Chapter 1: Q50E (page 10)
Express the surface area of a cube as a function of its volume.
The expression for the surface area of a cube as a function of its volume is \({\bf{A = 6}}{{\bf{V}}^{{\bf{2/3}}}}\) and domain of the function is\(\left( {0,\infty } \right)\).
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Prove the statement using the\(\varepsilon \), \(\delta \)definition of a limit and illustrate with a diagram like Figure 15.
\(\mathop {\lim }\limits_{x \to 4} \left( {2x - 5} \right) = 3\)
Find the domain and sketch the graph of the functions\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{3 - }}\frac{{\bf{1}}}{{\bf{2}}}{\bf{x}},{\bf{x}} \le {\bf{2}}\\{\bf{2x - 5}},{\bf{x}} > {\bf{2}}\end{array} \right.\).
Find the functions (a) \({\bf{f}} \circ {\bf{g}}\) ,(b) \({\bf{g}} \circ {\bf{f}}\) ,(c) \({\bf{f}} \circ {\bf{f}}\) and (d) \({\bf{g}} \circ {\bf{g}}\) and their domains.
\({\bf{f}}\left( {\bf{x}} \right){\bf{ = x - 2}}\) \({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}{{\bf{x}}^{\bf{2}}}{\bf{ + 3x + 4}}\)
Determine whether\({\bf{f}}\)is even, odd, or neither. If you have a graphing calculator, use it to check your answer visually.
64.\({\bf{f}}\left( {\bf{x}} \right){\bf{ = 1 + 3}}{{\bf{x}}^{\bf{3}}}{\bf{ - }}{{\bf{x}}^{\bf{5}}}\)
It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species of bats living in caves in central Mexico has been related to the surface area\({\bf{A}}\)of the caves by the equation\(S = 0.7{A^{0.3}}\).
(a) The cave called MisionImposible near Puebla, Mexico, has a surface area of\({\bf{A = 60}}\;{{\bf{m}}^{\bf{2}}}\). How many species of bats would you expect to find in that cave?
(b) If you discover that four species of bats live in a cave, estimate the area of the cave.
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