2011 Λύσεις Σχ. β&I

Fiïèô õìâû úôîùâè fiùè ùô μ âóè

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™¯fiÏÈÔ : ÕÌÂÛ· ÚÔ·ÙÂÈ fiÙÈ ÙÔ ∞μ°¢ Â›Ó·È ÚfiÌ‚Ô˜, ·ÊÔ‡ ÔÈ ‰È·ÁÒÓȘ ÙÔ˘ Ù¤ÌÓÔÓÙ·È Î¿ıÂÙ· Î·È ‰È¯ÔÙÔÌÔ‡ÓÙ·È. 7. ¶Ú¤ÂÈ i) f(2) = 6 2 2 + k = 6 k = 2. ii) g(–2) = 8 k(–2) 3 = 8 k = –1. iii) h(3) = 8 k k = 4. 8. i) ΔÔ ‰›Ô ÔÚÈÛÌÔ‡ Ù˘ f Â›Ó·È fiÏÔ ÙÔ . ñ °È· y = 0 ¤¯Ô˘Ì x = 4, ofiÙÂ Ë y = f(x) Ù¤ÌÓÂÈ ÙÔÓ xãx ÛÙÔ ÛËÌÂ›Ô ∞(4, 0). ñ °È· x = 0 ¤¯Ô˘Ì y = –4, ÔfiÙÂ Ë y = f(x) Ù¤ÌÓÂÈ ÙÔÓ yãy ÛÙÔ ÛËÌÂ›Ô μ(0, –4). √ÌÔ›ˆ˜ ii) ∏ g ¤¯ÂÈ ‰›Ô ÔÚÈÛÌÔ‡ fiÏÔ ÙÔ Î·È Ù¤ÌÓÂÈ ñ ÙÔÓ ¿ÍÔÓ· xãx ÛÙ· ÛËÌ›· ∞ 1 (2, 0) Î·È ∞ 2 (3, 0) Î·È ñ ÙÔÓ ¿ÍÔÓ· yãy ÛÙ· ÛËÌ›· B(0, 6). iii) H h ¤¯ÂÈ ‰›Ô ÔÚÈÛÌÔ‡ fiÏÔ ÙÔ Î·È ñ ¤¯ÂÈ Ì ÙÔÓ ¿ÍÔÓ· xãx ÎÔÈÓfi ÛËÌÂ›Ô ÙÔ ∞(1, 0). ñ Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· yãy ÛÙÔ ÛËÌÂ›Ô ÙÔ B(0, 1). iv) H q ¤¯ÂÈ ‰›Ô ÔÚÈÛÌÔ‡ fiÏÔ ÙÔ Î·È ñ ‰ÂÓ ¤¯ÂÈ ÎÔÈÓ¿ ÛËÌ›· Ì ÙÔÓ ¿ÍÔÓ· xãx. ñ Ù¤ÌÓÂÈ ÙÔÓ ¿ÍÔÓ· yãy ÛÙÔ ÛËÌÂ›Ô μ(0, 1). v) H Ê ¤¯ÂÈ ‰›Ô ÔÚÈÛÌÔ‡ ÙÔ Û‡ÓÔÏÔ [1, + ), ÔfiÙ ñ ¤¯ÂÈ Ì ÙÔÓ ¿ÍÔÓ· xãx ¤Ó· ÌfiÓÔ ÎÔÈÓfi ÛËÌÂ›Ô ÙÔ ∞(1, 0) Î·È ñ ‰ÂÓ ¤¯ÂÈ ÎÔÈÓ¿ ÛËÌ›· Ì ÙÔÓ ¿ÍÔÓ· yãy. vi) H „ ¤¯ÂÈ ‰›Ô ÔÚÈÛÌÔ‡ ÙÔ Û‡ÓÔÏÔ (– , –2] [2, + ), ÔfiÙ ñ ¤¯ÂÈ Ì ÙÔÓ ¿ÍÔÓ· xãx ‰‡Ô ÎÔÈÓ¿ ÛËÌ›·, Ù· ∞ 1 (–2, 0) Î·È ∞ 2 (2, 0). ñ ‰ÂÓ ¤¯ÂÈ ÎÔÈÓ¿ ÛËÌ›· Ì ÙÔÓ ¿ÍÔÓ· yãy. 4 = 8 (¢A) = (2 + 1) 2 + (5 – 1) 2 = 5. (°¢) = (–1 –2) 2 + (1 + 3) 2 = 5. (B°) = (2 – 5) 2 + (–3 –1) 2 = 5. ∫∂º∞§∞π√ 6: μ∞™π∫∂™ ∂¡¡√π∂™ Δø¡ ™À¡∞ƒΔ∏™∂ø¡ 82
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9. i) °È· x = 0 ¤¯Ô˘Ì f(0) = –1. ÕÚ· Ë C f Ù¤ÌÓÂÈ ÙÔÓ yãy ÛÙÔ ÛËÌÂ›Ô ∞(0, –1). °È· y = 0 ¤¯Ô˘Ì x 2 – 1 = 0 x = –1 x = 1. ÕÚ· Ë C f Ù¤ÌÓÂÈ ÙÔÓ xãx ÛÙ· ÛËÌ›· μ 1 (–1, 0) Î·È μ 2 (1, 0). ii) f(x) > 0 x 2 – 1 > 0 (x + 1)(x – 1) > 0 x < – 1 x > 1. 10. i) f(x) = g(x) x 2 – 5x + 4 = 2x – 6 x 2 – 7x + 10 = 0 x = ÕÚ· x = 5 x = 2. °È· x = 2, g(2) = 4 – 6 = –2. °È· x = 5, g(5) = 4. ÕÚ· Ù· ÎÔÈÓ¿ ÛËÌ›· ÙˆÓ C f Î·È C g Â›Ó·È Ù· ∞(2, –2) Î·È μ(5, 4). ii) f(x) < g(x) x 2 – 5x + 4 < 2x – 6 x 2 – 7x + 10 < 0 (x – 2)(x – 5) < 0 2 < x < 5. ¨ 6.3. ∏ Û˘Ó¿ÚÙËÛË f(x) = ·x + ‚ ∞ã √ª∞¢∞™ 1. Ÿˆ˜ Â›Ó·È ÁÓˆÛÙfi, ÁÈ· ÙÔ Û˘ÓÙÂÏÂÛÙ‹ ‰È‡ı˘ÓÛ˘ Ù˘ ¢ı›·˜ y = ·x + ‚ ÈÛ¯‡ÂÈ: · = Âʈ, fiÔ˘ ˆ Â›Ó·È Ë ÁˆÓ›· Ô˘ Û¯ËÌ·Ù›˙ÂÈ Ë y = ·x + ‚ Ì ÙÔÓ ¿ÍÔÓ· xãx. ∂Ô̤ӈ˜, ı· ¤¯Ô˘Ì i) Âʈ = 1, ÔfiÙ ˆ = 45Æ.
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