Question

What is the slope of the line tangent to the polar curve r = 2 cos θ − 1 at the point where θ = π?

Answer 1

To find the slope of the tangent line to a polar curve at a specific point, one must convert the polar equation into Cartesian coordinates, differentiate, and evaluate the derivative at the given point.

Step-by-step explanation:

To find the slope of the line tangent to the polar curve r = 2 cos θ − 1 at the point where θ = π, we first convert our polar equation to Cartesian coordinates and then differentiate it with respect to x. Using polar to Cartesian conversions (x = r cos(θ), y = r sin(θ)), we get new equations for x and y based on the given r. Then we find dy/dx and evaluate it at the angle θ = π to find the slope at this point.

Here is a step-by-step explanation:

1. Express the polar equation in terms of x and y.
2. Differentiate both x and y with respect to θ.
3. Calculate dy/dx by dividing the derivative of y by the derivative of x.
4. Substitute θ = π into dy/dx to find the slope of the tangent line.

Answer 2

The slope of the line tangent to the polar curve r = 2cos(θ) - 1 at the point where θ = π is 0.

Step-by-step explanation:

The slope of a line tangent to a polar curve can be found by taking the derivative of the polar equation with respect to θ and then evaluating it at the given value of θ. In this case, the polar equation is r = 2cos(θ) - 1. Taking the derivative of this equation with respect to θ gives:

dr/dθ = -2sin(θ)

Substituting θ = π into the derivative, we get:

dr/dθ = -2sin(π) = 0

Therefore, the slope of the line tangent to the polar curve r = 2cos(θ) - 1 at the point where θ = π is 0.