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The slope of a tangent line is the instantaneous rate of change of a function at a given point. It gives us an idea of how steep the curve is at that specific location.
To find this slope, we commonly use a mathematical tool called the derivative. The derivative represents how a function changes as its input changes. For the function \(y = x^3\), the derivative is \(\frac{dy}{dx} = 3x^{2}\).
When we want to know the slope at a specific point, like \((2, 8)\), we input the \(x\)-value from the point into the derivative. Substitute \(x = 2\) into \(3x^2\) to find the slope at the point.
After computing, we find that the slope is 12, indicating that the tangent line at the curve rises steeply at \((2, 8)\).

Analyzing curves involves examining how functions behave at different points.
Key aspects of curve analysis include looking at the slope to understand how a function increases or decreases. Using derivatives helps identify these changes. For polynomial functions like \(y = x^3\), the behavior of the curve can become steeper or less steep based on the degree of the polynomial.
Given \(y = x^3\) and its derivative \(3x^{2}\), it shows that the steepness of this curve increases as \(x\) moves away from zero. Positive slopes indicate the curve is rising, while negative slopes would suggest it is falling.
In the case of \((2,8)\), a slope of 12 means the curve is steeply rising at that point.

Function evaluation involves calculating the output of a function for a given input. For example, with the function \(y = x^3\), if \(x = 2\), then \(y\) can be evaluated by substituting \(x\) with 2.
Here’s how to do it:

• Substitute \(x = 2\) into the function \(y = x^3\).
• Compute \(y = 2^3\) which equals 8.

This shows that the point \((2, 8)\) is indeed on the curve defined by \(y = x^3\). Function evaluation confirms points on the curve and helps ensure accurate calculations when finding derivatives or slopes.