91Ó°ÊÓ

Finding Differentials Explain how to find a differential of a function.

Determining Concavity In Exercises \(5-16,\) determine the open intervals on which the graph of the function is concave upward or concave downward. $$f(x)=x^{2}-4 x+8$$

Critical Numbers Explain how to find the critical numbers of a function.

Using a Tangent Line Approximation In Exercises \(5-10\) , find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Then complete the table. $$f(x)=x^{2}, \quad(2,4)$$

Using Newton's Method In Exercises \(3-6\) calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. \(f(x)=\cos x, \quad x_{1}=1.6\)

Finding Numbers In find two positive numbers that satisfy the given requirements. The product is 185 and the sum is a minimum.

Determining Concavity In Exercises \(5-16,\) determine the open intervals on which the graph of the function is concave upward or concave downward. \(g(x)=3 x^{2}-x^{3}\)

Using Newton's Method In Exercises \(3-6\) calculate two iterations of Newton's Method to approximate a zero of the function using the given initial guess. \(f(x)=\tan x, \quad x_{1}=0.1\)

Using a Tangent Line Approximation In Exercises \(5-10\) , find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Then complete the table. \(f(x)=\frac{6}{x^{2}}, \quad\left(2, \frac{3}{2}\right)\)

Writing In Exercises \(3-6,\) explain why Rolle's Theorem does not apply to the function even though there exist \(a\) and \(b\) such that \(f(a)=f(b) .\) \(f(x)=\sqrt{\left(2-x^{2 / 3}\right)^{3}},\) \([-1,1]\)