Question

For the polar curve defined by the equation r=100/θ²​+1 for θ≥0, what is the instantaneous rate of change of r with respect to θ when θ=0?

A) -6
B) -6/5
C) 5/2
D) 26/5

Answer 1

The instantaneous rate of change of the radius with respect to the angle for the given polar curve does not exist when θ=0, as the derivative involves division by zero. Therefore, none of the given options are correct.

Step-by-step explanation:

The student has asked for the instantaneous rate of change of the radius r with respect to the angle θ for the polar curve defined by the equation r=100/θ²+1. To find this, we must take the derivative of r with respect to θ. Since we are looking for the instantaneous rate of change when θ=0, we will calculate the derivative and then evaluate it at θ=0.

First, we find the derivative using the power rule and the chain rule:

dr/dθ = d/dθ(100/θ²+1)

= d/dθ(100*θ^(-2) + 1)

= -200*θ^(-3)

However, when θ=0, this expression is undefined as it involves division by zero. Thus, the correct interpretation is that the instantaneous rate of change does not exist when θ=0. So, none of the given options A) -6, B) -6/5, C) 5/2, D) 26/5 are correct as they all assume the existence of a defined rate of change at that point.