91Ó°ÊÓ

Given \(\log 7=0.8451\), find the following: (a) \(\log 70\). Ans. \(1.8451\). (b) \(\log 700\). (c) \(\log 0.7 .\) Ans. \(-1+0.8451\). (d) \(\log 0.07\) (e) \(\log 0.007 .\) Ans. \(-3+0.8451\).

Find the slope of \(y=e^{x}\) at \(x=0\).

Suppose that the population of a town increases at the net rate per year of \(0.03466\) of its population at the beginning of the year. Find the formula that relates the population and time. Ans. \(P=\) \(P_{0}(1.03466)^{t}\).

Suppose that the population of a town was 5000 twenty years ago and that it increased continuously at a rate proportional to the existing population. Suppose that the population reached 15,000 at the end of the twenty years. What formula relates the population and the time? Ans. \(P=5000 e^{0.06 t}\).

Evaluate: (a) \(\int e^{4 x} d x . \quad\) Ans. \(=\frac{1}{4} e^{4 x}+C .\) (b) \(\int \frac{e^{1 / x^{4}}}{x^{3}} d x\). (c) \(\int e^{-x^{2}+3} x d x\) (d) \(\int\left(e^{x}+2\right)^{2} d x\). (e) \(\int\left(e^{x}+1\right)^{4} e^{x} d x\) (f) \(\int \frac{e^{2 x}}{e^{2 x}+5} d x\).

Sugar in water dissolves continuously at a rate proportional to the undissolved amount. If the amount of sugar is initially 200 grams and 100 grams are dissolved in 2 minutes, how long does it take to dissolve 150 grams? How long does it take to dissolve all 200 grams? Ans. \(4.1 \mathrm{~min} ; \infty\).

Graph the function \(y=e^{-x} \sin x\).

The population of the earth was about 3 billion in 1970 . It is estimated that the world's population is increasing continuously at a rate of \(2 \%\) per year of the existing population. When will the population of 20 billion be reached?

A drug is injected into the blood and is gradually absorbed and then eliminated. The concentration \(y\) of the drug at any time \(t\) is given by the formula \(y=\left[A /\left(C_{2}-C_{1}\right)\right]\left(e^{-C_{1} t}-e^{-C_{2} t}\right)\), where in \(A, C_{1}\) and \(C_{2}\) are positive constants. Find the time \(t\) at which the concentration is a maximum. Ans. \(t=\left[1 /\left(C_{1}-C_{2}\right)\right] \log \left(C_{1} / C_{2}\right)\).