91Ó°ÊÓ

Write each expression in the form \(e^{k x}\) for a suitable constant \(k.\) $$\left(e^{3}\right)^{x / 5},\left(\frac{1}{e^{2}}\right)^{x}$$

Simplify the function before differentiating. $$f(x)=\left(e^{3 x}\right)^{5}$$

Differentiate the following functions. $$y=3 e^{x}-7 x$$

Write the equation of the tangent line to the graph of \(y=\ln \left(x^{2}+e\right)\) at \(x=0\).

Find the extreme points on the graph of \(y=x^{2} e^{x}\), and decide which one is a maximum and which one is a minimum.

Differentiate. $$y=\ln \frac{x+1}{x-1}$$

(a) Find the point on the graph of \(y=e^{-x}\) where the tangent line has slope -2. (b) Plot the graphs of \(y=e^{-x}\) and the tangent line in part (a).

Allometric Equation Substantial empirical data show that, if \(x\) and \(y\) measure the sizes of two organs of a particular animal, then \(x\) and \(y\) are related by an allometric equation of the form $$\ln y-k \ln x=\ln c$$ where \(k\) and \(c\) are positive constants that depend only on the type of parts or organs that are measured and are constant among animals belonging to the same species. Solve this equation for \(y\) in terms of \(x, k,\) and \(c .\) (Source: Introduction to Mathematics for Life Scientists)

Graph the function \(y=\ln \left(e^{x}\right),\) and use trace to convince yourself that it is the same as the function \(y=x\). What do you observe about the graph of \(y=e^{\ln x} ?\)