Finalized by 문지호

Final OK by SGLee

p.229 Chapter 5.6 Exercise No.7(Old)

Evaluate .

Sol) 

[CAS] Draw the graph.

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

Answer :                                      █

1. Newton's method

 : 1차방정식은 쉽게, 2차 방정식도 간단한 근의 공식을 이용하여 쉽게 구해내고, 3차, 4차 방정식도 복잡하지만 근의 공식을 이용하여 근을 구해낼 수 있다. 5차 이상 넘어가게 되면 공식으로 또는 간단하게 방정식의 근을 구해낼 수 없다. 5차 이상 되는 고차 방정식의 근을 구하기 위하여 뉴튼은 근의 근사값을 구하는 방법을 생각해 냈다. 그래프적으로 보면 의 그래프가 그려져 있을 때 그 그래프위의 어느 한 점 에서의 접선을 그리고 그 접선의 x절편()을 구하여 에서 그래프 위의 점 를 구하여 점 B에서의 접선을 그린다. 이 과정을 계속 반복하다보면 이 되는 x의 근사값을 구해낼 수 있다.

 를 식으로 써서 구해보면

로 구할 수 있다.

이를 수열화 하면 공식의 틀을 갖춘 식이 나오게 된다.

이를 newton's method라 한다.

, 등을 가지고 연습해 보세요~ 계산이 복잡해 지면 sage를 쓰면 될 것입니다. 아니 어떻게 쓰는지만 알면 될 것입니다. 

4. State the Procedure for Newton’s Method. 

  Let us consider the graph of and we want to solve . We start with the (proper, 해에 충분히 가까운) initial approximation , which may be obtained by just guessing, or examining the graph of . Then we use the tangent line to the curve at the point to approximate the curve and look at the -intercept of , labeled . The equation of the tangent line is . Thus, we obtain . If , we can solve this equation for :

Under certain conditions, is usually a better approximation to the solution than . Then we repeat this procedure with replaced by , using the tangent line at . This gives a third approximation: . Continuing this process obtains a sequence of approximations , , , as shown in the Figure. In general, if then we have    .  The number becomes closer and closer to the solution if the sequence converges as . We note that if then the sequence may not converge. In this case, we have to choose a different initial .    ■

5. State what you know about the number :