Problem 6 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Suppose \(a, b,\) and \(c\) are numbers and $$ P_{t}=a t^{2}+b t+c $$ where \(t\) is any number. Show that $$ Q_{t}=P_{t+1}-P_{t} $$ is linearly related to \(t\) (that is, \(Q_{t}=\alpha t+\beta\) for some \(\alpha\) and \(\left.\beta\right)\), and that $$ R_{t}=Q_{t+1}-Q_{t} $$ is a constant.

Step-by-Step Solution

Answer

\( Q_t = 2at + a \) is linear; \( R_t = 2a \) is constant.

Step 1: Define and Simplify P(t+1)

First, substitute \( t+1 \) into the expression for \( P_t \). This gives us:\[ P_{t+1} = a(t+1)^2 + b(t+1) + c \]Expand the terms:\[ P_{t+1} = a(t^2 + 2t + 1) + b(t + 1) + c = at^2 + 2at + a + bt + b + c \]Combine like terms:\[ P_{t+1} = at^2 + (2a + b)t + (a + b + c) \]

Step 2: Calculate Q(t)

Evaluate \( Q_t = P_{t+1} - P_t \). We already have:\[ P_t = at^2 + bt + c \]Subtract:\[ Q_t = (at^2 + (2a + b)t + (a + b + c)) - (at^2 + bt + c) \]Simplify the expression:\[ Q_t = ((2a + b)t + (a + b + c)) - (bt + c) = (2a)t + (a + b + c - b - c) \]\[ Q_t = 2at + a \]Thus, \( Q_t = 2at + a \) which shows \( Q_t \) is linearly related to \( t \).

Step 3: Define and Calculate R(t)

Now calculate \( R_t = Q_{t+1} - Q_t \). First, express \( Q_{t+1} \):\[ Q_{t+1} = 2a(t+1) + a = 2at + 2a + a = 2at + 3a \]Subtract to find \( R_t \):\[ R_t = (2at + 3a) - (2at + a) = 3a - a \]\[ R_t = 2a \]Thus, \( R_t \) is a constant value.

Key Concepts

Linear Relationships

In the world of mathematics, linear relationships are essential for understanding how variables relate to one another through a linear equation. A linear relationship between two variables, say \( x \) and \( y \), can be expressed as \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. These relationships are characterized by a constant rate of change.

When dealing with polynomial functions like the example given, determining the linear relationship of a function involves showing how additional terms relate linearly to the variable of interest. Here, it's established through the calculation of \( Q_t = 2at + a \). Notice how the expression resolves into a linear form where \( \alpha = 2a \) and \( \beta = a \), making it clear that \( Q_t \) is linearly related to \( t \).

Understanding linear relationships can greatly simplify the process of analyzing changes, and is foundational in fields such as physics, economics, and statistical data modeling. In any situation where you encounter sequential or incremental changes to a function’s outputs, considering the linear relationship can offer a streamlined analytical approach.

Constant Difference

Constant difference is a concept that refers to the persistence of a constant value of change when you compare consecutive differences in a sequence. In the provided solution, the idea of constant difference is illustrated with \( R_t = 2a \), showing that each change from \( Q_{t+1} \) to \( Q_t \) results in a constant value of 2a.

A practical understanding of constant difference is vital when evaluating polynomial functions because it reveals consistent growth or shrinkage, thus helping us anticipate future outcomes of the function. In essence, if you discover that consecutive differences yield a constant, it tells you that the function is behaving predictably in terms of the rate of change.

This concept is particularly useful in arithmetic sequences and in the analysis of linear functions. With equations describing constant differences, one can infer a lot about the original polynomial's nature and behavior without having to solve for its explicit values.

Expansion of Expressions

The expansion of expressions is a critical process in algebra that involves rewriting a product of terms into a sum or difference to simplify an equation. This skill is especially useful in polynomial equations where factors like \((t+1)^2\) need to be expanded fully to realize the function's behavior.

For instance, taking \( P_{t+1} \) to the expression \( a(t+1)^2 + b(t+1) + c \), we expand \( (t+1)^2 \) into \( t^2 + 2t + 1 \). Hence, \( P_{t+1} \) becomes \( at^2 + 2at + a + bt + b + c \). Such expansions are vital because they transform products into a form that allows for direct subtraction or addition, crucial in defining \( Q_t \) and subsequently \( R_t \).

Mastering this expansion technique is foundational for solving many algebraic problems, as it simplifies complex equations and uncovers relationships between variables that simpler forms might obscure. Moreover, understanding expression expansion aids in breaking down polynomials into manageable linear components, as seen in this solution.