Chapter 2: Q27E (page 77)
Find an equation of the tangent line to the graph of \(y = B\left( x \right)\)at\(x = 6\),if\(B\left( {\bf{6}} \right) = {\bf{0}}\),and \(B'\left( 6 \right) = - \frac{1}{2}\).
The equation of the tangent line is \(y = - \frac{1}{2}x + 3\).
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Sketch the graph of the function f for which \(f\left( {\bf{0}} \right) = {\bf{0}}\), \(f'\left( {\bf{0}} \right) = {\bf{3}}\), \(f'\left( {\bf{1}} \right) = {\bf{0}}\), and \(f'\left( {\bf{2}} \right) = - {\bf{1}}\).
Use thegiven graph of f to state the value of each quantity, if it exists. If it does not exists, explain why.
(a)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ - }} f\left( x \right)\)
(b)\(\mathop {{\bf{lim}}}\limits_{x \to {{\bf{2}}^ + }} f\left( x \right)\)
(c) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{2}}} f\left( x \right)\)
(d) \(f\left( {\bf{2}} \right)\)
(e) \(\mathop {{\bf{lim}}}\limits_{x \to {\bf{4}}} f\left( x \right)\)
(f) \(f\left( {\bf{4}} \right)\)
43-45 Find the numbers at which f is discontinuous. At which of these numbers is f continuous from the right, from the left, or neither? Sketch the graph of f?
44. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{{\bf{2}}^x}}&{{\bf{if}}\,\,\,x \le {\bf{1}}}\\{{\bf{3}} - x}&{{\bf{if}}\,\,\,{\bf{1}} < x \le {\bf{4}}}\\{\sqrt x }&{{\bf{if}}\,\,\,x > {\bf{4}}}\end{array}} \right.\)
41-42 Show that f is continuous on \(\left( { - \infty ,\infty } \right)\).
\(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{{\bf{sin}}\,x}&{{\bf{if}}\,\,x < \frac{\pi }{{\bf{4}}}}\\{{\bf{cos}}\,x}&{{\bf{if}}\,\,\,x \ge \frac{\pi }{{\bf{4}}}}\end{array}} \right.\)
Each limit represents the derivative of some function f at some number a. State such as an f and a in each case.
\(\mathop {{\bf{lim}}}\limits_{x \to \frac{{\bf{1}}}{{\bf{4}}}} \frac{{\frac{{\bf{1}}}{x} - {\bf{4}}}}{{x - \frac{{\bf{1}}}{{\bf{4}}}}}\)
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