Problem 35 - Calculus

Question

Table 1.12 shows some values of a linear function \(f\) and an exponential function \(g .\) Find exact values (not decimal approximations) for each of the missing entries. $$\begin{array}{c|c|c|c|c|c} \hline x & 0 & 1 & 2 & 3 & 4 \\ \hline f(x) & 10 & ? & 20 & ? & ? \\ \hline g(x) & 10 & ? & 20 & ? & ? \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The missing values are: \(f(1) = 15\), \(f(3) = 25\), \(f(4) = 30\); \(g(1) = 10\sqrt{2}\), \(g(3) = 20\sqrt{2}\), \(g(4) = 40\).

Step 1: Identify the Linear Function

For the linear function \(f(x)\), use the given values of \(f(0) = 10\) and \(f(2) = 20\). Since it's linear, identify the pattern or the common difference. It appears \(f(x)\) increases by 5 for each increment of \(x\) by 1.

Step 2: Calculate Missing Values for f(x)

Using the identified common difference of 5, fill in the missing values. By observation and calculation: \(f(1) = f(0) + 5 = 15\); \(f(3) = f(2) + 5 = 25\); \(f(4) = f(3) + 5 = 30\).

Step 3: Identify the Exponential Function

For the exponential function \(g(x)\), use the given values \(g(0) = 10\) and \(g(2) = 20\). Find the growth factor by setting up the equation \(g(x) = 10 \, a^x\). Given \(g(2) = 20\), we have \(10 \, a^2 = 20\), which gives \(a^2 = 2\), and thus, \(a = \sqrt{2}\).

Step 4: Calculate Missing Values for g(x)

Using the base \(a = \sqrt{2}\), calculate the missing values. \(g(1) = 10 \times (\sqrt{2})^1 = 10 \sqrt{2}\); \(g(3) = 10 \times (\sqrt{2})^3 = 10 \times 2 \sqrt{2} = 20 \sqrt{2}\); \(g(4) = 10 \times (\sqrt{2})^4 = 10 \times 4 = 40\).

Key Concepts

Linear Functions

Linear functions are a fundamental concept in mathematics, often represented as a straight line when graphed. The general form of a linear function is given by the equation \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept, or the point where the line crosses the y-axis. The slope \(m\) indicates the rate at which \(f(x)\) increases or decreases as \(x\) changes.
Linear functions have a constant rate of change. This means that for every unit increase in \(x\), \(f(x)\) increases by the same amount.

• For example, if \(f(x)\) increases by 5 for each increase in \(x\), the function's pattern is predictable and straightforward.
• In the exercise provided, the linear function \(f(x)\) increases by 5 as \(x\) goes from 0 to 4, which makes this pattern easy to follow.
• The missing values in the table can be easily calculated by identifying this common difference or slope.

Exponential Functions

Exponential functions differ significantly from linear functions. They are described by an equation of the form \(g(x) = ab^x\), where \(a\) is the initial value (the value when \(x = 0\)), and \(b\) is the base or growth factor.
Exponential functions exhibit a multiplicative rate of change, meaning the function grows by a consistent factor, not a fixed amount. This growth results in a curve when plotted on a graph.

• In the given exercise, the values of \(g(x)\) double as \(x\) increases due to the base \(b = \sqrt{2}\).
• This means every two increments in \(x\) lead to the function's value doubling, as illustrated by the values \(g(2) = 20\), \(g(4) = 40\).
• Finding the correct growth factor is key to filling in the missing values of an exponential function.

Function Tables

Function tables are a useful tool for visualizing and calculating values in both linear and exponential functions. They list values of a function for different inputs, often presented in columns with \(x\) values and corresponding \(f(x)\) or \(g(x)\) values.

• In the exercise, a table is provided with specific \(x\) values and some missing function values.
• By understanding and applying the properties of linear and exponential functions, these tables can be completed accurately.
• For linear functions, identify the common difference, and for exponential functions, determine the growth factor or base.

Function tables are extremely helpful for identifying patterns and verifying solutions as they provide a clear way to compare changes in the function as \(x\) varies.

Pattern Recognition

Recognizing patterns is essential in solving problems with linear and exponential functions. It enables you to predict future values and fill in missing information easily.

• In linear functions, look for an arithmetic pattern, where the difference between consecutive \(f(x)\) values is constant.
• For exponential functions, identify a geometric pattern where the ratio between consecutive \(g(x)\) values remains the same.
• In this exercise, recognizing the arithmetic pattern of increasing by 5 helps solve \(f(x)\), while spotting the doubling pattern helps solve \(g(x)\).

Applying pattern recognition not only simplifies the problem-solving process but also deepens your understanding of how these functions behave and change with different inputs.