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Chapter 2: Problem 5

Calculate the average rate of change between adjacent points for the following functions and place the values in a third column in each table. (The first entry is "n.a.") $$ \begin{array}{rr} \hline x & f(x) \\ \hline 0 & 5 \\ 10 & 25 \\ 20 & 45 \\ 30 & 65 \\ 40 & 85 \\ 50 & 105 \\ \hline \end{array} $$ $$ \begin{array}{rl} \hline x & g(x) \\ \hline 0 & 270 \\ 10 & 240 \\ 20 & 210 \\ 30 & 180 \\ 40 & 150 \\ 50 & 120 \\ \hline \end{array} $$ a. Are the functions \(f(x)\) and \(g(x)\) increasing, decreasing, or constant throughout? b. Is the average rate of change of each function increasing, decreasing, or constant throughout?

Short Answer

Expert verified
f(x) is increasing and has a constant average rate of change; g(x) is decreasing and has a constant average rate of change.

Step by step solution

01

Calculate average rate of change for function f(x)

The average rate of change between two points is given by the formula \(\frac{f(x_2) - f(x_1)}{x_2 - x_1}\). For function f(x), calculate the average rate of change between adjacent points and fill the values in a third column: \[\begin{array}{|c|c|c|} \hline x & f(x) & \text{Average Rate of Change} \ \hline 0 & 5 & \text{n.a.} \ 10 & 25 & \frac{25-5}{10-0} = 2 \ 20 & 45 & \frac{45-25}{20-10} = 2 \ 30 & 65 & \frac{65-45}{30-20} = 2 \ 40 & 85 & \frac{85-65}{40-30} = 2 \ 50 & 105 & \frac{105-85}{50-40} = 2 \ \hline \end{array}\]
02

Calculate average rate of change for function g(x)

Similarly, calculate the average rate of change for the function g(x) and fill in the third column: \[\begin{array}{|c|c|c|} \hline x & g(x) & \text{Average Rate of Change} \ \hline 0 & 270 & \text{n.a.} \ 10 & 240 & \frac{240-270}{10-0} = -3 \ 20 & 210 & \frac{210-240}{20-10} = -3 \ 30 & 180 & \frac{180-210}{30-20} = -3 \ 40 & 150 & \frac{150-180}{40-30} = -3 \ 50 & 120 & \frac{120-150}{50-40} = -3 \ \hline \end{array}\]
03

Determine the behavior of function f(x)

Observe the values of function f(x). Since f(x) increases from 5 to 105 as x goes from 0 to 50, function f(x) is increasing.
04

Determine the behavior of function g(x)

Observe the values of function g(x). Since g(x) decreases from 270 to 120 as x goes from 0 to 50, function g(x) is decreasing.
05

Analyze the average rate of change for function f(x)

The average rate of change for function f(x) is consistently 2 for each interval. Therefore, the average rate of change for function f(x) is constant throughout.
06

Analyze the average rate of change for function g(x)

The average rate of change for function g(x) is consistently -3 for each interval. Hence, the average rate of change for function g(x) is constant throughout.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

increasing function
In mathematical terms, an **increasing function** is one where, as the input value (x) increases, the output value (f(x)) also increases. This means that the graph of the function moves upwards as you go from left to right.

Looking at the given function **f(x)**, we observe that it starts at 5 when x is 0 and goes up to 105 when x is 50. Here's a quick summary:
* When x is 10, f(x) is 25.
* At x = 20, f(x) is 45.
* At x = 30, f(x) is 65.
* At x = 40, f(x) is 85.
* Finally, at x = 50, f(x) is 105.

Because each successive value of f(x) is greater than the previous one, we can conclude that the function **f(x)** is an increasing function.
decreasing function
A **decreasing function** is the opposite of an increasing function. As the input value (x) increases, the output value (f(x)) decreases. So, the graph of the function moves downwards as you go from left to right.

Consider the given function **g(x)**. It starts at 270 when x is 0 and decreases to 120 when x is 50. Let's look at a few points in between:
* When x is 10, g(x) is 240.
* At x = 20, g(x) is 210.
* At x = 30, g(x) is 180.
* At x = 40, g(x) is 150.
* Finally, at x = 50, g(x) is 120.

Each successive value of g(x) is smaller than the previous one, which means that the function **g(x)** is a decreasing function.
constant rate of change
The **rate of change** of a function tells us how much the function's output changes for a given change in input. When this rate of change is constant, it means the output changes by the same amount for each unit increase in input.

For the function **f(x)**, we calculated the average rate of change as follows:
  • Between x = 0 and x = 10, \(\frac{25 - 5}{10 - 0} = 2\)
  • Between x = 10 and x = 20, \(\frac{45 - 25}{20 - 10} = 2\)
  • Between x = 20 and x = 30, \(\frac{65 - 45}{30 - 20} = 2\)
  • Between x = 30 and x = 40, \(\frac{85 - 65}{40 - 30} = 2\)
  • Between x = 40 and x = 50, \(\frac{105 - 85}{50 - 40} = 2\)

We see that for each interval, the rate of change is consistently 2. This means that the rate of change for **f(x)** is constant.

Similarly, for the function **g(x)**, the average rate of change is:
  • Between x = 0 and x = 10, \(\frac{240 - 270}{10 - 0} = -3\)
  • Between x = 10 and x = 20, \(\frac{210 - 240}{20 - 10} = -3\)
  • Between x = 20 and x = 30, \(\frac{180 - 210}{30 - 20} = -3\)
  • Between x = 30 and x = 40, \(\frac{150 - 180}{40 - 30} = -3\)
  • Between x = 40 and x = 50, \(\frac{120 - 150}{50 - 40} = -3\)

In this case, the rate of change is -3 for every interval, indicating that the rate of change for **g(x)** is also constant.

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Most popular questions from this chapter

Find the equation of the line in the form \(y=m x+b\) for each of the following sets of conditions. Show your work. a. Slope is \(\$ 1400 /\) year and line passes through the point \((10 \mathrm{yr}, \$ 12,000)\) b. Line is parallel to \(2 y-7 x=y+4\) and passes through the point (-1,2) c. Equation is \(1.48 x-2.00 y+4.36=0 .\) d. Line is horizontal and passes through ( 1.0,7.2 ). e. Line is vertical and passes through ( 275,1029 ). \(\mathbf{f}\). Line is perpendicular to \(y=-2 x+7\) and passes through (5,2)

(Graphing program optional.) Solve each equation for \(y\) in terms of \(x\), then identify the slope and the \(y\) -intercept. Graph each line by hand. Verify your answers with a graphing utility if available. a. \(-4 y-x-8=0\) b. \(\frac{1}{2} x-\frac{1}{4} y=3\) c. \(-4 x-3 y=9\) d. \(6 x-5 y=15\)

On the same axes, graph (and label with the correct equation) three lines that go through the point (0,2) and have the following slopes: a. \(m=-\frac{1}{2}\) b. \(m=-2\) c. \(m=-\frac{5}{6}\)

(Graphing program optional.) The equation \(K=4 F-160\) models the relationship between \(F\), the temperature in degrees Fahrenheit, and \(K,\) the number of chirps per minute for the snow tree cricket. a. Assuming \(F\) is the independent variable and \(K\) is the dependent variable, identify the slope and vertical intercept in the given equation. b. Identify the units for \(K, 4, F,\) and -160 . c. What is a reasonable domain for this model? d. Generate a small table of points that satisfy the equation. Be sure to choose realistic values for \(F\) from the domain of your model. e. Calculate the slope directly from two data points. Is this value what you expected? Why? f. Graph the equation, indicating the domain.

(Graphing program optional.) The accompanying table indicates the number of juvenile arrests (in thousands) in the United States for aggravated assault. $$ \begin{array}{ccc} \hline & \begin{array}{c} \text { Juvenile } \\ \text { Arrests } \\ \text { Year } \end{array} & \begin{array}{c} \text { Annual Average } \\ \text { (thousands) } \end{array} & \begin{array}{c} \text { Rate of Change } \\ \text { over Prior 5 Years } \end{array} \\ \hline 1985 & 36.8 & \text { n.a. } \\ 1990 & 54.5 & \\ 1995 & 68.5 & \\ 2000 & 49.8 & \\ 2005 & 36.9 & \\ \hline \end{array} $$ a. Fill in the third column in the table by calculating the annual average rate of change. b. Graph the annual average rate of change versus time. c. During what 5 -year period was the annual average rate of change the largest? d. Describe the change in aggravated assault cases during these years by referring both to the number and to the annual average rate of change.
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