Final answer:
To find the slope of the tangent line to the polar curve at the specified point, first find the Cartesian coordinates at that point using the equations x = r * cos(theta) and y = r * sin(theta). Then, differentiate the polar curve equation with respect to theta to find the rate of change of r with respect to theta. Substitute the value of theta into the derivative equation to find the slope of the tangent line.
Step-by-step explanation:
To find the slope of the tangent line to the polar curve at the specified point, we can use the relationship between polar and Cartesian coordinates. The Cartesian coordinates (x, y) can be derived from polar coordinates using the equations:
x = r * cos(theta)
y = r * sin(theta)
First, let's find the Cartesian coordinates at the point specified by theta = pi:
Substituting theta = pi into the equations, we get:
x = (1/pi) * cos(pi) = -1/pi
y = (1/pi) * sin(pi) = 0
So, the point on the polar curve corresponding to theta = pi is (-1/pi, 0)
The slope of the tangent line to the curve at this point can be found using the derivative of the polar curve equation. The derivative of r(theta) with respect to theta gives us the rate of change of r with respect to theta.
Let's differentiate r = 1/theta with respect to theta:
dr/dtheta = -1/theta^2
Now, let's substitute theta = pi into the derivative equation to find the slope:
slope = dr/dtheta(θ = pi) = -1/(pi^2)
Therefore, the slope of the tangent line to the polar curve at the point specified by theta = pi is -1/(pi^2).