91Ó°ÊÓ

In each pair of equations, one is true and one is false. Choose the correct one. $$ \ln 1=0 \quad \text { or } \quad \ln 0=1 $$

Find the equation for the tangent line to the curve \(y=f(x)\) at the given \(x\) -value. $$ f(x)=x \ln (x-1) \text { at } x=2 $$

In each pair of equations, one is true and one is false. Choose the correct one. $$ \ln (x+y)=\ln x \cdot \ln y \quad \text { or } \quad \ln (x \cdot y)=\ln x+\ln y $$

Does the graph of \(a^{x}\) for \(a<1\) have a horizontal asymptote? [Hint: Look at the graph on page 262.]

The following problems extend and augment the material presented in the text. a. Show that for a demand function of the form \(D(p)=c / p^{n}\), where \(c\) and \(n\) are positive constants, the elasticity is constant. b. What type of demand function has elasticity equal to 1 for every value of \(p\) ?

Find the equation for the tangent line to the curve \(y=f(x)\) at the given \(x\) -value. $$ f(x)=\frac{x^{2}}{1+\ln x} \text { at } x=1 $$

The following problems extend and augment the material presented in the text. Show that for a demand function of the form \(D(p)=a e^{-c p}\), where \(a\) and \(c\) are positive constants, the elasticity of demand is \(E(p)=c p .\)

Find the equation for the tangent line to the curve \(y=f(x)\) at the given \(x\) -value. $$ f(x)=e^{x^{2}-1} \text { at } x=1 $$

Which will be largest for very large values of \(x\) : \(x^{2}, e^{x}\), or \(x^{1000} ?\)

In each pair of equations, one is true and one is false. Choose the correct one. $$ \frac{\ln x}{\ln y}=\ln (x-y) \text { or } \ln \frac{x}{y}=\ln x-\ln y $$