Chapter 2: Limits and Derivatives

Find the derivative of the function using the definition of the derivative. State the domain of the function and the domain of the derivative.

22. \(f\left( x \right) = mx + b\)

Use equation 5 to find \(f'\left( a \right)\) at the given number \(a\).

\(f\left( x \right) = \frac{{\bf{1}}}{{\sqrt {{\bf{2}}x + {\bf{2}}} }}\), \(a = {\bf{1}}\)

Evaluate the limit, if it exists.

\(\mathop {{\bf{lim}}}\limits_{x \to {\bf{9}}} \frac{{{\bf{9}} - x}}{{{\bf{3}} - \sqrt x }}\)

19-32 Prove the statement using the \(\varepsilon \), \(\delta \) definition of a limit.

22. \(\mathop {{\bf{lim}}}\limits_{x \to - {\bf{1}}.{\bf{5}}} \frac{{{\bf{9}} - {\bf{4}}{x^{\bf{2}}}}}{{{\bf{3}} + {\bf{2}}x}} = {\bf{6}}\)

19-24Explain why the function is discontinuous at the given number\(a\). Sketch the graph of the function.

22. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{\frac{{{x^2} - x}}{{{x^2} - 1}}}&{if\;x \ne 1}\\1&{if\;x = 1}\end{array}} \right.,\,\,a = 1\)

19-24Explain why the function is discontinuous at the given number\(a\). Sketch the graph of the function.

23. \(f\left( x \right) = \left\{ {\begin{array}{*{20}{c}}{cosx}&{if\;x < 0}\\0&{if\;x = 0}\\{1 - {x^2}}&{if\;x > 0}\end{array}} \right.,\,a = 0\)

Evaluate the limit if it exists.

\(\mathop {{\bf{lim}}}\limits_{h \to {\bf{0}}} \frac{{\sqrt {{\bf{9}} + h} - {\bf{3}}}}{h}\)

Find \(f'\left( a \right)\).

\(f\left( x \right) = {\bf{2}}{x^2} - {\bf{5}}x + {\bf{3}}\)

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically.

\(\mathop {\lim }\limits_{x \to 4} \frac{{\ln x - \ln 4}}{{x - 4}}\)

Find the derivative of the function using the definition of the derivative. State the domain of the function and the domain of the derivative.

23. \(f\left( t \right) = 2.5{t^2} + 6t\)