Find an equation equivalent to xy = 1 in polar coordinates.

r²sinθcosθ = 1 OR r²sin2θ = 2

Step-by-step explanation:

Polar coordinates are represented in of the radius r and angle θ.

In the x axis:

x = rcosθ

Along the y-axis:

y = rsinθ

Rectangular coordinate to polar coordinate will be:

(x, y) = (rcosθ, rsinθ)

To convert the equation xy = 1 to polar coordinate, we will substitute the value of x and y in its polar form into the given equation to have:

(rcosθ)(rsinθ) = 1

r²sinθcosθ = 1...(1)

From the trigonometry identity

Sin2θ = 2sinθcosθ

Sinθcosθ = sin2θ/2...(2)

Substituting equation 2 into 1:

r²(sin2θ/2) = 1

r²sin2θ = 2

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50