Question

Find the slope of the tangent line to the given polar curve at the point specified by the value of θ. r = 6 cos (θ), θ = π/3.

Answer 1

To find the slope of the tangent line to the given polar curve at the point specified by the value of θ, first find the derivative of the polar curve with respect to θ. Substitute the value of θ into the derivative equation to calculate the slope of the tangent line. The slope of the tangent line to the polar curve r = 6 cos (θ) at θ = π/3 is -3√3.

Step-by-step explanation:

To find the slope of the tangent line to the given polar curve at the point specified by the value of θ, we can first find the derivative of the polar curve with respect to θ.

For the polar curve r = 6 cos (θ), the derivative is given by dr/dθ = -6 sin (θ).

Next, we can calculate the slope of the tangent line by substituting the value of θ into the derivative equation.

In this case, θ = π/3. So, substituting θ = π/3 into dr/dθ = -6 sin (θ), we get the slope of the tangent line as -6 sin (π/3) = -6 * (√3/2) = -3√3.

Therefore, the slope of the tangent line to the polar curve r = 6 cos (θ) at θ = π/3 is -3√3.

Answer 2

To find the slope of the tangent line to the given polar curve at the point specified by θ = π/3, we can find the derivative of the polar equation with respect to θ and evaluate it at the specified value.

Step-by-step explanation:

To find the slope of the tangent line to the polar curve at a given point, we need to determine the derivative of the polar equation with respect to θ and evaluate it at the specified value of θ. The given polar equation is r = 6 cos(θ) and the specified value of θ is π/3. To find the derivative, we can use the chain rule:

r' = -6 sin(θ)

Now, we can substitute π/3 into the derivative to find the slope of the tangent line:

r'(π/3) = -6 sin(π/3) = -6(√3/2) = -3√3

Therefore, the slope of the tangent line to the polar curve at the point specified by θ = π/3 is -3√3.