in the polar - coordinate system, the graph of a polar function r = f(θ) is shown with a domain of all real…
in the polar - coordinate system, the graph of a polar function r = f(θ) is shown with a domain of all real values of θ for 0 ≤ θ < 2π. on this interval of θ, the graph has no holes, passes through each point exactly one time, and as θ increases, the graph passes through the labeled points a, b, c, and d, in that order. 2 mark for review on which of the following intervals is the average rate of change of r with respect to θ greatest? a from a to c b from b to c c from b to d d from c to d
Answer
Explanation:
Step1: Recall average - rate - of - change formula
The average rate of change of $r = f(\theta)$ with respect to $\theta$ over the interval $[\theta_1,\theta_2]$ is $\frac{f(\theta_2)-f(\theta_1)}{\theta_2 - \theta_1}$. In polar coordinates, we can estimate this by looking at the change in $r$ values for a given change in $\theta$ values.
Step2: Analyze each interval visually
- For the interval from $A$ to $C$: The change in $r$ is relatively small over the change in $\theta$.
- For the interval from $B$ to $C$: The change in $r$ is also not very large over the change in $\theta$.
- For the interval from $B$ to $D$: The change in $r$ is significant. As $\theta$ changes from the value corresponding to $B$ to the value corresponding to $D$, $r$ changes from a value around $3$ to a value close to $0$.
- For the interval from $C$ to $D$: The change in $r$ is smaller compared to the change from $B$ to $D$.
Answer:
C. From $B$ to $D$