Lezione 5 - Esercizi e Piano Indefinito PDF

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>ĞnjŝŽŶĞ ŶĞϱ ϱ ϱͲ Ͳ ƐĞƌĐŝnjŝĞ ĞW W WŝŝĂŶŽ//Ŷ Ŷ Ě Ğ Ĩŝ Ŷ ŝ ƚ Ž ĚŽŵĞŶŝĐĂϰŽƚƚŽďƌĞϮϬϮϬ

ϭϲϮϵ

തࡱതത૙തሺ૙ሻ ൌǫ

ƐĞƌĐŝnjŝŽϭ

݀‫ܧ‬ଶ ൌ ݀‫ܧ‬ଵ

തതതோത ൌ ݀‫ܧ‬ തതതଵ ൅ ݀‫ܧ‬ തതതଶ ݀‫ܧ‬ ߙൌ

ߨ െߠ ʹ

݀ ݀‫ܧ‬ோ ൌ ʹ‫ܧ‬ଵ  ߙ ൌ ʹ ݈݀ ൌ ܴ ݀ߠ గோ

‫ܧ‬ோ ൌ න ݀‫ܧ‬ோ ݈݀ ൌ න ଴

ത‫ܧ‬തതோത ൌ െ

ߣ

ߤෞ ௬

ʹߨߝ଴ ܴ

଴

ߨ ߣ݈݀ ߣ݈݀ ߣ݈݀  ൬ െ ߠ൰ൌ ʹ  ߠ  ߙ ൌ ʹ Ͷߨߝ଴ ܴ ଶ Ͷߨߝ଴ ܴ ଶ ʹ Ͷߨߝ଴ ܴ ଶ

గோ

గ

గ ߣ ʹߣ݈݀ ଶ ʹ ߣ ܴ  ߠ ߣ ଶ ൌ ݀ߠ ൌ න ሾ ߠሿ  ߠ ൌ ଴ ଶ ʹߨߝ ܴ Ͷߨߝ଴ ܴ ଶ Ͷߨߝ ܴ ʹߨߝ ଴ ଴ ଴ܴ ଴

ƐĞƌĐŝnjŝŽϮ ďďŝĂŵŽƵŶĂŶĞůůŽĞƵŶĨŝůŽǣ ܿ ݁݅ ‫ܨ‬ത ൌǫ ൌ ݂‫݋݂݈݅ ݈ݑݏ ݁ݐܽ݉݁݊ݒݏݏ݌݉݋ܿ ݏ ݄ܿ݁ ܽݖݎ݋‬ ݈

ത ሺ‫ ݖ‬ሻ ൌ ‫ܧ‬

ߣଵ ʹ ߨ ܴ ‫ݖ‬

ෞ௭ ଷߤ

Ͷߨߝ଴ሺܴ ଶ ൅ ‫ ݖ‬ଶሻଶ

YƵĞƐƚŽƌĂƉƉƌĞƐĞŶƚĂŝůĐĂŵƉŽĞůĞƚƚƌŝĐŽƉƌŽĚŽƚƚŽĚĂůůΖĂŶĞůůŽĐŚĞĂŐŝƐĐĞƐƵƵŶŐĞŶĞƌŝĐŽĞůĞŵĞŶƚŽ ŝŶĨŝŶŝƚĞƐŝŵŽĚĞůĨŝůŽ

݀ ‫ ݍ‬ൌ ߣଶ ݀‫ݖ‬

݀‫ܨ‬ത തതത ‫ܧ‬଴ ሺ‫ݖ‬ሻ ൌ ݀ ‫ݍ‬

ஶ ݀ ‫ ܨ‬ൌන ‫ ܨ‬ൌ න ஶ

଴

ƐĞƌĐŝnjŝŽϯ

଴

ߣଵ ܴ ‫ݖ‬

ʹߝ଴ሺܴ ଶ ൅ ‫ ݖ‬ଶሻ

݀ തതത଴ ሺ‫ݖ‬ሻ ݀‫ܨ‬ത ൌ ‫ܧ ݍ‬଴തതതሺ‫ݖ‬ሻ ฺ ݀‫ܨ‬ത ‫ܧ צ‬

ߣଵ ߣଶ ܴ න ଷ ߣଶ ݀‫ ݖ‬ൌ ʹߝ ଴ ଶ

‫ݖ‬

ଷ ሺܴ ଶ ൅ ‫ ݖ‬ଶሻ ଶ

ͳ ߣଵ ߣଶ ܴ ൌ ൥െ ൩ ʹߝ଴ ଶ ඥሺܴ ൅ ‫ ݖ‬ଶሻ

ஶ ଴

ൌ

ߣଵ ߣଶ ʹߝ଴

ďďŝĂŵŽƵŶĚŝƐĐŽĞƵŶĨŝůŽǣ ݁ ܽ ܿ ݁݅ ݅ ത ሺ‫ ݖ‬ሻ ൌ ǫ ൌ ‫݋݌ܥ‬ ‫ܧ‬ ݈݁݁‫݋݂݈݅ ݈ݑݏ ݁ݐܽ݉݁݊ݒݏݏ݌݉݋ܿ ݏ ݄ܿ݁ ݋ݎݐݐ‬ ݈ ݀ ܿ ܿ ݀ ‫ݍ‬ ܳ ݅ ݅ ݅ ߪ ൌ ‫ݐݏ‬ ‫ ݎ ݀݅ ݎ݌ݑݏ‬ൌ ณ ൌ ܵ݀ ௦௘ ௨௡௜௙௢௥௠௘ ܵ ݂ ݈

WƌĞŶĚŝĂŵŽůĂƐƵƉĞƌĨŝĐŝĞĚĞůĚŝƐĐŽĞůĞĚŝǀŝĚŝĂŵŽŝŶƚĂŶƚĞƐƵƉĞƌĨŝĐŝĞůĞŵĞŶƚĂƌŝĂĨŽƌŵĂĚŝĂŶĞůůŽ ݀ ĐƵŝƌĂŐŐŝŽŝŶƚĞƌŶŽƐĂƌă‫ ݎ‬ĞƋƵĞůůŽĞƐƚĞƌŶŽƐĂƌă‫ ݎ‬൅ ݀ ‫ ݎ‬ĞĐŚĞƉƌĞƐĞŶƚĞƌăƵŶĂĐĞƌƚĂĐĂƌŝĐĂ‫ݍ‬Ǥ YƵŝŶĚŝƐŝĂŵŽŝŶŐƌĂĚŽĚŝĐĂůĐŽůĂƌĞŝůĐĂŵƉŽĞůĞƚƚƌŝĐŽŐĞŶĞƌĂƚŽĚĂƋƵĞƐƚĂƐƵƉĞƌĨŝĐŝĞŝŶĨŝŶŝƚĞƐŝŵĂ ŝŶƋƵĂŶƚŽğƉĂƌŝĂƋƵĞůůŽĚŝƵŶĂŶĞůůŽǀŝƐƚŽĐŚĞ݀ ‫ ݎ‬ğŵŽůƚŽƉŝĐĐŽůŽ

݀ ‫ ܧ‬ሺ‫ݖ‬ሻ ൌ

݀ ‫ݖݍ‬

ଷ

Ͷߨߝ଴ሺ ‫ ݎ‬ଶ ൅ ‫ ݖ‬ଶሻଶ ݀

ோ

‫ܧ‬ሺ ‫ ݖ‬ሻ ൌ න‫ ܧ‬ሺ‫ݖ‬ሻ ൌ න

଴

݀ ‫ݖݍ‬

ෞ௭ ଷߤ

Ͷߨߝ଴ሺ‫ ݎ‬ଶ ൅ ‫ ݖ‬ଶ ሻଶ

݀

݀ ᇱ ଶ ‫ͳ ۓ‬ሻሾ‫ ݕ‬ൌ ݂ሺ‫ ݔ‬ሻ ฺ ݀‫ݕ‬ ݀ ൌ ݂ ሺ‫ݔ‬ሻ݀‫ ݔ ฺ ݔ‬՜ ‫ ݔ‬൅݀݀‫ ݔ‬ሿ‫݅݊݀݅ݑݍ‬ ݀ଶ ܵ ൌ ߨ‫ ݀ܵ ฺ ݎ‬ൌ ʹߨ‫ݎ‬ ݀ ‫ݎ‬ ۖ ʹሻ ܵ݀ ൌ ߨሺ‫ ݎ‬൅ ‫ ݎ‬ሻ ଶ െ ߨ‫ ݎ‬ଶ ൌ ߨ ‫ ݎ‬ଶ ൅ ʹ‫ ݎ ݎ‬൅ ‫ݎ‬ต ൌ ʹߨ‫ݎ ݎ‬ ቆ ቇ ௧௘௥௠௜௡௘ ܵ݀ ൌ െ ‫ݎ‬ଶ ‫۔‬ ݁ ݁ ݁ ݀ ܽ ᇱ ͵ሻ ܵ‫݋݈݃݅ܽ݉݋ݒ‬ ݅݀ ݈ܽ‫ݎ ܽݖݖ݁ݐ‬ ܽ ݈ ݈݈݁݊ܽ‫݋‬ǡ ‫ݎݐ݁݊ݒ݀݅ ݋ݐݏܽݑݍ‬ ‫݋݈݋ܽ݊݃ݐݐݎ ݊ݑ‬ ௧௥௔௦௖௨௥௔௕௜௟௘ ۖ ݁ ݁ ܽ ݀ ᇱ ‫ݎߨʹ ݏ‬ǡ ‫ݎݏ ݈ܽ݁ݐ݋ݐ ݎ ݈ ݅݊݀݅ݑݍ‬ǣ ܵ݀ ൌ ʹߨ‫ݎ ݎ‬ ‫ە‬ ܾ ݀ ‫ݖݍ‬

ோ

‫ܧ‬ሺ ‫ ݖ‬ሻ ൌ න തതത ‫ܧ‬଴ ሺ‫ݖ‬ሻ ൌ

ෞ௭ ଷߤ ଴ Ͷߨߝ ሺ ଶ ଶ ଴ ‫ ݎ‬൅ ‫ ݖ‬ሻଶ ߪ

ʹߝ଴ ቈ

ͳെ

‫ݖ‬

ோ

ൌන

ߤ௭ ଵ቉ ෞ ሺ ܴ ଶ ൅ ‫ ݖ‬ଶሻ ଶ

ߪ ʹߨ‫ݎ‬

ଷ ଴ Ͷߨߝ ሺ‫ ݎ‬ଶ ൅ ‫ ݖ‬ଶ ሻଶ ଴

݀ ‫ݎ‬ൌ න

ோ

ߪ‫ݖݎ‬

݀

ଷ‫ݎ‬ ଴ ʹߝ଴ሺ‫ ݎ‬ଶ ൅ ‫ ݖ‬ଶ ሻଶ

ൌ

ߪ ‫ݖ‬

െ ʹߝ଴ ൥

ͳ

ோ

ଵ൩ ሺ‫ ݎ‬ଶ ൅ ‫ ݖ‬ଶሻଶ ଴

ൌ

ߪ ‫ͳ ݖ‬ െ ʹߝ ൥ ‫ݖ‬ ଴

ͳ

ଵ൩

ሺܴ ଶ ൅ ‫ ݖ‬ଶሻଶ

YƵĞƐƚĂĞƐƉƌĞƐƐŝŽŶĞŚĂƵŶΖŝŶƚĞƌĞƐƐĞƉƌĂƚŝĐŽŝŶƚĞƌĞƐƐĂŶƚĞƐĞܴ ൐൐ ‫ ݖ‬ŽƉƵƌĞܴ ฺ λ ŝůĨĂƚƚŽĐŚĞůĂƐŽƌŐĞŶƚĞĚŝƉĂƌƚĞŶnjĂĨŽƐƐĞƵŶĚŝƐĐŽƉĞƌĚĞĚŝƐŝŐŶŝĨŝĐĂƚŽ YƵĞƐƚŽĚŝƐĐŽƌƐŽğƉŽƐƐŝďŝůĞĞƐƚĞŶĚĞƌůŽĂƚƵƚƚĞůĞƐƵƉĞƌĨŝĐŝĐŚĞƐŝĞƐƚĞŶĚŽŶŽŝŶŵŽĚŽŝŶĚĞĨŝŶŝƚŽĞĐŚĞƐŝĂŶŽƵŶŝĨŽƌŵĞŵĞŶƚĞĐĂƌŝĐŚĞŽǀǀĞƌŽƋƵĞƐƚŽ ĚŝƐĐŽƌƐŽğǀĂůŝĚŽƉĞƌŝWŝĂŶŝ/ŶĚĞĨŝŶŝƚŝ YƵŝŶĚŝƉĞƌܴ ฺ λǣ ࣌

തതതത ࡱ૙ ሺࢠሻ ൌ

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