Use the tangent line approximation. Given f(3)=-4, f^'(3)=-2...

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Key Concepts

Derivative

A derivative represents the instantaneous rate of change of a function at a particular point and is graphically interpreted as the slope of the tangent line. In the context of tangent line approximation, the derivative at the point of tangency provides the necessary slope information to construct the linear approximation.

Linearization

Linearization is the process of approximating a function by its tangent line at a given point. This concept is closely related to the tangent line approximation, as it involves using the value of the function and its derivative at a specific point to create a linear function that closely mimics the function's behavior in the neighborhood of that point.

Tangent Line Approximation

The tangent line approximation is a fundamental concept in calculus that uses the tangent line at a given point on a function to estimate the function's value near that point. It is based on the assumption that, over a very small interval, the function behaves almost linearly, allowing for a simple linear equation to approximate its behavior.

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