Problem 17 - Applied Calculus

Question

Find a possible formula for the function represented by the data. $$\begin{array}{c|c|c|c|c} \hline t & 0 & 1 & 2 & 3 \\ \hline g(t) & 5.50 & 4.40 & 3.52 & 2.82 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The function is approximately \( g(t) = 5.5 \cdot 0.8^t \).

Step 1: Analyze the Pattern

Look at the given values to see if they form a recognizable sequence or pattern. Calculate the differences between successive values of \( g(t) \) to determine if there's a common rate of change or ratio.

Step 2: Check for a Linear Relationship

Calculate the differences between each successive term: for \( t=0 \) to \( t=1 \), \( g(t) \) decreases by \( 5.50 - 4.40 = 1.10 \); for \( t=1 \) to \( t=2 \), \( g(t) \) decreases by \( 4.40 - 3.52 = 0.88 \); and for \( t=2 \) to \( t=3 \), \( g(t) \) decreases by \( 3.52 - 2.82 = 0.70 \). These differences are not constant, so the data is not linear.

Step 3: Check for an Exponential Relationship

Verify if a common ratio exists. Calculate the ratios between successive terms: \( \frac{4.40}{5.50} \approx 0.8 \), \( \frac{3.52}{4.40} \approx 0.8 \), and \( \frac{2.82}{3.52} \approx 0.8 \). The ratios are roughly the same, indicating a possible exponential function with a common ratio of approximately 0.8.

Step 4: Formulate the Exponential Function

Since the pattern suggests an exponential relationship, use the general form \( g(t) = a \cdot r^t \). Given that \( g(0) = 5.5 \), we have \( a = 5.5 \). The ratio \( r \approx 0.8 \), as derived in the previous step. Thus, a possible formula for the function is \( g(t) = 5.5 \cdot 0.8^t \).

Key Concepts

Patterns in Data

When analyzing any set of data, spotting patterns is crucial to understanding the underlying relationships or rules that govern the data set. In our given example, the focus is on identifying whether the sequences in the data set follow a specific pattern. You start looking for clues by calculating the differences or ratios between successive data points.

Patterns can indicate different types of functions. Common patterns often show linear, quadratic, or exponential behaviors. In our case, the method involved examining the ratios of successive terms rather than the differences since differences weren't constant.

Looking at each step of change—whether in increase or decrease—can provide valuable insights. For instance, since we identified a consistent ratio between points, the data likely follows an exponential pattern, suggesting it represents an exponential function.

Rate of Change

Understanding the rate of change gives insight into how quickly or slowly values within a function or sequence are increasing or decreasing. With exponential functions, the rate of change is not constant; rather, it multiplies by a specific factor.

In our given data, you observe changes by calculating differences first, noticing they weren't constant, which ruled out a linear relationship. Further exploration through ratios revealed consistency close to 0.8.

The concept of a rate of change links directly to exponential growth or decay processes:

• In exponential growth, the quantity increases by a consistent ratio greater than 1.
• In exponential decay, the quantity decreases by a consistent ratio less than 1, as seen with our ratio of approximately 0.8.

Understanding these changes helps in formulating the correct function type that describes the observed data.

Sequence and Series

Sequences are lists of numbers arranged in a particular order depending on specific rules, often used to predict future outcomes. Each term in a sequence relates to its predecessors, guided by a pattern.

Our focus sequence, derived from the given data set, exhibited properties of a geometric sequence due to the common ratio of roughly 0.8.

To identify a likely formula, we used the pattern information to establish a rule:

• The first term, known as the constant multiplier in an exponential function, starts with 5.5.
• The common ratio, determined from the previous analysis, helps predict successive terms.

This ties into the idea of series, which considers sums of terms in a sequence and is reflected in the mathematical expressions, helping solve problems involving future predictions based on past data trends.