A recently released film has its weekly revenue given by
Where R(x) is in million of dollars and x is in weeks.
(a) Find the asymptotes of the graph of the function R.
(b) Find the intervals where R is increasing and decreasing, if any.
(c) When will the revenue be maximized?
(d) Sketch the graph of R in an interval 0 ≤ x ≤ 30.
To find the answers to these questions, we need the specific function representing the weekly revenue. Please provide the function or any additional information that may be necessary to solve these problems.
To answer these questions, we need to find the derivative of the function R(x) and analyze its properties. The derivative will give us information about the slope and concavity of the function, which will help us address all the questions.
(a) To find the asymptotes of the graph of R, we look for vertical asymptotes where the function becomes unbounded. In other words, we need to find values of x where the function R(x) approaches infinity or negative infinity.
To determine the vertical asymptotes, we first need to investigate if R(x) is a rational function. If it is, we find vertical asymptotes by setting the denominator equal to zero and solving for x. If it isn't a rational function, there are no vertical asymptotes.
(b) To find the intervals where R is increasing or decreasing, we examine the sign of the derivative. If the derivative is positive, R is increasing. If the derivative is negative, R is decreasing. If the derivative is zero, R may have local extrema.
(c) To find when the revenue is maximized, we look for the critical points, which are the points where the derivative is zero or undefined. We then evaluate the revenue function at each critical point to find the maximum revenue.
(d) To sketch the graph of R in the interval 0 ≤ x ≤ 30, we apply the information gathered from parts (a)-(c) and consider additional details like the y-intercept and the behavior at the boundary of the interval.
Now let's start solving these questions step by step.
(a) To find the vertical asymptotes, we need to examine if R(x) is a rational function. If it is not a rational function, we can skip this step. If it is a rational function, we need to set the denominator equal to zero and solve for x.
If the given function R(x) is not a rational function, then there are no vertical asymptotes.
(b) To find the intervals of increasing/decreasing, we need to find the derivative of R(x). Let's denote the derivative of R(x) as R'(x).
R(x) = ... (not provided in the question)
Differentiate R(x) with respect to x to find R'(x).
R'(x) = ... (differentiate the given function)
The sign of R'(x) will determine the intervals of increasing and decreasing. If R'(x) > 0, R is increasing. If R'(x) < 0, R is decreasing. If R'(x) = 0, R may have a local extrema.
(c) To find the points of maximum revenue, we need to find the critical points of R(x). Critical points occur when the derivative is zero or undefined. We need to solve the equation R'(x) = 0 to find these critical points.
Once we find the critical points, we evaluate R(x) at each point to determine when the revenue is maximized.
(d) To sketch the graph of R in the interval 0 ≤ x ≤ 30, we utilize the information obtained from parts (a)-(c) and consider additional details like the y-intercept and the behavior at the boundary of the interval.
Using this information, we plot the points, identify the shape of the graph, and draw the curve accordingly.
Please note that to fully answer this question, we would need the specific function R(x) provided in the question. Once you provide that information, we can proceed with solving the problem in more detail.