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A secant line is a straight line that intersects a curve at two or more points. In this context, it connects the endpoints of the function within the given interval. For the function \( y = 3x^3 + 2x + 1 \), the secant line is calculated using the values \( f(-1) \) and \( f(1) \).

• For \( x = -1 \): \( f(-1) = -4 \).
• For \( x = 1 \): \( f(1) = 6 \).

The slope of the secant line is determined by the formula: \[ \text{slope} = \frac{f(1) - f(-1)}{1 - (-1)} = 5 \].This slope represents the average rate of change of the function over the interval \([-1, 1]\). In graphical terms, it helps in comparing the function's steepness between two points. Calculating and understanding this line is crucial for applying the Mean Value Theorem.

A derivative represents the rate of change of a function at any given point. It gives insights into the function's slope at precise instants, rather than over an interval. For our function \( y = 3x^3 + 2x + 1 \), the derivative is derived as \( f'(x) = 9x^2 + 2 \).This derivative expression shows how the function's slope varies depending on the value of \( x \). To find where the instant rate of change matches the secant line's slope (5 in this case), we solve:\[ f'(c) = 5 \],leading to:\[ 9c^2 + 2 = 5 \].Solving gives:\[ c^2 = \frac{1}{3} \],resulting in:\[ c = \pm \frac{1}{\sqrt{3}} \].This tells us that at these points, the instantaneous rate of change equals the average rate of change, fulfilling the Mean Value Theorem requirements.

Interval notation is a mathematical notation used to describe a range of values along the number line. It is crucial when identifying feasible solutions for the Mean Value Theorem. In our case, the interval of interest is \([-1, 1]\), which means:

• The function is evaluated from \( x = -1 \) to \( x = 1 \).
• It includes both the endpoint values \(-1\) and \(1\).

Using interval notation helps clarify the domain we consider for solutions and computations. With respect to the Mean Value Theorem, it confirms the possible values of \( c \), ensuring that both solutions found, \( \frac{1}{\sqrt{3}} \) and \(-\frac{1}{\sqrt{3}} \), are indeed within the prescribed interval.

Function graphing involves plotting a mathematical function to visually analyze its properties and behavior. For the equation \( y = 3x^3 + 2x + 1 \), graphing helps us see changes over the interval and identify key features such as intercepts, turning points, and behavior at the extremes.By plotting this cubic function over \([-1, 1]\), you can:

• Observe the function's curvature and slope.
• Visualize where the secant line, connecting \( f(-1) = -4 \) and \( f(1) = 6 \), intersects the graph.
• Identify where the instantaneous slope (via the derivative) equals the secant slope.

Visualization is a powerful tool in comprehending the Mean Value Theorem, as it offers a tangible insight into how average and instantaneous rates of change relate.