91Ó°ÊÓ

(a) Use Newton's Method and the function \(f(x)=x^{n}-a\) to obtain a general rule for approximating \(x=\sqrt[n]{a}\). (b) Use the general rule found in part (a) to approximate \(\sqrt[4]{6}\) and \(\sqrt[3]{15}\) to three decimal places.

(a) find the critical numbers of \(f\) (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$f(x)=-\left(x^{2}+8 x+12\right)$$

(a) find the critical numbers of \(f\) (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$f(x)=x^{2}(3-x)$$

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=\sin x+\cos x, \quad[0,2 \pi]\)

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=x+2 \cos x, \quad[0,2 \pi]\)

Under what conditions will Newton's Method fail?

Analyze and sketch a graph of the function. Label any intercepts, relative extrema, points of inflection, and asymptotes. Use a graphing utility to verify your results. $$ y=-\frac{1}{3}\left(x^{3}-3 x+2\right) $$

Vertical Motion The height of a ball \(t\) seconds after it is thrown upward from a height of 32 feet and with an initial velocity of 48 feet per second is \(f(t)=-16 t^{2}+48 t+32\) (a) Verify that \(f(1)=f(2)\). (b) According to Rolle's Theorem, what must be the velocity at some time in the interval \((1,2) ?\) Find that time.

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{3}-3 x^{2}+3\)

A rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section) of 108 inches (see figure). Find the dimensions of the package of maximum volume that can be sent. (Assume the cross section is square.)