Question

Intel The sales per share \(S\) (in dollars) for Intel from 1992 to 2005 can be approximated by the function \(S=\left\\{\begin{array}{lr}-1.48+2.65 \ln t, & 2 \leq t \leq 10 \\ 0.1586 t^{2}-3.465 t+22.87, & 11 \leq t \leq 15\end{array}\right.\) where \(t\) represents the year, with \(t=2\) corresponding to 1992\. (Source: Intel)

Short answer

The sales per share of Intel for any given year between 1992 to 2000 can be calculated with the function \(S=-1.48+2.65 \ln t\), and for any given year between 2001 to 2005 can be calculated with the function \(S=0.1586 t^{2}-3.465 t+22.87\), where \(t\) is the year minus 1990.

Step-by-step solution

Analyzing the first function

The first function \(S=-1.48+2.65 \ln t\) is applicable between the years 1992 and 2000 (as \(t\) varies between 2 and 10 inclusive). Just substitute the year (minus 1990) into this function to get the sales for that year.

Analyzing the second function

The second function \(S=0.1586 t^{2}-3.465 t+22.87\) is applicable between the years 2001 and 2005 (as \(t\) varies between 11 and 15 inclusive). Similarly, just substitute the year (minus 1990) into this function to get the sales for that year.

Example Calculation

If one wants to find Intel's sales per share in 1995, since 1995 corresponds to \(t=5\), substitute 5 into the first function: \(S=-1.48+2.65\ln(5)\), which gives approximately \$3.31.

Key concepts

Piecewise Function Application

Understanding piecewise functions is crucial for analyzing scenarios where a single rule doesn't apply across an entire domain. A piecewise function is made up of several sub-functions,
each with its own formula and applicable for certain intervals of the independent variable. In real-world terms, this means defining different behaviors or values for different situations or periods.

Take the provided example of Intel's sales per share from 1992 to 2005. The exercise illustrates a piecewise function with two rules: one logarithmic and one quadratic, each representing the sales per share in different time frames. Students must note the distinct intervals \(2 \leq t \leq 10\) and \(11 \leq t \leq 15\) and apply the appropriate function for each time period. By identifying the correct interval for a given year and applying
the specific sub-function, you can calculate the sales per share for Intel in that year. This practical application demonstrates
how piecewise functions model situations that cannot be captured by a single rule.

Logarithmic Functions

In our example, logarithmic functions are used to model the first interval of Intel’s sales per share data. These functions have the general form \( y = a \ln(x) + b \), where \(a \) and \(b \) are constants,
and \(\ln(x)\) represents the natural logarithm of \(x\). Logarithmic functions are the inverses of exponential functions and are adept at capturing growth phenomena.

For instance, Intel’s sales per share from 1992 to 2000 grow at a rate that can be described by a logarithmic trend, showing rapid early growth that slows over time. When dealing with logarithmic functions, it’s important to remember that they're undefined for \(x\) values less than or equal to zero and that the rate of increase decreases as \(x\) increases. Students working with logarithms should be careful with their calculations and remember that logarithms have unique properties that can be utilized to simplify expressions, such as \(\ln(AB) = \ln(A) + \ln(B)\).

Quadratic Functions

The second part of Intel's sales data is modeled with a quadratic function. Quadratic functions are polynomial functions of degree two, typically written in
the form \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). They are characterized by their parabolic graphs, pointing either upward or downward depending on the sign of \(a\).

In the case of Intel's sales example, \(0.1586t^2-3.465t+22.87\) describes the period from 2001 to 2005. For students to understand this, visualization can greatly help; plotting the quadratic function often reveals its direction of opening and its vertex, providing a clear picture of how sales change over time. Quadratic functions are particularly useful in modeling situations involving acceleration, such as the trajectory of a projectile or, as in this exercise, periods of increasing or decreasing growth rates in business metrics.

Algebraic Models

The term algebraic models refers to using algebraic equations and functions to represent and solve real world problems.
In our Intel example, both logarithmic and quadratic functions form an algebraic model that describes sales per share over a period of time. Learning to construct and interpret algebraic models is a key skill in mathematics, helping students not only with textbook problems but also with reasoning and problem-solving in various fields.

When creating an algebraic model, one should identify the variables and constants at play, establish the relationships between them, and choose the most suitable mathematical forms to represent these relationships. This can include linear, quadratic, logarithmic, exponential, or other function types, and understanding the characteristics of each is vital for creating accurate models. For instance, careful analysis of past data
is needed to determine which algebraic model fits best, just as the creators of the piecewise function for Intel analyzed its sales data to choose logarithmic and quadratic models for different intervals.