Polar Curves and Tangents - One Line Questions
1.
For the circle r = a cos θ, find dr/dθ. —
-a sin θ
2.
For the curve r = 2 + cos θ, find dr/dθ. —
-sin θ
3.
For a polar curve r = f(θ), what is the formula for the slope dy/dx of the tangent line in terms of r and θ? —
(r' sin θ + r cos θ) / (r' cos θ - r sin θ)
4.
For the circle r = a cos θ, find the slope dy/dx at θ = 0. —
Undefined
5.
For the circle r = a sin θ, find the slope dy/dx at θ = π/2. —
Undefined
6.
For the curve r = 2 + cos θ, find the slope of the tangent at θ = π/2. —
-1
7.
For the curve r = 2 + cos θ, find the slope of the tangent at θ = 0. —
-1
8.
For the curve r = sin(2θ) (a four-petal rose), what is the slope of the tangent at θ = π/4? —
Undefined
9.
For the curve r = sin(2θ), what is the slope of the tangent at θ = 0? —
0
10.
For the cardioid r = a(1 - cos θ), find the slope dy/dx at θ = π/2. —
0
11.
For the curve r = 2 sin θ, find the slope of the tangent at θ = π/6. —
√3
12.
For the polar curve r = aθ (spiral of Archimedes), find dr/dθ. —
a
13.
The equation θ = α in polar coordinates represents which shape? —
A straight line through the origin
14.
What does the polar curve r = a(1 - cos θ) represent? —
A cardioid
15.
The polar curve r = a sec θ represents: —
A vertical line
16.
The polar curve r = a csc θ represents: —
A horizontal line
17.
For the circle r = a sin θ, find dr/dθ. —
a cos θ
18.
For the polar curve r = e^(aθ) (logarithmic spiral), find dr/dθ. —
a e^(aθ)
19.
What does the polar curve r = a cos θ represent? —
A circle passing through the origin
20.
What does the polar curve r = a sin θ represent? —
A circle passing through the origin
21.
For the cardioid r = a(1 - cos θ), find dr/dθ. —
a sin θ
22.
The equation r = a in polar coordinates represents which shape? —
A circle centered at the origin
23.
The polar curve r² = a² cos(2θ) is known as: —
Lemniscate of Bernoulli
24.
Consider the lemniscate r² = a² cos(2θ). What is the condition for the tangent to be perpendicular to the polar axis? —
cos(2θ) = 0
25.
Consider the lemniscate r² = a² cos(2θ). What is the condition for the tangent to be parallel to the polar axis? —
sin(2θ) = 0
26.
The condition r' cos θ - r sin θ = 0 for a polar curve means the tangent is: —
Parallel to the polar axis
27.
The condition r' sin θ + r cos θ = 0 for a polar curve means the tangent is: —
Perpendicular to the polar axis
28.
What is the polar equation of the tangent to the cardioid r = 1 - cos θ at θ = π/2? —
r = 1
29.
What is the polar equation of the tangent to the circle r = a cos θ at the point (a/2, π/3)? —
r = a cos(θ - π/3)
30.
The tangent to the polar curve r = a sin³θ is given by: —
r = a sin³θ sec(θ - θ₀)
31.
What is the equation of the tangent to the curve r = e^(aθ) at θ = 0? —
r = sec(θ)
32.
What is the equation of the tangent line to the polar curve r = f(θ) at the point (r₀, θ₀)? —
r = r₀ cos(θ - θ₀)
33.
What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) for r? —
r = sqrt(x^2 + y^2)
34.
What is the condition for a tangent to a polar curve r = f(θ) to be perpendicular to the initial line (polar axis)? —
r' sin θ + r cos θ = 0
35.
What is the condition for a tangent to a polar curve r = f(θ) to be parallel to the initial line (polar axis)? —
r' cos θ - r sin θ = 0
36.
What is the polar form of a complex number z = x + iy? —
r(cos θ + i sin θ)
37.
What is the angle (φ) between the radius vector and the tangent to the polar curve r = f(θ)? —
tan φ = r / (dr/dθ)
38.
What is the angle between the tangent and the initial line for r = aθ at θ = π/2? —
tan⁻¹(π/2)
39.
In polar coordinates (r, θ), what does 'r' represent? —
The distance from the origin
40.
In polar coordinates (r, θ), what does 'θ' represent? —
The angle from the positive x-axis
41.
In the formula for dy/dx of a polar curve, what does r' represent? —
The derivative of r with respect to θ (dr/dθ)
42.
If tan φ = 0 for a polar curve, what is the relationship between the radius vector and the tangent? —
The tangent is parallel to the radius vector.
43.
If tan φ is undefined for a polar curve, what is the relationship between the radius vector and the tangent? —
The tangent is perpendicular to the radius vector.
44.
A tangent to a polar curve is a line that intersects the curve at a point and has the same direction as the curve at that point. —
True
45.
The tangent to the polar curve r = f(θ) at the origin (r=0) is given by the line θ = θ₀, where θ₀ is the smallest positive value for which f(θ₀) = 0. —
True
46.
What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ)? —
x = r cos θ, y = r sin θ
47.
At what point on the cardioid r = 1 + cos θ is the tangent perpendicular to the polar axis? —
θ = π
48.
At what point on the cardioid r = 1 + cos θ is the tangent parallel to the polar axis? —
θ = 0
49.
What is the relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) for θ? —
θ = tan⁻¹(y/x)