Find the points on the given curve where the tangent line is horizontal or vertical. (assume 0 ≤ θ < π. enter your answers as a comma-separated list of ordered pairs.) r = 6 cos(θ)

We have two points with horizontal tangents at (6, 0) and (-6, π).

Here we have the equation:

r = 6*cos(θ).

We want to find the points, of the form (r, θ), such that the tangent line is horizontal or vertical.

If we differentiate, we will get:

r' = -6*sin(θ).

If r' is 0, then we have a horizontal tangent, this happens for:

θ = 0 and θ = π

And if r' tends to infinnity, then we have a vertical line, we can see that it does not happen.

Then we only have two horizontal lines when θ = 0 and θ = π, evaluating that in our function we get:

r = 6*Cos(0) = 6

r = 6*cos(π) = -6

So the two points are (6, 0) and (-6, π)

If you want to learn more about differentiation, you can read:

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The given equation describes a circle of radius 3 centered at (x, y) = (3, 0). The critical points in (x, y) coordinates are

... (x, y) = {(0, 0), (3, -3), (6, 0), (3, 3)}

In (r, θ) coordinates, they are

... (r, θ) = {(6, 0), (3√2, π/4), (0, π/2), (-3√2, 3π/4)}