Chapter 2: Q33E (page 127)
Find the points on the lemniscate in Exercise 23 where the tangent is horizontal.
The points on the lemniscate curve in Exerice 23 where the tangent is horizontal are \(\left( { \pm \frac{{{\rm{5}}\sqrt {\rm{3}} }}{{\rm{4}}}, \pm \frac{{\rm{5}}}{{\rm{4}}}} \right)\).
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Sketch the graph of the function fwhere the domain is \(\left( { - {\bf{2}},{\bf{2}}} \right)\),\(f'\left( {\bf{0}} \right) = - {\bf{2}}\), \(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} f\left( x \right) = \infty \), f is continuous at all numbers in its domain except \( \pm {\bf{1}}\), and f is odd.
19-32 Prove the statement using the \(\varepsilon \), \(\delta \)definition of a limit.
25. \(\mathop {{\bf{lim}}}\limits_{x \to 0} {x^{\bf{2}}} = {\bf{0}}\)
A warm can of soda is placed in a cold refrigerator. Sketch the graph of the temperature of the soda as a function of time. Is the initial rate of change of temperature greater or less than the rate of change after an hour?
Explain the meaning of each of the following.
(a)\(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{3}}} f\left( x \right) = \infty \)
(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{4}}^ + }} f\left( x \right) = - \infty \)
(a) The van der Waals equation for \({\rm{n}}\) moles of a gas is \(\left( {P + \frac{{{n^{\rm{2}}}a}}{{{V^{\rm{2}}}}}} \right)\left( {V - nb} \right) = nRT\) where \(P\)is the pressure,\(V\) is the volume, and\(T\) is the temperature of the gas. The constant\(R\) is the universal gas constant and\(a\)and\(b\)are positive constants that are characteristic of a particular gas. If \(T\) remains constant, use implicit differentiation to find\(\frac{{dV}}{{dP}}\).
(b) Find the rate of change of volume with respect to pressure of \({\rm{1}}\) mole of carbon dioxide at a volume of \(V = {\rm{10}}L\) and a pressure of \(P = {\rm{2}}{\rm{.5atm}}\). Use \({\rm{a}} = {\rm{3}}{\rm{.592}}{{\rm{L}}^{\rm{2}}}{\rm{ - atm}}/{\rm{mol}}{{\rm{e}}^{\rm{2}}}\)and \(b = {\rm{0}}{\rm{.04267}}L/mole\).
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