Evolutes, Envelopes and Asymptotes - One Line Questions
1.
For the curve y = x^3, the center of curvature at x=0 is: —
(0, 0)
2.
For the curve y = x^3 - x, what is the limit of y/x as x approaches infinity? —
∞
3.
The radius of curvature of y = x^2 at (0,0) is: —
1/2
4.
The envelope of the family of lines x cos α + y sin α = p (constant) is: —
A circle
5.
Which of the following is NOT a property of asymptotes? —
A curve can intersect its asymptote multiple times.
6.
If a curve has a vertical asymptote at x = a, it means that as x approaches a, the value of y tends to: —
Infinity (positive or negative)
7.
If a curve has a horizontal asymptote at y = b, it means that as x tends to infinity (positive or negative), the value of y tends to: —
The value of b
8.
What is the envelope of the family of ellipses x^2/a^2 + y^2/b^2 = 1, where a + b = k (a constant)? —
A parabola
9.
The evolute of the cardioid r = a(1 - cos θ) is: —
A nephroid
10.
The evolute of a parabola is a: —
Parabola
11.
Which curve has the evolute as itself? —
Straight line
12.
The evolute of a cycloid is a: —
Cycloid
13.
The evolute of an ellipse is a: —
Astroid
14.
What is the locus of the centers of curvature of a given curve called? —
Evolute
15.
An asymptote is a line such that the distance between the curve and the line approaches zero as they tend to infinity. This statement defines: —
Asymptote
16.
An envelope of a family of curves is a curve that is tangent to each member of the family. This statement defines: —
Envelope
17.
The envelope of a family of curves F(x, y, a) = 0 is found by solving F = 0 and ∂F/∂a = 0 for x and y in terms of a, and then eliminating a. This is the method for finding the: —
Envelope
18.
An asymptote is a tangent to the curve at infinity. This is a conceptual definition of: —
Asymptote
19.
If the family of curves depends on two parameters, say F(x, y, a, b) = 0, the envelope is found by solving F=0 and ∂F/∂a=0 (or ∂F/∂b=0) and eliminating the parameter. This is a method for finding the: —
Envelope
20.
For a curve y = f(x), the center of curvature (h, k) is given by: —
h = x + y'(1 + y'^2)/y'' , k = y - (1 + y'^2)/y''
21.
The center of curvature (h, k) for a curve defined parametrically x = x(t), y = y(t) is: —
h = x - y'(x'^2 + y'^2)/(x'y'' - y'x''), k = y + x'(x'^2 + y'^2)/(x'y'' - y'x'')
22.
For a rational function f(x) = P(x)/Q(x), if the degree of P(x) is one greater than the degree of Q(x), the curve has a: —
Oblique (slant) asymptote
23.
Which of the following is a property of the evolute? —
It is the locus of the centers of curvature.
24.
What is the equation of the tangent to the evolute of a curve at a point? —
It is the normal to the original curve at the corresponding point.
25.
After finding m, c for the oblique asymptote y = mx + c is calculated as: —
lim (x->inf) (y - mx)
26.
To find the oblique asymptote y = mx + c for a curve y = f(x), m is calculated as: —
lim (x->inf) y/x
27.
The evolute of a curve is the envelope of its: —
Normals
28.
For the curve y = e^x, what is the nature of its asymptotes? —
One horizontal asymptote
29.
The envelope of the family of lines y = mx + c, where c = -m^2, is a: —
Parabola
30.
The evolute of a curve is also known as the locus of the: —
Centers of the osculating circles
31.
The evolute of a curve is the locus of the intersection of two adjacent: —
Normals
32.
The evolute of the curve x = a cos^3 t, y = a sin^3 t (astroid) is: —
Another astroid (scaled and rotated)
33.
The evolute of the circle x^2 + y^2 = a^2 is: —
The origin (0,0)
34.
The envelope of the family of chords of an ellipse that are normal to the ellipse is: —
The evolute of the ellipse
35.
What is the relationship between the evolute and the involute of a curve? —
The involute of the evolute is the original curve.
36.
Consider the family of circles (x-a)^2 + y^2 = r^2. The envelope of this family (for fixed r) is: —
The lines y = r and y = -r
37.
Consider the family of lines tangent to the parabola y^2 = 4ax. The envelope of this family is: —
The parabola itself
38.
The evolute of the curve r = e^(aθ) (logarithmic spiral) is: —
The same logarithmic spiral
39.
The envelope of the family of circles passing through the origin and having their centers on the line y = x is: —
The line y = x
40.
The envelope of the family of circles with radius r and center on the x-axis is: —
The lines y = r and y = -r
41.
What is the relationship between the radius of curvature and the distance from the center of curvature to the curve? —
They are equal.
42.
The curve y = ln(x) has an asymptote at: —
x = 0
43.
The curve xy = c^2 has asymptotes at: —
x = 0 and y = 0
44.
The evolute of the rectangular hyperbola xy = c^2 is: —
(x^2 + y^2)^2 = 4c^2(x^2 - y^2)
45.
The equation of the evolute of the curve x = a(cos t + t sin t), y = a(sin t - t cos t) is: —
x^2 + y^2 = a^2
46.
The asymptotes of the hyperbola x^2/a^2 - y^2/b^2 = 1 are given by: —
y = ±(b/a)x
47.
The asymptote of the curve y = sqrt(x^2 + 1) as x approaches infinity is: —
y = x
48.
The curve y = x + 1/x has an oblique asymptote. What is it? —
y = x
49.
The radius of curvature (ρ) for a curve y = f(x) is given by: —
ρ = (1 + y'^2)^3/2 / |y''|
50.
If a curve is represented parametrically by x = x(t) and y = y(t), the radius of curvature is given by: —
ρ = (x'^2 + y'^2)^3/2 / |x'y'' - y'x''|