91Ó°ÊÓ

Bacteria grown in a certain culture increase at a rate proportional to the amount present. If there are 1000 bacteria present initially and the amount doubles in \(1 \mathrm{hr}\), how many bacteria will there be in \(3 \frac{1}{2} \mathrm{hr}\) ?

Prove the given property if \(a\) is any positive. number and \(x\) and \(y\) are any real numbers. $$ \left(a^{x}\right)^{y}=a^{x y} $$

In a certain chemical reaction the rate of conversion of a substance is proportional to the amount of the substance still untransformed at that time. After 10 min one-third of the original amount of the substance has been converted, and \(20 \mathrm{~g}\) has been converted after \(15 \mathrm{~min}\). What was the original amount of the substance?

A tank contains 100 gal of fresh water and brine containing \(2 \mathrm{lb}\) of salt per gallon flows into the tank at the rate of 3 gal/min. If the mixture, kept uniform by stirring, flows out at the same rate, how many pounds of salt are there in the tank at the end of 30 min?

If the purchasing power of a dollar is decreasing at the rate of \(8 \%\) annually, compounded continuously, how long will it take for the purchasing power to be 50 cents?

Given that the function \(f\) is continuous and increasing on the closed interval \([a, b]\). Assuming Theorem 9.3.1(i) and (ii), prove \(f^{-1}\) is continuous from the right at \(f(a)\) and continuous from the left at \(f(b)\).

$$ \int \frac{4^{\ln (1 / x)}}{x} d x $$

Prove: \(\lim _{x \rightarrow+\infty} e^{x}=+\infty\), by showing that for any \(N>0\) there exists an \(M>0\) such that \(e^{x}>N\) whenever \(x>M\)