91Ó°ÊÓ

In Problems 33– 44, find the one-sided limit.

$\underset{x→0^{+}}{\lim }\frac{x^3-x^2}{x^4+x^2}$

In Problems 41 and 42, a function f is defined over an interval [a,b]

(a) Graph f, indicating the areaA under f from a tob.

(b) Approximate the area A by partitioning [a, b] into three subintervals of equal length and choosing u as the left endpoint of each subinterval.

(c) Approximate the areaA by partitioning [a,b] into six subintervals of equal length and choosingu as the left endpoint of each subinterval.

(d) Express areaA as an integral.

(e) Use a graphing utility to approximate the integral.

$f\left(x\right)=\frac{1}{x^2},\left[1,4\right]$

In Problems 7– 42, find each limit algebraically.

$\underset{x→3}{\lim }\frac{x^3-3x^2+4x-12}{x^4-3x^3+x-3}.$

In Problems 33–42, use a graphing utility to find the derivative of each function at the given number.
$f\left(x\right)=e^{-x}\sin x$at$2$

The volume V of a right circular cylinder of height$3$ feet and radius r feet isrole="math" localid="1647029593559" $V=V\left(r\right)=3{\mathrm{Ï€°ù}}^2$. Find the instantaneous rate of change of

the volume with respect to the radius r at$r=3$.

In Problems 43–52, find the limit as x approaches c of the average rate of change of each function from c to x.

$c=2;f\left(x\right)=5x-3.$

In Problems 43 and 44, an integral is given.

(a) What area does the integral represent?

(b) Provide a graph that illustrates this area.

(c) Use a graphing utility to approximate this area.

${∫}_{-1}^3(9-x^2)dx$

Use a graphing utility to find the indicated limit rounded to two decimal places.

$\underset{x→1}{\lim }\frac{x^3-x^2+x-1}{x^4-x^3+2x-2}$

In Problems $33-44$, find the one-sided limit.

$\underset{x→-2^{+}}{\lim }\frac{x^2+x-2}{x^2+2x}$

Use a graphing utility to find the indicated limit rounded to two decimal places.

$\underset{x→-1}{\lim }\frac{x^3+x^2+3x+3}{x^4+x^3+2x+2}$