FM GDD 0715
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8/19/2019 FM GDD 0715
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VIDEOJUEGOS
GDD 0715
Gu’a Pr‡ctica
Fundamentos Matem‡ticos
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8/19/2019 FM GDD 0715
2/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
■ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥
▲ ❛ ♣ ❝ ✐ ❝ ❛ ♦ ♥ ❛ ❝ ✐ ✈ ✐ ❞ ❛ ❞ ❡ ✉ ❡ ❡ ❛ ② ✉ ❞ ❛ ♥ ❛ ❡ ❛ ✜ ♠ ❛ ❡ ❧ ❝ ♦ ♥ ♦ ❝ ✐ ♠ ✐ ❡ ♥ ♦ ❛ ❞ ✉ ✐ ✲
✐ ❞ ♦ ❡ ♥ ❧ ❛ ❤ ♦ ❛ ❢ ❡ ♥ ❡ ❛ ♣ ♦ ❢ ❡ ♦ ✱ ♣ ❛ ❛ ❧ ♦ ❣ ❛ ❧ ❧ ❡ ✈ ❛ ❧ ♦ ❛ ✉ ♥ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥
❡ ✜ ❝ ✐ ❡ ♥ ❡ ✳ ❊ ❛ ❣ ✉ ❛ ❡ ♣ ❡ ♠ ✐ ✐ ❝ ♦ ♥ ♦ ❝ ❡ ❧ ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ② ♣ ♦ ❝ ❡ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛ ❡ ❛ ❧ ✐ ✲
③ ❛ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳
❉ ❡ ❝ ✐ ♣ ❝ ✐ ♥
▲ ❛ ♣ ❝ ✐ ❝ ❛ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ♥ ❞ ✐ ❢ ❡ ❡ ♥ ❡ ❡ ❛ ♣ ❛ ❞ ❡ ❡ ❛ ❧ ✐ ③ ❛ ❝ ✐ ♥ ✳ ❉ ❡ ❜ ❡ ❛ ❡ ♥ ❞ ❡ ❝ ❛ ❞ ❛
✉ ♥ ♦ ❞ ❡ ❧ ♦ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ✐ ♥ ❞ ✐ ❝ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ♥ ✉ ❝ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ✉ ❡ ❛ ❡ ❛ ♥ ❝ ♦ ♥ ✐ ❞ ❡ ❛ ✲
❞ ❛ ♣ ❛ ❛ ❡ ✈ ❛ ❧ ✉ ❛ ❝ ✐ ♥ ✳
▲ ❛ ♣ ❝ ✐ ❝ ❛ ♦ ♥ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ♠ ❡ ♥ ❡ ❞ ✐ ❡ ❛ ❞ ❛ ❡ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ♣ ♦ ❡ ❧ ❡ ❛ ❛ ❝ ❛ ❞ ♠ ✐ ❝ ❛ ② ❡ ❧
❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ♣ ❛ ❛ ❡ ❡ ✈ ❛ ❧ ✉ ❛ ❞ ❛ ♣ ♦ ❡ ❛ ❧ ✐ ♠ ❛ ✳ ▲ ♦ ✉ ♣ ❡ ✈ ✐ ♦ ❡ ✜ ♠ ❛ ♥
✉ ❘ ❡ ❣ ✐ ♦ ❞ ❡ ♦ ❣ ❡ ♦ ❆ ❝ ❛ ❞ ♠ ✐ ❝ ♦ ✭ ❘ ❆ ✮ ❡ ♥ ❡ ❧ ❡ ❝ ✉ ❛ ❞ ♦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ❡ ♥ ❡ ✱ ♣ ♦
❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ♥ ❡ ❣ ❛ ❞ ❛ ❝ ♦ ♥ ❧ ♦ ❡ ✉ ✐ ✐ ♦ ❞ ❡ ❝ ✐ ♦ ❡ ♥ ❡ ❛ ❣ ✉ ❛ ✳
❊ ♥ ❡ ❣ ❛
▲ ❛ ❢ ❡ ❝ ❤ ❛ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ✉ ❡ ❡ ♦ ♠ ❛ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ♣ ❛ ❛ ❧ ❛ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ✐ ♥ ❡ ❧ ❛ ❞ ❡ ❧ ❛ ❡ ♥ ✲
❡ ❣ ❛ ❞ ❡ ❧ ❘ ❆ ✳ ❊ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❡ ♥ ❢ ♦ ♠ ❛ ♦ ❞ ✐ ❣ ✐ ❛ ❧ ② ✈ ❛ ✐ ♥ ❡ ♥ ❡ ✱ ❧ ❛ ❢ ❡ ❝ ❤ ❛
❝ ♦ ♥ ✐ ❞ ❡ ❛ ❞ ❛ ♣ ❛ ❛ ❡ ♥ ❡ ❣ ❛ ❡ ❧ ❛ ❞ ❡ ❧ ❘ ❆ ② ♥ ♦ ❧ ❛ ❞ ❡ ❧ ❡ ♥ ✈ ♦ ❞ ❡ ❧ ❛ ❝ ❤ ✐ ✈ ♦ ✳ ❙ ✐ ✉ ♥
❛ ❧ ✉ ♠ ♥ ♦ ✉ ❜ ❡ ✉ ♥ ❛ ❝ ❤ ✐ ✈ ♦ ❛ ● ♦ ♦ ❣ ❧ ❡ ❉ ✐ ✈ ❡ ✱ ♣ ♦ ❡ ❥ ❡ ♠ ♣ ❧ ♦ ✱ ♣ ❡ ♦ ♥ ♦ ❡ ❧ ❧ ❛ ❡ ❧ ❘ ❆ ❛ ♥ ❡
❞ ❡ ❧ ❛ ❢ ❡ ❝ ❤ ❛ ② ❤ ♦ ❛ ❧ ♠ ✐ ❡ ✱ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ♦ ❜ ❡ ✼ ✳
❊ ❡ ♣ ♦ ♥ ❛ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ❡ ♥ ❡ ♦ ❞ ♦ ❧ ♦ ♠ ❛ ❡ ✐ ❛ ❧ ❡ ❛ ✐ ❡ ♠ ♣ ♦ ② ❡ ♥ ❡ ✉ ♥ ❡ ✲
♣ ❛ ❧ ❞ ♦ ❞ ❡ ✉ ❛ ❜ ❛ ❥ ♦ ✱ ♦ ❜ ❡ ♦ ❞ ♦ ❡ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ✐ ❛ ❞ ❛ ✳
❈ ✉ ❛ ❧ ✉ ✐ ❡ ❞ ✉ ❞ ❛ ♦ ❜ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ❞ ❡ ❜ ❡ ❡ ❝ ✉ ✐ ❛ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✲
✐ ♥ ✳ ❚ ❡ ❡ ❝ ♦ ♠ ❡ ♥ ❞ ❛ ♠ ♦ ✉ ❡ ❧ ♦ ❤ ❛ ❣ ❛ ❝ ♦ ♥ ❛ ♥ ✐ ❝ ✐ ♣ ❛ ❝ ✐ ♥ ② ❛ ❞ ♠ ✐ ♥ ✐ ❡ ✉
✐ ❡ ♠ ♣ ♦ ✳
❉ ❡ ❜ ❡ ❡ ♣ ❡ ❛ ❧ ♦ ❢ ♦ ♠ ❛ ♦ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❞ ❡ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳ ❈ ✉ ❛ ❧ ✉ ✐ ❡ ❞ ✉ ❞ ❛ ❛ ❧ ❡ ✲
♣ ❡ ❝ ♦ ✱ ♣ ✉ ❡ ❞ ❡ ❡ ❝ ✉ ✐ ❛ ❧ ▼ ❛ ♥ ✉ ❛ ❧ ❞ ❡ ❋ ♦ ♠ ❛ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛ ❡ ❧ ❝ ✉ ❛ ❧ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ❡ ♥
❡ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ✳
❉ ❡ ❜ ❡ ❡ ❝ ♦ ❣ ❡ ✉ ❛ ❜ ❛ ❥ ♦ ❡ ♥ ❡ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ❛ ♠ ❛ ❞ ❛ ✉ ♥ ❛ ❡ ♠ ❛ ♥ ❛
❞ ❡ ♣ ✉ ❞ ❡ ✉ ❡ ❧ ❛ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❤ ❛ ② ❛ ♥ ✐ ❞ ♦ ❛ ✐ ❣ ♥ ❛ ❞ ❛ ✳ ❉ ❡ ♥ ♦ ❡ ❛ ✱ ❡ ❧ ❡ ❛ ❞ ❡
✉ ♣ ❡ ✈ ✐ ✐ ♥ ♥ ♦ ❡ ❤ ❛ ❡ ♣ ♦ ♥ ❛ ❜ ❧ ❡ ❞ ❡ ❝ ♦ ♥ ❡ ✈ ❛ ❞ ✐ ❝ ❤ ♦ ❛ ❜ ❛ ❥ ♦ ✳
✶
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8/19/2019 FM GDD 0715
3/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
❖ ❜ ❥ ❡ ✐ ✈ ♦
❊ ♥ ❡ ❛ ❡ ✐ ❡ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ♥ ❧ ♦ ♣ ✐ ♥ ❝ ✐ ♣ ✐ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✈ ✐ ♦ ❡ ♥ ❝ ❧ ❛ ❡
❡ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ♦ ✐ ❡ ♥ ❛ ❞ ♦ ❛ ❧ ❛ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ❞ ❡ ✈ ✐ ❞ ❡ ♦ ❥ ✉ ❡ ❣ ♦ ✳
❈ ❛ ❧ ❡ ♥ ❞ ❛ ✐ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛
❆ ❝ ♦ ♥ ✐ ♥ ✉ ❛ ❝ ✐ ♥ ❡ ❡ ❛ ❧ ❛ ❧ ❛ ❢ ❡ ❝ ❤ ❛ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❞ ❡ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳ ❚ ♦ ❞ ❛ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ✱
✐ ♥ ❡ ① ❝ ❡ ♣ ❝ ✐ ♥ ✱ ❡ ❡ ❛ ❧ ✐ ③ ❛ ♥ ❝ ♦ ♠ ♦ ♠ ① ✐ ♠ ♦ ❛ ❧ ❛ ✷ ✶ ✿ ✹ ✺ ❤ ♦ ❛ ✳
▲ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ♥ ❡ ❣ ❛ ❞ ❛ ❢ ✉ ❡ ❛ ❞ ❡ ❧ ♣ ❡ ✐ ♦ ❞ ♦ ❛ ❝ ♦ ❞ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ❛ ❣ ❡ ♥ ❞ ❛ ✱ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ♥
♦ ❜ ❡ ✼ ❞ ✉ ❛ ♥ ❡ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❡ ♠ ❛ ♥ ❛ ✱ ❝ ♦ ♠ ♦ ✐ ♥ ❞ ✐ ❝ ❛ ❡ ❧ ▼ ❛ ♥ ✉ ❛ ❧ ❞ ❡ ❧ ❊ ✉ ❞ ✐ ❛ ♥ ❡ ✳
◆ ♦ ✳ ❞ ❡ ❝ ✐ ❝ ❛ ◆ ♦ ♠ ❜ ❡ ❞ ❡ ❧ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❊ ♥ ❡ ❣ ❛
✶ ❈ ♦ ♥ ❝ ❡ ♣ ♦ ❇ ✐ ❝ ♦ ✷ ✽ ❊ ♥ ❡ ♦ ✷ ✵ ✶ ✻
✷ ❈ ♦ ♥ ❥ ✉ ♥ ♦ ◆ ✉ ♠ ✐ ❝ ♦ ✹ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻
✸ ❚ ✐ ❣ ♦ ♥ ♦ ♠ ❡ ❛ ✶ ✶ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻
✹ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ✶ ✽ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻
✺ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❡ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✷ ✺ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻
✻ ➪ ❧ ❣ ❡ ❜ ❛ ❱ ❡ ❝ ♦ ✐ ❛ ❧ ✸ ▼ ❛ ③ ♦ ✷ ✵ ✶ ✻
▲ ❛ ❢ ❡ ❝ ❤ ❛ ♣ ✉ ❞ ❛ ♥ ❝ ❛ ♠ ❜ ✐ ❛ ❡ ❛ ❧ ♦ ❧ ❛ ❣ ♦ ❞ ❡ ❧ ✐ ♠ ❡ ❡ ✱ ♣ ❡ ♦ ❡ ♦ ❡ ❧ ❡ ♥ ♦ ✐ ✜ ❝ ❛ ❝ ♦ ♥
❛ ♥ ❡ ❧ ❛ ❝ ✐ ♥ ✳
❋ ♦ ♠ ❛ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛
▲ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ❞ ❡ ❜ ❡ ♥ ❡ ♥ ❡ ❣ ❛ ❛ ♠ ❛ ♥ ♦ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ❤ ♦ ❥ ❛ ❜ ❧ ❛ ♥ ❝ ❛ ❞ ❡ ❛ ♠ ❛ ♦ ❝ ❛ ✲
❛ ✱ ❛ ♣ ♦ ✈ ❡ ❝ ❤ ♥ ❞ ♦ ❧ ❛ ♣ ♦ ❛ ♠ ❜ ♦ ❧ ❛ ❞ ♦ ✳ ❙ ❡ ❞ ❡ ❜ ❡ ❝ ♦ ♣ ✐ ❛ ❡ ❧ ❡ ♥ ✉ ♥ ❝ ✐ ❛ ❞ ♦ ❞ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦
❝ ♦ ♠ ♣ ❧ ❡ ♦ ② ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❧ ♦ ❝ ♦ ♥ ❧ ❛ ♠ ❛ ② ♦ ❛ ♠ ♣ ❧ ✐ ✉ ❞ ♣ ♦ ✐ ❜ ❧ ❡ ✳ ❚ ♦ ❞ ❛ ② ❝ ❛ ❞ ❛ ✉ ♥ ❛ ❞ ❡ ❧ ❛
❤ ♦ ❥ ❛ ❞ ❡ ❜ ❡ ♥ ❛ ❡ ♥ ♦ ♠ ❜ ❡ ② ♥ ♠ ❡ ♦ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ✱ ❛ ❝ ♦ ♠ ♦ ❝ ❞ ✐ ❣ ♦ ❞ ❡ ❧ ❝ ✉ ♦ ✳
▲ ❛ ❡ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❞ ❡ ❜ ❡ ❡ ❝ ❧ ❛ ❛ ② ❝ ♦ ♥ ❧ ✐ ♠ ♣ ✐ ❡ ③ ❛ ✳ ❊ ♣ ❡ ♣ ♦ ♥ ❞ ❡ ❛ ♥ ❡
♠ ❛ ♥ ❡ ♥ ❡ ✉ ♥ ♦ ❞ ❡ ♥ ❞ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ② ❡ ✈ ✐ ❛ ♦ ❞ ♦ ✐ ♣ ♦ ❞ ❡ ❛ ② ♦ ♥ ❡ ✳
❙ ✐ ❡ ♣ ❡ ❡ ♥ ❛ ✉ ♥ ❣ ❛ ❞ ♦ ❞ ❡ ✐ ♠ ✐ ❧ ✐ ✉ ❞ ✐ ♥ ❛ ❝ ❡ ♣ ❛ ❜ ❧ ❡ ❡ ♥ ❧ ❛ ♠ ❡ ♦ ❞ ♦ ❧ ♦ ❣ ❛ ♦ ❧ ♦ ❡ ✉ ❧ ❛ ❞ ♦
❡ ♥ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❞ ❡ ❞ ♦ ♦ ♠ ❛ ❧ ✉ ♠ ♥ ♦ ✱ ✉ ❡ ❞ ❛ ❛ ❝ ✐ ❡ ✐ ♦ ❞ ❡ ❧ ❡ ✉ ✐ ♣ ♦ ❞ ❡ ✉ ♣ ❡ ✲
✈ ✐ ✐ ♥ ❡ ❧ ❛ ♥ ✉ ❧ ❛ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ❛ ♦ ❧ ♦ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ♥ ❝ ✉ ❡ ✐ ♥ ✳
❊ ❡ ❝ ♦ ♠ ❡ ♥ ❞ ❛ ❜ ❧ ❡ ♠ ❛ ♥ ❡ ❥ ❛ ❝ ♦ ♠ ♦ ❢ ❛ ❝ ❝ ✐ ♦ ♥ ❡ ♦ ❞ ❛ ❧ ❛ ❝ ✐ ❢ ❛ ✉ ❡ ♥ ♦ ❡ ❛ ♥ ♥ ♠ ❡ ♦
❡ ♥ ❡ ♦ ✱ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❛ ✐ ♦ ♣ ✉ ❡ ❞ ❡ ❡ ① ✐ ✐ ♣ ❞ ✐ ❞ ❛ ❞ ❡ ♣ ❡ ❝ ✐ ✐ ♥ ❡ ♥ ❡ ✉ ❧ ❛ ❞ ♦ ✉ ❡ ❡ ♥
❝ ♦ ♥ ✐ ❞ ❡ ❛ ❞ ♦ ❝ ♦ ♠ ♦ ✐ ♥ ❝ ♦ ❡ ❝ ♦ ✳ ❉ ❡ ❡ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ❡ ❧ ✉ ♦ ❞ ❡ ♥ ♠ ❡ ♦ ❞ ❡ ❝ ✐ ♠ ❛ ❧ ❡ ✱ ❡
❡ ❝ ♦ ♠ ✐ ❡ ♥ ❞ ❛ ✉ ✐ ❧ ✐ ③ ❛ ❝ ✉ ❛ ♦ ❝ ✐ ❢ ❛ ❞ ❡ ♣ ✉ ❞ ❡ ❧ ♣ ✉ ♥ ♦ ✳
✷
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8/19/2019 FM GDD 0715
4/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
◆ ♦ ❡ ❛ ❝ ❡ ♣ ❛ ♥ ❞ ♦ ❝ ✉ ♠ ❡ ♥ ♦ ✉ ❡ ♥ ♦ ❡ ♥ ❣ ❛ ♥ ❡ ❧ ❢ ♦ ♠ ❛ ♦ ❡ ❛ ❜ ❧ ❡ ❝ ✐ ❞ ♦ ② ♥ ♦ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ✲
♥ ❛ ✉ ❡ ❧ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✉ ❡ ♥ ♦ ❝ ✉ ♠ ♣ ❧ ❛ ♥ ❝ ♦ ♥ ❧ ❛ ❝ ❛ ❛ ❝ ❡ ✐ ❝ ❛ ✐ ♥ ❞ ✐ ❝ ❛ ❞ ❛ ✳ ❙ ❡ ♣ ♦ ❤ ❜ ❡
❡ ❧ ✉ ♦ ❞ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❞ ♦ ❛ ❝ ♦ ♥ ❧ ❛ ❡ ① ❝ ❡ ♣ ❝ ✐ ♥ ❞ ❡ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✉ ❡ ❡ ❧ ❡ ✉ ✐ ♣ ♦ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥
✐ ♥ ❞ ✐ ✉ ❡ ✭ ❡ ♥ ❧ ♦ ✉ ❡ ❡ ❡ ❛ ❧ ♠ ❡ ♥ ❡ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ✮ ✳
❈ ✐ ❡ ✐ ♦ ❞ ❡ ❊ ✈ ❛ ❧ ✉ ❛ ❝ ✐ ♥
▲ ❛ ♣ ❝ ✐ ❝ ❛ ❞ ❡ ♠ ❛ ❡ ♠ ✐ ❝ ❛ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ♥ ♣ ♦ ♣ ♦ ♠ ❡ ❞ ✐ ♦ ❜ ❛ ♥ ❞ ♦ ❡ ❡ ♥ ✉ ❡ ❝ ❛ ❞ ❛
❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✐ ❡ ♥ ❡ ✉ ♥ ♦ ❧ ♦ ❡ ✉ ❧ ❛ ❞ ♦ ♣ ♦ ✐ ❜ ❧ ❡ ✳ ◗ ✉ ❡ ❞ ❛ ❛ ❝ ✐ ❡ ✐ ♦ ❞ ❡ ❧ ❡ ✉ ✐ ♣ ♦ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥
❡ ❧ ❝ ♦ ♥ ✐ ❞ ❡ ❛ ✐ ✱ ♣ ❛ ❛ ❝ ❛ ❞ ❛ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✉ ② ♦ ❡ ✉ ❧ ❛ ❞ ♦ ❡ ❛ ✐ ♥ ❝ ♦ ❡ ❝ ♦ ✱ ✉ ❡ ♣ ❡ ❝ ✐ ✈ ♦
❞ ❡ ❛ ♦ ❧ ❧ ♦ ❡ ❡ ✈ ❛ ❧ ✉ ❛ ❜ ❧ ❡ ✳
❙ ✐ ♥ ✐ ♠ ♣ ♦ ❛ ❡ ❧ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❞ ✐ ✜ ❝ ✉ ❧ ❛ ❞ ✱ ♦ ❞ ♦ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✐ ❡ ♥ ❡ ♥ ❡ ❧ ♠ ✐ ♠ ♦ ✈ ❛ ❧ ♦ ♣ ❛ ❛
❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ✐ ♥ ✳
❉ ✐ ♥ ♠ ✐ ❝ ❛ ● ❡ ♥ ❡ ❛ ❧
▲ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❞ ❡ ❡ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ♥ ❡ ① ♣ ❡ ❛ ❞ ♦ ❝ ♦ ♥ ❡ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✿ A
✱ B
② C
❀ ❧ ❛
❝ ✉ ❛ ❧ ❡ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❞ ❡ ❧ ♦ ❡ ❧ ✐ ♠ ♦ ♥ ♠ ❡ ♦ ❞ ❡ ❧ ❛ ♠ ❛ ❝ ✉ ❧ ❛ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ❡ ❣ ♥ ❧ ❛
❢ ♠ ✉ ❧ ❛ ✿ ★ ★ ✭ A − 1
✮ ✭ B − 1
✮ ✭ C − 1
✮ ✳
❊ ❞ ❡ ❝ ✐ ✉ ❡ ❧ ♦ ✈ ❛ ❧ ♦ ❡ ❞ ❡ A
✱ B
② C
✱ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ♥ ❞ ❡ ♠ ❛ ♥ ❡ ❛ ✉ ❡ ❛ ❧ ❡ ❛ ❧ ❡ ✶ ❞ ❡ ♥
❧ ♦ ❧ ✐ ♠ ♦ ❡ ♥ ♠ ❡ ♦ ❞ ❡ ✉ ♠ ❛ ❝ ✉ ❧ ❛ ✳ ❖ ❞ ✐ ❝ ❤ ♦ ❞ ❡ ♦ ❛ ❢ ♦ ♠ ❛ ✱ ✉ ♠ ❛ ✶ ❛ ❧ ♦ ❡
❧ ✐ ♠ ♦ ♥ ♠ ❡ ♦ ② ♦ ❜ ❡ ♥ ❞ A
✱ B
② C
✳
❈ ❛ ♦ ❞ ❡ ❡ ❥ ❡ ♠ ♣ ❧ ♦ ✿
◆ ♠ ❡ ♦ ❞ ❡ ❛ ❧ ✉ ♠ ♥ ♦ ✿ ✶ ✵ ✶ ✻ ✼ ✳
❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❞ ❡ ♣ ❝ ✐ ❝ ❛ ✿ ❘ ❡ ♦ ❧ ✈ ❡ ❧ ❛ ❡ ❝ ✉ ❛ ❝ ✐ ♥ ✿ Ax2 + Bx + c = 0
❊ ✉ ✐ ✈ ❛ ❧ ❡ ♥ ❡ ❡ ❣ ♥ ❧ ❛ ♠ ❛ ❝ ✉ ❧ ❛ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ✿ 2x2 + 7x + 8 = 0
❙ ✐ ✉ ♥ ♦ ❞ ❡ ❡ ♦ ♥ ♠ ❡ ♦ ✭ A
✱ B
♦ C
✮ ❡ ✐ ❣ ✉ ❛ ❧ ❛ ✾ ✱ ♥ ♦ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❧ ❛ ❢ ♠ ✉ ❧ ❛ ❛ ♥ ❡
✐ ♥ ❞ ✐ ❝ ❛ ❞ ❛ ✱ ❡ ✉ ❡ ❞ ❛ ❝ ♦ ♠ ♦ ✾ ✳
✸
-
8/19/2019 FM GDD 0715
5/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❛ ❉ ❡ ❛ ♦ ❧ ❧ ❛
✶ ✳ ❈ ♦ ♥ ❝ ❡ ♣ ♦ ❇ ✐ ❝ ♦
✶ ✳ ✶ ✳ ❯ ♥ ❛ ♦ ❧ ✉ ❝ ✐ ♥ ✉ ♠ ✐ ❝ ❛ ❝ ♦ ♥ ✐ ❡ ♥ ❡ ✉ ♥ 10A
❣ ❛ ♠ ♦ ❞ ❡ ✉ ♥ ❝ ♦ ♠ ♣ ✉ ❡ ✲
♦ ♣ ♦ ❝ ❛ ❞ ❛ 100B
❝ ❡ ♥ ♠ ❡ ♦ ❝ ❜ ✐ ❝ ♦ ❞ ❡ ♦ ❧ ✉ ❝ ✐ ♥ ✳ ❙ ✐ ❡ ❡ ① ❛ ❡ ♥
15C ❣ ❛ ♠ ♦ ❞ ❡ ♦ ❧ ✉ ❝ ✐ ♥ ❡ ♥ ✉ ♥ ❡ ♥ ✈ ❛ ❡ ✈ ❛ ❝ ♦ ✱ ➽ ❈ ✉ ♥ ♦ ❣ ❛ ♠ ♦ ❞ ❡
❝ ♦ ♠ ♣ ✉ ❡ ♦ ❤ ❛ ❜ ❛ ❡ ♥ ❞ ✐ ❝ ❤ ♦ ❝ ♦ ♥ ❡ ♥ ❡ ❞ ♦ ❄
✶ ✳ ✷ ✳ ▲ ❛ ♠ ✉ ✐ ♥ ❛ X
♣ ♦ ❞ ✉ ❝ ❡ 100A
♣ ✐ ❡ ③ ❛ ❡ ❧ ❞ ♦ ❜ ❧ ❡ ❞ ❡ ♣ ✐ ❞ ♦ ✉ ❡ ❧ ❛
♠ ✉ ✐ ♥ ❛ Y
✳ ▲ ❛ ♠ ✉ ✐ ♥ ❛ Y
♣ ♦ ❞ ✉ ❝ ❡
100A♣ ❛ ❡ ❡ ♥
B♠ ✐ ♥ ✉ ♦ ✳
❙ ✐ ❝ ❛ ❞ ❛ ♠ ✉ ✐ ♥ ❛ ❢ ❛ ❜ ✐ ❝ ❛ ❛ ✉ ♥ ❛ ❛ ❛ ❝ ♦ ♥ ❛ ♥ ❡ ✱ ➽ ❈ ✉ ♥ ❛ ♣ ✐ ❡ ③ ❛
♣ ♦ ❞ ✉ ❝ ❡ ❧ ❛ ♠ ✉ ✐ ♥ ❛
X ❡ ♥
C ♠ ✐ ♥ ✉ ♦ ❄
✶ ✳ ✸ ✳ ➽ ❈ ✉ ❧ ❡ ♥ ♠ ❡ ♦ ♣ ✐ ♠ ♦ ❡ ♥ ❡ 2
② 100
♦ ♥ ❢ ❛ ❝ ♦ ❡ ❞ ❡ 4898A
❄
✶ ✳ ✹ ✳ ❉ ❡ ❛ ♦ ❧ ❧ ❛ ✿
✶ ✳ ✹ ✳ ✶ ✳ (Ax + By)2
✶ ✳ ✹ ✳ ✷ ✳ (Cx − Ay)2
✶ ✳ ✹ ✳ ✸ ✳ (Bx + Ay)3
✶ ✳ ✹ ✳ ✹ ✳ (Cx + By)4
✶ ✳ ✹ ✳ ✺ ✳ (Ax − Cy + Bz )5
✶ ✳ ✺ ✳ ❉ ✐ ✈ ✐ ❞ ✐
−3x2 + 14x − 8x − 2
✶ ✳ ✻ ✳ ❘ ❡ ♦ ❧ ✈ ❡ ❧ ❛ ❙ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❊ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✿
✶ ✳ ✻ ✳ ✶ ✳
5 + Az
Bz + C = A − B + C
✶ ✳ ✻ ✳ ✷ ✳
BC x2 + (A − C )x +√
BC = (A + B + C )3
✹
-
8/19/2019 FM GDD 0715
6/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✷ ✳ ❈ ♦ ♥ ❥ ✉ ♥ ♦ ◆ ✉ ♠ ✐ ❝ ♦
✷ ✳ ✶ ✳ ❛ ❛ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ✿
X = {3, 5, 8, 35, 21, 2, A}Y = {1, 2, 3, 4, 8, B, 15, 21}
W = {1, 4, 6, 9, 15}Z = {1, 5, C, 6, 9, 11}
❘ ❡ ✉ ❡ ❧ ✈ ❡ ✿
✷ ✳ ✶ ✳ ✶ ✳ X
∪Y
✷ ✳ ✶ ✳ ✷ ✳ X ∩ Y
✷ ✳ ✶ ✳ ✸ ✳ (X ∪Y ) ∩ W
✷ ✳ ✶ ✳ ✹ ✳ (Z ∩W ) ∪ X
✷ ✳ ✶ ✳ ✺ ✳ XY (investigar)
✷ ✳ ✶ ✳ ✻ ✳ W − Z
✷ ✳ ✶ ✳ ✼ ✳ X c
✷ ✳ ✶ ✳ ✽ ✳
(Z −Y ) ∩ (W − X )✷ ✳ ✶ ✳ ✾ ✳
(Z ∪Y ) − (W ∩ X )
✷ ✳ ✷ ✳ ❉ ❡ ❡ ♠ ✐ ♥ ❡ ❧ ❛ ❝ ❛ ❞ ✐ ♥ ❛ ❧ ✐ ❞ ❛ ❞ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❞ ❡ ❧ ❡ ❥ ❡ ✲
❝ ✐ ❝ ✐ ♦ ✷ ✳ ✶ ✳
✷ ✳ ✸ ✳ ❉ ❡ ❡ ♠ ✐ ♥ ❡ ❧ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ♣ ♦ ❡ ♥ ❝ ✐ ❛ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❞ ❡ ❧
❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✷ ✳ ✶ ❝ ✉ ② ❛ ❝ ❛ ❞ ✐ ♥ ❛ ❧ ✐ ❞ ❛ ❞ ❡ ❛ ♠ ❡ ♥ ♦ ❛ ✺ ✳
✷ ✳ ✹ ✳ ❉ ❡ ✜ ♥ ❡ ♣ ♦ ❡ ① ❡ ♥ ✐ ♥ ❡ ❧ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ✱ ❞ ♦ ♥ ❞ ❡ Z
❡ ♣ ❡ ❡ ♥ ❛
❡ ❧ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ♥ ♠ ❡ ♦ ❡ ♥ ❡ ♦ ✿
W = {X |AX − Y = B, Y ∈ (−5A, 2C ], Y ∈ Z}
✷ ✳ ✺ ✳ ❘ ❡ ❛ ❧ ✐ ❝ ❡ ❧ ♦ ❞ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❱ ❡ ♥ ♥ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❞ ❡ ❧
❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✷ ✳ ✶ ✳ ❉ ✐ ❝ ❤ ♦ ❞ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❜ ❡ ♥ ❡ ♣ ❡ ❡ ♥ ❛ ❛ ♥ ♦ ❡ ❧ ❝ ♦ ♥ ✲
❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❝ ♦ ♠ ♦ ❧ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ❧ ❧ ❡ ❣ ❛ ❛ ❧ ✳
✺
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8/19/2019 FM GDD 0715
7/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✸ ✳ ❚ ✐ ❣ ♦ ♥ ♦ ♠ ❡ ❛
✸ ✳ ✶ ✳ ❈ ♦ ♥ ✈ ✐ ❡ ❡ ❛ ❘ ❛ ❞ ✐ ❛ ♥ ❡ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ♥ ❣ ✉ ❧ ♦
✸ ✳ ✶ ✳ ✶ ✳ A × 30◦
✸ ✳ ✶ ✳ ✷ ✳ B × 10◦
✸ ✳ ✶ ✳ ✸ ✳ C × 45◦
✸ ✳ ✶ ✳ ✹ ✳
A × 23◦
✸ ✳ ✶ ✳ ✺ ✳ B × 7◦
✸ ✳ ✶ ✳ ✻ ✳ C × 90◦
✸ ✳ ✶ ✳ ✼ ✳ A × 110◦
✸ ✳ ✶ ✳ ✽ ✳ B × 360◦
✸ ✳ ✶ ✳ ✾ ✳
C × 126◦
✸ ✳ ✶ ✳ ✶ ✵ ✳ (A + B + C )◦
✸ ✳ ✷ ✳ ❈ ♦ ♥ ✈ ✐ ❡ ❡ ❛ ● ❛ ❞ ♦ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ♥ ❣ ✉ ❧ ♦ ❞ ❛ ❞ ♦ ❡ ♥ ❘ ❛ ❞ ✐ ❛ ♥ ❡
✸ ✳ ✷ ✳ ✶ ✳ π/A
✸ ✳ ✷ ✳ ✷ ✳ π/B
✸ ✳ ✷ ✳ ✸ ✳ π/C
✸ ✳ ✷ ✳ ✹ ✳ 6Aπ
✸ ✳ ✷ ✳ ✺ ✳ 10Bπ
✸ ✳ ✷ ✳ ✻ ✳
5Cπ
✸ ✳ ✷ ✳ ✼ ✳
7π/C
✸ ✳ ✷ ✳ ✽ ✳ 3π/A
✸ ✳ ✷ ✳ ✾ ✳ Bπ/2
✸ ✳ ✷ ✳ ✶ ✵ ✳ 2πC
✸ ✳ ✸ ✳ ❉ ❛ ❞ ♦ ❡ ❧ ✐ ♥ ❣ ✉ ❧ ♦ ❡ ♣ ❡ ❡ ♥ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✜ ❣ ✉ ❛ ✿
A
B
A + B 2
2 α
β
❈ ❛ ❧ ❝ ✉ ❧ ❛ ✿
✸ ✳ ✸ ✳ ✶ ✳ sen α
✸ ✳ ✸ ✳ ✷ ✳ cos α
✸ ✳ ✸ ✳ ✸ ✳ sen β
✸ ✳ ✸ ✳ ✹ ✳ tan β
✸ ✳ ✸ ✳ ✺ ✳ tan α
✸ ✳ ✸ ✳ ✻ ✳ cos β
✸ ✳ ✹ ✳ ❈ ♦ ♠ ♣ ✉ ❡ ❜ ❛ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ♣ ♦ ♣ ✐ ❡ ❞ ❛ ❞ ❡ ❡ ♥ ❡ ❧ ✐ ♥ ❣ ✉ ❧ ♦ ❞ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ✲
❝ ✐ ♦ ❛ ♥ ❡ ✐ ♦ ✿
✸ ✳ ✹ ✳ ✶ ✳ sen2 α + cos2 α = 1
✸ ✳ ✹ ✳ ✷ ✳ 1 + tan2 β = sec2 β
✻
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8/19/2019 FM GDD 0715
8/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✸ ✳ ✺ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❧ ✈ ❛ ❧ ♦ ❞ ❡ ♦ ❞ ♦ ❧ ♦ ❧ ❛ ❞ ♦ ✱ ♥ ❣ ✉ ❧ ♦ ② ❡ ❧ ❡ ❛ ❞ ❡ ❧ ✐ ♥ ❣ ✉ ❧ ♦
❡ ♣ ❡ ❡ ♥ ❛ ❞ ♦ ♣ ♦ ❧ ❛ ✜ ❣ ✉ ❛ ✿
A+B+C
A
(10B-5)º
✸ ✳ ✻ ✳ ❉ ❡ ♠ ✉ ❡ ❛ ❧ ❛ ❧ ❧ ❛ ♠ ❛ ❞ ❛ ❵ ❞ ❡ ✐ ❣ ✉ ❛ ❧ ❞ ❛ ❞ ❞ ❡ ❧ ✐ ♥ ❣ ✉ ❧ ♦ ✬ ♣ ❛ ❛ ❧ ❛ ✜ ❣ ✉ ❛
❞ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❛ ♥ ❡ ✐ ♦ ✳
✸ ✳ ✼ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ✱ ✉ ❛ ♥ ❞ ♦ ✐ ❣ ♦ ♥ ♦ ♠ ❡ ❛ ✱ ❡ ❧ ❡ ❛ ❞ ❡ ❧ ♣ ❡ ♥ ❣ ♦ ♥ ♦ ❡ ♣ ❡ ❡ ♥ ✲
❛ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✜ ❣ ✉ ❛ ✿
Radio=AC
✸ ✳ ✽ ✳ ❯ ♥ ❢ ❛ ♥ ❝ ♦ ✐ ❛ ❞ ♦ ✈ ❡ ❛ ✉ ♥ ♦ ❧ ❞ ❛ ❞ ♦ ❡ ♥ ❡ ♠ ✐ ❣ ♦ ❛ ❛ ✈ ❞ ❡ ✉ ♠ ✐ ❛
❡ ❧ ❡ ❝ ♣ ✐ ❝ ❛ ✳ ❙ ✐ ❧ ❛ ♠ ✐ ❛ ✐ ❡ ♥ ❡ ✉ ♥ ♥ ❣ ✉ ❧ ♦ ❞ ❡ ✈ ✐ ✐ ♥ ❞ ❡ C ◦
② ❡ ❧ ❡ ♥ ❡ ♠ ✐ ✲
❣ ♦ ❝ ✉ ❜ ❡ ❧ ❛ ♠ ✐ ❛ ❞ ❞ ❡ ❧ ❛ ♠ ✐ ❛ ✱ ➽ ❆ ✉ ❞ ✐ ❛ ♥ ❝ ✐ ❛ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ✱ ✐
✉ ♣ ♦ ♥ ❡ ♠ ♦ ✉ ❡ ♠ ✐ ❞ ❡ 180 + B
♠ ❞ ❡ ❛ ❧ ✉ ❛ ❄
✸ ✳ ✾ ✳ ❉ ❡ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦ ❞ ❡ ❧ ❛ ❝ ❛ ❧ ❧ ❡ ❡ ♦ ❜ ❡ ✈ ❛ ✉ ♥ ❡ ❞ ✐ ✜ ❝ ✐ ♦ ❜ ❛ ❥ ♦ ✉ ♥ ♥ ❣ ✉ ❧ ♦
❞ ❡ 30◦
② ✐ ♥ ♦ ❛ ❝ ❡ ❝ ❛ ♠ ♦ 10A ♠ ✱ ❜ ❛ ❥ ♦ ✉ ♥ ♥ ❣ ✉ ❧ ♦ ❞ ❡ 60◦
✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛
❧ ❛ ❛ ❧ ✉ ❛ ❞ ❡ ❧ ❡ ❞ ✐ ✜ ❝ ✐ ♦ ✳
✸ ✳ ✶ ✵ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✿
✸ ✳ ✶ ✵ ✳ ✶ ✳ ❙ ✐
sen α = 4/5② α
❡ ✉ ♥ ♥ ❣ ✉ ❧ ♦ ❛ ❣ ✉ ❞ ♦ ✱ ➽ ❈ ✉ ❧ ❡ ❡ ❧ ✈ ❛ ❧ ♦ ❞ ❡
tan α❄
✸ ✳ ✶ ✵ ✳ ✷ ✳ ❙ ✐
tan α = 1.25✱ ②
α❡ ✉ ♥ ♥ ❣ ✉ ❧ ♦ ❡ ♥ ❡ ❧ ❡ ❝ ❡ ❝ ✉ ❛ ❞ ❛ ♥ ❡ ✱ ➽ ❈ ✉ ❧
❡ ❡ ❧ ✈ ❛ ❧ ♦ ❞ ❡ cos α
❄
✸ ✳ ✶ ✵ ✳ ✸ ✳ ❙ ✐ cos α =
√ 2/2
② α
❡ ✉ ♥ ♥ ❣ ✉ ❧ ♦ ❛ ❣ ✉ ❞ ♦ ✱ ➽ ❈ ✉ ❧ ❡ ❡ ❧ ✈ ❛ ❧ ♦ ❞ ❡
cos(180 − α) ❄
✼
-
8/19/2019 FM GDD 0715
9/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✹ ✳ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❈ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛
❛ ❛ ♦ ❞ ♦ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❞ ❡ ❡ ❡ ❡ ♠ ❛ ✱ ❡ ❞ ❡ ❜ ❡ ♥ ❣ ❛ ✜ ❝ ❛ ❧ ♦ ♣ ✉ ♥ ♦ ❡ ♥ ❝ ✉ ❡ ✐ ♥ ❡ ♥
✉ ❡ ♣ ❡ ❝ ✐ ✈ ♦ ✐ ❡ ♠ ❛ ❞ ❡ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ✳ ❙ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ❧ ❛ ✐ ❞ ❛ ❞ ❡ ♥ ✉ ❡ ♣ ❡ ❡ ♥ ❛ ✲
❝ ✐ ♦ ♥ ❡ ✳ ▲ ❛ ♥ ♦ ♠ ❡ ♥ ❝ ❧ ❛ ✉ ❛ ❞ ❡ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❢ ✐ ❝ ❛ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ❡ ❛ ❧ ♥ ❣ ✉ ❧ ♦ ❵ ❤ ❡ ❛ ✬
✭ θ
✮ ❝ ♦ ♠ ♦ ♥ ❣ ✉ ❧ ♦ ♣ ♦ ❧ ❛ ② ❵ ♣ ❤ ✐ ✬ ✭ ϕ
✮ ❝ ♦ ♠ ♦ ♥ ❣ ✉ ❧ ♦ ❛ ③ ✐ ♠ ✉ ❛ ❧ ✳
✹ ✳ ✶ ✳ ❯ ♥ ③ ♦ ♠ ❜ ✐ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ❡ ♥ ❡ ❧ ♣ ✉ ♥ ♦ X (−A,−B) ② ✉ ♥ ♦ ❜ ❡ ✈ ✐ ✈ ✐ ❡ ♥ ✲
❡ ❡ ♥ ❡ ❧ ♣ ✉ ♥ ♦ Y (C, A + B)
✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❧ ❛ ❞ ✐ ❛ ♥ ❝ ✐ ❛ ❡ ♥ ❡ ❡ ❧ ❧ ♦ ✳
✹ ✳ ✷ ✳ ❙ ✐ (r, θ) = (C,Aπ/B)
♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ♣ ♦ ❧ ❛ ❡ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦ P
✱ ❡ ♥ ✲
❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡ P
✳
✹ ✳ ✸ ✳ ❙ ✐ (x, y) = (AC,BC )
♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦
P ✱ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ♣ ♦ ❧ ❛ ❡ ❞ ❡
P ✳
✹ ✳ ✹ ✳ ❙ ✐ (r,θ,z ) = (A,Cπ,B/2)
♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❝ ✐ ❧ ♥ ❞ ✐ ❝ ❛ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦
P ✱ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡
P ✳
✹ ✳ ✺ ✳ ❙ ✐
(x,y,z ) = (A,B,C ) ♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦
P ✱ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❝ ✐ ❧ ♥ ❞ ✐ ❝ ❛ ❞ ❡
P ✳
✹ ✳ ✻ ✳ ❙ ✐ (ρ,θ,ϕ) = (B,Cπ,Aπ)
♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❢ ✐ ❝ ❛ ❞ ❡ ✉ ♥ ♣ ✉ ♥ ♦
P ✱ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡
P ✳
✹ ✳ ✼ ✳ ❙ ✐ (x,y,z ) = (11C,−23B, 7A) ♦ ♥ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ❞ ❡ ✉ ♥
♣ ✉ ♥ ♦ P
✱ ❡ ♥ ❝ ♦ ♥ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❢ ✐ ❝ ❛ ❞ ❡ P
✳
✹ ✳ ✽ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❢ ✐ ❝ ❛ ❞ ❡ ❧ ♣ ✉ ♥ ♦
(r,θ,z ) = (AB − C,BCπ,C + A)
✹ ✳ ✾ ✳ ❯ ♥ ❛ ✈ ✐ ♥ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ❡ ♥ ❧ ❛ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛
(ρ,θ,ϕ) = (1348A, B × 45◦, C × 18◦)
❈ ❛ ❧ ❝ ✉ ❧ ❛ ✉ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❝ ❛ ♥ ❣ ✉ ❧ ❛ ❡ ✳
✹ ✳ ✶ ✵ ✳ ➽ ◗ ✉ ✐ ❡ ♠ ❛ ❞ ❡ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ❡ ❛ ❡ ❧ ♠ ❛ ❞ ❡ ❝ ✉ ❛ ❞ ♦ ♣ ❛ ❛ ❡ ✲
♣ ❡ ❡ ♥ ❛ ❧ ❛ ♣ ♦ ✐ ❝ ✐ ♥ ❞ ❡ ✉ ♥ ❡ ① ♣ ❧ ♦ ❛ ❞ ♦ ✉ ❡ ✈ ❛ ✉ ❜ ✐ ❡ ♥ ❞ ♦ ✉ ♥ ❛
♠ ♦ ♥ ❛ ❛ ❄ ❊ ① ♣ ❧ ✐ ❝ ❛ ♣ ♦ ✉ ✳
✽
-
8/19/2019 FM GDD 0715
10/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✺ ✳ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❊ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡
✺ ✳ ✶ ✳ ❘ ❡ ♦ ❧ ✈ ❡ ❡ ❧ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✐ ❡ ♠ ❛ ♣ ♦ ❡ ❧ ♠ ♦ ❞ ♦ ❞ ❡ ● ❛ ✉ ✭ ✐ ♥ ❞ ✐ ❝ ❛
❧ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ❡ ♠ ❛ ✐ ❝ ❡ ✮ ✳
Ax + By + Cz = 5A
−Bx + (A + B)y − z = 10(A − C )Cx + Ay + C (A/B)z = −3B
✺ ✳ ✷ ✳ ❘ ❡ ♦ ❧ ✈ ❡ ❡ ❧ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✐ ❡ ♠ ❛ ♣ ♦ ✉ ♥ ♠ ♦ ❞ ♦ ❛ ♥ ❛ ❧ ✐ ❝ ♦ ✳
13x + 8By + 10Cz = 2
(4/A)x + 7By + 2Cz = 6
2x + BAy − Cz = 25
✺ ✳ ✸ ✳ ❘ ❡ ♦ ❧ ✈ ❡ ❡ ❧ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✐ ❡ ♠ ❛ ❞ ❡ ❡ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ♦ ❡ ❧ ♠ ♦ ❞ ♦ ❞ ❡
● ❛ ✉ ✭ ✐ ♥ ❞ ✐ ❝ ❛ ❧ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ❡ ♥ ❡ ♠ ❛ ✐ ❝ ❡ ✮ ✳
73.Ax + 22.By + 11.Cz = 14A.18x + B.26y + C.44z = 10
2.5x + 3.7By + 7.6z = 22
✺ ✳ ✹ ✳ ❘ ❡ ♦ ❧ ✈ ❡ ❡ ❧ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✐ ❡ ♠ ❛ ❞ ❡ ❡ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ♦ ✉ ♥ ♠ ♦ ❞ ♦ ❣ ✲
✜ ❝ ♦ ✳
(A + B)x˘y = C 3
(A/C )x + By = 45
✾
-
8/19/2019 FM GDD 0715
11/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✻ ✳ ➪ ❧ ❣ ❡ ❜ ❛ ❱ ❡ ❝ ♦ ✐ ❛ ❧
✻ ✳ ✶ ✳ ❉ ❡ ♠ ♦ ❛ ✱ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ A
✱ B
② C
✱ ✐ ❡ ❧ ♠ ❞ ✉ ❧ ♦ ❞ ❡ ❧ ❛
✉ ♠ ❛ ❞ ❡ ❞ ♦ ✈ ❡ ❝ ♦ ❡ ❞ ❛ ❞ ♦ ✐ ❡ ♠ ♣ ❡ ❡ ♠ ❡ ♥ ♦ ✉ ❡ ❡ ❧ ♠ ❞ ✉ ❧ ♦
❞ ❡ ❧ ❛ ❞ ✐ ❢ ❡ ❡ ♥ ❝ ✐ ❛ ❞ ❡ ❡ ♦ ✈ ❡ ❝ ♦ ❡ ✳
✻ ✳ ✷ ✳ ❉ ❡ ♠ ♦ ❛ ✱ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ❧ ❛ ✈ ❛ ✐ ❛ ❜ ❧ ❡ A
✱ B
② C
✱ ❧ ♦ ❝ ❛ ♦ ❡ ♥ ✉ ❡ ❡ ❧
♠ ❞ ✉ ❧ ♦ ❞ ❡ ❧ ❛ ✉ ♠ ❛ ❞ ❡ ❞ ♦ ✈ ❡ ❝ ♦ ❡ ❝ ♦ ✐ ♥ ❝ ✐ ❞ ❡ ❝ ♦ ♥ ❧ ❛ ✉ ♠ ❛ ❞ ❡ ❧ ♦
♠ ❞ ✉ ❧ ♦ ❞ ❡ ❧ ♦ ✈ ❡ ❝ ♦ ❡ ✉ ❡ ❡ ✉ ♠ ❛ ♥ ✳
✻ ✳ ✸ ✳ ❉ ❛ ❞ ♦ ❧ ♦ ✈ ❡ ❝ ♦ ❡ V = Ai +Bj−Ck
② U = −i+7 j +3k
✱ ❞ ❡ ❡ ♠ ✐ ♥ ❛
(2V /3)
−5U
✳ ● ❛ ✜ ❝ ❛ ❧ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♥ ② ❡ ❧ ❡ ✉ ❧ ❛ ❞ ♦ ✳
✻ ✳ ✹ ✳ ❍ ❛ ❧ ❧ ❛ ❡ ❧ ✈ ❡ ❝ ♦ ✉ ♥ ✐ ❛ ✐ ♦ ❞ ❡
V = C i + 3Bj + 5k✳
✻ ✳ ✺ ✳ ❉ ❡ ❡ ♠ ✐ ♥ ❛ ❡ ❧ ♥ ❣ ✉ ❧ ♦ ✉ ❡ ❢ ♦ ♠ ❛ ❡ ❧ ✈ ❡ ❝ ♦ ❛ ♥ ❡ ✐ ♦ ❝ ♦ ♥ ❡ ❧ ❡ ❥ ❡ X
② ❧ ❛ ♠ ❛ ❣ ♥ ✐ ✉ ❞ ❞ ❡ ✉ ♣ ♦ ② ❡ ❝ ❝ ✐ ♥ ♦ ❜ ❡ ❞ ✐ ❝ ❤ ♦ ❡ ❥ ❡ ✳
✻ ✳ ✻ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❧ ❛ ♣ ♦ ② ❡ ❝ ❝ ✐ ♦ ♥ ❡ ❞ ❡ ❧ ♦ ✈ ❡ ❝ ♦ ❡ R(A, B)
✱ E (C, A
✮ ② V (B, C )
♦ ❜ ❡ ❧ ♦ ❡ ❥ ❡ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ♦ x
✫ y
✳
✻ ✳ ✼ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❧ ❛ ❝ ♦ ♠ ♣ ♦ ♥ ❡ ♥ ❡ ✈ ❡ ❝ ♦ ✐ ❛ ❧ ② ❧ ❛ ❝ ♦ ♠ ♣ ♦ ♥ ❡ ♥ ❡ ❡ ❝ ❛ ❧ ❛ ❞ ❡ ❧
✈ ❡ ❝ ♦ (B, 7)
♦ ❜ ❡ (C, 6)
✳
✻ ✳ ✽ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❧ ♣ ♦ ❞ ✉ ❝ ♦ ♣ ✉ ♥ ♦ ❞ ❡ ❧ ♦ ✈ ❡ ❝ ♦ ❡ ✿
✻ ✳ ✽ ✳ ✶ ✳ v(A,C,−3), u(B, 5, 7)
✻ ✳ ✽ ✳ ✷ ✳ v(C,B, 3), u(−A, 2, B)
✻ ✳ ✽ ✳ ✸ ✳
v(2,−6, A), u(−B, −C, 8)
✻ ✳ ✾ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❧ ♥ ❣ ✉ ❧ ♦ ❡ ♥ ❡ ❧ ♦ ✈ ❡ ❝ ♦ ❡ ✿
✻ ✳ ✾ ✳ ✶ ✳ v(A, C ), u(C, B)
✻ ✳ ✾ ✳ ✷ ✳ v(−B, C ), u(A, B)
✻ ✳ ✾ ✳ ✸ ✳ v(−1, 13), u(A, 0)
✶ ✵
-
8/19/2019 FM GDD 0715
12/13
❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊
▼ ➱ ❳ ■ ❈ ❖
● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦
✻ ✳ ✶ ✵ ✳ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❧ ♣ ♦ ❞ ✉ ❝ ♦ ✈ ❡ ❝ ♦ ✐ ❛ ❧ ❞ ❡ ❧ ♦ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ✈ ❡ ❝ ♦ ❡ ✿
✻ ✳ ✶ ✵ ✳ ✶ ✳
v(5, 8,−1), u(2, A, 6)✻ ✳ ✶ ✵ ✳ ✷ ✳
v(4,−B, 7), u(3, A, 2)✻ ✳ ✶ ✵ ✳ ✸ ✳
v(C,−6, 9), u(9,−A,−1)
✻ ✳ ✶ ✶ ✳ ❍ ❛ ❧ ❧ ❛ ❡ ❧ ✈ ❡ ❝ ♦ ✉ ♥ ✐ ❛ ✐ ♦ ♣ ❡ ♣ ❡ ♥ ❞ ✐ ❝ ✉ ❧ ❛ ❛ ❧ ♦ ✈ ❡ ❝ ♦ ❡ V (A,B,−C ✮
② W (−1, 0, 2)
✳
✶ ✶
-
8/19/2019 FM GDD 0715
13/13
Fundamentos Matem‡ticos
Pr‡ctica 1 Pr‡ctica2
Pr‡ctica 3 Pr‡ctica 4
Pr‡ctica 5 Pr‡ctica 6
Pr‡ctica 7
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