91Ó°ÊÓ

Find the given limit. $$ \lim _{t \rightarrow \infty} \frac{(t-1)(2 t+1)}{(3 t-2)(t+4)} $$

Find all antiderivatives of the given function. $$ 4 x^{2}+6 x-1 $$

Suppose the population \(f(t)\) of a given species grows exponentially, so that \(f(t)=f(0) e^{k t}\) for some positive constant \(k\) a. Show that the population doubles during any time interval of duration \((\ln 2) / k\). Thus \((\ln 2) / k\) is the doubling time \(d\). b. Show that \(f(t)=f(0) 2^{t / d}\).

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(t)=\frac{1}{\sqrt{t-t^{2}}} $$

Find all numbers \(c\) in the interval \((a, b)\) for which the line tangent to the graph of \(f\) is parallel to the line joining \((a, f(a))\) and \((b, f(b))\). $$ f(x)=x^{3}-2 ; a=-3, b=3 $$

According to one model, the time rate \(R\) at which a tumor grows is given by $$ R=A x \ln \frac{B}{x} \text { for } 0

Find all critical numbers of the given function. $$ g(x)=x-4 / x^{2} $$

Find all antiderivatives of the given function. $$ \sin x $$

Suppose the amount \(f(t)\) of a radioactive substance decays exponentially, so that \(f(t)=f(0) e^{k t}\) for some negative constant \(k\). a. Show that the amount decreases by half in any time interval of duration \(-(\ln 2) / k .\) Thus \(-(\ln 2) / k\) is the half-life \(h\) b. Show that \(f(t)=f(0)\left(\frac{1}{2}\right)^{t / / h}\).

Determine the values of \(c\) at which \(f^{\prime}\) changes from positive to negative, or from negative to positive. $$ f(t)=\sin t+\frac{1}{2} t $$