Exercise 6.1 [Ste10, p. 641] Eliminate the parameter to find a Cartesian equation of the curve. Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases.
(i) x = 1 − t2, y = t − 2, −2 ≤ t ≤ 2. (ii) x = t − 1, y = t3 + 1, −2 ≤ t ≤ 2.
(iii) x = sint, y = csct, 0 < t < π/2. (iv) x = tan2θ, y = secθ, −π/2 < θ < π/2.
Exercise 6.2 (4pts) [Ste10, p. 651] Find dy/dx and d2y/dx2. For which values of t is the curve convex?
(i) x = 2sint, y = 3cost, 0 < t < 2π. (ii) x = cos2t, y = cost, 0 < t < π.
Exercise 6.3 [Ste10, p. 651] Given the astroid x = acos3θ, y = asin3θ, a > 0, 0 ≤ θ < 2π.
(a) Find the area of the region enclosed by the astroid.
(b) Find the total length of the astroid.
Exercise 6.4 [Ste10, p. 651] The curvature at a point P of a curve is defined as
where ϕ is the angle of inclination of the tangent line at P. Thus the curvature is the absolute value of the rate of change of ϕ with respect to arc length.
(a) For a parametric curve x = x(t), y = y(t), show that
where the dots indicate derivatives with respect to t, i.e., x˙ = dx/dt.
(b) By regarding a curve y = f(x) as the parametric curve x = x, y = f(x), with parameter x, show that
Exercise 6.5 [Ste10, p. 664] Find the points on the given polar curve where the tangent line is horizontal or vertical.
(i) r = 1 + cosθ. (ii) r = eθ.
Exercise 6.6 [Ste10, p. 669] Find the area enclosed by the loop of the strophoid r = 2cosθ − secθ.
Exercise 6.7 [Ste10, p. 669] Find the area of the region that lies inside the first (polar) curve and outside the second (polar) curve.
(i) r = 2cosθ, r = 1. (ii) r = 1 − sinθ, r = 1.
Exercise 6.8 [Ste10, p. 669] Find the exact length of the polar curve.
(i) r = 2cosθ, 0 ≤ θ ≤ π. (ii) r = 5θ, 0 ≤ θ ≤ 2π. (iii) r = θ2, 0 ≤ θ ≤ 2π. (iv) r = 2(1 + cosθ). References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on page 1).
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