Chapter 3: Q19E (page 173)
19-22: (a) Find the differential \(dy\) and (b) evaluate \(dy\) for the given values of x and \(dx\).
19. \(y = {e^{\frac{x}{{10}}}},{\rm{ }}x = 0,{\rm{ }}dx = 0.1\)
a) The differential of the function is \(dy = \frac{1}{{10}}{e^{\frac{x}{{10}}}}dx\).
b) The value of \(dy\) is 0.01.
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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.
2. \(\mathop {lim}\limits_{x \to 2} \frac{{{x^2} + x - 6}}{{x - 2}}\).
Find the derivative of the function:
27. \(g\left( u \right) = {\left( {\frac{{{u^3} - 1}}{{{u^3} + 1}}} \right)^8}\)
Find \(f'\left( x \right)\) and \(f''\left( x \right)\).
32.\(f\left( x \right) = \sqrt x {e^x}\)
Find equations of the tangent line to the given curve at the specific point.
36. \(y = \frac{{1 + x}}{{1 + {e^x}}}\), \(\left( {0,\frac{1}{2}} \right)\)
53-56 Find \(y'\) and \(y''\).
54. \(y = {\left( {{\bf{1}} + \sqrt x } \right)^{\bf{3}}}\)
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