91Ó°ÊÓ

How do you determine the absolute maximum and minimum values of a continuous function on a closed interval?

Is it possible for a function to satisfy \(f(x)>0, f^{\prime}(x)>0\), and \(f^{\prime \prime}(x)<0\) on an interval? Explain.

Sketch a curve with the following properties. $$f(x)=3 x-x^{3}$$

Determine whether Rolle's Theorem applies to the following functions on the given interval. If so, find the point(s) that are guaranteed to exist by Rolle's Theorem. $$f(x)=1-|x| ;[-1,1]$$

In terms of limits, what does it mean for the rates of growth of \(f\) and \(g\) to be comparable as \(x \rightarrow \infty ?\)

Use a calculator or program to compute the first 10 iterations of Newton's method when it is applied to the following functions with the given initial approximation. Make a table similar to that in Example 1. $$f(x)=x^{3}+x^{2}+1 ; x_{0}=-2$$

What two nonnegative real numbers \(a\) and \(b\) whose sum is 23 maximize \(a^{2}+b^{2} ?\) Minimize \(a^{2}+b^{2} ?\)

Explain how a function can have an absolute minimum value at an endpoint of an interval.

For a given function \(f\), explain the steps used to solve the initial value problem \(F^{\prime}(t)=f(t), F(0)=10\)

Use a calculator or program to compute the first 10 iterations of Newton's method when it is applied to the following functions with the given initial approximation. Make a table similar to that in Example 1. $$f(x)=\sin x+x-1 ; x_{0}=1.5$$