91Ó°ÊÓ

Find a number \(w\) such that $$ \frac{4-\ln w}{3-5 \ln w}=3.6 $$

Suppose a colony of bacteria has tripled in two hours. What is the continuous growth rate of this colony of bacteria?

(a) Using a calculator or computer, verify that $$ 2^{t}-1 \approx 0.693147 t $$ for some small numbers \(t\) (for example, try \(t=0.001\) and then smaller values of \(t\) ). (b) Explain why \(2^{t}=e^{t \ln 2}\) for every number \(t\). (c) Explain why the approximation in part (a) follows from the approximation \(e^{t} \approx 1+t\)

For Exercises \(33-36,\) find a formula for \((f \circ g)(x)\) assuming that \(f\) and \(g\) are the indicated functions. $$ f(x)=\ln x \quad \text { and } \quad g(x)=e^{5 x} $$

Find the equation of the circle centered at the origin in the \(x y\) -plane that has circumference \(9 .\)

Find the area inside the ellipse in the \(x y\) -plane determined by the given equation. $$ \frac{x^{2}}{7}+\frac{y^{2}}{16}=1 $$

For Exercises \(33-36,\) find a formula for \((f \circ g)(x)\) assuming that \(f\) and \(g\) are the indicated functions. $$ f(x)=\ln x \text { and } g(x)=e^{4-7 x} $$

Suppose \(x\) is a positive number. (a) Explain why \(x^{t}=e^{t \ln x}\) for every number \(t\) (b) Explain why $$ \frac{x^{t}-1}{t} \approx \ln x $$ if \(t\) is close to 0 [Part (b) of this problem gives another illustration of why the natural logarithm deserves the title "natural".]

Find the area inside the ellipse in the \(x y\) -plane determined by the given equation. $$ \frac{x^{2}}{9}+\frac{y^{2}}{5}=1 $$

For Exercises \(33-36,\) find a formula for \((f \circ g)(x)\) assuming that \(f\) and \(g\) are the indicated functions. $$ f(x)=e^{2 x} \quad \text { and } \quad g(x)=\ln x $$