2026 Introduction to University Math 08

Transcript

1. pineapple.cc ௕࡚େֶ৘ใσʔλՊֶ෦ େֶ਺ֶೖ໳
    Introduction to University Mathematics
2. pineapple.cc How to take notes w ܗࣜΛܾΊΔ
   w ݟग़͠Λ͚ͭΔ
   w ϖʔδʹςʔϚ
   w Օ৚ॻ͖ʹ͢Δ
   w ༨നΛ࢒͢ ϊʔτͷऔΓํ Life Hacker
3. pineapple.cc ߨٛͷਐΊํʢୈճʙୈճʣ Agenda (Day 2 to Day 8) w େֶ਺ֶ΁ͷೖ໳
   Introduction to University Mathematics ⭐⭐
   w ਺ֶΛ޷͖ʹͳΔ࿩
   Stories about math that makes you fun ⭐
   w ܭࢉػՊֶʢ৘ใՊֶʣ΁ͷڮ౉͠
   A bridge to computer science ⭐⭐⭐
4. pineapple.cc ධՁ Credits w ग़੮
   Attendance
   w ΫΠζͱϛχϨϙʔτ
   Quiz and mini report
   w Ϩϙʔτʢ೚ҙʣ
   Report (optional) w ఺ʢ2ʣ఺ʢ2ʣ
   50 points (Q1) + 50 points (Q2)
5. pineapple.cc ࣭໰
   ʢ೥݄೔ʣ Question 07 (26th May 2025) w ҐஔΛ࣌ؒͰඍ෼͢ΔͱԿ͕ಘΒΕΔʁ
   
   What do you get when you di ff erentiate position with respect to time?
   w 1⃣ڑ཭
   Distance.
   w 2⃣໘ੵ
   Area.
   w 3⃣଎౓
   Velocity.
   w 4⃣ͦͷଞɾΘ͔Βͳ͍
   Other/Don’t know.
6. pineapple.cc ࣭໰
   ʢ೥݄೔ʣ Question 07 (26th May 2025) w ҐஔΛ࣌ؒͰඍ෼͢ΔͱԿ͕ಘΒΕΔʁ
   
   What do you get when you di ff erentiate position with respect to time?
   w 1⃣ڑ཭
   Distance.
   w 2⃣໘ੵ
   Area.
   w 3⃣଎౓
   Velocity.
   w 4⃣ͦͷଞɾΘ͔Βͳ͍
   Other/Don’t know.
7. pineapple.cc 🚗 t x v = dx dt
8. pineapple.cc 🎾 ⚽ m1 v m2 p ≜ mv
9. pineapple.cc p ≜ mv F = d dt p
10. pineapple.cc ඍ෼
    differentiation
16. pineapple.cc f′  (a) ≜ lim h→0 f(a + h) − f(a) h
18. pineapple.cc f′  (x) ≜ lim h→0 f(x + h) − f(x) h
26. pineapple.cc χϡʔτϯ๏
    Newton’s method
27. pineapple.cc ٻࠜ Root Finding f(x) = 0 զʑ͕ѻ͏໰୊ʹ͋Γ͕ͪͳ͜ͱ • ؔ਺ͷத਎͕Θ͔Βͳ͍ • ؔ਺͕ෳࡶ͗ͯ͢ղ͚ͳ͍
28. pineapple.cc ೋ෼๏ Bisection method ූ߸͕ҟͳΔ
29. pineapple.cc χϡʔτϯɾϥϑιϯ๏ʢχϡʔτϯ๏ʣ Newton-Raphson method (Newton’s method)
30. pineapple.cc ෳૉ਺
    complex
31. pineapple.cc x − 1 = 0
32. pineapple.cc x = 1
33. pineapple.cc x + 1 = 0
34. pineapple.cc x = − 1
37. pineapple.cc x2 − 1 = 0
38. pineapple.cc x = ± 1
39. pineapple.cc x2 + 1 = 0
40. pineapple.cc x = ± −1
41. pineapple.cc i2 ≡ − 1
42. pineapple.cc (x1 + iy1) + (x2 + iyr) = (x1 + x2) + i (y1 + y2) (x1 + iy1) (x2 + iy2) = (x1 x2 − y1 y2) + i (x1 y2 + x2 y1)
43. pineapple.cc z1 + z2 = z3 , (z1 , z2 , z3 ∈ ℤ) z1 z2 = z3 , (z1 , z2 , z3 ∈ ℤ) (z1 + z2) + z3 = z1 + (z2 + z3) (z1 z2) z3 = z1 (z2 z3) 0 + z = z + 0 = z 1z = z1 = z
44. pineapple.cc z*z = x2 + y2 where z = x + iy, z* = x − iy
45. pineapple.cc −z + z = 0 ( z* z*z) z = 1
55. pineapple.cc z = x + iy Z = [ 1 0 0 1] x + [ 0 −1 1 0 ] y = [ x −y y x ]
56. pineapple.cc z* = x − iy where z = x + iy Z* = [ x y −y x] where Z = [ x −y y x ]
57. pineapple.cc
58. pineapple.cc exp(iθ) = cos(θ) + i sin(θ) eiθ = cos(θ) + i sin(θ)
59. pineapple.cc 0 x y 1 θ p p = [ cos(θ) sin(θ)]
60. pineapple.cc 0 x y 1 θ p q α q = [ cos(α) −sin(α) sin(α) cos(α) ] p p = [ cos(θ) sin(θ)] q = R(α)p
61. pineapple.cc 0 x y 1 p = cos(θ) + i sin(θ) θ ℜ ℑ p
62. pineapple.cc 0 x y 1 θ p q α q = (cos(α) + i sin(α)) p p = cos(θ) + i sin(θ) ℜ ℑ
63. pineapple.cc 0 x y 1 θ p q α q = exp(iα)p = exp(iα)exp(iθ) = exp (i(α + θ)) p = exp(θ) ℜ ℑ
64. pineapple.cc exp(iα) = cos(α) + i sin(α)
65. pineapple.cc exp ([ 0 −α α 0 ]) = [ cos(α) 0 0 cos(α)] + [ 0 −sin(α) sin(α) 0 ] = [ cos(α) −sin(α) sin(α) cos(α) ] iα cos(α) i sin(α) R(α)
66. pineapple.cc exp ([ 0 −α α 0 ]) ≃ [ 1 0 0 1] + [ 0 −α α 0 ] = [ 1 −α α 1 ] where α ≃ 0
67. pineapple.cc ௒ෳૉ਺
    hyper complex
68. pineapple.cc z = x + iy, where i2 = − 1 q = s + iu + jv + kw where i2 = j2 = k2 = − 1, ijk = − 1, ij = k, jk = i, ki = j, ji = − k, kj = − i, ik = − j
69. pineapple.cc q = ( s + ti u + vi −u + vi s − ti ) = s ( 1 0 0 1) + t ( i 0 0 −i) + u ( 0 1 −1 0) + v ( 0 i i 0)
72. William Rowan Hamilton
73. pineapple.cc ࣭໰
    ʢ೥݄೔ʣ Question 06 (4th June 2024) w ڏ਺୯Ґ͸ͳͥJͱॻ͘ͷʁ
    Why is the imaginary unit written as i?
    w 1⃣*NBHJOBSZ
    VOJU
    ͷ಄จࣈ͔ͩΒ
    Because it is an imaginary unit.
    w 2⃣*NQPSUBOU
    OVNCFS
    ͷ಄จࣈ͔ͩΒ
    Because it is an important number.
    w 3⃣*NQPTTJCMF
    WFDUPS
    ͷ಄จࣈ͔ͩΒ
    Because it is an impossible vector.
    w 4⃣ͦͷଞɾΘ͔Βͳ͍
    Other/Don’t know.
74. pineapple.cc ೚ҙϨϙʔτ
    Optional report
75. pineapple.cc ೚ҙϨϙʔτ Optional report w FYQ Y ΛYͰඍ෼͠ͳ͍͞
    w ΦΠϥʔͷެ͔ࣜΒࡾ֯ؔ਺ʢTJO
    DPTʣͷഒ֯ͷެࣜΛಋग़͠ͳ͍͞
    w ೚ҙͷෳૉ਺ΛY࣮ߦྻͰදݱ͠ɼ଍͠ࢉɼֻ͚ࢉͷنଇ͕Ұக͢Δ͜ͱ Λࣔ͠ͳ͍͞ PDFͱͯ͠LACS΁ఏग़͢Δ͜ͱʢఏग़͸೚ҙʣ
76. pineapple.cc 🔎45&".
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