Find the slope of the tangent line to the graph of the polar equation at the point specified by the value of 𝜃.
r = 1/𝜃
𝜃 = 𝜋
dy/dx = (dr/d𝜃) / (dx/d𝜃) = (r' sin𝜃 + r cos𝜃) / (r' cos𝜃 - r sin𝜃)
= (-sin𝜃/𝜃^2 + cos𝜃/𝜃) / (-cos𝜃/𝜃^2 - sin𝜃/𝜃)
= (-sin𝜃 + r cos𝜃)/(-cos𝜃 - r sin𝜃) if 𝜃≠0
at 𝜃=𝜋, that means
dy/dx = -𝜋
oops. did you catch my typo?
To find the slope of the tangent line to the graph of the polar equation at the given point, we need to find the derivative of the polar equation with respect to θ, and then substitute the value of θ to find the slope.
1. First, let's find the derivative of the polar equation r = 1/θ with respect to θ. To do this, we need to convert the polar equation into Cartesian coordinates using the following relations:
x = r * cos(θ)
y = r * sin(θ)
Substituting r = 1/θ in these equations, we get:
x = (1/θ) * cos(θ)
y = (1/θ) * sin(θ)
2. Now, differentiate both x and y with respect to θ using the chain rule:
dx/dθ = -(1/θ^2) * cos(θ) + (1/θ) * (-sin(θ))
dy/dθ = -(1/θ^2) * sin(θ) + (1/θ) * cos(θ)
3. Therefore, the derivative of the polar equation r = 1/θ with respect to θ is given by:
dr/dθ = sqrt[(dx/dθ)^2 + (dy/dθ)^2]
= sqrt[(-(1/θ^2) * cos(θ) + (1/θ) * (-sin(θ)))^2 + (-(1/θ^2) * sin(θ) + (1/θ) * cos(θ))^2]
4. Now, substitute the value of θ = π into the derivative equation to find the slope of the tangent line at the point (x, y):
slope = dr/dθ (θ = π)
Compute the derivative and evaluate it at θ = π to find the slope.
By following these steps, you should be able to find the slope of the tangent line to the graph of the polar equation at the specified point.