Problem 36 - Calculus

Question

Match the functions \(h(s), f(s),\) and \(g(s),\) whose values are in Table \(1.13,\) with the formulas $$ y=a(1.1)^{s}, \quad y=b(1.05)^{s}, \quad y=c(1.03)^{s} $$ assuming \(a, b,\) and \(c\) are constants. Note that the function values have been rounded to two decimal places. $$\begin{array}{c|c||c|c||c|c} \hline s & h(s) & s & f(s) & s & g(s) \\ \hline 2 & 1.06 & 1 & 2.20 & 3 & 3.47 \\ 3 & 1.09 & 2 & 2.42 & 4 & 3.65 \\ 4 & 1.13 & 3 & 2.66 & 5 & 3.83 \\ 5 & 1.16 & 4 & 2.93 & 6 & 4.02 \\ 6 & 1.19 & 5 & 3.22 & 7 & 4.22 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

h(s) matches with \(a(1.1)^s\), f(s) with \(b(1.05)^s\), and g(s) with \(c(1.03)^s\).

Step 1: Identify Growth Factors

The growth factors in the formulas are given as 1.1, 1.05, and 1.03, corresponding to the exponential forms of functions. Observe how the values increase with each successive increase in \(s\) for \(h(s), f(s),\) and \(g(s)\).

Step 2: Compare Increase Patterns

For each table of function values: \(h(s), f(s),\) and \(g(s)\), notice the increase from one \(s\) value to the next. Compare this increase to the given growth factors to match functions.

Step 3: Match Function h(s)

For \(h(s)\), each subsequent value increases at a factor closest to 1.1, i.e., \(h(2) = 1.06, h(3) = 1.09, h(4) = 1.13, h(5) = 1.16, h(6) = 1.19\). Thus, \(h(s)\) matches with the growth factor of 1.1, so \(h(s) = a(1.1)^s\).

Step 4: Match Function f(s)

For \(f(s)\), each subsequent value increases at a factor closest to 1.05, i.e., \(f(1) = 2.20, f(2) = 2.42, f(3) = 2.66, f(4) = 2.93, f(5) = 3.22\). Thus, \(f(s)\) matches with the growth factor of 1.05, so \(f(s) = b(1.05)^s\).

Step 5: Match Function g(s)

For \(g(s)\), each subsequent value increases at a factor closest to 1.03, i.e., \(g(3) = 3.47, g(4) = 3.65, g(5) = 3.83, g(6) = 4.02, g(7) = 4.22\). Thus, \(g(s)\) matches with the growth factor of 1.03, so \(g(s) = c(1.03)^s\).

Key Concepts

Exponential Functions

When we talk about exponential functions, we refer to mathematical expressions where a constant base is raised to a variable exponent. These functions have a distinct property. They exhibit rapid growth or decay, depending on whether the base is greater or less than 1.
For instance, the formulas in our exercise:

• \( y = a(1.1)^{s} \)
• \( y = b(1.05)^{s} \)
• \( y = c(1.03)^{s} \)

all represent exponential growth functions. The base numbers, 1.1, 1.05, and 1.03, are crucial here. They determine the factor by which the function values multiply with each incremental increase in the variable \( s \).
Exponential functions can be recognized by their "J-shaped" curve when plotted, which signifies how rapidly they grow as \( s \) increases. This behavior applies to many natural processes, such as population growth or interest compounding.

Patterns of Increase

Patterns of increase in exponential functions are key to understanding their behavior and applications. This pattern refers to the consistent ratio or factor by which the values of a function grow as the input increases.
In our exercise, observing the tables of \( h(s), f(s), \) and \( g(s) \) can help identify this increase. Each of these function's values increases as \( s \) grows, in a way that corresponds to their respective base growth factors.

• \( h(s) \) rises by a factor close to 1.1
• \( f(s) \) enhances with a factor around 1.05
• \( g(s) \) expands with a factor nearing 1.03

These distinct increase patterns help identify which function matches which formula. Consistently seeing these patterns enhances your ability to infer and recognize exponential behaviors in real-world contexts.

Matching Functions to Formulas

Matching functions to formulas involves identifying which exponential expression aligns best with observed data or given patterns. In the exercise, we had to associate the functions \( h(s), f(s), \) and \( g(s) \) in the table to specific exponential formulas based on their growth factors.
This requires a step-by-step analysis:

• Look at \( h(s) \), which increases with a factor close to 1.1. It matches with \( y = a(1.1)^s \).
• For \( f(s) \), the growth factor appears akin to 1.05. Therefore, it aligns with \( y = b(1.05)^s \).
• The function \( g(s) \) grows by a factor near 1.03, which corresponds with \( y = c(1.03)^s \).

This concept of matching helps in practical scenarios where one has to interpret data, predict trends, or even create models of predictions using exponential functions. It showcases the importance of understanding the basis and effects of growth factors and how they apply to different scenarios.