91Ó°ÊÓ

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=1, F^{\prime}(0)=3, F(0)=4$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=\cos x, F^{\prime}(0)=3, F(\pi)=4$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=4 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=1, F(0)=3$$

Consider the function \(f(x)=\left(a b^{x}+(1-a) c^{x}\right)^{1 / x},\) where \(a, b,\) and \(c\) are positive real numbers with \(0

Prove that \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}=e^{a},\) for \(a \neq 0\).

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=672 x^{5}+24 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$

A mass oscillates up and down on the end of a spring. Find its position \(s\) relative to the equilibrium position if its acceleration is \(a(t)=\sin (\pi t),\) and its initial velocity and position are \(v(0)=3\) and \(s(0)=0,\) respectively.

Show that \(x^{x}\) grows faster than \(b^{x}\) as \(x \rightarrow \infty,\) for \(b>1\).

a. For what values of \(b>0\) does \(b^{x}\) grow faster than \(e^{x}\) as \(x \rightarrow \infty ?\) b. Compare the growth rates of \(e^{x}\) and \(e^{a x}\) as \(x \rightarrow \infty,\) for \(a>0\).

A large tank is filled with water when an outflow valve is opened at \(t=0 .\) Water flows out at a rate, in gal/min, given by \(Q^{\prime}(t)=0.1\left(100-t^{2}\right),\) for \(0 \leq t \leq 10\). a. Find the amount of water \(Q(t)\) that has flowed out of the tank after \(t\) minutes, given the initial condition \(Q(0)=0\) b. Graph the flow function \(Q,\) for \(0 \leq t \leq 10\) c. How much water flows out of the tank in 10 min?