A generator produces electrical power, P, in watts, according to the function:
P(R)= 120/ (0.4 + R)^2
where R is the resistance, in ohms. Determine the intervals on which the power is increasing.
To determine the intervals on which the power is increasing, we need to find the derivative of the power function with respect to the resistance, and then examine where the derivative is positive.
First, let's find the derivative of the power function:
P(R) = 120 / (0.4 + R)^2
To do this, we can use the quotient rule. Let's rewrite the function as:
P(R) = 120 * (0.4 + R)^(-2)
Now, let's find the derivative using the quotient rule:
dP/dR = [0 * (0.4 + R)^(-2) - 120 * (-2) * (0.4 + R)^(-3)] / (0.4 + R)^4
= [240 * (0.4 + R)^(-3)] / (0.4 + R)^4
= 240 / (0.4 + R)^3 * (0.4 + R)^(-4)
= 240 / (0.4 + R)^4
Now, let's examine where the derivative is positive to determine the intervals on which the power is increasing. Since the derivative is a constant, the power function is always increasing. Therefore, the power increases on all intervals.
In conclusion, the power is increasing on all intervals.