(a) Find the equation of the tangent line to f(x)=x^3 at the...

(a) Find the equation of the tangent line to $f(x)=x^{3}$ at the point where $x=2$
(b) Graph the tangent line and the function on the same axes. If the tangent line is used to estimate values of the function near $x=2,$ will the estimates be overestimates or underestimates?

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Key Concepts

Linear Approximation

Linear approximation (or local linearization) uses the tangent line to estimate the values of a function near a given point. This is based on the idea that if a function is differentiable, the tangent line will closely mirror the behavior of the function in a small interval around that point.

Concavity

Concavity, determined by the second derivative of a function, indicates whether the function curves upward or downward. This concept is important because it shows whether the tangent line lies above or below the function: for a concave upward function, the tangent line typically underestimates the function's value near the point of tangency.

Derivative

The derivative represents the instantaneous rate of change of a function at a given point. It is computed using differentiation rules and indicates the slope of the tangent line to the function’s graph, which is key in finding the equation of that tangent line.

Tangent Line Equation

The tangent line at a specific point on a function is given by the point-slope form, using both the value of the function and its derivative at that point. This linear equation approximates the function near the point of tangency.

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