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# Ââè èúâùè fi ùô ûëìâô 1 2 ôè

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ÂÂÈ‰‹ ‰È¤Ú¯ÂÙ·È ·fi ÙÔ ÛËÌÂ›Ô ∞(1, 2), ÔÈ Û˘ÓÙÂÙ·ÁÌ¤ÓÂ˜ ÙÔ˘ ÛËÌÂ›Ô˘ ∞ ı· Â·ÏËıÂ‡Ô˘Ó ÙËÓ ÂÍ›ÛˆÛ‹ ÙË˜. ÕÚ· ı· ÈÛ¯‡ÂÈ 2 = · Ø 1 2 · = 2. √fiÙÂ, Ë ˙ËÙÔ‡ÌÂÓË ÂÍ›ÛˆÛË Â›Ó·È Ë y = 2x 2 . 2. i) H ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ Ê(x) = 0,5x 2 Â›Ó·È ÌÈ· ·Ú·‚ÔÏ‹ ·ÓÔÈ¯Ù‹ ÚÔ˜ Ù· ¿Óˆ ÌÂ ÎÔÚ˘Ê‹ ÙËÓ ·Ú¯‹ ÙˆÓ ·ÍfiÓˆÓ Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙÔÓ yãy (Û¯.). ∏ ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙˆÓ Û˘Ó·ÚÙ‹ÛÂˆÓ f(x) = 0,5x 2 + 2 Î·È g(x) = 0,5x 2 – 3 ÚÔÎ‡ÙÔ˘Ó ·fi Î·Ù·ÎfiÚ˘ÊË ÌÂÙ·Ùfi- ÈÛË ÙË˜ ·Ú·‚ÔÏ‹˜ y = 0,5x 2 , ÙË˜ ÌÂÓ ÚÒÙË˜ Î·Ù¿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ¿Óˆ, ÙË˜ ‰Â ‰Â‡ÙÂÚË˜ Î·Ù¿ 3 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· Î¿Ùˆ. ii) ∏ ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ „(x) = –0,5x 2 Â›Ó·È ÌÈ· ·Ú·‚ÔÏ‹ ·ÓÔÈ¯Ù‹ ÚÔ˜ Ù· Î¿- Ùˆ ÌÂ ÎÔÚ˘Ê‹ ÙËÓ ·Ú¯‹ ÙˆÓ ·ÍfiÓˆÓ Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙÔÓ ¿ÍÔÓ· yãy (Û¯.). √È ÁÚ·ÊÈÎ¤˜ ·Ú¿ÛÙ·ÛÂÈ˜ ÙˆÓ Û˘Ó·ÚÙ‹ÛÂ- ˆÓ h(x) = –0,5x 2 –2 Î·È q(x) = –0,5x 2 + 3 ÚÔÎ‡ÙÔ˘Ó ·fi Î·Ù·ÎfiÚ˘ÊÂ˜ ÌÂÙ·ÙÔ- ›ÛÂÈ˜ ÙË˜ ·Ú·‚ÔÏ‹˜ y = –0,5x 2 , ÙË˜ ÌÂÓ ÚÒÙË˜ Î·Ù¿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· Î¿Ùˆ, ÙË˜ ‰Â ‰Â‡ÙÂÚË˜ Î·Ù¿ 3 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ¿Óˆ. ¶·Ú·Ù‹ÚËÛË : ∂ÂÈ‰‹ ÔÈ Û˘Ó·ÚÙ‹ÛÂÈ˜ „, h Î·È q Â›Ó·È ·ÓÙ›ıÂÙÂ˜ ÙˆÓ Û˘Ó·ÚÙ‹ÛÂˆÓ Ê, f Î·È g ·ÓÙÈÛÙÔ›¯ˆ˜, ÁÈ· Ó· ¯·Ú¿ÍÔ˘ÌÂ ÙÈ˜ ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ ÙÔ˘˜ ·ÚÎÂ› Ó· ·›ÚÓ·ÌÂ ÙÈ˜ Û˘ÌÌÂÙÚÈÎ¤˜ ÙˆÓ ÁÚ·ÊÈÎÒÓ ·Ú·ÛÙ¿ÛÂˆÓ ÙˆÓ Ê, f Î·È g ˆ˜ ÚÔ˜ ÙÔÓ ¿ÍÔÓ· xãx.

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3. i) X·Ú¿ÛÛÔ˘ÌÂ ÙË ÁÚ·ÊÈÎ‹ ·- Ú¿ÛÙ·ÛË ÙË˜ Ê(x) = 0,5x 2 , fiˆ˜ ÛÙËÓ ¿ÛÎËÛË 2. i). √È ÁÚ·ÊÈ- Î¤˜ ·Ú¿ÛÙ·ÛÂÈ˜ ÙˆÓ Û˘Ó·Ú- Ù‹ÛÂˆÓ f(x) = 0,5(x – 2) 2 Î·È g(x) = 0,5(x + 2) 2 ÚÔÎ‡ÙÔ˘Ó ·fi ÔÚÈ˙fiÓÙÈÂ˜ ÌÂÙ·ÙÔ›ÛÂÈ˜ ÙË˜ ·Ú·‚ÔÏ‹˜ y = 0,5x 2 , ÙË˜ ÌÂÓ ÚÒÙË˜ Î·Ù¿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ‰ÂÍÈ¿, ÙË˜ ‰Â ‰Â‡ÙÂÚË˜ Î·Ù¿ 2 ÌÔ- Ó¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿. ii) Ã·Ú¿ÛÛÔ˘ÌÂ ÙË ÁÚ·ÊÈÎ‹ ·- Ú¿ÛÙ·ÛË ÙË˜ „(x) = –0,5x 2 , fiˆ˜ ÛÙËÓ ¿ÛÎËÛË 2. ii). √È ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ ÙˆÓ Û˘- Ó·ÚÙ‹ÛÂˆÓ h(x) = –0,5(x – 2) 2 Î·È q(x) = –0,5(x + 2) 2 ÚÔ- Î‡ÙÔ˘Ó ·fi ÔÚÈ˙fiÓÙÈÂ˜ ÌÂ- Ù·ÙÔ›ÛÂÈ˜ ÙË˜ ·Ú·‚ÔÏ‹˜ y = –0,5x 2 , ÙË˜ ÚÒÙË˜ Î·Ù¿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ‰ÂÍÈ¿, ÙË˜ ‰Â ‰Â‡ÙÂÚË˜ Î·Ù¿ ‰‡Ô ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿.
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