91Ó°ÊÓ

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\cos x, \quad x_{1}=1.6\)

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ y=x^{2}-x-2 $$

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ g(x)=x+\frac{4}{x^{2}} $$

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=x^{5}, \quad(2,32) $$

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=\sqrt{x}, \quad(2, \sqrt{2}) $$

Determining Concavity In Exercises \(3-14\) , determine the open intervals on which the graph is concave upward or concave downward. $$ g(x)=3 x^{2}-x^{3} $$

Finding the Value of the Derivative at Relative Extrema In Exercises \(1-6,\) find the value of the derivative (if it exists) at each indicated extremum. $$ f(x)=-3 x \sqrt{x+1} $$

Complete two iterations of Newton’s Method to approximate a zero of the function using the given initial guess. \(f(x)=\tan x, \quad x_{1}=0.1\)

Finding Numbers In Exercises \(3-8,\) find two positive numbers that satisfy the given requirements. The product is 185 and the sum is a minimum.

Explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b) .\) \(f(x)=\sqrt{\left(2-x^{2 / 3}\right)^{3}},[-1,1]\)