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Chapter 3: Problem 17

\(17-28\) A function is given. Determine the average rate of change of the function between the given values of the variable. $$ f(x)=3 x-2 ; \quad x=2, x=3 $$

Short Answer

Expert verified
The average rate of change is 3.

Step by step solution

01

Understand the formula for the average rate of change

The average rate of change of a function between two points is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \), where \( a \) and \( b \) are the given values of \( x \).
02

Identify the given values

In the problem, the given values of \( x \) are \( x = 2 \) and \( x = 3 \). Thus, \( a = 2 \) and \( b = 3 \).
03

Evaluate the function at \( x = 2 \)

Substitute \( x = 2 \) into the function: \[ f(2) = 3(2) - 2 = 6 - 2 = 4 \]
04

Evaluate the function at \( x = 3 \)

Substitute \( x = 3 \) into the function: \[ f(3) = 3(3) - 2 = 9 - 2 = 7 \]
05

Apply the average rate of change formula

Use the values calculated: \[ \frac{f(3) - f(2)}{3 - 2} = \frac{7 - 4}{3 - 2} = \frac{3}{1} = 3 \].
06

Conclude the solution

The average rate of change of the function from \( x = 2 \) to \( x = 3 \) is 3.

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Key Concepts

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

Linear Functions
Linear functions are fundamental concepts in algebra that describe a straight line when graphed on a coordinate plane. A linear function is expressed in the form: \( f(x) = mx + b \) Here, \( m \) represents the slope, and \( b \) is the y-intercept. The slope \( m \) indicates how steep the line is, and the y-intercept \( b \) is the point where the line crosses the y-axis.
  • In our function \( f(x) = 3x - 2 \), the slope \( m = 3 \) means the line rises 3 units for every 1 unit increase in \( x \).
  • The y-intercept \( b = -2 \) means the line crosses the y-axis at the point (0, -2).
Understanding linear functions helps in predicting how changes in one variable affect another, which is crucial for evaluating the average rate of change.
Evaluating Functions
Evaluating functions involves finding the output value of a function for a particular input value. It is akin to solving for \( y \) in a given function \( f(x) \). To evaluate a function, substitute the given input value into the function and perform the necessary calculations. In our problem, we evaluated the function \( f(x) = 3x - 2 \) at two different points:
  • First, for \( x = 2 \): substitute 2 in place of \( x \) to get \( f(2) = 3(2) - 2 = 4 \).
  • Next, for \( x = 3 \): substitute 3 in place of \( x \) to get \( f(3) = 3(3) - 2 = 7 \).
By evaluating the function at these specific points, you gain the necessary values to calculate the average rate of change.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols. They are the building blocks of algebra and present various quantities in mathematical statements. Understanding how to manipulate algebraic expressions is key to solving equations and evaluating functions. For our function \( f(x) = 3x - 2 \), substitute different values for \( x \) to calculate the expression's value at those points. This step is crucial for calculating changes between different states, such as finding the average rate of change.
  • An algebraic expression like \( 3x - 2 \) incorporates a variable term \( 3x \) and a constant term \(-2\), revealing linearity.
  • Understanding how to substitute and simplify algebraic expressions ensures accurate function evaluation.
Practicing the process of substituting and simplifying helps in recognizing patterns and changes in the function's output.

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Most popular questions from this chapter

Even and Odd Power Functions What must be true about the integer \(n\) if the function $$f(x)=x^{n}$$ is an even function? If it is an odd function? Why do you think the names "even" and "odd" were chosen for these function properties?

Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose \(g(x)=\sqrt{f(x)},\) where \(f(x) \geq 0\) for all \(x\) . Explain why the local minima and maxima of \(f\) and \(g\) occur at the same values of \(x .\) (b) Let \(g(x)\) be the distance between the point \((3,0)\) and the point \(\left(x, x^{2}\right)\) on the graph of the parabola \(y=x^{2}\) Express \(g\) as a function of \(x .\) (c) Find the minimum value of the function \(g\) that you found in part (b). Use the principle described in part (a) to simplify your work.

Coughing When a foreign object lodged in the trachea (windpipe) forces a person to cough, the diaphragm thrusts upward causing an increase in pressure in the lungs. At the same time, the trachea contracts, causing the expelled air to move faster and increasing the pressure on the foreign object. According to a mathematical model coughing, the velocity \(v\) of the airstream through an average-sized person's trachea is related to the radius \(r\) of the trachea (in centimeters) by the function $$ v(r)=3.2(1-r) r^{2}, \quad \frac{1}{2} \leq r \leq 1 $$ Determine the value of \(r\) for which \(v\) is a maximum.

Determine whether the equation defines y as a function of x. (See Example 10.) $$ 3 x+7 y=21 $$

Changing Temperature Scales The temperature on a certain afternoon is modeled by the function $$C(t)=\frac{1}{2} t^{2}+2$$ where \(t\) represents hours after 12 noon \((0 \leq t \leq 6),\) and \(C\) is measured in "C. (a) What shifting and shrinking operations must be performed on the function \(y=t^{2}\) to obtain the function \(y=C(t) ?\) (b) Suppose you want to measure the temperature in \(^{\circ} \mathrm{F}\) instead. What transformation would you have to apply to the function \(y=C(t)\) to accomplish this? (Use the fact that the relationship between Celsius and Fahrenheit degrees is given by \(F=\frac{9}{5} C+32 .\) ) Write the new function \(y=F(t)\) that results from this transformation.
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