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Chapter 2: Problem 101

The revenues \(R\) (in millions of dollars) for a company from 2003 through 2010 can be modeled by \(R=6.212 t^{3}-132.87 t^{2}+863.2 t-1115,3 \leq t \leq 10\) where \(t\) represents the year, with \(t=3\) corresponding to 2003 (a) Use a graphing utility to approximate any relative extreme of the model over its domain. (b) Use the graphing utility to approximate the intervals on which the revenue for the company is increasing and decreasing over its domain. (c) Use the results of parts (a) and (b) to describe the company's revenue during this time period.

Short Answer

Expert verified
The location and value of relative extremes, as well as intervals of increase and decrease, should be obtained from a graphing utility. By combining this information, a general description of the company's revenue from 2003-2010 may be made.

Step by step solution

01

Searching for relative extremes

Relative extremes will occur at points where the first derivative of the function is zero or undefined. Identifying these points can be done with a graphing utility, which graphically represents the function. The relative extremes will be the highest and lowest points within the domain.
02

Defining intervals of increase and decrease

Intervals of increase and decrease can be approximated by observing the graph. When the function is rising, it is increasing; when it's falling, it's decreasing. These intervals can be denoted with ranges of \(t\) on the x-axis.
03

Describing the company's revenue

Use the information from above and the graph to describe if and when the revenue was increasing or decreasing. Any relative extremes, or turning points, can be particularly noteworthy. Comparing the revenue between different years can provide meaningful insights into the financial trends during the given period.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a digital tool that helps visualize mathematical functions. In this case, a graphing utility can be used to represent polynomial functions like the one given for the revenues of a company from 2003 to 2010.
By plotting the function, one can easily identify critical features such as peaks, troughs, and intersections with the axes. These critical features assist in analyzing the overall behavior of the polynomial function.
  • To start plotting, input the polynomial equation: \(R=6.212 t^{3}-132.87 t^{2}+863.2 t-1115\) where \(3 \leq t \leq 10\).
  • Set the x-axis to represent the time (years) and the y-axis to depict revenue in millions of dollars.
Using a graphing utility simplifies complex calculations and provides a visual interpretation, enhancing comprehension of the revenue's trends over the specified period.
Relative Extrema
Relative extrema are points where a function reaches a high or low value relative to the surrounding values. More formally, a relative extremum is a point where the first derivative of the function is zero or undefined. These points are crucial in understanding how revenue fluctuates over time.
For the revenue polynomial, relative extrema can be identified directly from the graph:
  • Look for the peaks, which are local maxima where the revenue is highest.
  • Look for the troughs, which are local minima where the revenue is lowest.
These points help to pinpoint critical periods of financial performance, such as high profits or significant downturns. In practical terms, when analyzing a business's revenue model, such extrema inform strategic business decisions.
Increasing and Decreasing Intervals
Identifying increasing and decreasing intervals provides insight into when a company's revenue is rising or falling.
  • An increasing interval is one in which the revenue consistently rises over time, reflected by a positive slope on the graph between two points. Revenue growth during this interval is positive.
  • A decreasing interval is where the revenue falls over time, shown by a negative slope. This indicates a reduction in revenue over that specific time.
  • The intervals can be identified by observing how the graph behaves as it travels from left to right along the x-axis.
Quantifying these intervals depends on analyzing the sign of the first derivative throughout the domain. This method provides a precise mathematical description of when the revenue is increasing or decreasing, helping decision-makers implement effective strategies to enhance growth.
Revenue Modeling
Revenue modeling involves creating mathematical representations of a company's revenue over a specified period.
  • The polynomial function \(R=6.212 t^{3}-132.87 t^{2}+863.2 t-1115\) models the company's revenue.
  • This function uses \(t\) for time, where \(t=3\) corresponds to 2003 and extends to \(t=10\), representing 2010.
The polynomial used in a revenue model provides a predictive view and allows analysts to forecast future revenues based on past performance. It captures essential trends and shifts, offering a quantitative look into the company's financial health during the specified timeframe. Such a model is invaluable for strategic planning, facilitating better-informed business decisions. Exploring how variables interact and change with time enhances a company's ability to anticipate future challenges and opportunities.

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Most popular questions from this chapter

A page that is \(x\) inches wide and \(y\) inches high contains 30 square inches of print. The top and bottom margins are each 1 inch deep, and the margins on each side are 2 inches wide (see figure). (a) Write a function for the total area \(A\) of the page in terms of \(x\) (b) Determine the domain of the function based on the physical constraints of the problem. (c) Use a graphing utility to graph the area function and approximate the page size for which the least amount of paper is used.

Use long division to verify that \(y_{1}=y_{2}.\) $$y_{1}=\frac{x^{4}-3 x^{2}-1}{x^{2}+5}, \quad y_{2}=x^{2}-8+\frac{39}{x^{2}+5}$$

A company that produces calculators estimated that the profit \(P\) (in dollars) from selling a particular model of calculator was $$P=-152 x^{3}+7545 x^{2}-169,625, \quad 0 \leq x \leq 45$$ where \(x\) was the advertising expense (in tens of thousands of dollars). For this model of calculator, the advertising expense was \(\$ 400,000(x=40)\) and the profit was \(\$ 2,174,375.\) (a) Use a graphing utility to graph the profit function. (b) Use the graph from part (a) to estimate another amount the company could have spent on advertising that would have produced the same profit. (c) Use synthetic division to confirm the result of part (b) algebraically.

You want to make an open box from a rectangular piece of material, 15 centimetres by 9 centimetres, by cutting equal squares from the corners and turning up the sides. (a) Let \(x\) represent the side length of each of the squares removed. Draw a diagram showing the squares removed from the original piece of material and the resulting dimensions of the open box. (b) Use the diagram to write the volume \(V\) of the box as a function of \(x .\) Determine the domain of the function. (c) Sketch the graph of the function and approximate the dimensions of the box that will yield a maximum volume. (d) Find values of \(x\) such that \(V=56 .\) Which of these values is a physical impossibility in the construction of the box? Explain.

Proof Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.
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