Problem 47
Question
As \(x\) moves from left to right through the point \(c=2,\) is the graph of \(f(x)=x^{3}-3 x+2\) rising, or is it falling? Give reasons for your answer.
Step-by-Step Solution
Verified Answer
The graph of \(f(x)\) is rising as it passes through \(c=2\).
1Step 1: Determine the Derivative
To understand if the function is rising or falling at a specific point, we need to calculate the derivative of the function. The derivative of a function gives us the slope of the tangent line at any point. Given the function \(f(x) = x^3 - 3x + 2\), differentiate it with respect to \(x\): \(f'(x) = 3x^2 - 3\).
2Step 2: Evaluate the Derivative at c=2
Substitute \(x = 2\) into the derivative to find the slope of the tangent line at that point. Compute \(f'(2) = 3(2)^2 - 3 = 3 \times 4 - 3 = 12 - 3 = 9\).
3Step 3: Analyze the Sign of the Derivative
The sign of the derivative at a particular point tells us whether the function is rising or falling. If \(f'(c) > 0\), then the function is rising at \(x = c\). If \(f'(c) < 0\), the function is falling. Here, \(f'(2) = 9\) which is positive.
Key Concepts
DerivativeTangent LineRising and Falling Functions
Derivative
A derivative represents how a function changes as its input changes. In simpler terms, it tells us the rate of change or the slope of a function at a specific point. For any given function, its derivative can be calculated, allowing us to understand the behavior of the function in detail.
When you take the derivative of a function, you essentially find an equation that gives you the slope of a tangent line to any point on the function's graph. In our example, we had a function:
When you take the derivative of a function, you essentially find an equation that gives you the slope of a tangent line to any point on the function's graph. In our example, we had a function:
- \(f(x) = x^3 - 3x + 2\)
- \(f'(x) = 3x^2 - 3\)
Tangent Line
The tangent line of a graph at a specific point is a straight line that just "touches" the graph at that point. The slope of this line is given by the derivative of the function at that point. This concept is important because it gives us a linear approximation of the function's behavior in the neighborhood of the point.
In our example, by finding the derivative \(f'(x) = 3x^2 - 3\), we learned the slope of the tangent line at any point \(x\). Specifically, when we evaluated the derivative at \(x = 2\), the slope of the tangent line was found to be \(f'(2) = 9\). This means that at \(x = 2\), the tangent line is rising steeply, because the slope (9) indicates a rapid increase of the function.
The tangent line often plays a key role in understanding instantaneous rates of change, and in approximating a curve locally as a straight line. It helps us to approximate values of the function near the point \(c\). This approximation is usually highly accurate because the tangent line is the best linear approximation at the given point.
In our example, by finding the derivative \(f'(x) = 3x^2 - 3\), we learned the slope of the tangent line at any point \(x\). Specifically, when we evaluated the derivative at \(x = 2\), the slope of the tangent line was found to be \(f'(2) = 9\). This means that at \(x = 2\), the tangent line is rising steeply, because the slope (9) indicates a rapid increase of the function.
The tangent line often plays a key role in understanding instantaneous rates of change, and in approximating a curve locally as a straight line. It helps us to approximate values of the function near the point \(c\). This approximation is usually highly accurate because the tangent line is the best linear approximation at the given point.
Rising and Falling Functions
When we describe functions as "rising" or "falling," we're talking about how the function behaves as we move along the x-axis. This behavior is closely linked to the derivative of the function. If the derivative is positive at a point, the function is rising, indicating that as we move from left to right, the function values increase.
Conversely, if the derivative is negative at a certain point, the function is falling, which means the function values decrease as \(x\) increases.
In the context of our example, we calculated the derivative at a specific point \(c = 2\). The result, \(f'(2) = 9\), was positive, telling us that the function \(f(x) = x^3 - 3x + 2\) is rising when \(x=2\). Hence, the function behaves like an upward slope as \(x\) crosses this point.
Understanding whether a function is rising or falling can be useful in many real-world applications such as predicting trends or understanding physical phenomena. In calculus, knowing the sign of the derivative helps quickly determine these behaviors.
Conversely, if the derivative is negative at a certain point, the function is falling, which means the function values decrease as \(x\) increases.
In the context of our example, we calculated the derivative at a specific point \(c = 2\). The result, \(f'(2) = 9\), was positive, telling us that the function \(f(x) = x^3 - 3x + 2\) is rising when \(x=2\). Hence, the function behaves like an upward slope as \(x\) crosses this point.
Understanding whether a function is rising or falling can be useful in many real-world applications such as predicting trends or understanding physical phenomena. In calculus, knowing the sign of the derivative helps quickly determine these behaviors.
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Problem 47
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