Problem 29 - Applied Calculus

Question

(a) Which (if any) of the functions in the following table could be linear? Find formulas for those functions. (b) Which (if any) of these functions could be exponential? Find formulas for those functions. $$\begin{array}{r|c|c|c} \hline x & f(x) & g(x) & h(x) \\ \hline-2 & 12 & 16 & 37 \\ -1 & 17 & 24 & 34 \\ 0 & 20 & 36 & 31 \\ 1 & 21 & 54 & 28 \\ 2 & 18 & 81 & 25 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

(a) Only \( h(x) \) is linear: \( h(x) = -3x + 31 \). (b) Only \( g(x) \) is exponential: \( g(x) = 36 \cdot 1.5^x \).

Step 1: Analyze Each Function for Linearity

A function is linear if it has a constant rate of change. For each of the given functions \( f(x), g(x), \) and \( h(x) \), compute the differences between consecutive outputs. - For \( f(x) \): - \( 17 - 12 = 5 \) - \( 20 - 17 = 3 \) - \( 21 - 20 = 1 \) - \( 18 - 21 = -3 \) - The rate of change is not constant.- For \( g(x) \): - \( 24 - 16 = 8 \) - \( 36 - 24 = 12 \) - \( 54 - 36 = 18 \) - \( 81 - 54 = 27 \) - The rate of change is not constant.- For \( h(x) \): - \( 34 - 37 = -3 \) - \( 31 - 34 = -3 \) - \( 28 - 31 = -3 \) - \( 25 - 28 = -3 \) - The rate of change is constant.Thus, \( h(x) \) is the only linear function.

Step 2: Determine Formula for Linear Function

Since \( h(x) \) is linear and its rate of change between consecutive terms is \(-3\), we can express it in the form \( h(x) = mx + c \), where \( m \) is the slope.The slope \( m = -3 \). Now, find \( c \) using the point (0, 31):\[ 31 = -3 \cdot 0 + c \]\[ c = 31 \]Thus, the linear function is:\[ h(x) = -3x + 31 \]

Step 3: Analyze Each Function for Exponential Behavior

A function is exponential if its outputs are multiplied by a constant factor as \( x \) increases by 1. Compare the ratios of consecutive terms for each function. - For \( f(x) \): - Ratio \( \frac{17}{12} \approx 1.42 \) - Ratio \( \frac{20}{17} \approx 1.18 \) - Ratio \( \frac{21}{20} = 1.05 \) - Ratio \( \frac{18}{21} \approx 0.86 \) - The ratios are not constant.- For \( g(x) \): - Ratio \( \frac{24}{16} = 1.5 \) - Ratio \( \frac{36}{24} = 1.5 \) - Ratio \( \frac{54}{36} = 1.5 \) - Ratio \( \frac{81}{54} = 1.5 \) - The ratios are constant.- For \( h(x) \): - Ratio \( \frac{34}{37} \approx 0.92 \) - Ratio \( \frac{31}{34} \approx 0.91 \) - Ratio \( \frac{28}{31} \approx 0.90 \) - Ratio \( \frac{25}{28} \approx 0.89 \) - The ratios are not constant.Only \( g(x) \) could be exponential.

Step 4: Determine Formula for Exponential Function

Since \( g(x) \) has a constant ratio of 1.5, it follows the pattern \( g(x) = a \cdot r^x \), where \( r = 1.5 \).To find \( a \), use the point at \( x = 0 \):\[ 36 = a \cdot 1.5^0 \]\[ a = 36 \]Thus, the exponential function is:\[ g(x) = 36 \cdot 1.5^x \]

Key Concepts

Linear Functions

Linear functions are fundamental in understanding many mathematical concepts. These functions are characterized by a constant rate of change, which means their graph is a straight line. In simpler terms, a linear function grows or shrinks by the same amount with each step of the variable.

If you imagine climbing a hill where every step you take has the same height, that's like following a linear function.

Mathematically, linear functions are expressed in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. The slope, \( m \), represents how steep the line is and the y-intercept, \( b \), indicates where the line crosses the y-axis.

In our example, function \( h(x) \) is linear because it has a constant rate of change of \(-3\). Thus, its formula can be expressed as \( h(x) = -3x + 31 \), showing the consistent reduction by 3 as \( x \) increases by 1.

Exponential Functions

Exponential functions differ significantly from linear functions. Instead of having a constant difference between values, exponential functions have a constant ratio. This means each output is obtained by multiplying the previous output by the same number, referred to as the base.

If you've ever seen something grow quickly, starting small and getting exponentially larger, you've witnessed an exponential function in action.

The general form of an exponential function is \( y = a \cdot r^x \), where \( a \) is a constant representing the initial value, and \( r \) is the base or growth factor.

In the exercise, \( g(x) \) is exponential with a constant ratio of 1.5. Thus, its formula is \( g(x) = 36 \cdot 1.5^x \). Here, 36 is the initial value and 1.5 is the constant that each output is multiplied by to obtain the next.

Rate of Change

The rate of change is an essential concept when analyzing functions. In simple terms, it measures how fast a function's output values change concerning its input values. It's a critical aspect to determine whether a function is linear or exponential.

For linear functions, the rate of change is constant, which means the function increases or decreases by the same amount as the input changes. This creates a straight line when plotted on a graph.

For example, in function \( h(x) \), the rate of change is \(-3\), consistently decreasing by the same amount.

For exponential functions, though, we look at a constant ratio instead. The outputs grow by multiplying by a constant, rather than adding. In \( g(x) \), the constant ratio of 1.5 determines how outputs increase rapidly as inputs increase.

Constant Ratio

A constant ratio is a key feature that distinguishes exponential functions from other types. It indicates that as the independent variable increases, the function's value multiplies by the same factor each time.

Imagine you have a plant that triples its height every day. Here, 3 is the constant ratio because every day the height is 3 times what it was the previous day.

This constant growth factor can be found by taking the ratio of consecutive terms in the function's sequence. If these ratios are constant, the function is exponential.

In the original exercise, \( g(x) \) shows a constant ratio of 1.5 between consecutive outputs. This characteristic helps us identify it as an exponential function and find its expression, which is \( g(x) = 36 \cdot 1.5^x \).