91Ó°ÊÓ

Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N_{0} 2^{t} $$ where \(N_{0}\) is the population size at time \(0 .\) Find the rate of growth \(d N / d t .\) Express your solution in terms of \(N(t) .\) Show that the growth rate of the population is proportional to the population size.

Use the identity $$\cos (\alpha+\beta)=\cos \alpha \cos \beta-\sin \alpha \sin \beta$$ and the definition of the derivative to show that $$ \frac{d}{d x} \cos x=-\sin x $$

Find the normal line to $$ f(x)=\frac{a x^{2}}{a+1} $$ at \(x=2 .\) Assume that \(a\) is a positive constant.

Let \(f(x)=\ln x\). We know that \(f^{\prime}(x)=\frac{1}{x}\). We will use th fact and the definition of derivatives to show that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$ (a) Use the definition of the derivative to show that $$ f^{\prime}(1)=\lim _{h \rightarrow 0} \frac{\ln (1+h)}{h} $$ (b) Show that (a) implies that $$ \ln \left[\lim _{h \rightarrow 0}(1+h)^{1 / h}\right]=1 $$ (c) Set \(h=\frac{1}{n}\) in (b) and let \(n \rightarrow \infty\). Show that this implies that $$ \lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e $$

Differentiate with respect to the independent variable. \(f(x)=\sqrt{3 x}\left(x^{2}-1\right)\)

Assume that \(f(x)\) is differentiable with respect to \(x\). Show that $$ \frac{d}{d x} \ln \left[\frac{f(x)}{x}\right]=\frac{f^{\prime}(x)}{f(x)}-\frac{1}{x} $$

Find the normal line to $$ f(x)=\frac{x^{3}}{a+1} $$ at \(x=2 a\). Assume that \(a\) is a positive constant.

Use the quotient rule to show that $$ \frac{d}{d x} \cot x=-\csc ^{2} x $$

Differentiate with respect to the independent variable. \(f(x)=\frac{\sqrt{5 x}\left(1+x^{2}\right)}{\sqrt{2}}\)

Suppose that a bacterial colony grows in such a way that at time \(t\) the population size is $$ N(t)=N_{0} 2^{t} $$ where \(N_{0}\) is the population size at time \(0 .\) Find the per capita growth rate. $$ \frac{1}{N} \frac{d N}{d t} $$