MATH
2011 &Icirc;›&Iuml;&Iuml;ƒ&Icirc;&micro;&Icirc;&sup1;&Iuml;‚ &Icirc;&pound;&Iuml;‡. &Icirc;&sup2;&I

# 5 h ëùôìâóë âíûˆûë âóè ùë ìôúê

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5. H ˙ËÙÔ‡ÌÂÓË ÂÍ›ÛˆÛË Â›Ó·È ÙË˜ ÌÔÚÊ‹˜ C = · Ø F + ‚ ÂÂÈ‰‹ ÙÔ ÓÂÚfi ·- ÁÒÓÂÈ ÛÙÔ˘˜ 0ÆC ‹ ÛÙÔ˘˜ 32ÆF, ı· ÈÛ¯‡ÂÈ 0 = · Ø 32 + ‚. (1) ∂ÂÈ‰‹, ÂÈÏ¤ÔÓ, ÙÔ ÓÂÚfi ‚Ú¿˙ÂÈ ÛÙÔ˘˜ 100ÆC ‹ ÛÙÔ˘˜ 212ÆF, ı· ÈÛ¯‡ÂÈ 100 = · Ø 212 + ‚. (2) ∞Ó ·Ê·ÈÚ¤ÛÔ˘ÌÂ Î·Ù¿ Ì¤ÏË ÙÈ˜ (1) Î·È (2) ‚Ú›ÛÎÔ˘ÌÂ 100 = · Ø 180, ÔfiÙÂ Î·È ÂÔÌ¤Óˆ˜ . ÕÚ·, Ë ˙ËÙÔ‡ÌÂÓË ÂÍ›ÛˆÛË Â›Ó·È ∞Ó ˘¿Ú¯ÂÈ ıÂÚÌÔÎÚ·Û›· Ô˘ Ó· ÂÎÊÚ¿˙ÂÙ·È Î·È ÛÙÈ˜ ‰‡Ô ÎÏ›Ì·ÎÂ˜ ÌÂ ÙÔÓ ·ÚÈıÌfi Δ, ÙfiÙÂ ı· ÈÛ¯‡ÂÈ Δ = (Δ – 32) 9Δ = 5Δ –5 Ø 32 4Δ = –5 Ø 32 Δ = –40. ÕÚ· ÔÈ –40ÆF ·ÓÙÈÛÙÔÈ¯Ô‡Ó ÛÙÔ˘˜ –40ÆC. 6. ∏ ÁÚ·ÊÈÎ‹ ·Ú¿ÛÙ·ÛË ÙË˜ f ·ÔÙÂÏÂ›Ù·È: ∞fi ÙÔ ÙÌ‹Ì· ÙË˜ Â˘ıÂ›·˜ y = –x + 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ÙÂÙÌËÌ¤ÓË x (– ∞, 0]. ∞fi ÙÔ ÙÌ‹Ì· ÙË˜ Â˘ıÂ›·˜ y = 2 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ÙÂÙÌËÌ¤ÓË x [ 0, 1] Î·È 5 9 C = 5 9 (F – 32). C = 5 9 F – 5 9 32 ‚ = – 5 9 32 · = 5 9 ∫∂º∞§∞π√ 6: μ∞™π∫∂™ ∂¡¡√π∂™ Δø¡ ™À¡∞ƒΔ∏™∂ø¡ 84

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∞fi ÙÔ ÙÌ‹Ì· ÙË˜ Â˘ıÂ›·˜ y = x + 1 ÙÔ˘ ÔÔ›Ô˘ Ù· ÛËÌÂ›· ¤¯Ô˘Ó ÙÂÙÌË- Ì¤ÓË x [1, + ∞). 7. i) OÈ Ú›˙Â˜ ÂÍ›ÛˆÛË˜ f(x) = 1 Â›Ó·È ÔÈ ÙÂÙÌËÌ¤ÓÂ˜ ÎÔÈÓÒÓ ÛËÌÂ›ˆÓ ÙË˜ y = f(x) Î·È ÙË˜ Â˘ıÂ›·˜ y = 1, ‰ËÏ·‰‹ ÔÈ ·ÚÈıÌÔ› –1 Î·È 1. √È Ú›˙Â˜ ÙË˜ ÂÍ›ÛˆÛË f(x) = x Â›Ó·È ÙÂÙÌËÌ¤ÓÂ˜ ÙˆÓ ÎÔÈÓÒÓ ÛËÌÂ›ˆÓ ÙË˜ y = f(x) Î·È ÙË˜ Â˘ıÂ›·˜ y = x, ‰ËÏ·‰‹ ÔÈ ·ÚÈıÌÔ› –2, 0 Î·È 1. ii) √È Ï‡ÛÂÈ˜ ÙË˜ ·Ó›ÛˆÛË˜ f(x) < 1 Â›Ó·È ÔÈ ÙÂÙÌËÌ¤ÓÂ˜ ÙˆÓ ÛËÌÂ›ˆÓ ÙË˜ y = f(x) Ù· ÔÔ›· ‚Ú›ÛÎÔÓÙ·È Î¿Ùˆ ·fi ÙËÓ Â˘ıÂ›· y = 1, ‰ËÏ·‰‹ ÔÈ ·ÚÈıÌÔ› x (– ∞, 1) – {–1}. √È Ï‡ÛÂÈ˜ ÙË˜ ·Ó›ÛˆÛË˜ f(x) ≥ x Â›Ó·È ÔÈ ÙÂÙÌËÌ¤ÓÂ˜ ÙˆÓ ÛËÌÂ›ˆÓ ÙË˜ y = f(x) Ù· ÔÔ›· ‚Ú›ÛÎÔÓÙ·È ¿Óˆ ·fi ÙËÓ Â˘ıÂ›· y = x ‹ ÛÙËÓ Â˘ıÂ›· ·˘Ù‹, ‰ËÏ·‰‹ Ù· ÛËÌÂ›· x [– 2, 0] [1, + ∞). 8. i) √È ÁÚ·ÊÈÎ¤˜ ·Ú·ÛÙ¿ÛÂÈ˜ ÙˆÓ f(x) = |x| Î·È g(x) = 1 ‰›ÓÔÓÙ·È ÛÙÔ ‰ÈÏ·Ófi Û¯‹Ì·. OÈ Ï‡ÛÂÈ˜ ÙË˜ ·Ó›ÛˆÛË˜ |x| ≤ 1 Â›Ó·È ÔÈ ÙÂÙÌËÌ¤ÓÂ˜ ÙˆÓ ÛËÌÂ›ˆÓ ÙË˜ y = |x| Ô˘ ‚Ú›ÛÎÔÓÙ·È Î¿Ùˆ ·fi ÙËÓ Â˘ıÂ›· y = 1 ‹ ÛÙËÓ Â˘ıÂ›· ·˘Ù‹, ‰ËÏ·‰‹ Ù· x [– 1, 1 ].
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