91Ó°ÊÓ

First find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. (See Example 2.) \(\frac{d y}{d x}=\frac{x}{y} ; y=1\) at \(x=1\)

What number exceeds its square by the maximum amount? Begin by convincing yourself that this number is on the interval \([0,1]\).

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(h(z)=\frac{z^{4}}{4}-\frac{4 z^{3}}{6}\)

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ \Psi(x)=x^{2}+3 x ; I=[-2,1] $$

First find the general solution (involving a constant C) for the given differential equation. Then find the particular solution that satisfies the indicated condition. (See Example 2.) \(\frac{d y}{d x}=\sqrt{\frac{x}{y}} ; y=4\) at \(x=1\)

In Problems 1-10, use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. \(f(x)=e^{-x}\)

The population of the United States was \(3.9\) million in 1790 and 178 million in 1960 . If the rate of growth is assumed proportional to the number present, what estimate would you give for the population in 2000 ? (Compare your answer with the actual 2000 population, which was 275 million.)

Show that for a rectangle of given perimeter \(K\) the one with maximum area is a square.

In Problems 5-26, identify the critical points and find the maximum value and minimum value on the given interval. $$ G(x)=\frac{1}{5}\left(2 x^{3}+3 x^{2}-12 x\right) ; I=[-3,3] $$

In each of the Problems 1-21, 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. $$ F(t)=\frac{1}{t-1} ;[0,2] $$