91Ó°ÊÓ

Use a differential to estimate the value of the indicated expression. Then compare your estimate with the result given by a calculator. $$(33)^{3 / 5}$$

Describe the concavity of the graph and find the points of inflection (if any). $$f(x)=x^{3}-3 x+2$$

A heap of rubbish in the shape of a cube is being compacted into a smaller cube. Giver that the volume decreases at the rate of 2 cubic meters per minute, find the rate of change of an edge of the cube when the volume is exactly 2 cubic meters. What is the rate of change of the surface area of the cube at that instant?

Set \(f(x)=x^{2 / 3}-1 .\) Note that \(f(-1) \quad f(1) \cdot 0 .\) Verify that there does not exist a number \(c\) in (-1,1) for which \(f^{\prime}(c)=0 .\) Explain how this does not violate Rolle's theorem.

Find the intervals on which \(f\) increases and the intervals on which \(f\) decreases. $$f(x)=(x+1)^{4}$$

Sketch the graph of the function using the approach presented in this section. $$f(x)=\frac{2}{3} x^{3}-\frac{1}{2} x^{2}-10 x-1$$

Find the critical points. Then find and classify all the extreme values. $$f(x)=x^{2}+\frac{1}{x} . \quad x \in\left[\frac{1}{10}, 2\right]$$

Find the critical points and the local extreme values. $$f(x)=\frac{1+x}{1-x}$$.

An object moves along the \(x\) -axis, its position at each time \(l \geq 0\) given by \(x(t)\). Determine the times, if any, at which (a) the velocity is zero, (b) the acceleration is zero. $$x(t)=5 t+1$$.

Find the vertical and horizontal asymptotes. $$f(x)=\frac{2 x}{x^{2}-9}$$