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).

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