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

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?