91Ó°ÊÓ

Suppose you want to travel \(D\) miles at a constant speed of \((60+x)\) mi/hr, where \(x\) could be positive or negative. The time in minutes required to travel \(D\) miles is \(T(x)=60 D(60+x)^{-1}\) Show that the linear approximation to \(T\) at the point \(x=0\) is \(T(x) \approx L(x)=D\left(1-\frac{x}{60}\right)\)

What two nonnegative real numbers with a sum of 23 have the largest possible product?

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. $$f(x)=\cos 4 x ;[\pi / 8,3 \pi / 8]$$

Suppose \(f\) is continuous on an interval containing a critical point \(c\) and \(f^{\prime \prime}(c)=0 .\) How do you determine whether \(f\) has a local extreme value at \(x=c ?\)

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. $$f(x)=1-|x| ;[-1,1]$$

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

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 they are applied to the following functions with the given initial approximation. Make a table similar to that in Example 1 $$f(x)=x^{3}+x^{2}+1 ; x_{0}=-2$$

Sketch a graph of the following polynomials. Identify local extrema, inflection points, and \(x-\) and \(y\) -intercepts when they exist. $$f(x)=3 x-x^{3}$$

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