Advanced Calculus (Solutions Manual) by Gerald B. Folland

By Gerald B. Folland

This can be the entire options handbook to Gerald Folland's Advanced Calculus.

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Then ❽ € ✂ ✂ ✟ ✄❈❛ ✙ ✎✢✜ ❍☞❊➇ ✍◆➁❬✒ ✟ ☞❊➁✫✒ ➁✘✡❲❛ ☞✎✍ ✏✓✒ ✎ ✟ ✙ ☞❊➁❬✒ ➁ ❽☎ ✍ ❵ ✌ ✍ ❵ ✌ € € ✎ ✟ ✄ ★▲❛ ✂ ☞❊➇ ✍✇➁✫✒ ✟ ✎✢✜ ☞❊➁✫✒ ➁❧✍ ✙▲❛ ✂ ☞❊➁✫✒ ✯➁ ✄ ✍ ❵ ✌ ❵ ✌ ✍ (b) 2. 3. 4. 5. 6. 7. 6. Improper Integrals 31 € ✚ ✄ ✎ € ☞❊➁❬✒ ➁ ✡✥✙▲❛ ✟ € ✎☞ ✍✑✏✓✒ ✎ ✟ ☞❊➁✫✒ ➁ ✍ ❵ ✌ ✍ ❵ ✌ , so On the other hand, ❽ ✝ ✝ € € € ✟✚ ✄❈❛ ✂ ☞✎✍✑✏✓✒ ✎ ✟ ✙▲❛ ➁ ✍ ▼❛ ✂ ✎ ✟ ☞❊➁❬✒ ➁❧⑩ ✍ ✙▲❛ ✟ ✂ ✎☞ ✍❀✙✜✒ ✎✢✜ ☞ ✙▲❛❜✒ ☞❊➁✫✒ ➁ ❽ ✍ ✌ ❵ ✌ ✍ ❵ ✌ ✍ € € ✎✢✜ ✎ ✟ ✄ ★▲❛ ✜ ✂ ☞❊➁✫✒ ➁⑧✍⑩❅▲❛ ✂ ☞❊➁✫✒ ➁✯✄ ❵ ✌ ❵ ✌ ✍ ✍ € ✎ ✟ ☞❊➁❬✒ ➁ ✟☎ ✡ ✚ ✟ ✎✢✜ ★▲❛ ✎ ✟ Thus, in ❽ ❽ , the terms involving ✍ add up to ✝ ✍ ❵ ✌ , and the terms involving ✍ add up to the negative of this quantity.

Near✓€ ✒ infinity; hence converges. € ✟ ❈➇ ✎ ✜ ☛ ☞✎✏❝✍✇➇❫✒ ✎ ✟ ✓✒ € (b) The integrand is near 0 and near 1; hence converges but ✝ ✟ diverges. ✝ € ✓✒ € ✟ ☎ ☎ ➇ ✂ ✍t✏ ☛✌➇ ❫ ➇ ✣ ✽ ☞ ✂ t ✍ ✓ ✏ ✒ ✌ ☛ ➇ ✎ ➇ ✛ (c) For near✒ 0, , so and ✝ converges. For large, the integrand is ✂ ✎ ☎ ✟ ; hence ✝ € ✒ converges. less than €✓✒ ✟ € ☛✳✍✑✏✓✽✩➇ ➇ ✓✒ € €✟ (d) The integrand is for near 0; hence ✝ diverges. ) ➇ ✎ ✓€ ✒☎✄ ✄✼ ✔ ✕✂➇ ✎ € ☛q ➇ ✎ ✓€ ✒☎✄ ➇ ➇✡✎ €✓✒☎✄ ✼✄✔ ✕❤➇ ✎ € ☛❞➇ ✎✔✓ ✒☎✄ ➇ 3. (a) The integrand is comparable to €✓✒ ☛ (e) verges.

It☞ ✄ follows ✟ ✒ . On that ☛ , and taking the supremum over all gives ✙ € ✙ ✟ ✞ ✕ ☞ ✄ ❯ ✒ ✤ ✞❀☞ ✄ ❯ ✒❄✍ ✌❧✄✳✏✠➂✹✙ ✞✭☞ ✄ ✒ ✤ ✙ ✰ the hand,✘ ☞ given that for ✞ ☞ other ✄ ✒☛✄ ✞ ✄✤€✮✒❙✡ ✞ ☞ ,✄ choose ✒✟ ✤ ✞✭☞ ✄✤and €✹✒❄✡ ✞❀☞ so ✄ ✒✟ ■✍ ✙ ☛ ✞❀☞ ✄✲✒ ✤ ✞✭☞ ;✄✭then € ✒❙✡ ✞❀☞ ✄ ✟ ✒ ☛ ☛ ☛✖ . Since is arbitrary, , ✝ ✎ 6. ); ➂✱✍❦is✼✄✔ oriented ✕✂❛❨✒ ✁ ✧ ✁ then✄ ■☞❊❛❨✒✭✄ ☞❊❛ ➂❨❛ ✟ ✒ ✍✤✙★q✚❛✑q✳✙ ✁ ✌ ✁❋by✄ ✟ ✟ ☞❊❛ ✠➂❨❛ ✒❫✧✩☞✎,✏✠➂✹✙▲❛❨✒ ✌ ❛❴✡ ✟ ✍ ✟ ✒❁✄ ✓. € ☞❊➇ ✂ ☎✄✖ ✡ ☞♦✼ ✯❃❴➇❴✒✎✙▲➇❫✒ ➇ ✄ ☎ ✌ (d) Parametrize the parabolic portion of ⑩✧ ☛●☞❊❛❜✒☛✄♠☞✎✍✯❛ ➂✚ ☎✒ ✍❀✙❧q✥❛✂qt✙ , .

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