Let F (X) = $\cos ^ { 2 } x$ (A) Find a Linear Approximation of F at X =

question 33

Essay

Let f (x) = $\cos ^ { 2 } x$ .(a) Find a linear approximation of f at x = $\frac { \pi } { 3 } .$ (b) Use this linear approximation to predict the value of the function at $\frac { \pi } { 3 } - 1 , \frac { \pi } { 3 } - 0.1 , \frac { \pi } { 3 } + 1 \text {, }$ and $\frac { \pi } { 3 } + 0.1$ (c) Make a table comparing your estimates with the actual function values. Discuss what this tells you about the linear approximation.(d) Graph the original function and its linear approximation over the interval $\left[ \frac { \pi } { 3 } - 1 , \frac { \pi } { 3 } + 1 \right]$ What does the graph tell you about the size of the difference between the function values and the linear approximation values?

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Definitions:

Perceived Behavioral Control

An individual's belief in their capability to exert control over their own behavior.

Transtheoretical Model

A theory of behavioral change that describes five stages individuals go through to change their behavior: precontemplation, contemplation, preparation, action, and maintenance.

Maintenance

The ongoing care or upkeep of systems, equipment, or property to ensure continued functioning and prevention of degradation or failure.

Precontemplation

The stage in which individuals are not yet considering a change in their behavior or are unaware of the need for change.