91Ó°ÊÓ

Differentiate. $$ g(x)=x^{5} \ln (3 x) $$

The Rule of 70. The relationship between doubling time \(T\) and growth rate \(k\) is the basis of the Rule of \(70 .\) Since $$ T=\frac{\ln 2}{k}=\frac{0.693147}{k}=\frac{69.3147}{100 k} \approx \frac{70}{100 k} $$ we can estimate the length of time needed for a quantity to double by dividing the growth rate \(k\) (expressed as a percentage) into \(70 .\) Estimate the time needed for an amount of money to double, if the interest rate is \(7 \%,\) compounded.

The Rule of 70. The relationship between doubling time \(T\) and growth rate \(k\) is the basis of the Rule of \(70 .\) Since $$ T=\frac{\ln 2}{k}=\frac{0.693147}{k}=\frac{69.3147}{100 k} \approx \frac{70}{100 k} $$ we can estimate the length of time needed for a quantity to double by dividing the growth rate \(k\) (expressed as a percentage) into \(70 .\)

Differentiate. $$ g(x)=x^{2} \ln (7 x) $$

Differentiate. $$ g(x)=x^{4} \ln |6 x| $$

The Rule of 70. The relationship between doubling time \(T\) and growth rate \(k\) is the basis of the Rule of \(70 .\) Since $$ T=\frac{\ln 2}{k}=\frac{0.693147}{k}=\frac{69.3147}{100 k} \approx \frac{70}{100 k} $$ we can estimate the length of time needed for a quantity to double by dividing the growth rate \(k\) (expressed as a percentage) into \(70 .\) Describe two situations where it would be preferable to use the Rule of 70 instead of the formula \(T=\ln (2) / k\). Explain why it would be acceptable to use this rule in these situations.

Differentiate. $$ g(x)=x^{9} \ln |2 x| $$

The revenue of Red Rocks, Inc., in millions of dollars, is given by the function $$ R(t)=\frac{4000}{1+1999 e^{-0.5 t}} $$ where \(t\) is measured in years a) What is \(R(0),\) and what does it represent? b) Find \(\lim _{t \rightarrow \infty} R(t) .\) Call this value \(R_{\max },\) and explain what it means c) Find the value of \(t\) (to the nearest integer) for which \(R(t)=0.99 R_{\max }\)

Differentiate. $$ y=\frac{\ln x}{x^{5}} $$

Differentiate. $$ y=\frac{\ln x}{x^{4}} $$