Problem 36 - Applied Calculus

Question

Values of \(F(t), G(t),\) and \(H(t)\) are in Table \(1.23 .\) Which graph is concave up and which is concave down? Which function is linear? $$\begin{array}{c|c|c|c} \hline \ell & F(t) & G(t) & H(t) \\ \hline 10 & 15 & 15 & 15 \\ 20 & 22 & 18 & 17 \\ 30 & 28 & 21 & 20 \\ 40 & 33 & 24 & 24 \\ 50 & 37 & 27 & 29 \\ 60 & 40 & 30 & 35 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

G(t) is linear, F(t) is concave down, H(t) is concave up.

Step 1: Understand the problem

We need to determine the concavity of the graphs for functions \(F(t)\), \(G(t)\), and \(H(t)\) based on the values given in the table. Additionally, we need to identify if any of these functions is linear.

Step 2: Check for Linearity

A function is linear if it exhibits a constant rate of change, which means the difference between consecutive function values is constant. Let's compute these differences for each function:- \( F(t): 22-15 = 7, 28-22 = 6, 33-28 = 5, 37-33 = 4, 40-37 = 3 \)- \( G(t): 18-15 = 3, 21-18 = 3, 24-21 = 3, 27-24 = 3, 30-27 = 3 \)- \( H(t): 17-15 = 2, 20-17 = 3, 24-20 = 4, 29-24 = 5, 35-29 = 6 \)Only \( G(t) \) has constant differences, indicating it is linear.

Step 3: Determine Concavity of F(t)

To determine the concavity, examine the second differences (i.e., the difference of differences) of \( F(t) \):- First differences: \( 7, 6, 5, 4, 3 \)- Second differences: \( 6-7 = -1, 5-6 = -1, 4-5 = -1, 3-4 = -1 \)Since all second differences are negative, \( F(t) \) is concave down.

Step 4: Determine Concavity of H(t)

For \( H(t) \), let's evaluate the second differences:- First differences: \( 2, 3, 4, 5, 6 \)- Second differences: \( 3-2 = 1, 4-3 = 1, 5-4 = 1, 6-5 = 1 \)All second differences are positive, indicating that \( H(t) \) is concave up.

Key Concepts

Linear Functions

Linear functions are a crucial concept in mathematics, especially when examining graphs and their behaviors. A linear function is characterized by a constant rate of change. This means that when you plot the function on a graph, it forms a straight line. In mathematical terms, if you have a function with values at regular intervals, the difference between consecutive values remains the same. For example, looking at the data provided for the functions, when we calculate the first differences of function \( G(t) \), we find:

• \( G(t): 18-15 = 3 \)
• \( 21-18 = 3 \)
• \( 24-21 = 3 \)
• \( 27-24 = 3 \)
• \( 30-27 = 3 \)

All these differences are equal, confirming that \( G(t) \) is a linear function. If you are given a table of values, you can use this method to check for linearity by ensuring the first differences are constant. This simple test directly tells us if a function is linear. Understanding linear functions helps in identifying trends and making predictions based on the linear relationships.

Second Differences

The concept of second differences is essential when analyzing the curvature of a graph, which tells us about concavity. The first differences tell us how much each value of the function changes from one point to the next. Second differences then measure how these first differences change. Here's how it works:

• First, calculate the first differences between consecutive function values.
• Then, calculate the differences between these first differences—these are the second differences.

Analyzing second differences helps us determine whether a function is concave up or concave down. For instance, let's consider \( F(t) \), which shows the following first differences:

• \( 7, 6, 5, 4, 3 \)

The second differences are:

• \( 6 - 7 = -1 \)
• \( 5 - 6 = -1 \)
• \( 4 - 5 = -1 \)
• \( 3 - 4 = -1 \)

All second differences are negative, indicating that \( F(t) \) is concave down. If the second differences are positive, like in \( H(t) \), this indicates concave up.

Rate of Change

The rate of change is a fundamental concept that measures how a function's value changes concerning its input. This concept is analogous to the slope in linear functions and is applicable across various types of functions to understand how they behave. To evaluate the rate of change, calculate the difference between successive outputs of a function for successive input values. For example, in \( G(t) \), the rate of change is constant, as discussed earlier, which gives us the first differences such as:

• \( 3, 3, 3, 3, 3 \)

This constancy tells us that for every unit increase in "t", the function value increases by the same amount, signifying a constant rate of change. In contrast, \( F(t) \) and \( H(t) \) have varying rates of change, which we infer from their changing first differences. A variable rate of change often signals that the graph may not be linear, and it usually exhibits some form of concavity, like concave up or concave down based on the second differences. Understanding the rate of change helps us predict and explain the dynamic behavior of functions in practical applications.