Question

Use calculus to determine the exact y-coordinate of the highest points on the curve r=sin(2θ). y-coordinate of highest point:

Answer 1

To find the exact y-coordinate of the highest points on the given curve, we can use calculus. By finding the derivative of the curve, setting it equal to 0, and solving for the corresponding values of θ, we can determine the y-coordinate. In this case, the highest point on the curve has a y-coordinate of 1.

Step-by-step explanation:

Solution:

To determine the exact y-coordinate of the highest points on the curve r = sin(2θ), we need to find the maximum value of the y-coordinate.

1. First, find the derivative of r with respect to θ. The derivative of sin(2θ) is 2cos(2θ).
2. Set the derivative equal to 0 and solve for θ. 2cos(2θ) = 0 → cos(2θ) = 0 → 2θ = π/2 +nπ, where n is an integer.
3. Divide both sides by 2 to solve for θ. θ = π/4 + nπ/2.
4. Now, substitute the values of θ into the equation r = sin(2θ) to find the corresponding y-coordinates. For example, when θ = π/4, y = sin(2(π/4)) = sin(π/2) = 1.

Therefore, the y-coordinate of the highest point on the curve r = sin(2θ) is 1.

Answer 2

The highest points on the curve r=sin(2θ) occur when the sine function reaches its maximum value of 1. The y-coordinate of these highest points can be found by converting to Cartesian coordinates and is √2/2.

Step-by-step explanation:

To find the exact y-coordinate of the highest points on the polar curve r=sin(2θ), we first acknowledge that the polar equation relates the radius r to the angle θ. The maximum value of r, and thus the farthest point from the pole (origin), occurs when sin(2θ) is at its maximum, which is 1. Consequently, the maximum radius r will be 1.

Converting the polar coordinates to Cartesian coordinates at this maximum radius:

• x = r*cos(θ) = cos(θ)
• y = r*sin(θ) = sin(θ)

Since the maximum r corresponds to points where 2θ equals π/2, 5π/2, etc., the y-coordinate of these high points will be sin(π/4), sin(5π/4), etc. Specifically, sin(π/4) = √2/2 is the y-coordinate for one of the highest points, while sin(5π/4) = -√2/2 (which is not the highest point since we are looking for maximum y).

Thus, the y-coordinate of the highest point on the curve is √2/2.