## 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.

### mathematics

##### Description

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.