Finding the Slope of a Graph In Exercises 31-38 , find the...

Finding the Slope of a Graph In Exercises $31-38$ , find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results.
$$f(x)=2(x-4)^{2}$$ (2,8)

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Key Concepts

Tangent Line and Slope

The slope of the tangent line to the graph of a function at a given point reflects the function’s instantaneous rate of change at that point. This concept is essential in understanding how the function behaves locally and forms the basis for many applications in mathematics and physics, such as motion and optimization.

Derivative

The derivative of a function represents the instantaneous rate of change at any point on the graph of the function. It is defined as the limit of the difference quotient and gives the slope of the tangent line to the curve at a specific point. Calculating the derivative is a fundamental concept in calculus, enabling the analysis of how functions change.

Graphing Utility for Derivatives

Graphing utilities aid in visualizing functions and their derivatives by providing numerical and graphical approximations of slopes at various points. Using these tools can confirm analytical results by allowing users to compute and observe the derivative, thereby reinforcing the understanding of instantaneous change and the behavior of functions.

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