91Ó°ÊÓ

Writing In Exercises 23 and 24 , give a short explanation of why the approximation is valid. $$ \sqrt{4.02} \approx 2+\frac{1}{4}(0.02) $$

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{2 x+1}{\sqrt{x^{2}-x}} $$

Find all relative extrema. Use the Second Derivative Test where applicable. \(g(x)=x^{2}(6-x)^{3}\)

Use a graphing utility to graph the function on the closed interval [a,b]. Determine whether Rolle's Theorem can be applied to \(f\) on the interval and, if so, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=\frac{x}{2}-\sin \frac{\pi x}{6}, \quad[-1,0] $$

Use a graphing utility to graph the function on the closed interval [a,b]. Determine whether Rolle's Theorem can be applied to \(f\) on the interval and, if so, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$ f(x)=2+\arcsin \left(x^{2}-1\right), \quad[-1,1] $$

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow-\infty} \frac{-3 x+1}{\sqrt{x^{2}+x}} $$

Writing In Exercises 23 and 24 , give a short explanation of why the approximation is valid. $$ \tan 0.05 \approx 0+1(0.05) $$

Find all relative extrema. Use the Second Derivative Test where applicable. \(f(x)=x^{3}-9 x^{2}+27 x\)

Locate the absolute extrema of the function on the closed interval. $$ h(t)=\frac{t}{t-2},[3,5] $$

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=(x+2)^{2}(x-1) $$