8/19/2019 FM GDD 0715

1/13

 VIDEOJUEGOS

GDD 0715

Gu’a Pr‡ctica

Fundamentos Matem‡ticos

8/19/2019 FM GDD 0715

2/13

❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊  

▼ ➱ ❳ ■ ❈ ❖  

● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦  

■ ♥ ♦ ❞ ✉ ❝ ❝ ✐ ♥      

▲ ❛ ♣ ❝ ✐ ❝ ❛ ♦ ♥ ❛ ❝ ✐ ✈ ✐ ❞ ❛ ❞ ❡ ✉ ❡ ❡ ❛ ② ✉ ❞ ❛ ♥ ❛ ❡ ❛ ✜ ♠ ❛ ❡ ❧ ❝ ♦ ♥ ♦ ❝ ✐ ♠ ✐ ❡ ♥ ♦ ❛ ❞ ✉ ✐ ✲  

✐ ❞ ♦ ❡ ♥ ❧ ❛ ❤ ♦ ❛ ❢ ❡ ♥ ❡ ❛ ♣ ♦ ❢ ❡ ♦ ✱ ♣ ❛ ❛ ❧ ♦ ❣ ❛ ❧ ❧ ❡ ✈ ❛ ❧ ♦ ❛ ✉ ♥ ♥ ✐ ✈ ❡ ❧ ❞ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ❝ ✐ ♥  

❡ ✜ ❝ ✐ ❡ ♥ ❡ ✳ ❊ ❛ ❣ ✉ ❛ ❡ ♣ ❡ ♠ ✐ ✐ ❝ ♦ ♥ ♦ ❝ ❡ ❧ ♦ ♦ ❜ ❥ ❡ ✐ ✈ ♦ ② ♣ ♦ ❝ ❡ ❞ ✐ ♠ ✐ ❡ ♥ ♦ ♣ ❛ ❛ ❡ ❛ ❧ ✐ ✲  

③ ❛ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳  

❉ ❡ ❝ ✐ ♣ ❝ ✐ ♥      

▲ ❛ ♣ ❝ ✐ ❝ ❛ ❝ ♦ ♠ ♣ ❡ ♥ ❞ ❡ ♥ ❞ ✐ ❢ ❡ ❡ ♥ ❡ ❡ ❛ ♣ ❛ ❞ ❡ ❡ ❛ ❧ ✐ ③ ❛ ❝ ✐ ♥ ✳ ❉ ❡ ❜ ❡ ❛ ❡ ♥ ❞ ❡ ❝ ❛ ❞ ❛  

✉ ♥ ♦ ❞ ❡ ❧ ♦ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ✐ ♥ ❞ ✐ ❝ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ✐ ♥ ✉ ❝ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ✉ ❡ ❛ ❡ ❛ ♥ ❝ ♦ ♥ ✐ ❞ ❡ ❛ ✲  

❞ ❛ ♣ ❛ ❛ ❡ ✈ ❛ ❧ ✉ ❛ ❝ ✐ ♥ ✳  

▲ ❛ ♣ ❝ ✐ ❝ ❛ ♦ ♥ ❡ ♣ ❡ ❝ ✐ ❛ ❧ ♠ ❡ ♥ ❡ ❞ ✐ ❡ ❛ ❞ ❛ ❡ ♥ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ♣ ♦ ❡ ❧ ❡ ❛ ❛ ❝ ❛ ❞ ♠ ✐ ❝ ❛ ② ❡ ❧  

❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ♣ ❛ ❛ ❡ ❡ ✈ ❛ ❧ ✉ ❛ ❞ ❛ ♣ ♦ ❡ ❛ ❧ ✐ ♠ ❛ ✳ ▲ ♦ ✉ ♣ ❡ ✈ ✐ ♦ ❡ ✜ ♠ ❛ ♥  

✉ ❘ ❡ ❣ ✐ ♦ ❞ ❡ ♦ ❣ ❡ ♦ ❆ ❝ ❛ ❞ ♠ ✐ ❝ ♦ ✭ ❘ ❆ ✮ ❡ ♥ ❡ ❧ ❡ ❝ ✉ ❛ ❞ ♦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ❡ ♥ ❡ ✱ ♣ ♦  

❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ♥ ❡ ❣ ❛ ❞ ❛ ❝ ♦ ♥ ❧ ♦ ❡ ✉ ✐ ✐ ♦ ❞ ❡ ❝ ✐ ♦ ❡ ♥ ❡ ❛ ❣ ✉ ❛ ✳  

❊ ♥ ❡ ❣ ❛    

▲ ❛ ❢ ❡ ❝ ❤ ❛ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ✉ ❡ ❡ ♦ ♠ ❛ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ♣ ❛ ❛ ❧ ❛ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ✐ ♥ ❡ ❧ ❛ ❞ ❡ ❧ ❛ ❡ ♥ ✲  

❡ ❣ ❛ ❞ ❡ ❧ ❘ ❆ ✳ ❊ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❡ ♥ ❢ ♦ ♠ ❛ ♦ ❞ ✐ ❣ ✐ ❛ ❧ ② ✈ ❛ ✐ ♥ ❡ ♥ ❡ ✱ ❧ ❛ ❢ ❡ ❝ ❤ ❛  

❝ ♦ ♥ ✐ ❞ ❡ ❛ ❞ ❛ ♣ ❛ ❛ ❡ ♥ ❡ ❣ ❛ ❡ ❧ ❛ ❞ ❡ ❧ ❘ ❆ ② ♥ ♦ ❧ ❛ ❞ ❡ ❧ ❡ ♥ ✈ ♦ ❞ ❡ ❧ ❛ ❝ ❤ ✐ ✈ ♦ ✳ ❙ ✐ ✉ ♥  

❛ ❧ ✉ ♠ ♥ ♦ ✉ ❜ ❡ ✉ ♥ ❛ ❝ ❤ ✐ ✈ ♦ ❛ ● ♦ ♦ ❣ ❧ ❡ ❉ ✐ ✈ ❡ ✱ ♣ ♦ ❡ ❥ ❡ ♠ ♣ ❧ ♦ ✱ ♣ ❡ ♦ ♥ ♦ ❡ ❧ ❧ ❛ ❡ ❧ ❘ ❆ ❛ ♥ ❡  

❞ ❡ ❧ ❛ ❢ ❡ ❝ ❤ ❛ ② ❤ ♦ ❛ ❧ ♠ ✐ ❡ ✱ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ♦ ❜ ❡ ✼ ✳  

❊ ❡ ♣ ♦ ♥ ❛ ❜ ✐ ❧ ✐ ❞ ❛ ❞ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ❡ ♥ ❡ ♦ ❞ ♦ ❧ ♦ ♠ ❛ ❡ ✐ ❛ ❧ ❡ ❛ ✐ ❡ ♠ ♣ ♦ ② ❡ ♥ ❡ ✉ ♥ ❡ ✲  

♣ ❛ ❧ ❞ ♦ ❞ ❡ ✉ ❛ ❜ ❛ ❥ ♦ ✱ ♦ ❜ ❡ ♦ ❞ ♦ ❡ ♥ ❡ ❧ ❝ ❛ ♦ ❞ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ✐ ❛ ❞ ❛ ✳  

❈ ✉ ❛ ❧ ✉ ✐ ❡ ❞ ✉ ❞ ❛ ♦ ❜ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ❞ ❡ ❜ ❡ ❡ ❝ ✉ ✐ ❛ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✲  

✐ ♥ ✳ ❚ ❡ ❡ ❝ ♦ ♠ ❡ ♥ ❞ ❛ ♠ ♦ ✉ ❡ ❧ ♦ ❤ ❛ ❣ ❛ ❝ ♦ ♥ ❛ ♥ ✐ ❝ ✐ ♣ ❛ ❝ ✐ ♥ ② ❛ ❞ ♠ ✐ ♥ ✐ ❡ ✉  

✐ ❡ ♠ ♣ ♦ ✳  

❉ ❡ ❜ ❡ ❡ ♣ ❡ ❛ ❧ ♦ ❢ ♦ ♠ ❛ ♦ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❞ ❡ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳ ❈ ✉ ❛ ❧ ✉ ✐ ❡ ❞ ✉ ❞ ❛ ❛ ❧ ❡ ✲  

♣ ❡ ❝ ♦ ✱ ♣ ✉ ❡ ❞ ❡ ❡ ❝ ✉ ✐ ❛ ❧ ▼ ❛ ♥ ✉ ❛ ❧ ❞ ❡ ❋ ♦ ♠ ❛ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛ ❡ ❧ ❝ ✉ ❛ ❧ ❡ ❡ ♥ ❝ ✉ ❡ ♥ ❛ ❡ ♥  

❡ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ✳  

❉ ❡ ❜ ❡ ❡ ❝ ♦ ❣ ❡ ✉ ❛ ❜ ❛ ❥ ♦ ❡ ♥ ❡ ❧ ❡ ❛ ❞ ❡ ✉ ♣ ❡ ✈ ✐ ✐ ♥ ❛ ♠ ❛ ❞ ❛ ✉ ♥ ❛ ❡ ♠ ❛ ♥ ❛  

❞ ❡ ♣ ✉ ❞ ❡ ✉ ❡ ❧ ❛ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ❝ ✐ ♦ ♥ ❡ ❤ ❛ ② ❛ ♥ ✐ ❞ ♦ ❛ ✐ ❣ ♥ ❛ ❞ ❛ ✳ ❉ ❡ ♥ ♦ ❡ ❛ ✱ ❡ ❧ ❡ ❛ ❞ ❡  

✉ ♣ ❡ ✈ ✐ ✐ ♥ ♥ ♦ ❡ ❤ ❛ ❡ ♣ ♦ ♥ ❛ ❜ ❧ ❡ ❞ ❡ ❝ ♦ ♥ ❡ ✈ ❛ ❞ ✐ ❝ ❤ ♦ ❛ ❜ ❛ ❥ ♦ ✳  

✶ 

8/19/2019 FM GDD 0715

3/13

❙ ❆ ❊ ■ ◆ ❙ ❚ ■ ❚ ❯ ❚ ❊  

▼ ➱ ❳ ■ ❈ ❖  

● ✉ ❛ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❝ ✐ ❝ ♦  

❖ ❜ ❥ ❡ ✐ ✈ ♦      

❊ ♥ ❡ ❛ ❡ ✐ ❡ ❞ ❡ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ❛ ♣ ❧ ✐ ❝ ❛ ♥ ❧ ♦ ♣ ✐ ♥ ❝ ✐ ♣ ✐ ♦ ♠ ❛ ❡ ♠ ✐ ❝ ♦ ✈ ✐ ♦ ❡ ♥ ❝ ❧ ❛ ❡  

❡ ♥ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ♣ ❡ ❝ ✜ ❝ ♦ ♦ ✐ ❡ ♥ ❛ ❞ ♦ ❛ ❧ ❛ ♣ ♦ ❣ ❛ ♠ ❛ ❝ ✐ ♥ ❞ ❡ ✈ ✐ ❞ ❡ ♦ ❥ ✉ ❡ ❣ ♦ ✳  

❈ ❛ ❧ ❡ ♥ ❞ ❛ ✐ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛    

❆ ❝ ♦ ♥ ✐ ♥ ✉ ❛ ❝ ✐ ♥ ❡ ❡ ❛ ❧ ❛ ❧ ❛ ❢ ❡ ❝ ❤ ❛ ❞ ❡ ❡ ♥ ❡ ❣ ❛ ❞ ❡ ❝ ❛ ❞ ❛ ♣ ❝ ✐ ❝ ❛ ✳ ❚ ♦ ❞ ❛ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ✱  

✐ ♥ ❡ ① ❝ ❡ ♣ ❝ ✐ ♥ ✱ ❡ ❡ ❛ ❧ ✐ ③ ❛ ♥ ❝ ♦ ♠ ♦ ♠ ① ✐ ♠ ♦ ❛ ❧ ❛ ✷ ✶ ✿ ✹ ✺ ❤ ♦ ❛ ✳  

▲ ❛ ♣ ❝ ✐ ❝ ❛ ❡ ♥ ❡ ❣ ❛ ❞ ❛ ❢ ✉ ❡ ❛ ❞ ❡ ❧ ♣ ❡ ✐ ♦ ❞ ♦ ❛ ❝ ♦ ❞ ❛ ❞ ♦ ❡ ♥ ❧ ❛ ❛ ❣ ❡ ♥ ❞ ❛ ✱ ❡ ❝ ❛ ❧ ✐ ✜ ❝ ❛ ♥  

♦ ❜ ❡ ✼ ❞ ✉ ❛ ♥ ❡ ❧ ❛ ✐ ❣ ✉ ✐ ❡ ♥ ❡ ❡ ♠ ❛ ♥ ❛ ✱ ❝ ♦ ♠ ♦ ✐ ♥ ❞ ✐ ❝ ❛ ❡ ❧ ▼ ❛ ♥ ✉ ❛ ❧ ❞ ❡ ❧ ❊ ✉ ❞ ✐ ❛ ♥ ❡ ✳  

◆ ♦ ✳ ❞ ❡ ❝ ✐ ❝ ❛ ◆ ♦ ♠ ❜ ❡ ❞ ❡ ❧ ❊ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❊ ♥ ❡ ❣ ❛  

✶ ❈ ♦ ♥ ❝ ❡ ♣ ♦ ❇ ✐ ❝ ♦ ✷ ✽ ❊ ♥ ❡ ♦ ✷ ✵ ✶ ✻  

✷ ❈ ♦ ♥ ❥ ✉ ♥ ♦ ◆ ✉ ♠ ✐ ❝ ♦ ✹ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻  

✸ ❚ ✐ ❣ ♦ ♥ ♦ ♠ ❡ ❛ ✶ ✶ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻  

✹ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❝ ♦ ♦ ❞ ❡ ♥ ❛ ❞ ❛ ✶ ✽ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻  

✺ ❙ ✐ ❡ ♠ ❛ ❞ ❡ ❡ ❝ ✉ ❛ ❝ ✐ ♦ ♥ ❡ ✷ ✺ ❋ ❡ ❜ ❡ ♦ ✷ ✵ ✶ ✻  

✻ ➪ ❧ ❣ ❡ ❜ ❛ ❱ ❡ ❝ ♦ ✐ ❛ ❧ ✸ ▼ ❛ ③ ♦ ✷ ✵ ✶ ✻  

▲ ❛ ❢ ❡ ❝ ❤ ❛ ♣ ✉ ❞ ❛ ♥ ❝ ❛ ♠ ❜ ✐ ❛ ❡ ❛ ❧ ♦ ❧ ❛ ❣ ♦ ❞ ❡ ❧ ✐ ♠ ❡ ❡ ✱ ♣ ❡ ♦ ❡ ♦ ❡ ❧ ❡ ♥ ♦ ✐ ✜ ❝ ❛ ❝ ♦ ♥  

❛ ♥ ❡ ❧ ❛ ❝ ✐ ♥ ✳  

❋ ♦ ♠ ❛ ♦ ❞ ❡ ❊ ♥ ❡ ❣ ❛      

▲ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ❞ ❡ ❜ ❡ ♥ ❡ ♥ ❡ ❣ ❛ ❛ ♠ ❛ ♥ ♦ ✉ ✐ ❧ ✐ ③ ❛ ♥ ❞ ♦ ❤ ♦ ❥ ❛ ❜ ❧ ❛ ♥ ❝ ❛ ❞ ❡ ❛ ♠ ❛ ♦ ❝ ❛ ✲  

❛ ✱ ❛ ♣ ♦ ✈ ❡ ❝ ❤ ♥ ❞ ♦ ❧ ❛ ♣ ♦ ❛ ♠ ❜ ♦ ❧ ❛ ❞ ♦ ✳ ❙ ❡ ❞ ❡ ❜ ❡ ❝ ♦ ♣ ✐ ❛ ❡ ❧ ❡ ♥ ✉ ♥ ❝ ✐ ❛ ❞ ♦ ❞ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦  

❝ ♦ ♠ ♣ ❧ ❡ ♦ ② ❞ ❡ ❛ ♦ ❧ ❧ ❛ ❧ ♦ ❝ ♦ ♥ ❧ ❛ ♠ ❛ ② ♦ ❛ ♠ ♣ ❧ ✐ ✉ ❞ ♣ ♦ ✐ ❜ ❧ ❡ ✳ ❚ ♦ ❞ ❛ ② ❝ ❛ ❞ ❛ ✉ ♥ ❛ ❞ ❡ ❧ ❛  

❤ ♦ ❥ ❛ ❞ ❡ ❜ ❡ ♥ ❛ ❡   ♥ ♦ ♠ ❜ ❡ ② ♥ ♠ ❡ ♦ ❞ ❡ ❧ ❛ ❧ ✉ ♠ ♥ ♦ ✱ ❛ ❝ ♦ ♠ ♦ ❝ ❞ ✐ ❣ ♦ ❞ ❡ ❧ ❝ ✉ ♦ ✳  

▲ ❛ ❡ ♦ ❧ ✉ ❝ ✐ ♥ ❞ ❡ ❧ ♦ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❞ ❡ ❜ ❡ ❡ ❝ ❧ ❛ ❛ ② ❝ ♦ ♥ ❧ ✐ ♠ ♣ ✐ ❡ ③ ❛ ✳ ❊ ♣ ❡ ♣ ♦ ♥ ❞ ❡ ❛ ♥ ❡  

♠ ❛ ♥ ❡ ♥ ❡ ✉ ♥ ♦ ❞ ❡ ♥ ❞ ❡ ❞ ❡ ❛ ♦ ❧ ❧ ♦ ② ❡ ✈ ✐ ❛ ♦ ❞ ♦ ✐ ♣ ♦ ❞ ❡ ❛ ② ♦ ♥ ❡ ✳  

❙ ✐ ❡ ♣ ❡ ❡ ♥ ❛ ✉ ♥ ❣ ❛ ❞ ♦ ❞ ❡ ✐ ♠ ✐ ❧ ✐ ✉ ❞ ✐ ♥ ❛ ❝ ❡ ♣ ❛ ❜ ❧ ❡ ❡ ♥ ❧ ❛ ♠ ❡ ♦ ❞ ♦ ❧ ♦ ❣ ❛ ♦ ❧ ♦ ❡ ✉ ❧ ❛ ❞ ♦  

❡ ♥ ❡ ❧ ❛ ♣ ❝ ✐ ❝ ❛ ❞ ❡ ❞ ♦ ♦ ♠ ❛ ❧ ✉ ♠ ♥ ♦ ✱ ✉ ❡ ❞ ❛ ❛ ❝ ✐ ❡ ✐ ♦ ❞ ❡ ❧ ❡ ✉ ✐ ♣ ♦ ❞ ❡ ✉ ♣ ❡ ✲  

✈ ✐ ✐ ♥ ❡ ❧ ❛ ♥ ✉ ❧ ❛ ❧ ❛ ❡ ♥ ❡ ❣ ❛ ❝ ♦ ♠ ♣ ❧ ❡ ❛ ♦ ❧ ♦ ❡ ❧ ❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ❡ ♥ ❝ ✉ ❡ ✐ ♥ ✳  

❊ ❡ ❝ ♦ ♠ ❡ ♥ ❞ ❛ ❜ ❧ ❡ ♠ ❛ ♥ ❡ ❥ ❛ ❝ ♦ ♠ ♦ ❢ ❛ ❝ ❝ ✐ ♦ ♥ ❡ ♦ ❞ ❛ ❧ ❛ ❝ ✐ ❢ ❛ ✉ ❡ ♥ ♦ ❡ ❛ ♥ ♥ ♠ ❡ ♦  

❡ ♥ ❡ ♦ ✱ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❛ ✐ ♦ ♣ ✉ ❡ ❞ ❡ ❡ ① ✐ ✐ ♣ ❞ ✐ ❞ ❛ ❞ ❡ ♣ ❡ ❝ ✐ ✐ ♥ ❡ ♥ ❡ ✉ ❧ ❛ ❞ ♦ ✉ ❡ ❡ ♥  

❝ ♦ ♥ ✐ ❞ ❡ ❛ ❞ ♦ ❝ ♦ ♠ ♦ ✐ ♥ ❝ ♦ ❡ ❝ ♦ ✳ ❉ ❡ ❡ ♥ ❡ ❝ ❡ ❛ ✐ ♦ ❡ ❧ ✉ ♦ ❞ ❡ ♥ ♠ ❡ ♦ ❞ ❡ ❝ ✐ ♠ ❛ ❧ ❡ ✱ ❡  

❡ ❝ ♦ ♠ ✐ ❡ ♥ ❞ ❛ ✉ ✐ ❧ ✐ ③ ❛ ❝ ✉ ❛ ♦ ❝ ✐ ❢ ❛ ❞ ❡ ♣ ✉ ❞ ❡ ❧ ♣ ✉ ♥ ♦ ✳  

✷ 

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}

✷ ✳ ✺ ✳  ❘ ❡ ❛ ❧ ✐ ❝ ❡ ❧ ♦ ❞ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❱ ❡ ♥ ♥ ❞ ❡ ❧ ♦ ❝ ♦ ♥ ❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❞ ❡ ❧  

❡ ❥ ❡ ❝ ✐ ❝ ✐ ♦ ✷ ✳ ✶ ✳ ❉ ✐ ❝ ❤ ♦ ❞ ✐ ❛ ❣ ❛ ♠ ❛ ❞ ❡ ❜ ❡ ♥ ❡ ♣ ❡ ❡ ♥ ❛ ❛ ♥ ♦ ❡ ❧ ❝ ♦ ♥ ✲  

 ❥ ✉ ♥ ♦ ❡ ✉ ❧ ❛ ♥ ❡ ❝ ♦ ♠ ♦ ❧ ❛ ♦ ♣ ❡ ❛ ❝ ✐ ♦ ♥ ❡ ♣ ❛ ❛ ❧ ❧ ❡ ❣ ❛ ❛ ❧ ✳ 

✺ 

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 

✻ 

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

!"#$%&

'()*+,-./+&

!"#$%&

'()*+,-./+&

!"#$%&'()*+,-./+&

!"#$%&

'()*+,-./+&

!"#$%&

'()*+,-./+&

!"#$%&'()*+,-./+&

!"#$%&

'()*+,-./+&