Download Advanced Calculus (Solutions Manual) by Gerald B. Folland PDF

By Gerald B. Folland

This can be the whole strategies handbook to Gerald Folland's Advanced Calculus.

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C) The integrand equals ✓€ ➈ ✒ , which is roughly ✄✄✒ for near 1; hence converges. ✓€ ✒ ✄✄✒ ✏✓✽✩➇ ✟ ☞❊➇ ✟ ✡❲➇❴✒ ✜❀✄✳✏✓✽✩➇ ✓✾☞❊➇❧✡✕✏✓✒ ☛❈➇ ✎ ✓ ➇ (d) Converges since for near 0. ➇ ✏❝✍✪✸✻✺✜✼❬➇ ☛ ✟€ ➇ ✟ ✼ ✔ ✕ ✜ ✙▲➇ ☛ ☞ ✙▲➇❴✒ ✜ ✄ ☛✳✏✓✽✈✏✱❅▲➇ (e) Diverges by comparision to ➇ ☞❊➇ ✟ ✍⑥➇⑧✍❦✙✜✒ ✎ € ➇ ✟ and ✕ ➇✞✎✢✜ , so the integrand is ✒ ✪ ; hence diverges. ✂ ☎ ✯✎ near✓€ ✒ infinity; hence converges. € ✟ ❈➇ ✎ ✜ ☛ ☞✎✏❝✍✇➇❫✒ ✎ ✟ ✓✒ € (b) The integrand is near 0 and near 1; hence converges but ✝ ✟ diverges.

A) 2. ✝ have ✂ . Hence the arc length is ✸✻✺✜✼ ✟❴❛ ✂ ✆ ✒ ✟ ✏❑✡ ✸✻✺✜✼ ✟➃❛ ❛ ✄ ✼✙✔ ✕ ✟ ❛❫✡❲✸✻✺✜✼ ✟ ❛❫✡ ☞ ✙✑✍⑩ ❁ ❛ ✄ ✙■✼✄✔ ✕❤❛❜✒ ✌ ☞ ✙✑✍⑩✙■✼✄✔ ✕❤❛❜✒ ✌ ✒ ✒ ✎ ✆ ✟ ✎ ✆ ✟ ✒ ✒ ✆ ✟ ✧✗✧✗✧ ❛❝✄✳✙ ✆ ✟ ✧✗✧✗✧ ✂ ✙ ◗ ✂✲✄ € ☞ € ❃❦✍⑩❛❨✒ ❛❤✄ ✟€ ❃❦✍◗✙✄✂ ✍ ✙■✼✄✔ ✕✯❛✯✄✳✙ ✍◗✙■✸✻✺✜✼❾✙✄✂⑧✄ ✒ ✟ ✟ Let , so . Then ✝ ✎ ✆ ✟ ; ✌ ✝ ✌ ❁✼✄✔ ✕ ✟ ✂ ✸✻✺✜✼ ✟ ❛ ✄ ✄✼ ✔ ✕ ✟ ✄✙ ✂ ✄ ❁✄✼ ✔ ✕ ✟ ✂✬✸✻✺✜✼ ✟ ✂ ✞ ✄ ✙ ✆ ✒ ✟ ✛ ✏❑✡❲✸✻✺✜✼ ✟ ✂ ✂ ✄ , and , so ✝ ✌ ✙ ✆ ✒ ✟ ✆ ✙✘✍✪✼✄✔✖✕ ✟ ✂ ✂✦✄✕✙ ✜ ✒ ✟ ☞ ✙ ✎ ✓€ ✒ ✟ ✒ ✞◗✄ ✒ ✆ ✟ ✌ ✝ ✄ ✟ ✝ ✎ 4. 5. ✌ ✂✑✄ ✙■✼ ✆ ✁ ✘✏❑✡❲✏ ✼ ✟ ➇ ➇❋✄⑨✸✻✺✜✼✄ ❝➇ ➇ ✌ ➁ ✌ .

14. We have 15. 5 Functions Defined by Integrals ✄ € ➇ ➁ ☞♦✰✣➂❨➁❬✒ ✄ ✰ ➇✖✄ ❶ ✰ as✣ a function of for each , and ❵ . For fixed , one☞❊➇❄studies 1. 1. ❵ behavior of as as in Exercise 9, First one verifies (by l’Hˆ o that ✰ ➁ ✣ ✰✾✡ ✶ ❊ ☞ ❄ ➇ ❨ ➂ ❬ ➁ ❑ ✒ ✄ ✬ ✙ ✣ ❘ ❊ ☞ ❄ ➇ ❨ ➂ ➁ ✎ ✒ ✩ ✎ ✍ ➁ ❈ ✙ ✰ ■ ✙ ❘ ✖ ✗ as for every and (by induction) that ❽ ❵ ✶✥✤ ✰ ✗ ❘ ☞❊➇❄➂❨➁❬✒❨✽✩➁✮✣ ✰ for➁✮✣ ✰✾✡ where is a polynomial. induction again, it follows that for all ,❽ ❵ as , and hence ❘✗ ✆☛ € ☞❊➇❄➂❜By ✰✾✒ ❵ € exists and equals zero.

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