Chapter 3

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{2}-x}$$

Find the exact value of each expression. (a) \(\sin ^{-1}(\sqrt{3} / 2)\) (b) \(\cos ^{-1}(-1)\)

Find the numerical value of each expression. (a) \(\sinh 0\) (b) \(\cosh 0\)

A population of protozoa develops with a constant relative growth rate of 0.7944 per member per day. On day zero the population consists of two members. Find the population size after six days.

Differentiate the function. $$f(x)=\log _{10}\left(x^{3}+1\right)$$

(a) What is a one-to-one function'? (b) How can you tell from the graph of a function whether it is one-to-one?

(a) Write an equation that defines the exponential function with base \(a>0\) . (b) What is the domain of this function? (c) If \(a \neq 1,\) what is the range of this function? (d) Sketch the general shape of the graph of the exponential function for each of the following cases. (i) \(a>1\)

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}$$

Find the exact value of each expression. (a) \(\tan ^{-1}(1 / \sqrt{3})\) (b) \(\sec ^{-1} 2\)

Find the numerical value of each expression. (a) \(\tanh 0\) (b) tanh 1

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 60 cells. (a) Find the relative growth rate. (b) Find an expression for the number of cells after \(t\) hours. (c) Find the number of cells after 8 hours. (d) Find the rate of growth after 8 hours. (e) When will the population reach \(20,000\) cells?

Differentiate the function. $$ f(x)=x \ln x-x $$

(a) How is the number \(e\) defined? (b) What is an approximate value for \(e ?\) (c) What is the natural exponential function?

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why.$$\lim _{x \rightarrow(\pi / 2)^{+}} \frac{\cos x}{1-\sin x}$$

Find the exact value of each expression. (a) arctan 1 (b) \(\sin ^{-1}(1 / \sqrt{2})\)

Find the numerical value of each expression. (a) \(\sinh (\ln 2)\) (b) \(\sinh 2\)

A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to \(420 .\) (a) Find an expression for the number of bacteria after \(t\) hours. (b) Find the number of bacteria after 3 hours. (c) Find the rate of growth after 3 hours. (d) When will the population reach \(10,000 ?\)

Differentiate the function. $$ f(x)=\sin (\ln x) $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {1.5} & {2.0} & {3.6} & {5.3} & {2.8} & {2.0} \\\ \hline\end{array} $$

Graph the given functions on a common screen. How are these graphs related? $$y=2^{x}, \quad y=e^{x}, \quad y=5^{x}, \quad y=20^{x}$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{\sin 4 x}{\tan 5 x}$$

Find the exact value of each expression. (a) \(\cot ^{-1}(-\sqrt{3})\) (b) \(\arccos \left(-\frac{1}{2}\right)\)

Find the numerical value of each expression. (a) \(\cosh 3\) (b) \(\cosh (\ln 3)\)

A bacteria culture grows with constant relative growth rate. The bacteria count was 400 after 2 hours and \(25,600\) after 6 hours. (a) What is the relative growth rate? Express your answer as a percentage. (b) What was the intitial size of the culture? (c) Find an expression for the number of bacteria after thours. (d) Find the number of cells after 4.5 hours. (e) Find the rate of growth after 4.5 hours. (f) When will the population reach \(50,000 ?\)

Differentiate the function. $$ f(x)=\ln \left(\sin ^{2} x\right) $$

A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {1} & {2} & {3} & {4} & {5} & {6} \\\ \hline f(x) & {1.0} & {1.9} & {2.8} & {3.5} & {3.1} & {2.9} \\\ \hline\end{array} $$

Graph the given functions on a common screen. How are these graphs related? $$y=e^{x}, \quad y=e^{-x}, \quad y=8^{x}, \quad y=8^{-x}$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{t \rightarrow 0} \frac{e^{2 t}-1}{\sin t}$$

Find the numerical value of each expression. (a) \(\operatorname{sech} 0\) (b) \(\cosh ^{-1} 1\)

Find the exact value of each expression. (a) \(\tan (\arctan 10)\) (b) \(\sin ^{-1}(\sin (7 \pi / 3))\)

Differentiate the function. $$ f(x)=\ln \frac{1}{x} $$

Graph the given functions on a common screen. How are these graphs related? $$y=3^{x}, \quad y=10^{x}, \quad y=\left(\frac{1}{3}\right)^{x}, \quad y=\left(\frac{1}{10}\right)^{x}$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0} \frac{x^{2}}{1-\cos x}$$

Find the exact value of each expression. (a) \(\sinh 1\)

Find the exact value of each expression. (a) \(\tan \left(\sec ^{-1} 4\right)\) (b) \(\sin \left(2 \sin ^{-1}\left(\frac{3}{5}\right)\right)\)

The table gives the population of India, in millions, for the second half of the 20 th century. $$\begin{array}{|c|c|}\hline \text { Year } & {\text { Population }} \\\ \hline 1951 & {361} \\ \hline 1961 & {439} \\ \hline 1971 & {548} \\ \hline 1981 & {683} \\ \hline 1901 & {846} \\ \hline 2001 & {1029} \\\ \hline\end{array}$$ (a) Use the exponential model and the census figures for 1951 and 1961 to predict the population in \(2001 .\) Com- pare with the actual figure. (b) Use the exponential model and the census figures for 1961 and 1981 to predict the population in \(2001 .\) Com- pare with the actual population. Then use this model to predict the population in the years 2010 and 2020 . (c) Graph both of the exponential functions in parts (a) and (b) together with a plot of the actual population. Are these models reasonable ones?

Differentiate the function. $$ y=\frac{1}{\ln x} $$

Graph the given functions on a common screen. How are these graphs related? $$y=0.9^{x}, \quad y=0.6^{x}, \quad y=0.3^{x}, \quad y=0.1^{x}$$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{1+\cos 2 \theta}$$

$$\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}$$

Prove that \(\cos \left(\sin ^{-1} x\right)=\sqrt{1-x^{2}}\)

Experiments show that if the chemical reaction $$\mathrm{N}_{2} \mathrm{O}_{5} \rightarrow 2 \mathrm{NO}_{2}+\frac{1}{2} \mathrm{O}_{2}$$ takes place at \(45^{\circ} \mathrm{C},\) the rate of reaction of dinitrogen pentoxide is proportional to its concentration as follows: $$-\frac{d\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]}{d t}=0.0005\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]$$ (a) Find an expression for the concentration \(\left[\mathrm{N}_{2} \mathrm{O}_{5}\right]\) after \(t\) seconds if the initial concentration is \(C .\) (b) How long will the reaction take to reduce the concen- tration of \(\mathrm{N}_{2} \mathrm{O}_{5}\) to 90\(\%\) of its original value?

Differentiate the function. $$ f(x)=\sin x \ln (5 x) $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{\csc \theta}$$

Prove the identity. \(\cosh (-x)=\cosh x\) (This shows that cosh is an even function.)

Simplify the expression. \(\tan \left(\sin ^{-1} x\right)\)

Strontium-90 has a half-life of 28 days. (a) A sample has a mass of 50 \(\mathrm{mg}\) initially. Find a formula for the mass remaining after \(t\) days. (b) Find the mass remaining after 40 days. (c) How long does it take the sample to decay to a mass of 2 \(\mathrm{mg} ?\) (d) Sketch the graph of the mass function.

Differentiate the function. $$ f(x)=\log _{5}\left(x e^{x}\right) $$

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{x \rightarrow 0^{+}} \frac{\ln x}{x}$$

Prove the identity. $$ \cosh x+\sinh x=e^{x} $$