Chapter 2: Q33E (page 77)
If an equation of the tangent line to the curve \(y = f\left( x \right)\) at the point where \(a = {\bf{2}}\) is \(y = {\bf{4}}x - {\bf{5}}\), find \(f\left( {\bf{2}} \right)\) and \(f'\left( {\bf{2}} \right)\).
The value of \(f\left( 2 \right)\) is 3.
The value of \(f'\left( 2 \right)\) is 4.
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Each limit represents the derivative of some function f at some number a. State such as an f and a in each case.
\(\mathop {{\bf{lim}}}\limits_{h \to {\bf{0}}} \frac{{{\bf{tan}}\left( {\frac{\pi }{{\bf{4}}} + h} \right) - {\bf{1}}}}{h}\)
Describe the intervals on which each function f is continuous.
(a). Prove Theorem 4, part 3.
(b). Prove Theorem 4, part 5.
If \(f\left( x \right) = 3{x^2} - {x^3}\), find \(f'\left( 1 \right)\) and use it to find an equation of the tangent line to the curve \(y = 3{x^2} - {x^3}\) at the point \(\left( {1,2} \right)\).
Sketch the graph of the function gthat is continuous on its domain \(\left( { - {\bf{5}},{\bf{5}}} \right)\) and where\(g\left( {\bf{0}} \right) = {\bf{1}}\), \(g'\left( {\bf{0}} \right) = {\bf{1}}\), \(g'\left( { - {\bf{2}}} \right) = {\bf{0}}\), \(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{5}}^ + }} g\left( x \right) = \infty \), and \(\mathop {{\bf{lim}}}\limits_{x \to - {{\bf{5}}^ - }} g\left( x \right) = {\bf{3}}\).
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