Vv156 Honors Calculus II  Assignment 6

Vv156 Honors Calculus II 
Assignment 6

This assignment has a total of (30 points).
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.
x = 1 − t
2
(i) , y = t − 2, −2 ≤ t ≤ 2. x = t − 1, y = t
3
(ii) + 1, −2 ≤ t ≤ 2.
(iii) x = sin t, y = csc t, 0 < t < π/2. x = tan2
(iv) θ, y = sec θ, −π/2 < θ < π/2.
Exercise 6.2 (4 pts) [Ste10, p. 651] Find dy/ dx and d
2
y/ dx
2
. For which values of t is the curve convex?
(i) x = 2 sin t, y = 3 cost, 0 < t < 2π. (ii) x = cos 2t, y = cost, 0 < t < π.
Exercise 6.3 (4 pts) [Ste10, p. 651] Given the astroid x = a cos3
θ, y = a sin3
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
κ =
 
 
 
 
dϕ
ds
 
 
 
 
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
κ =
 
  d
2y/ dx
2
 
 
[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. r = e
θ
(ii) .
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. r = 5θ
(ii) , 0 ≤ θ ≤ 2π. r = θ
2
(iii) , 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).
Page 1 of 1