91Ó°ÊÓ

Let \(y=f(x)=x^{2}+x\). a. Find the average rate of change of \(y\) with respect to \(x\) in the interval from \(x=2\) to \(x=3\), from \(x=2\) to \(x=2.5\), and from \(x=2\) to \(x=2.1\). b. Find the (instantaneous) rate of change of \(y\) at \(x=2\). c. Compare the results obtained in part (a) with that of part (b).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 0^{+}} \frac{x-1}{x^{2}+1}\)

Find the indicated limit. \(\lim _{x \rightarrow 1}\left(1-2 x^{2}\right)\)

Let \(y=f(x)=x^{2}-4 x\). a. Find the average rate of change of \(y\) with respect to \(x\) in the interval from \(x=3\) to \(x=4\), from \(x=3\) to \(x=3.5\), and from \(x=3\) to \(x=3.1\). b. Find the (instantaneous) rate of change of \(y\) at \(x=3\). c. Compare the results obtained in part (a) with that of part (b).

Find the indicated one-sided limit, if it exists. \(\lim _{x \rightarrow 2^{+}} \frac{x+1}{x^{2}-2 x+3}\)

Find the indicated limit. \(\lim _{t \rightarrow 3}\left(4 t^{2}-2 t+1\right)\)

Find the derivative of each function. \(f(x)=\left(3 x^{2}-1\right)\left(x^{2}-\frac{1}{x}\right)\)

Find the derivative of each function. \(f(u)=(2 u+1)^{3 / 2}+\left(u^{2}-1\right)^{-3 / 2}\)

Find the derivative of the function \(f\) by using the rules of differentiation. \(f(x)=-\frac{1}{3}\left(x^{-3}-x^{6}\right)\)

Find the derivative of each function. \(f(x)=\frac{x}{x^{2}-4}-\frac{x-1}{x^{2}+4}\)