B A Curve Has Equation Y = X^3+ 3x^2- 6. A) Obtain Dy/dx And Hence Find The X Co-ordinates Of Any Turning
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Understanding the B A Curve Equation and Its Derivatives
The B A curve described by the equation \( y = x^3 + 3x^2 - 6 \) is a classic example of a cubic function. Such curves are fundamental in calculus, serving as models for various real-world phenomena, including physics, engineering, and economics. A critical aspect of analyzing these curves involves determining their slopes at different points, which is achieved through differentiation. Specifically, calculating the derivative \( \frac{dy}{dx} \) provides insight into the curve's increasing or decreasing nature, as well as identifying points of local maxima, minima, or points of inflection—collectively known as turning points.
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Step 1: Differentiating the Equation \( y = x^3 + 3x^2 - 6 \)
The primary goal here is to find the derivative \( \frac{dy}{dx} \), which represents the slope of the tangent line to the curve at any point \( x \). The process involves applying basic differentiation rules to each term of the polynomial.
Differentiation Rules Used:
- Power Rule: \( \frac{d}{dx} x^n = n x^{n-1} \)
- Constant Rule: \( \frac{d}{dx} c = 0 \), where \( c \) is a constant
Derivation of \( \frac{dy}{dx} \):
Applying the power rule term-by-term:- Derivative of \( x^3 \) is \( 3x^2 \)
- Derivative of \( 3x^2 \) is \( 6x \)
- Derivative of \( -6 \) is \( 0 \)
\[
\boxed{
\frac{dy}{dx} = 3x^2 + 6x
}
\]
This expression signifies how the slope of the B A curve varies with \( x \).
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Step 2: Finding the X-Coordinates of Turning Points
Turning points are points on the curve where the slope of the tangent line is zero, indicating a local maximum or minimum. To find these points, set the derivative \( \frac{dy}{dx} \) to zero and solve for \( x \).
Setting the Derivative to Zero:
\[
3x^2 + 6x = 0
\]
Factor out common terms:
\[
3x(x + 2) = 0
\]
This yields two solutions:
\[
\boxed{
x = 0 \quad \text{or} \quad x = -2
}
\]
These are the potential x-coordinates of the turning points on the curve.
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Step 3: Confirming the Nature of the Turning Points
While setting the derivative to zero gives potential turning points, it's essential to determine whether these are maxima, minima, or points of inflection. This involves analyzing the second derivative.
Calculating the Second Derivative \( \frac{d^2 y}{dx^2} \):
Differentiate \( \frac{dy}{dx} = 3x^2 + 6x \):
\[
\frac{d^2 y}{dx^2} = 6x + 6
\]
Now, evaluate at the critical points:
- For \( x = 0 \):
\[
\frac{d^2 y}{dx^2} = 6(0) + 6 = 6 > 0
\]
Since the second derivative is positive, the function is concave upward at \( x = 0 \), indicating a local minimum.
- For \( x = -2 \):
\[
\frac{d^2 y}{dx^2} = 6(-2) + 6 = -12 + 6 = -6 < 0
\]
Since the second derivative is negative, the function is concave downward at \( x = -2 \), indicating a local maximum.
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Step 4: Calculating the Corresponding Y-Coordinates
To find the actual points, substitute the \( x \)-values back into the original equation:
- For \( x = 0 \):
\[
y = (0)^3 + 3(0)^2 - 6 = -6
\]
Point of local minimum: \( (0, -6) \)
- For \( x = -2 \):
\[
y = (-2)^3 + 3(-2)^2 - 6 = -8 + 3(4) - 6 = -8 + 12 - 6 = -2
\]
Point of local maximum: \( (-2, -2) \)
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Summary of Findings
- The derivative \( \frac{dy}{dx} = 3x^2 + 6x \) indicates the slope of the B A curve at any point \( x \).
- The curve has turning points at \( x = 0 \) and \( x = -2 \).
- At \( x = 0 \), the point \( (0, -6) \) is a local minimum.
- At \( x = -2 \), the point \( (-2, -2) \) is a local maximum.
Additional Insights into the B A Curve
Understanding the behavior of the B A curve involves analyzing its derivatives and the nature of its turning points. Here are some key points:
- Increasing and decreasing intervals:
- When \( \frac{dy}{dx} > 0 \), the curve is increasing.
- When \( \frac{dy}{dx} < 0 \), the curve is decreasing.
- Concavity:
- Determined by the second derivative \( \frac{d^2 y}{dx^2} \), which indicates whether the curve is concave up or down.
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Practical Applications of Derivatives in Curve Analysis
The ability to find derivatives and turning points has numerous practical applications:
- Optimizing functions:
- Finding maximum profit, minimum cost, or optimal resource allocation.
- Analyzing motion, where derivatives represent velocity and acceleration.
- Designing structures with specific curvature or stress points.
- Understanding marginal cost and revenue functions.
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Conclusion and Final Remarks
The process of deriving \( \frac{dy}{dx} \) from the given cubic equation \( y = x^3 + 3x^2 - 6 \) demonstrates fundamental calculus techniques essential for analyzing the behavior of curves. Identifying the critical points at \( x = 0 \) and \( x = -2 \), and classifying them as a minimum and maximum respectively, provides valuable insights into the shape and characteristics of the B A curve. Whether applied in academic settings or real-world scenarios, understanding how to differentiate functions and interpret their derivatives remains a cornerstone of mathematical analysis.
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