Linear approximation

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After completing this section, students should be able to do the following.

Define linear approximation as an application of the tangent to a curve.
Find the linear approximation to a function at a point and use it to approximate the function value.
Identify when a linear approximation can be used.
Label a graph with the appropriate quantities used in linear approximation.
Find the error of a linear approximation.
Compute differentials.
Use the second derivative to discuss whether the linear approximation over or underestimates the actual function value.
Contrast the notation and meaning of $\d y$ versus $\Delta y$ .
Understand that the error shrinks faster than the displacement in the input.
Justify the chain rule via the composition of linear approximations.