Learning Objectives

The problems in this assignment seek to develop both geometric intuition and analytic techniques for determining whether the linearization of a function f centered at a is a good approximation with specified accuracy over a given interval.

Some Preliminaries

Let be a specified error bound. A linear approximation, L, of a function f at center a is considered an “-good”
approximation of f over a given interval, (b, c), if |f(x) − L(x)| < for all x values in that interval.
If the error bound is not met on this interval, so the function is more than away from the linearization at some
point in (b, c), then the approximation is considered “-bad.

Use the Matlab tool lineartool2 to complete the following assignment.
Given a function f and a center a, lineartool2 calculates the linearization function L. When you run lineartool2,
four windows will open: three graphs and a control panel.

• The graph titled “Size of Second Derivative” shows the graph of |f 00|. You will use this graph to solve problem 13.

• The graph “Overview” shows the function f is blue and a tangent line in L in pink, over a fixed large scale view.

• The graph “Zoom” shows the same things as “Overview” but zoomed into the interval (a − δ, a + δ), for the
value of δ specified in the control panel. The interval (a − δ, a + δ) is also marked on “Overview” and “Size
of Second Derivative” graphs with a red rectangle.

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