Problem 46
Question
Use the methods of this section to sketch the curve \(y=x^{3}-3 a^{2} x+2 a^{3},\) where \(a\) is a positive constant. What do the members of this family of curves have in common? How do they differ from each other?
Step-by-Step Solution
Verified Answer
Curves share critical points but differ in width and position based on \(a\).
1Step 1: Identify the Function Type and Key Parameters
The function given is a cubic polynomial: \(y = x^3 - 3a^2x + 2a^3\). Identify the key parameter \(a\), which is a constant and positive, and affects the position and shape of the curve.
2Step 2: Find the Derivative
Find the first derivative \(y'\) to determine the critical points: \[y' = \frac{d}{dx}(x^3 - 3a^2x + 2a^3) = 3x^2 - 3a^2.\]
3Step 3: Solve for Critical Points
Set the derivative equal to zero to find critical points: \(3x^2 - 3a^2 = 0\). Simplifying, \(x^2 = a^2\), gives critical points \(x = a\) and \(x = -a\).
4Step 4: Determine Nature of Critical Points
Use the second derivative test to determine whether these points are maxima, minima, or points of inflection. The second derivative is: \[y'' = \frac{d}{dx}(3x^2 - 3a^2) = 6x.\] Substitute \(x = a\), \(y'' = 6a > 0\), indicating a local minimum. Substitute \(x = -a\), \(y'' = -6a < 0\), indicating a local maximum.
5Step 5: Analyze End Behavior
As \(x\to \infty\), \(y = x^3 - 3a^2x + 2a^3\) will grow positively, and as \(x\to -\infty\), it will grow negatively, as the leading term \(x^3\) dominates the behavior.
6Step 6: Sketch the Curve
Based on previous analysis, sketch the curve: a cubic graph with local maximum at \(x = -a\), local minimum at \(x = a\), and behavior extending to infinity in opposite directions.
7Step 7: Compare Family of Curves
All curves have a similar shape due to their cubic nature, sharing the property of turning points at \(x = -a\) and \(x = a\). They differ by the specific \(a\) value, altering their width and position along the x-axis.
Key Concepts
Critical PointsDerivative TestEnd BehaviorCurve Sketching
Critical Points
When working with cubic polynomials like the function given in the exercise, understanding critical points is key to analyzing the behavior of the curve. Critical points occur where the first derivative of the function, which represents the slope of the tangent, is zero. This helps us identify where the function changes direction, forming peaks and troughs.
For the function \(y = x^3 - 3a^2x + 2a^3\), the first derivative is \(y' = 3x^2 - 3a^2\). To find the critical points, set this derivative equal to zero: \(3x^2 - 3a^2 = 0\). Solving this gives \(x = a\) and \(x = -a\), marking the points where the function's slope is zero. These points are crucial in curve sketching because they indicate where the curve will have a local maximum or minimum.
For the function \(y = x^3 - 3a^2x + 2a^3\), the first derivative is \(y' = 3x^2 - 3a^2\). To find the critical points, set this derivative equal to zero: \(3x^2 - 3a^2 = 0\). Solving this gives \(x = a\) and \(x = -a\), marking the points where the function's slope is zero. These points are crucial in curve sketching because they indicate where the curve will have a local maximum or minimum.
Derivative Test
The derivative test, particularly the second derivative test, helps us determine the nature of the critical points. Once we have our critical points, we apply this test to classify each as a maximum, minimum, or point of inflection.
The second derivative of our function, \(y'' = 6x\), tells us about the concavity of the function at the critical points. For \(x = a\), \(y'' = 6a\), which is positive, indicating a local minimum at this point. Conversely, at \(x = -a\), \(y'' = -6a\), which is negative, meaning there is a local maximum.
This analysis using the second derivative is particularly useful as it gives us structural insights into the curve's peaks and troughs, allowing for more accurate graphing.
The second derivative of our function, \(y'' = 6x\), tells us about the concavity of the function at the critical points. For \(x = a\), \(y'' = 6a\), which is positive, indicating a local minimum at this point. Conversely, at \(x = -a\), \(y'' = -6a\), which is negative, meaning there is a local maximum.
This analysis using the second derivative is particularly useful as it gives us structural insights into the curve's peaks and troughs, allowing for more accurate graphing.
End Behavior
The end behavior of a polynomial function describes how the function behaves as \(x\) approaches infinity or negative infinity. For cubic polynomials like \(y = x^3 - 3a^2x + 2a^3\), the term \(x^3\) predominates, significantly influencing the shape of the graph.
As \(x\) approaches infinity, the \(x^3\) term causes \(y\) to approach infinity as well. Similarly, as \(x\) approaches negative infinity, the \(x^3\) term causes \(y\) to approach negative infinity.
Understanding this behavior is crucial because it provides boundary conditions for sketching the graph. Knowing the end behavior allows us to properly align the cubic curve's tails in both directions.
As \(x\) approaches infinity, the \(x^3\) term causes \(y\) to approach infinity as well. Similarly, as \(x\) approaches negative infinity, the \(x^3\) term causes \(y\) to approach negative infinity.
Understanding this behavior is crucial because it provides boundary conditions for sketching the graph. Knowing the end behavior allows us to properly align the cubic curve's tails in both directions.
Curve Sketching
Curve sketching is the process of creating a visual representation of a function based on its algebraic form and mathematical characteristics. In this exercise, with our understanding of critical points, derivative tests, and end behavior, sketching the curve becomes straightforward.
Begin by plotting the critical points \(x = a\) (a local minimum) and \(x = -a\) (a local maximum) on the graph. Then, utilize the end behavior to extend the graph's tails—rising towards infinity on the right and falling towards negative infinity on the left.
Begin by plotting the critical points \(x = a\) (a local minimum) and \(x = -a\) (a local maximum) on the graph. Then, utilize the end behavior to extend the graph's tails—rising towards infinity on the right and falling towards negative infinity on the left.
- Identify and mark the x-intercepts if necessary.
- Draw a smooth, continuous curve that passes through these critical points and follows the determined end behavior.
- Consider symmetry or repetitive patterns, if applicable.
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