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Chapter 13: Problem 25

By considering the graph of the function but not calculating any limits, give the value of \(f^{\prime}(2)\) for each function. $$f(x)=-x$$

Short Answer

Expert verified
\(f'(2) = -1\)

Step by step solution

01

Identify Function Type

The given function is a linear function: \(f(x) = -x\). Linear functions have constant slopes except at discontinuities, which do not exist for this function.
02

Determine the Slope from the Linear Function

For a linear function \(f(x) = mx + b\), the derivative \(f'(x)\) is simply the constant \(m\), representing the slope. Here, \(f(x) = -x\) implies that \(m = -1\).
03

Evaluate the Derivative at the Given Point

Since the derivative \(f'(x)\) is constant at \(-1\) for any \(x\), it follows that \(f'(2) = -1\).

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Key Concepts

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

Linear Functions
A linear function is one of the simplest and most fundamental types of functions represented by the formula \( f(x) = mx + b \). In this equation:
  • \( m \) is the slope of the line
  • \( b \) is the y-intercept, which is the point where the line crosses the y-axis
Linear functions create straight lines when graphed, making them easy to analyze. Since they form a straight line, they have a constant rate of change. This simplicity is why they serve as a perfect introduction to other mathematical concepts like differentiation.
Slope
The slope of a line is a measure of its steepness. It tells us how much the y-value of a function changes for a change in the x-value. For a linear function represented by \( f(x) = mx + b \), the slope is the constant \( m \).
Here's how slope is helpful:
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope means the line falls as it moves from left to right.
  • If the slope is zero, the line is horizontal, indicating no vertical change.
In the function \( f(x) = -x \), the slope is \( -1 \). This means that for every 1 unit increase in \( x \), the y-value decreases by 1.
Graphical Analysis
Graphical analysis involves examining the graph of a function to understand its behavior. With linear functions, this means observing the line, its slope, and its y-intercept.
Some key points include:
  • Linear functions yield straight lines.
  • The steepness of the line is determined by the slope.
  • The position where the line crosses the y-axis is the y-intercept.
In our example, the function \( f(x) = -x \) results in a line with a negative slope. If we plotted this, the line would descend from left to right, perfectly matching the mathematical description that its slope is \(-1\). Analyzing graphs is an intuitive way to grasp the underlying properties of functions.
Differentiation
Differentiation is a process in calculus that helps us find the derivative of a function. The derivative is a tool that measures how a function's output changes as its input changes. For a linear function, the derivative is straightforward鈥攊t is simply the slope \( m \).
For any function \( f(x) = mx + b \):
  • The derivative \( f'(x) = m \).
  • This derivative is constant because linear functions have a uniform rate of change.
In our problem, since \( f(x) = -x \), the derivative is \( f'(x) = -1 \). This tells us that the function changes at a constant rate of \(-1\) no matter the value of \( x \). Differentiation provides a powerful way to quantify change, forming the foundation for more complex calculus concepts.

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Most popular questions from this chapter

Find the exact value of each integral, using formulas from geometry. Do not use a calculator. $$\int_{-4}^{0} \sqrt{16-x^{2}} d x$$

Find the equation of the tangent line to the function \(f\) at the given point. Then graph the function and the tangent line together. $$f(x)=\frac{1}{2} x^{2}-2 \text { at }(2,0)$$

Find \(f^{\prime}(x)\) using the alternative definition. $$f(x)=2 x^{2}-x$$

Approximate the area under the graph of \(f(x)\) and above the \(x\) -axis, using each of the following methods with \(n=4\). (a) Use left endpoints. (b) Use right endpoints. (c) Average the answers in parts ( \(a\) ) and ( \(b\) ). (d) Use midpoints. $$f(x)=\frac{2}{x} \text { from } x=1 \text { to } x=9$$

Solve each problem. A helicopter is gradually rising straight up in the air. Its distance from the ground \(t\) seconds after takeoff is \(s(t)\) feet, where $$s(t)=t^{2}+t$$. (a) How long will it take for the helicopter to rise 20 feet? (b) Find the vertical velocity of the helicopter when it is 20 feet above the ground.
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