Final answer:
To find the exact y-coordinate of the highest points on the curve described by r=sin(2θ), convert the equation to Cartesian coordinates, differentiate, find critical points, and use these to calculate the y-coordinates.
Step-by-step explanation:
The problem requires us to use calculus to determine the exact y-coordinate of the highest points on a polar curve given by r = sin(2θ). To find the y-coordinate of the highest points, we need to do the following:
- Convert the polar equation to Cartesian coordinates using the relationship y = r sin(θ).
- Differentiate the resulting Cartesian equation with respect to θ to get dy/dθ.
- Identify the critical points by setting dy/dθ equal to zero and solving for θ.
- Determine which of these critical points correspond to maximum values of y by using the second derivative test or by considering the nature of the function.
- Substitute these θ values back into the original polar equation to find the corresponding r values, then calculate the y-coordinates with y = r sin(θ).
It is important to note that the y-coordinate in polar coordinates is associated with the sine function and the radial distance from the origin.