2011 Λύσεις Σχ. β&I

Ââè èúâùè fi ùô ûëìâô 1 2 ôè

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ÂÂȉ‹ ‰È¤Ú¯ÂÙ·È ·fi ÙÔ ÛËÌÂ›Ô ∞(1, 2), ÔÈ Û˘ÓÙÂÙ·Á̤Ó˜ ÙÔ˘ ÛËÌ›Ԣ ∞ ı· Â·ÏËıÂ‡Ô˘Ó ÙËÓ Â͛ۈۋ Ù˘. ÕÚ· ı· ÈÛ¯‡ÂÈ 2 = · Ø 1 2 · = 2. √fiÙÂ, Ë ˙ËÙÔ‡ÌÂÓË Â͛ۈÛË Â›Ó·È Ë y = 2x 2 . 2. i) H ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ Ê(x) = 0,5x 2 Â›Ó·È ÌÈ· ·Ú·‚ÔÏ‹ ·ÓÔȯً ÚÔ˜ Ù· ¿Óˆ Ì ÎÔÚ˘Ê‹ ÙËÓ ·Ú¯‹ ÙˆÓ ·ÍfiÓˆÓ Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙÔÓ yãy (Û¯.). ∏ ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË ÙˆÓ Û˘Ó·ÚÙ‹ÛÂˆÓ f(x) = 0,5x 2 + 2 Î·È g(x) = 0,5x 2 – 3 ÚÔ·ÙÔ˘Ó ·fi ηٷÎfiÚ˘ÊË ÌÂÙ·Ùfi- ÈÛË Ù˘ ·Ú·‚ÔÏ‹˜ y = 0,5x 2 , Ù˘ ÌÂÓ ÚÒÙ˘ ηٿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ¿Óˆ, Ù˘ ‰Â ‰Â‡ÙÂÚ˘ ηٿ 3 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· οو. ii) ∏ ÁÚ·ÊÈ΋ ·Ú¿ÛÙ·ÛË Ù˘ „(x) = –0,5x 2 Â›Ó·È ÌÈ· ·Ú·‚ÔÏ‹ ·ÓÔȯً ÚÔ˜ Ù· ο- Ùˆ Ì ÎÔÚ˘Ê‹ ÙËÓ ·Ú¯‹ ÙˆÓ ·ÍfiÓˆÓ Î·È ¿ÍÔÓ· Û˘ÌÌÂÙÚ›·˜ ÙÔÓ ¿ÍÔÓ· yãy (Û¯.). √È ÁÚ·ÊÈΤ˜ ·Ú¿ÛÙ·ÛÂȘ ÙˆÓ Û˘Ó·ÚÙ‹ÛÂ- ˆÓ h(x) = –0,5x 2 –2 Î·È q(x) = –0,5x 2 + 3 ÚÔ·ÙÔ˘Ó ·fi ηٷÎfiÚ˘Ê˜ ÌÂÙ·ÙÔ- ›ÛÂȘ Ù˘ ·Ú·‚ÔÏ‹˜ y = –0,5x 2 , Ù˘ ÌÂÓ ÚÒÙ˘ ηٿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· οو, Ù˘ ‰Â ‰Â‡ÙÂÚ˘ ηٿ 3 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ¿Óˆ. ¶·Ú·Ù‹ÚËÛË : ∂Âȉ‹ ÔÈ Û˘Ó·ÚÙ‹ÛÂȘ „, h Î·È q Â›Ó·È ·ÓÙ›ıÂÙ˜ ÙˆÓ Û˘Ó·ÚÙ‹ÛÂˆÓ Ê, f Î·È g ·ÓÙÈÛÙÔ›¯ˆ˜, ÁÈ· Ó· ¯·Ú¿ÍÔ˘Ì ÙȘ ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ÙÔ˘˜ ·ÚΛ Ó· ·›ÚÓ·Ì ÙȘ Û˘ÌÌÂÙÚÈΤ˜ 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˘Ì ÙË ÁÚ·ÊÈ΋ ·- Ú¿ÛÙ·ÛË Ù˘ Ê(x) = 0,5x 2 , fiˆ˜ ÛÙËÓ ¿ÛÎËÛË 2. i). √È ÁÚ·ÊÈ- Τ˜ ·Ú¿ÛÙ·ÛÂȘ ÙˆÓ Û˘Ó·Ú- Ù‹ÛÂˆÓ f(x) = 0,5(x – 2) 2 Î·È g(x) = 0,5(x + 2) 2 ÚÔ·ÙÔ˘Ó ·fi ÔÚÈ˙fiÓÙȘ ÌÂÙ·ÙÔ›ÛÂȘ Ù˘ ·Ú·‚ÔÏ‹˜ y = 0,5x 2 , Ù˘ ÌÂÓ ÚÒÙ˘ ηٿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ‰ÂÍÈ¿, Ù˘ ‰Â ‰Â‡ÙÂÚ˘ ηٿ 2 ÌÔ- Ó¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿. ii) ÷ڿÛÛÔ˘Ì ÙË ÁÚ·ÊÈ΋ ·- Ú¿ÛÙ·ÛË Ù˘ „(x) = –0,5x 2 , fiˆ˜ ÛÙËÓ ¿ÛÎËÛË 2. ii). √È ÁÚ·ÊÈΤ˜ ·Ú·ÛÙ¿ÛÂȘ ÙˆÓ Û˘- Ó·ÚÙ‹ÛÂˆÓ h(x) = –0,5(x – 2) 2 Î·È q(x) = –0,5(x + 2) 2 ÚÔ- ·ÙÔ˘Ó ·fi ÔÚÈ˙fiÓÙȘ ÌÂ- Ù·ÙÔ›ÛÂȘ Ù˘ ·Ú·‚ÔÏ‹˜ y = –0,5x 2 , Ù˘ ÚÒÙ˘ ηٿ 2 ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ‰ÂÍÈ¿, Ù˘ ‰Â ‰Â‡ÙÂÚ˘ ηٿ ‰‡Ô ÌÔÓ¿‰Â˜ ÚÔ˜ Ù· ·ÚÈÛÙÂÚ¿.
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