Revenue. Revenues from sales of a soft drink over a 2-year...

Revenue. Revenues from sales of a soft drink over a 2-year period are given approximately by $$ R(t)=4-3 \cos \frac{\pi t}{6} \quad 0 \leq t \leq 24 $$ where $R(t)$ is revenue (in millions of dollars) for a month of sales $t$ months after February $1 .$ (A) What is the rate of change of revenue $t$ months after February $1 ?$ (B) What is the rate of change of revenue 1 month after February $1 ? 6$ months after February $1 ? 11$ months after February $1 ?$ (C) Find all local maxima and minima for $0<t<24$. (D) Find the absolute maximum and minimum for $0 \leq t \leq 24$ (E) Repeat part ( $\mathrm{C}$ ), using a graphing calculator.

Revenue. Revenues from sales of a soft drink over a 2-year period are given approximately by
$$
R(t)=4-3 \cos \frac{\pi t}{6} \quad 0 \leq t \leq 24
$$
where $R(t)$ is revenue (in millions of dollars) for a month of sales $t$ months after February $1 .$
(A) What is the rate of change of revenue $t$ months after February $1 ?$
(B) What is the rate of change of revenue 1 month after February $1 ? 6$ months after February $1 ? 11$ months after February $1 ?$
(C) Find all local maxima and minima for $0<t<24$.
(D) Find the absolute maximum and minimum for $0 \leq t \leq 24$
(E) Repeat part ( $\mathrm{C}$ ), using a graphing calculator.

Key Concepts

Differentiation

Differentiation is the process of finding the derivative of a function, which measures how the function's output changes as its input changes. In the context of revenue analysis, differentiation is used to determine the rate of change of revenue with respect to time.

Rate of Change

The rate of change is a measure of how quickly a function's value is increasing or decreasing at a particular point. It provides insights into how revenue is evolving over time, indicating periods of growth or decline.

Trigonometric Functions

Trigonometric functions, such as cosine, are periodic functions that model cyclic behavior. Analyzing these functions involves understanding their amplitude, period, and phase shift, which are essential when studying revenue trends that repeat over a specified time interval.

Critical Points

Critical points are the values in the domain of a function where the derivative is zero or undefined. These points are important because they are potential candidates for local maxima or minima, which help identify significant turning points in the revenue function.

Local and Absolute Extrema

Local extrema refer to points where the function has a local maximum or minimum, meaning the function's value is higher or lower than nearby points. Absolute extrema are the highest or lowest values over the entire interval of interest. Distinguishing between the two helps in making comprehensive assessments of the revenue performance over time.

Graphical Analysis

Graphical analysis involves using tools like graphing calculators to visualize functions. This is useful for confirming analytical findings and understanding the overall behavior of complex functions, especially when they involve periodic and oscillatory components.

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