91Ó°ÊÓ

What is the most important characteristic of exponential growth?

\(55-58\). For each function: a. Find \(f^{\prime}(x)\) b. Evaluate the given expression and approximate it to three decimal places. $$ f(x)=\frac{e^{x}}{x}, \text { find and approximate } f^{\prime}(3) $$

Why can't we define logs of negative numbers, such as \(\ln (-2)\) ? [Hint: If \(\ln (-2)=x\), what is the equivalent exponential statement? What is the sign of \(e^{x}\) ?]

Assuming the same fixed nominal interest rate, put the following types of compounding in order of increasing benefit to the depositor: daily, continuously, quarterly, semiannually. compared to someone making \(\$ 40,000\) who had taken no calculus, a comparable person who had taken \(x\) years of calculus would be earning \(\$ 40,000 e^{0.195 x} .\) Find the salary of a person who has taken \(x=2\) years of calculus. [Note: Other mathematics courses were included in the study, but calculus courses brought the greatest increase in salary.]

\(55-58\). For each function: a. Find \(f^{\prime}(x)\) b. Evaluate the given expression and approximate it to three decimal places. $$ f(x)=\ln \left(e^{x}-1\right), \text { find and approximate } f^{\prime}(3) . $$

Why can't we define the logarithm of zero?

Which type of compounding would give the shortest doubling time for a fixed interest rate: daily, continuous, or annual? Which would give the longest?

Explain why \(5 \%\) compounded continuously is better than \(5 \%\) compounded monthly.

\(59-60 .\) Find the second derivative of each function. $$ f(x)=e^{-x^{5} / 5} $$

\(59-60 .\) Find the second derivative of each function. $$ f(x)=e^{-x^{6} / 6} $$