Question

Horizontal and vertical tangents Find the points at which the following polar curves have horizontal or vertical tangent lines.

1.. r = 4 cos e
2. r= 2 + 2 sin 0 1
3. r = sin 20 T
4.. r = 3 + 6 sin e

Answer 1

To find the points at which the given polar curves have horizontal or vertical tangent lines, differentiate the polar equation with respect to the angle and solve for the angles that result in a horizontal tangent or undefined first derivative. For the given polar curves, the points with horizontal tangents are (4, 0), (4, pi), (2, pi/2), (2, 3pi/2), (0, pi/40), (0, 3pi/40), ..., (0, 39pi/40), (3, pi/2), and (3, 3pi/2).

Step-by-step explanation:

To find the points at which the given polar curves have horizontal or vertical tangent lines, we need to differentiate the polar equation with respect to the angle θ. A horizontal tangent occurs when the first derivative of r with respect to θ is equal to 0, and a vertical tangent occurs when the first derivative of r with respect to θ is undefined. Let's find the points for each of the given polar curves:

1. r = 4cos(θ):
   First derivative: dr/dθ = -4sin(θ)
   Horizontal tangent: -4sin(θ) = 0 => θ = 0, π
   Vertical tangent: The first derivative is never undefined.
   Points with horizontal tangents: (4, 0), (4, π)
2. r = 2 + 2sin(θ):
   First derivative: dr/dθ = 2cos(θ)
   Horizontal tangent: 2cos(θ) = 0 => θ = π/2, 3π/2
   Vertical tangent: The first derivative is never undefined.
   Points with horizontal tangents: (2, π/2), (2, 3π/2)
3. r = sin(20θ):
   First derivative: dr/dθ = 20cos(20θ)
   Horizontal tangent: 20cos(20θ) = 0 => θ = π/40, 3π/40, 5π/40, ..., 39π/40
   Vertical tangent: The first derivative is never undefined.
   Points with horizontal tangents: (0, π/40), (0, 3π/40), (0, 5π/40), ..., (0, 39π/40)
4. r = 3 + 6sin(θ):
   First derivative: dr/dθ = 6cos(θ)
   Horizontal tangent: 6cos(θ) = 0 => θ = π/2, 3π/2
   Vertical tangent: The first derivative is never undefined.
   Points with horizontal tangents: (3, π/2), (3, 3π/2)