91Ó°ÊÓ

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 x^{5}-5 x^{9}\right) d x$$

Linear approximation Find the linear approximation to the following functions at the given point a. $$f(x)=e^{3 x-6} ; a=2$$

Use the guidelines of this section to make a complete graph of \(f\). $$f(x)=x^{3}-6 x^{2}-135 x$$

Increasing and decreasing functions Find the intervals on which \(f\) is increasing and the intervals on which it is decreasing. $$f(x)=-\frac{x^{3}}{3}+\frac{x^{2}}{2}+2 x$$

Linear approximation Find the linear approximation to the following functions at the given point a. $$f(x)=9(4 x+11)^{2 / 3} ; a=4$$

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant. $$f(x)=\frac{1}{8} x^{3}-\frac{1}{2} x$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(3 u^{-2}-4 u^{2}+1\right) d u$$

Mean Value Theorem Consider the following functions on the given interval \([a, b]\) a. Determine whether the Mean Value Theorem applies to the following functions on the given interval \([a, b]\). b. If so, find the point(s) that are guaranteed to exist by the Mean Value Theorem. $$f(x)=\frac{1}{(x-1)^{2}} ;[0,2]$$

Limits Evaluate the following limits. Use l'Hópital's Rule when it is comvenient and applicable. $$\lim _{x \rightarrow \infty} \frac{4 x^{3}-2 x^{2}+6}{\pi x^{3}+4}$$

Use the guidelines of this section to make a complete graph of \(f\). $$f(x)=x^{4}+8 x^{3}-270 x^{2}+1$$