Question

A polar curve is given by the equation r = 10θ/θ² + 1 for θ > 0. What is the instantaneous rate of change of r with respect to o when 0 = 2 ?

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

Answer 1

The instantaneous rate of change of r with respect to θ when θ = 2 is B. -6/5.

Step-by-step explanation:

The given polar curve equation is r = 10θ/(θ² + 1). To find the instantaneous rate of change of r with respect to θ when θ = 2, we need to calculate the derivative of r with respect to θ and substitute θ = 2 into the derivative.

First, let's differentiate the equation r = 10θ/(θ² + 1) with respect to θ.

Using the quotient rule, we have:

r' = [10(θ² + 1) - 10θ(2θ)]/(θ² + 1)²

Simplifying the numerator, we get:

r' = [10(θ² + 1) - 20θ²]/(θ² + 1)²

Now, substitute θ = 2 into r' to find the instantaneous rate of change:

r'(2) = [10(2² + 1) - 20(2²)]/(2² + 1)²

Simplifying further, we have:

r'(2) = [10(4 + 1) - 20(4)]/(4 + 1)²

r'(2) = (10(5) - 20(4))/25

r'(2) = (50 - 80)/25

r'(2) = -30/25

Therefore, the instantaneous rate of change of r with respect to θ when θ = 2 is -6/5.