a data set of coordinate pairs is modeled by the function f(θ) = sin θ. if the model f is consistent with…

a data set of coordinate pairs is modeled by the function f(θ) = sin θ. if the model f is consistent with the data set, which of the following should describe the dependent variable values on the interval π < θ < 3π/2? a the values should be decreasing at a decreasing rate on the interval. b the values should be decreasing at an increasing rate on the interval. c the values should be increasing at a decreasing rate on the interval. d the values should be increasing at an increasing rate on the interval.

a data set of coordinate pairs is modeled by the function f(θ) = sin θ. if the model f is consistent with the data set, which of the following should describe the dependent variable values on the interval π < θ < 3π/2? a the values should be decreasing at a decreasing rate on the interval. b the values should be decreasing at an increasing rate on the interval. c the values should be increasing at a decreasing rate on the interval. d the values should be increasing at an increasing rate on the interval.

Answer

Explanation:

Step1: Recall the derivative of sine function

The derivative of $y = f(\theta)=\sin\theta$ is $f'(\theta)=\cos\theta$, and the second - derivative is $f''(\theta)=-\sin\theta$.

Step2: Analyze the sign of the first - derivative on the given interval

On the interval $\pi<\theta<\frac{3\pi}{2}$, $\cos\theta<0$. Since $f'(\theta)=\cos\theta<0$ on the interval $\pi<\theta<\frac{3\pi}{2}$, the function $y = f(\theta)=\sin\theta$ is decreasing on this interval.

Step3: Analyze the sign of the second - derivative on the given interval

On the interval $\pi<\theta<\frac{3\pi}{2}$, $\sin\theta<0$, so $f''(\theta)=-\sin\theta>0$. A positive second - derivative means the function is concave up, and the rate of change of the function (the first - derivative) is increasing. Since the function is decreasing (first - derivative is negative) and the rate of the decrease is increasing (second - derivative is positive), the function is decreasing at an increasing rate.

Answer:

B. The values should be decreasing at an increasing rate on the interval.