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Chapter 3: Problem 56

Consider the line \(f(x)=m x+b,\) where \(m\) and \(b\) are constants. Show that \(f^{\prime}(x)=m\) for all \(x .\) Interpret this result.

Short Answer

Expert verified
Answer: The derivative of a linear function, given by \(f(x) = mx + b\), is equal to its slope, which is represented by the constant \(m\). The derivative indicates that the rate of change or the gradient of the function is constant for all values of \(x\). This means that the slope remains the same for any point on the line defined by the function.

Step by step solution

01

Identify the function and its type

The given function is \(f(x) = mx + b\), which is a linear function with slope \(m\) and a y-intercept \(b\).
02

Apply the rules of differentiation

To find the derivative of \(f(x)\) with respect to \(x\), we will apply the power rule of differentiation. The power rule states that if \(f(x) = ax^n\), where \(a\) and \(n\) are constants, then the derivative is \(f'(x) = anx^{n-1}\). Considering the function \(f(x) = mx + b\), the powers of \(x\) in both terms are \(1\) for the first term, and \(0\) for the second term.
03

Differentiate the first term

Following the power rule, let's first differentiate the term \(mx\). We have: \(f'(x) = m \cdot 1x^{1-1} = m \cdot 1x^0 = m \cdot 1 = m\)
04

Differentiate the second term

Now, we differentiate the term \(b\). We have: \(f'(x) = 0 \cdot bx^{0-1} = 0\), as the power (\(0-1\)) is now negative in case of constants.
05

Combine the results

Combine the results from Steps 3 and 4 to find the derivative of the complete function: \(f'(x) = m + 0 = m\)
06

Interpret the result

The result \(f'(x) = m\) indicates that the derivative of a linear function is equal to its slope \(m\). This means that the rate of change or the gradient of the function is constant for all values of \(x\). In other words, for any point on the line \(f(x) = mx + b\), the slope will always be the same, which is \(m\).

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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 among the simplest types of functions you will encounter in mathematics. They are typically defined in the form \(f(x) = mx + b\), where \(m\) represents the slope of the line, and \(b\) denotes the y-intercept.
The slope \(m\) tells us how steep the line is, or how much \(y\) changes as \(x\) increases. A positive slope means the line ascends as you move from left to right, while a negative slope means it descends.
  • The y-intercept \(b\) is the point where the line crosses the y-axis. This occurs when \(x = 0\).
  • Linear functions produce straight lines when graphed on a coordinate plane.
Linear functions are fundamental in understanding concepts of rates of change and are used extensively in various real-life applications, such as predicting trends and analyzing financial data.
Derivative
The derivative of a function provides us with the rate at which the function's value changes with respect to change in its variable. It is a core concept in calculus, denoted by \(f'(x)\), or sometimes \(\frac{df}{dx}\).
For linear functions like \(f(x) = mx + b\), the derivative measures the function's slope, which is constant across the line.
  • The derivative tells us the function's instantaneous rate of change at any given point.
  • For a linear function, the derivative remains the same everywhere on the graph, equivalent to the slope \(m\).
Understanding derivatives is crucial, as they are foundational in topics such as motion, growth rates, and optimization problems.
Derivatives give insight into the behavior of functions, allowing mathematicians and scientists to predict and analyze changes in various contexts.
Power Rule
The power rule is a helpful tool in calculus for differentiating functions of the form \(ax^n\), where \(a\) and \(n\) are constants. It simplifies the process of finding derivatives by providing a quick formula: \(f'(x) = anx^{n-1}\).
This rule is crucial for understanding how algebraic expressions change and is widely applied when dealing with polynomial functions.
  • When differentiating a linear term like \(mx\), the power rule simplifies to \(m\), since \(n = 1\) for \(x\).
  • Constant terms like \(b\) (with \(n = 0\)) differentiate to zero, providing no change or slope contribution.
By mastering the power rule, one can efficiently compute derivatives, a skill essential for advanced studies in mathematics, natural sciences, and engineering disciplines.

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Most popular questions from this chapter

A lighthouse stands 500 m off of a straight shore, the focused beam of its light revolving four times each minute. As shown in the figure, \(P\) is the point on shore closest to the lighthouse and \(Q\) is a point on the shore 200 m from \(P\). What is the speed of the beam along the shore when it strikes the point \(Q ?\) Describe how the speed of the beam along the shore varies with the distance between \(P\) and \(Q\). Neglect the height of the lighthouse.

Use any method to evaluate the derivative of the following functions. $$f(x)=\frac{4-x^{2}}{x-2}$$

Suppose your graphing calculator has two functions, one called sin \(x,\) which calculates the sine of \(x\) when \(x\) is in radians, and the other called \(s(x),\) which calculates the sine of \(x\) when \(x\) is in degrees. a. Explain why \(s(x)=\sin \left(\frac{\pi}{180} x\right)\) b. Evaluate \(\lim _{x \rightarrow 0} \frac{s(x)}{x} .\) Verify your answer by estimating the limit on your calculator.

An observer is \(20 \mathrm{m}\) above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is \(20 \mathrm{m}\) horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer's line of sight makes with the horizontal (it may be positive or negative). Assuming that the elevator rises at a rate of \(5 \mathrm{m} / \mathrm{s}\), what is the rate of change of the angle of elevation when the elevator is \(10 \mathrm{m}\) above the ground? When the elevator is \(40 \mathrm{m}\) above the ground?

Find \(f^{\prime}(x), f^{\prime \prime}(x),\) and \(f^{\prime \prime \prime}(x)\) \(f(x)=\frac{x^{2}-7 x}{x+1}\)
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