91Ó°ÊÓ

, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of \(c ;\) if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$ S(\theta)=\sin \theta ;[-\pi, \pi] $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{3}{x^{2}}-\frac{2}{x^{3}} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ q(x)=x^{4}-6 x^{3}-24 x^{2}+3 x+1 $$

Find the general antiderivative \(F(x)+C\) for each of the following. $$ f(x)=\frac{\sqrt{2 x}}{x}+\frac{3}{x^{5}} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ f(x)=x^{4}+8 x^{3}-2 $$

Use Newton's Method to calculate \(\sqrt[4]{47}\) to five decimal places.

Find the critical points and use the test of your choice to decide which critical points give a local maximum value and which give a local minimum value. What are these local maximum and minimum values? $$ r(s)=3 s+s^{2 / 5} $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=\frac{x}{1+x^{2}} ; I=[-1,4] $$

Find the \(x y\) -equation of the curve through \((1,2)\) whose slope at any point is three times the square of its \(y\) -coordinate.

Approximate the values of \(x\) that give maximum and minimum values of the function on the indicated intervals. $$ f(x)=x^{4}+x^{3}+x^{2}+x ;[-1,1] $$