Description
Note: Unless specified otherwise, you must show the details of your work via logical reasoning for each exercise. Simply writing a final result (whether correct or not) will receive 0 point.
Exercise 6.1 (8 pts) [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 = sin t, y = csc t, 0 < t < π/2. (iv) x = tan2 θ, y = sec θ, −π/2 < θ < π/2.
Exercise 6.2 (4 pts) [Ste10, p. 651] Find dy/ dx and d2y/ dx2. For which values of t is the curve convex? (i) x = 2 sin t, y = 3 cos t, 0 < t < 2π. (ii) x = cos 2t, y = cos t, 0 < t < π.
Exercise 6.3 (4pts) [Ste10, p. 651] Given the astroid x = acos3 θ, y = asin3 θ, a > 0, 0 ≤ θ < 2π. (a) (2 pts) Find the area of the region enclosed by the astroid.
(b) (2 pts) Find the total length of the astroid.
Exercise 6.4 (4 pts) [Ste10, p. 651] The curvature at a point P of a curve is defined as
d φ κ = d s
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) (2 pts) For a parametric curve x = x(t), y = y(t), show that κ = | x ̇ y ̈ − x ̈ y ̇ |
[ x ̇ 2 + y ̇ 2 ] 3 / 2
where the dots indicate derivatives with respect to t, i.e., x ̇ = dx/ dt.
(b) (2 pts) By regarding a curve y = f (x) as the parametric curve x = x, y = f (x), with parameter x, show that d2y/ dx2
κ= [1+(dy/dx)2]3/2
Exercise 6.5 (2 pts) [Ste10, p. 664] Find the points on the given polar curve where the tangent line is horizontal or
vertical.
(i) r = 1 + cos θ. (ii) r = eθ .
Exercise 6.6 (2 pts) [Ste10, p. 669] Find the area enclosed by the loop of the strophoid r = 2 cos θ − sec θ. Exercise 6.7 (2 pts) [Ste10, p. 669] Find the area of the region that lies inside the first (polar) curve and outside
the second (polar) curve.
(i) r = 2 cos θ, r = 1. (ii) r = 1 − sin θ, r = 1.
Exercise 6.8 (4 pts) [Ste10, p. 669] Find the exact length of the polar curve.
(i) r = 2 cos θ, 0 ≤ θ ≤ π. (ii) r = 5θ, 0 ≤ θ ≤ 2π. (iii) r = θ2, 0 ≤ θ ≤ 2π. (iv) r = 2(1 + cos θ).
References
[Ste10] J. Stewart. Calculus: Early Transcendentals. 7th ed. Cengage Learning, 2010 (Cited on page 1).
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