2011 Λύσεις Σχ. β&I

Œùûè fi ùè èûfiùëùâ 1 úôîùâè fiùè

This preview shows 15 out of 17 pages.

ŒÙÛÈ, ·fi ÙȘ ÈÛfiÙËÙ˜ (1) ÚÔ·ÙÂÈ fiÙÈ · = ‚ = Á. ™¯fiÏÈÔ: √ Û˘ÁÎÂÎÚÈ̤ÓÔ˜ ÙÚfiÔ˜ ÌÔÚ› Ó· ÂÊ·ÚÌÔÛı› Î·È fiÙ·Ó Ù· ·, ‚, Á Â›Ó·È ÔÔÈÔȉ‹ÔÙ Ú·ÁÌ·ÙÈÎÔ› ·ÚÈıÌÔ›, ‰È·ÊÔÚÂÙÈÎÔ› ÙÔ˘ ÌˉÂÓfi˜, ÂÓÒ ÁÈ· ÙÔ˘˜ ‰‡Ô ÚÒÙÔ˘˜ ÙÚfiÔ˘˜ ··ÈÙÂ›Ù·È ÛÙËÓ ÂÚ›ÙˆÛË ·˘Ù‹ Ó· ·Ô‰ÂȯÙ› fiÙÈ · + ‚ + Á ≠ 0. · = Á = Á · = k, · = Á = Á · = · + ‚ + Á ‚ + Á + · = 1, = x 2 – xy + y 2 x – y x – y x 2 – xy + y 2 = 1 x 3 + y 3 x 2 – y 2 : x 2 x – y – y = (x + y) (x 2 – xy + y 2 ) (x – y) (x + y) : x 2 – xy + y 2 x – y = x + y x – y 1 y + x xy = x + y x – y xy x + y = xy x – y x + y x – y x –1 – y –1 x –2 – y –2 = x + y x – y 1 x 1 y 1 x 2 1 y 2 = x + y x – y 1 x 1 y 1 x 1 y 1 x + 1 y 2.1. √È Ú¿ÍÂȘ Î·È ÔÈ È‰ÈfiÙËÙ¤˜ ÙÔ˘˜ 15
Image of page 15

Subscribe to view the full document.

ii) ·ã ÙÚfiÔ˜: Œ¯Ô˘Ì · – ‚ = ‚ – Á (1) Î·È · – ‚ = Á – · (2), ÔfiÙÂ, ·Ó ÚÔÛı¤ÙÔ˘Ì ηٿ ̤ÏË ÙȘ ÈÛfiÙËÙ˜ (1) Î·È (2) ÚÔ·ÙÂÈ fiÙÈ 2· – 2‚ = ‚ – · 3· = 3‚ · = ‚ ŒÙÛÈ, ·fi ÙËÓ ÈÛfiÙËÙ· (1) ‚Ú›ÛÎÔ˘Ì fiÙÈ Î·È ‚ = Á. ÕÚ· · = ‚ = Á ÔfiÙ ÙÔ ÙÚ›ÁˆÓÔ Â›Ó·È ÈÛfiÏ¢ÚÔ. ‚ã ÙÚfiÔ˜: £¤ÙÔ˘Ì · – ‚ = ‚ – Á = Á – · = k, ÔfiÙ ¤¯Ô˘Ì · – ‚ = k, ‚ – Á = k Î·È Á – · = k (2) ∞Ó ÙÒÚ· ÚÔÛı¤ÛÔ˘Ì ηٿ ̤ÏË ÙȘ ÈÛfiÙËÙ˜ (2), ‚Ú›ÛÎÔ˘Ì fiÙÈ k = 0, ÔfiÙÂ, ÏfiÁˆ ÙˆÓ ÈÛÔÙ‹ÙˆÓ ·˘ÙÒÓ, Â›Ó·È · = ‚ = Á Î·È ¿Ú· ÙÔ ÙÚ›ÁˆÓÔ Â›- Ó·È ÈÛfiÏ¢ÚÔ. 6. ∞Ó x Î·È y Â›Ó·È ÔÈ ‰È·ÛÙ¿ÛÂȘ ÙÔ˘ ÔÚıÔÁˆÓ›Ô˘, ÙfiÙ ı· ÈÛ¯‡ÂÈ L = 2x + 2y Î·È ∂ = xy ÔfiÙÂ, ÏfiÁˆ Ù˘ ˘fiıÂÛ˘, ı· ¤¯Ô˘Ì 2x + 2y = 4· Î·È xy = · 2 Î·È ¿Ú· y = 2· – x (1) Î·È xy = · 2 (2) §fiÁˆ Ù˘ (1), Ë (2) ÁÚ¿ÊÂÙ·È ÈÛÔ‰‡Ó·Ì·: x(2· – x) = · 2 2·x – x 2 = · 2 x 2 – 2·x + · 2 = 0 (x – ·) 2 = 0 x – · = 0 x = · ŒÙÛÈ ·fi ÙËÓ (1) ¤¯Ô˘Ì fiÙÈ Î·È y = · Î·È ¿Ú· ÙÔ ÔÚıÔÁÒÓÈÔ Â›Ó·È ÙÂÙÚ¿- ÁˆÓÔ. 7. £· ÂÚÁ·Ûıԇ̠̠ÙË Ì¤ıÔ‰Ô Ù˘ ··ÁˆÁ‹˜ Û ¿ÙÔÔ. i) ∞˜ ˘Ôı¤ÛÔ˘Ì fiÙÈ · + ‚ = Á . TfiÙ ı· Â›Ó·È ‚ = Á – · (ˆ˜ ‰È·ÊÔÚ¿ ÚËÙÒÓ), Ô˘ Â›Ó·È ¿ÙÔÔ. ii) ∞˜ ˘Ôı¤ÛÔ˘Ì fiÙÈ ·‚ = Á . TfiÙ ı· Â›Ó·È ‚ = Á · (ˆ˜ ËÏ›ÎÔ ÚËÙÒÓ), Ô˘ Â›Ó·È ¿ÙÔÔ.
Image of page 16
Image of page 17
You've reached the end of this preview.
  • Winter '09
  • Nikos

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern