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Chapter 4: Problem 7

For Activities 7 through \(18,\) write the first and second derivatives of the function. \(f(x)=-3 x+7\)

Short Answer

Expert verified
First derivative: \( f'(x) = -3 \), Second derivative: \( f''(x) = 0 \).

Step by step solution

01

Understand the Function

The function given is a linear function: \( f(x) = -3x + 7 \). A linear function in the form \( ax + b \) has a constant rate of change, which will assist in finding its derivatives.
02

Calculate the First Derivative

The first derivative of a function, \( f'(x) \), represents the rate of change of the function. For a linear function like \( f(x) = -3x + 7 \), the derivative is simply the coefficient of \( x \). Thus, \( f'(x) = -3 \).
03

Calculate the Second Derivative

The second derivative, \( f''(x) \), represents the rate of change of the first derivative. Since the first derivative \( f'(x) = -3 \) is a constant, its derivative is \( 0 \). Therefore, \( f''(x) = 0 \).

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Key Concepts

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

Understanding the First Derivative
The first derivative of a function, commonly denoted as \( f'(x) \), provides us with the rate of change or the slope of the function at any given point. Imagine you are standing on a hill; the steeper the hill, the faster you're moving up or down. In mathematical terms, this steepness is described by the first derivative.

For linear functions like \( f(x) = ax + b \), the derivative is straightforward. Linear functions have a constant rate of change, which means they have a straight-line graph. So, the steepness or slope of a linear function is constant everywhere along the x-axis.
  • For \( f(x) = -3x + 7 \), the slope is \(-3\), which means for every unit increase in \( x \), \( f(x) \) decreases by 3 units.
  • This is why the first derivative \( f'(x) = -3 \). It indicates a constant decrease in \( f(x) \) as \( x \) increases.
Exploring the Second Derivative
The second derivative of a function, denoted as \( f''(x) \), tells us how the rate of change itself is changing. In other words, it offers us the rate of change of the first derivative. You can think of it as the acceleration if the first derivative is the velocity. This is particularly insightful when dealing with functions that curve or have varying slopes.

In the case of a linear function like \( f(x) = -3x + 7 \), the first derivative \( f'(x) \) is a constant. This constancy implies that there is no acceleration or further change in slope, which leads us to:
  • The second derivative \( f''(x) = 0 \). A zero second derivative in linear functions highlights that the graph is a straight line without any bends or curvatures.
  • This result reinforces the nature of linear functions, which are defined by consistent rates of change.
What Makes a Function Linear?
Linear functions are among the simplest forms of functions in mathematics, characterized by their straight-line graphs. The general form of a linear function is \( f(x) = ax + b \), where \( a \) and \( b \) are constants.

Key characteristics of linear functions include:
  • Constant slope: The value of \( a \) in the function represents the slope, which does not change regardless of \( x \). This means that each step along the x-axis results in a proportional change in \( f(x) \).
  • Intercept: The \( b \) is the y-intercept, which indicates where the line crosses the y-axis.
  • Predictability: Because of their linear nature, these functions allow easy predictions of the output, making them practical for modeling relationships in many fields.
Understanding these features clarifies why derivatives of linear functions are simple to calculate and constant.

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Most popular questions from this chapter

a. Graph \(g, g^{\prime}\), and \(g^{\prime \prime}\) between \(x=0\) and \(x=15 .\) Indicate the relationships among points on the three graphs that correspond to maxima, minima, and inflection points. b. Calculate the input and output of the inflection point on the graph of \(g .\) Is it a point of most rapid decline or least rapid decline? \(g(x)=0.04 x^{3}-0.88 x^{2}+4.81 x+12.11\)

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A set of 24 duplexes ( 48 units) was sold to a new owner. The previous owner charged 700 per month rent and had a 100 % occupancy rate. The new owner, aware that this rental amount was well below other rental prices in the area, immediately raised the rent to 750 Over the next 6 months, an average of 47 units were occupied. The owner raised the rent to 800 and found that 2 of the units were unoccupied during the next 6 months. a. What was the monthly rental income i. for the previous owner? ii. after the first rent increase? iii. after the second rent increase? b. Construct an equation for the monthly rental income as a function of the number of \(\$ 50\) increases in the rent. c. If the occupancy rate continues to decrease as the rent increases in the same manner as it did for the first two increases, what rent amount will maximize the owner's rental income from these duplexes? How much will the owner collect in rent each month at this rental amount? d. What other considerations besides rental income should the owner take into account when determining an optimal rental amount?
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