91Ó°ÊÓ

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d x}=-y^{2} x\left(x^{2}+2\right)^{4} ; y=1 \text { at } x=0 $$

, 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. $$ g(x)=x^{5 / 3} ;[-1,1] $$

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? $$ f(x)=(x-2)^{5} $$

Use the Concavity Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. $$ f(z)=z^{2}-\frac{1}{z^{2}} $$

If a radioactive substance loses \(15 \%\) of its radioactivity in 2 days, what is its half-life?

Use Newton's Method to approximate the indicated root of the given equation accurate to five decimal places. Begin by sketching a graph. $$ \text { The smallest positive root of } 2 \cot x=x $$

Identify the critical points and find the maximum value and minimum value on the given interval. $$ f(x)=x^{5}-\frac{25}{2} x^{3}+20 x-1 ; I=[-3,2] $$

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

, 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 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? $$ g(t)=\pi-(t-2)^{2 / 3} $$