To find the gradient of the curve given by the equation \( y = 9 - 3x^2 \) at the point where \( x = -2 \), we need to follow these steps:
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Understand the function: Here, we have a function of \( x \), which means we are dealing with a curve. The gradient (or slope) of the curve at any point can be found by calculating the derivative of the function.
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Differentiate the function: The gradient is found by calculating the derivative of \( y \) with respect to \( x \).
Starting with our function: \[ y = 9 - 3x^2 \]
We differentiate this with respect to \( x \): \[ \frac{dy}{dx} = 0 - 6x = -6x \]
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Evaluate the derivative at \( x = -2 \): We now substitute \( x = -2 \) into the derivative we found: \[ \frac{dy}{dx} = -6(-2) = 12 \]
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Interpret the result: The value \( 12 \) represents the gradient of the curve \( y = 9 - 3x^2 \) at the point where \( x = -2 \). This means that at \( x = -2 \), the slope of the curve is 12, indicating that the curve is rising steeply at that point.
Summary:
- The gradient of the curve \( y = 9 - 3x^2 \) at \( x = -2 \) is 12.
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