SalasSV_08_09_ex_ans

# SalasSV_08_09_ex_ans - A-64 ANSWERS TO ODD-NUMBERED...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: A-64 ANSWERS TO ODD-NUMBERED EXERCISES 1 dx = 2 cos x du and the result follows. 1 − u2 du where u = cos x. The result follows. 1 − u2 x 37. Let u = tan . Then 2 39. csc x dx = sin x dx = sin2 x x +C 2 43. sin x dx = − 1 − cos2 x −2 +C 1 + tanh (x/2) 41. 2 tan−1 tanh SECTION 8.7 1. (a) 506 (b) 650 5. (a) π ∼ 3. 1312 = (c) 572 (d) 578 (e) 576 3. (a) 1.394 (b) 1.7915 (b) 0.9122 (c) 1.8090 (c) 1.1776 (d) 1.1533 (e) 1.1614 (b) π ∼ 3. 1416 = 7. (a) 1.8440 9. (a) 0.8818 (b) 0.8821 11. Such a curve passes through the three points (a1 , b1 ), (a2 , b2 ), (a3 , b3 ) iff b1 = a2 A + a1 B + C , 1 which happens iff A= b1 (a2 − a3 ) − b2 (a1 − a3 ) + b3 (a1 − a2 ) , (a1 − a3 )(a1 − a2 )(a2 − a3 ) B=− b1 (a2 − a2 ) − b2 (a2 − a2 ) + b3 (a2 − a2 ) 2 3 1 3 1 2 , (a1 − a3 )(a1 − a2 )(a2 − a3 ) b2 = a2 A + a2 B + C , 2 b3 = a2 A + a3 B + C , 3 C= a2 (a2 b3 − a3 b2 ) − a2 (a1 b3 − a3 b1 ) + a2 (a1 b2 − a2 b1 ) 1 2 3 . (a1 − a3 )(a1 − a2 )(a2 − a3 ) (b) n ≥ 2 15. (a) n ≥ 238 (b) n ≥ 10 17. (a) n ≥ 51 (b) n ≥ 4 19. (a) n ≥ 37 (b) n ≥ 3 13. (a) n ≥ 8 21. (a) 78 (b) 7 1 0 23. f (4) (x) = 0 for all x; therefore by (8.7.3) the theoretical error is zero 1 1 3 T −= = E2 8 3 24 (b) S1 − 0 b a 1 25. (a) T2 − x2 dx = x4 dx = 1 1 5 S −= = E1 24 5 120 29. (a) 49. 4578 (b) 1280.56 31. error ≤ 4. 01 × 10−7 27. Using the hint, Mn = area ABCD = area AEFD ≤ 1 f (x) dx ≤ Tn . 33. 0 4 dx = 4 tan−1 x 1 + x2 1 0 =4 π 4 −0 =π 3. 14159 (a) 3.14141 (b) 3.14159 SECTION 8.8 1. y1 is; y2 is not 11. y = x + C e2x 23. y = 2 e−x + x − 1 3. y1 and y2 are solutions 13. y = 2 3 5. y1 and y2 are solutions 15. y = C ee x 7. y = − 1 + C e2x 2 9. y = 2 2 5 + C e−(5/2)x 21. y = C (x + 1)−2 nx + Cx4 17. y = 1 + C (e−x + 1) 27. y = x2 (ex − e) 19. y = e−x 12 x 2 +C 25. y = e−x ln(1 + ex ) + e − ln 2 29. y = C1 ex + C2 x ex 41. (a) 200 ( 4 )t /5 5 (b) 200 2 ( 4 )t /25 5 35. T (1) ∼ 40. 10◦ ; 1. 62 min = dP = k (M − P ) dt (b) P (t ) = M (1 − e−0.0357t ) (c) 65 days 37. (a) v(t ) = 32 (1 − e−kt ) k (b) 1 − e−kt < 1; e−kt → 0 as t → ∞ 39. (a) i(t ) = E [1 − e−(R/L)t ] R E (b) i(t ) → (amps) as t → ∞ R L (c) t = ln 10 seconds R 43. (a) liters 45. (a) P (t ) = 1000 e( sin 2π t )/π (b) P (t ) = 2000 e( sin 2π t )/π − 1000 SECTION 8.9 1. y = C e−(1/2) cos (2x+3) 9. ln | y + 1| + 3. x4 + 2 =C y2 5. y sin y + cos y = − cos 1 x +C √ 7. e−y = ex − xex + C 1 − x2 15. y + ln | y| = x3 −x−5 3 1 = ln | ln x| + C y+1 11. y2 = C ( ln x)2 − 1 13. sin−1 y = 1 − ANSWERS TO ODD-NUMBERED EXERCISES 17. x2 1 + x + ln ( y2 + 1) − tan−1 y = 4 2 2 19. y = ln 3e2x − 2 23. x2 − y2 = C y y= x y 3 2 A-65 21. y = 3 x + C 2 25. y2 = −2x + C y x2 – y 2 = 1 y 2 = –2 x y = ex xy = 1 1 x xy = –1 x x y=– 2 3 y 2 – x2 = 1 x 27. A differential equation for the given family is y2 = 2xyy + y2 (y )2 . Now replace y by − which simpliﬁes to y2 = 2xyy + y2 ( y )2 . Thus the given family is self-orthogonal. 29. (a) C (t ) = 31. (a) v(t ) = 33. (a) y(t ) = kA2 t 0 1 + kA0 t α c e(α/m)t −β (b) C (t ) = A0 B0 (ekA0 t − ekB0 t ) A0 ekA0 t − B0 ekB0 t (b) v(t ) = 1 2xy y2 . Then the resulting differential equation is y2 = − + , y y (y )2 , where C is an arbitrary constant. α v0 α v0 e−(a/m)t = (α/m)t − β v ( α + β v0 ) e α + β v0 − β v0 e−(α/m)t 0 (b) v(t ) = 15. 65(1 + 0. 70 e 1 − 0. 70 e−1.25t −1.25t (c) lim v(t ) = 0 t →∞ 25, 000 , y(20) ∼ 1544 = 1 + 249 e−0.1398t (b) 40 days 35. (a) 88. 82 m/sec ) (c) 15.65 m/sec CHAPTER 9 SECTION 9.1 1. (a) 2 13 (b) 29 13 3. (0, 1) is the closest point ( − 1, 1) the farthest away 5. 17 2 7. Adjust the sign of A and B so that the equation reads Ax + By = |C |. Then we have B |C | A +y√ =√ . x√ A2 + B2 A2 + B 2 A2 + B 2 Now set A = cos α , √ A2 + B 2 B = sin α , √ A2 + B2 |C | = p. √ A2 + B2 p is the length of OQ, the distance between the line and the origin; α is the angle from the positive x-axis to the line segment OQ. y Q P α O x 9. y2 = 8x y 11. (x + 1)2 = −12(y − 3) y (–1,3) 13. 4y = (x − 1)2 y 15. ( y − 1)2 = −2(x − 3 ) 2 y ( , 1) 3 2 x x (1,0) x x ...
View Full Document

• Spring '10
• SMITH
• Prime number, xy, Pallavolo Modena, Sisley Volley Treviso, Associazione Sportiva Volley Lube, Piemonte Volley

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern