Howard, Anton - Cálculo - 6ª Edição - Exercicios Resolvidos

Howard, Anton - Cálculo - 6ª Edição - Exercicios Resolvidos

(Parte 1 de 6)

CHAPTER 1 Analytic Geometry in Calculus

EXERCISE SET 1.1 1.

cosθ sinθ

r = 2+2 sinθ

(b) From (8-9), symmetry about the y-axis and Theorem 1.1.1(b), the equation is of the form r = a ± bsinθ. The cartesian points (3,0) and (0,5) give a =3 and 5= a + b,s o b = 2 and r = 3+2 sinθ.

Exercise Set 1.1 449 21.

Line

Line (

Circle

Circle

2Cardioid 27. Circle

Cardioid 29.

Cardioid

10Cardioid

8 Cardioid

1 Limaçon

Cardioid

Limaçon

Limaçon

Limaçon 37.

Limaçon

Limaçon

Limaçon

7 Limaçon

Lemniscate

Lemniscate 43. Lemniscate

Spiral

Spiral

Spiral

Four-petal rose

Four-petal rose

Eight-petal rose

Three-petal rose 51.

58. In I, along the x-axis, x = r grows ever slower with θ.I n I x = r grows linearly with θ.

59. (a) r = a/cosθ, x = r cosθ = a, a family of vertical lines (b) r = b/sinθ, y = r sinθ = b, a family of horizontal lines

Exercise Set 1.1 451

curve if and only if r = f(θ − α).

is a circle of radius 12

dy/dθ = 0 if cosθ =1 /2o ri fc osθ = −1; θ = π/3o r π (or θ = −π/3, which leads to the minimum point).

An alternate proof follows directly from the Law of Cosines.

sinθ cosθ

sinθ sinθ

, so lim θ→0 y does not exist.

71. Note that r →± ∞ as θ approaches odd multiples of π/2; x = rcosθ = 4tanθcosθ = 4sinθ, y = rsinθ = 4tanθsinθ so x →± 4 and y →± ∞ as θ approaches

so x = 2 is a vertical asymptote.

73. Let r = asinnθ (the proof for r = acosnθ is similar). If θ starts at 0, then θ would have to increase by some positive integer multiple of π radians in order to reach the starting point and begin to retrace the curve. Let (r,θ) be the coordinates of a point P on the curve for 0 ≤ θ< 2π.N ow asinn(θ +2 π)= asin(nθ +2 πn)= asinnθ = r so P is reached again with coordinates (r,θ +2 π) thus the curve is traced out either exactly once or exactly twice for 0 ≤ θ< 2π. If for 0 ≤ θ< π, P(r,θ) is reached again with coordinates (−r,θ + π) then the curve is traced out exactly once for 0 ≤ θ< π, otherwise exactly once for 0 ≤ θ< 2π. But

{ asinnθ, n even

−asinnθ, n odd so the curve is traced out exactly once for 0 ≤ θ< 2π if n is even, and exactly once for 0 ≤ θ< π if n is odd.

EXERCISE SET 1.2

Exercise Set 1.2 453

(Parte 1 de 6)

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