Problem 2
Question
Find the derivative at the indicated point from the graph of each function. $$ f(x)=-3 x ; x=-2 $$
Step-by-Step Solution
Verified Answer
The derivative at \( x = -2 \) for the function \( f(x) = -3x \) is \( -3 \).
1Step 1: Review the Derivative Concept
The derivative of a function at a point gives the slope of the tangent line to the function's graph at that particular point. For a linear function like \( f(x) = -3x \), the derivative is constant since the graph is a straight line.
2Step 2: Calculate the Derivative
For the function \( f(x) = -3x \), the derivative can be found using the basic rule that the derivative of \( mx + b \) is \( m \). Therefore, the derivative \( f'(x) = -3 \) for all \( x \).
3Step 3: Evaluate the Derivative at the Given Point
According to the problem, we need to find the derivative at \( x = -2 \). Since \( f'(x) = -3 \) for all \( x \), then \( f'(-2) = -3 \).
4Step 4: Conclusion and Interpretation
The slope of the tangent line to the function \( f(x) = -3x \) at \( x = -2 \) is \( -3 \). This means that at any point on this linear function's graph, the slope is always \( -3 \), indicating a consistently decreasing function.
Key Concepts
Understanding DerivativesExploring Tangent LinesUnderstanding the Slope of Linear Functions
Understanding Derivatives
A derivative in calculus is a key concept used to understand the rate of change of a function with respect to one of its variables. Think of it as a tool that tells you how steep the graph of a function is at any given point. The derivative of a function at a particular point gives us the slope of the tangent line at that point.
When dealing with linear functions, the derivative is particularly straightforward. For a function like \( f(x) = -3x \), you'll notice that its graph is a straight line. This implies that the slope, and therefore the derivative, is constant everywhere on the graph.
To find the derivative, we often rely on derivative rules that simplify the process. For instance, the derivative of a linear function in the form \( mx + b \) is simply \( m \), the coefficient of \( x \). Therefore, the derivative of \( f(x) = -3x \) is \( -3 \), independent of \( x \).
When dealing with linear functions, the derivative is particularly straightforward. For a function like \( f(x) = -3x \), you'll notice that its graph is a straight line. This implies that the slope, and therefore the derivative, is constant everywhere on the graph.
To find the derivative, we often rely on derivative rules that simplify the process. For instance, the derivative of a linear function in the form \( mx + b \) is simply \( m \), the coefficient of \( x \). Therefore, the derivative of \( f(x) = -3x \) is \( -3 \), independent of \( x \).
Exploring Tangent Lines
Tangent lines are straight lines that touch a curve at only one point and have the same slope as the curve at that point. Imagining this can help you visualize how derivatives work in calculus.
When you compute a derivative at a point, you are effectively finding out the slope of the tangent line at that specific point on the graph.
For the linear function \( f(x) = -3x \), the tangent line at any point \( x \) on the graph is simply the line itself (since it's already straight). This means the slope of the tangent line is the same as the slope of the function everywhere, which is \( -3 \). Because of this trait, the tangent line of a linear function doesn't change its slope; it remains constant across the entire graph.
When you compute a derivative at a point, you are effectively finding out the slope of the tangent line at that specific point on the graph.
For the linear function \( f(x) = -3x \), the tangent line at any point \( x \) on the graph is simply the line itself (since it's already straight). This means the slope of the tangent line is the same as the slope of the function everywhere, which is \( -3 \). Because of this trait, the tangent line of a linear function doesn't change its slope; it remains constant across the entire graph.
Understanding the Slope of Linear Functions
The slope is an essential concept in understanding linear functions.
It measures the rate of change of the function. In linear functions, the slope is constant, meaning it doesn't change regardless of where you look on the function's graph.
For the function \( f(x) = -3x \), the slope is \( -3 \). This tells us that for every one unit increase in \( x \), \( f(x) \) decreases by 3 units. Consequently, the graph of this function slants downwards from left to right at a constant rate.
It measures the rate of change of the function. In linear functions, the slope is constant, meaning it doesn't change regardless of where you look on the function's graph.
For the function \( f(x) = -3x \), the slope is \( -3 \). This tells us that for every one unit increase in \( x \), \( f(x) \) decreases by 3 units. Consequently, the graph of this function slants downwards from left to right at a constant rate.
- Positive slope: the line inclines upwards.
- Negative slope: the line inclines downwards, as seen in our function \( f(x) = -3x \).
- Zero slope: results in a horizontal line, indicating no change as \( x \) varies.
Other exercises in this chapter
Problem 1
Find the inverse of each function and differentiate each inverse in two ways: (i) Differentiate the inverse function directly, and (ii) use (4.12) to find the d
View solution Problem 2
Use the formula $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ to approximate the value of the given function. Then compare your result with the value you get from a
View solution Problem 2
Differentiate the functions with respect to the independent variable. \(f(x)=(4 x+5)^{3}\)
View solution Problem 2
In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=3 \cos x-2 \sin x $$
View solution