3 \u03c0 4 1292 \u03c0 \u03c0 2 1 1294 a e 2 \u03c0 c 2 \u03c0 ln 2 1 2 1296 4 3 1297 a 0968 b 7199 1298

3 π 4 1292 π π 2 1 1294 a e 2 π c 2 π ln 2 1 2

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3 π 4 1292. π ( π 2 + 1) 1294. (a) ( e 2) π (c) 2 π (ln 2 1) 2 1296. 4 3 1297. (a) 0.968 (b) 7.199 1298. (a) 7 B (b) 3 4 1309. 3 2 1311. 1 1312. d.n.e. 1314. 1 1315. e 1316. 1 1317. 1 1318. e 1319. e 1320. 1 1321. 1 1322. 0 1323. ln 2 1324. 1 1325. 1 1326. 1 1327. 27 10 1344. 1 1347. divergent 1348. 1 1349. 1 4 1351. ln 4 1352. π 1353. 2 π 2 1354. ln 2 1359. 0.5 ml 1360. A 1361. (b) negative (c) 1 2 ln 3 1362. B 1364. only one is divergent 1365. only one is false 1366. (b) 3 . 75( x 1 . 4) = y 2 . 6 (d) 2.2 1367. (a) 1 (b) 2, 6, 24 1372. (a) 13 6 (b) use the Inter- mediate Value Theorem (c) y = 21 2 3 x (d) 21 6 1373. (a) 1 (b) π 2 (c) π 1374. 3 4 1375. 0.726 1376. (a) x ( t ) = a a akt + 1 (b) x = a (1 e ( a b ) kt ) b ae ( a b ) kt 1377. a = 1 2 , converges to 1 4 ln 2 1378. (a) π (b) π (2 e 5) (c) π 2 ( e 2 3) (d) π 2 ( e 2 4 e +5) 1379. (b) 8 π 27 (9(ln 2) 2 6 ln 2+2) 1380. (a) x = n ( e ( n +1) kt 1) n + e ( n +1) kt 1381. (a) π 1382. 100 π 1384. (a) 1 (b) 2 π (1 e k ke k ) (c) 1 2 (1 e 2 k ) 1386. C 1387. C 1388. B 1389. D 1390. C
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338 The AP CALCULUS PROBLEM BOOK 1391. E 1392. E 1393. E 1394. A 1395. A 1396. D 1397. B 1398. A 1399. C 1400. A 1401. C 1402. D 1403. D 1410. 0 1411. 0 1412. 1 1413. 1 1414. 0 1415. divergent 1416. e 2 / 3 1417. 1 1418. π 2 1419. 1 1420. unbounded, divergent 1421. bounded, convergent 1426. 1 3 1427. divergent 1428. divergent 1429. divergent 1430. 0 1431. e 1432. 0 1433. divergent 1434. divergent 1435. 0 1436. 5 1437. 0 1438. increasing 1439. increasing 1440. decreasing 1441. increasing 1443. oscillating 1447. 1 1448. three are false 1458. 1 < x < 1 1459. 1 < x < 1 1463. x < 1 and x > 1 1464. 1 < x < 5 1465. 1 e < x < e 1466. 23 99 1469. 140 99 1470. 41333 33300 1471. 22 7 1472. 28 1473. 8 1477. divergent 1478. divergent 1479. divergent 1485. divergent 1486. divergent 1487. divergent 1488. convergent 1489. divergent 1490. convergent 1491. divergent 1492. convergent 1493. convergent 1494. a = 1 1501. 5 6 1502. converges absolutely 1503. converges absolutely 1504. diverges 1505. converges conditionally 1506. converges condintionally 1507. converges absolutely 1508. converges absolutely 1509. diverges 1510. converges conditionally 1511. diverges 1513. 0.2 1514. 0.00001 1529. 1 x 1 1530. 1 x 1 1531. 1 4 x < 1 4 1532. 2 < x 4 1533. 2 < x 8 1534. 1 e x < 1 e 1535. x = 0 1537. 1 < x < 3, 4 4 ( x 1) 2 1538. 1 e < x < e , 1 1 ln x 1546. (b) ( 1) n +1 ( 1 3 ) n ( x 3) n 1 (c) 0 < x < 6 1547. (b) ( 1) n +1 ( x π 2 ) 2 n 1 (2 n 1)! (c) R 1550. (b) ( 1) n π 2 n ( x 1 2 ) 2 n (2 n )! (c) R 1552. (b) e 2 ( x 2) n n ! (c) R 1555. 1 + x 2 x 2 8 + x 3 16 5 x 4 128 + − · · · 1557. divergent 1558. 9 + 5 x 2 x 2 + 6 x 3 1559. 1 + 1 2 (3 x π ) 2 1560. 1 1 8 (4 x π ) 2 1561. (a) 1 < x < 1 (b) 1 + x 2 x 2 8 + x 3 16 + · · · 1572. first and last; second and third 1573. a = 0 1574. G (0) = i , G ( π 3 ) = 1 2 i + 3 2 j 1575. || F || = | t | t 2 + 4, || G || = 1
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APPENDIX C. ANSWERS 339 1578. yes, for t = 0 1585. (a) ( 2 t, 9 t 2 ) (b) ( 2 , 18 t ) (c) ( 2 , 9 ) (d) 85 1588. (a) ( π cos πt, π sin πt ) (b) (− π 2 sin πt, π 2 cos πt ) (c) (− π, 0 ) (d) π 1591. 3 i + (4 2 2) j 1593. R ( t ) = [( t + 1) 3 / 2 1] i ( e t 1) j 1594. R ( t ) = (8 t +100) i +(8 t 16 t 2 ) j 1595. 3 13 1596. 3 π 1597. 1 27 (85 3 / 2 72 3 / 2 ) 1598. 4 13 1599. (a) ( 3 sin t ) i + (2 sin t ) j (b) ( 3 cos t ) i (2 sin t ) j (c) 2 3 (d) x 2 9 + y 2 4 = 1 (e) 2 3 ( x 3 2 2) = y 2 1600. (a) R ( t ) = i + 18 t j , R ′′ ( t ) = 18 j (c) y = 18 x 13 1601. (a) 1 3(1 + 1 2 3) (b) (2 3 4)( x π ) = y 2 3 1605.
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