91影视

A functionf that is defined on [鈭�2, 2] with f (鈭�2) = f (2) = 0 such that f is continuous everywhere, differentiable everywhere except at x = 鈭�1, and fails the conclusion of Rolle鈥檚 Theorem .

Find the locations and values of any global extrema of each function f in Exercises 11鈥�20 on each of the four given intervals. Do all work by hand by considering local extrema and endpoint behavior. Afterwards, check your answers with graphs.

$x^{\frac{3}{2}}(3x-5)$on the interval

$$\begin{gathered} \left(a\right)[0,4] \\ \left(b\right)[0,4) \\ \left(c\right)[0,1) \\ \left(d\right)(0,1) \end{gathered}$$

For Exercises 15鈥�20, sketch the graph of a function fthat has the indicated characteristics. If a graph is not possible, explain why.

f positive, f' negative, and f'' positive on$\mathrm{鈩�}$.

Determine the graph of a function f from the graph of its derivative f'.

Calculate each of the limits in Exercises 15鈥�20

(a) using L鈥橦opital鈥檚 rule and (b) without using L鈥橦opital鈥檚 rule.

$\underset{x鈫�鈭�}{\lim }\frac{3^x}{1-4^x}$

For each function f that follows, construct sign charts forf, f',and f '', if possible. Examine function values or limits at any interesting values and at 卤鈭�. Then interpret this information to sketch a labeled graph of f.

$f\left(x\right)=\frac{1}{3x+1}$

Find the locations and values of any global extrema of each function f in Exercises 11鈥�20 on each of the four given intervals. Do all work by hand by considering local extrema and endpoint behavior. Afterwards, check your answers with graphs.

$f\left(x\right)=\sin \left(\frac{\mathrm{蟺}}{2}x\right)$on the interval

$$\begin{gathered} \left(a\right)[-2,2] \\ \left(b\right)(-2,2) \\ \left(c\right)[-1,1) \\ \left(d\right)[0,鈭�) \end{gathered}$$

Suppose the radius r, height h, and volume V of a cylinder are functions of time t, and further suppose that the volume of the cylinder is always constant. Write $\frac{dr}{dt}$ in terms of r, h, and $\frac{dh}{dt}$.

A function f defined on [1, 5] with f (1) = f (5) = 0 such that f is continuous everywhere except for x = 2, differentiable everywhere except at x = 2, and fails the conclusion of Rolle鈥檚 Theorem .

For Exercises 15鈥�20, sketch the graph of a function f that has the indicated characteristics. If a graph is not possible, explain why.

f negative, f' negative, and f'' negative on $\mathrm{鈩�}.$