Mathematics for Management -- Supplementary Electronic Materials

Quiz: Exponential & Logarithmic Functions - incl. Inverse Functions

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1 Which of the following is identical to \(ln \left( \frac{ {\rm{e}}^x \sqrt{x} }{ (5x-1)^{\cos(x)} } \right) \).

\( x - \frac{1}{2} \ln(x) - \cos(x) \ln(5x-1) \) \( \frac{(5x-1)^{\cos(x)}}{{\rm{e}}^{x} \sqrt{x}}\) \( \frac{\ln(x - \tfrac{1}{2} \ln(x))}{\ln(\cos(x) \ln(5x-1))}\) \( \ln \left( \frac{\frac{1}{2} {\rm{e}}^x x^{-1/2}}{-5 \sin(x) (5x-1)^{\cos(x)}} \right)\)

2 Let \( f(x) = \sqrt[5]{\sin({\rm{e}}^x)} \) then \(f'(x)\) is.

\( \frac{1}{5}\left( \sin({\rm{e}}) \right)^{-4/5}\) \( \frac{{\rm{e}}^x \left( \sin( {\rm{e}}^x ) \right)^{-4/5}}{5 \sqrt{1 - {\rm{e}}^{2x}}} \) \( \frac{{\rm{e}}^x}{5 \sqrt{1 - {\rm{e}}^{2x}}} \) \( \frac{1}{5} \left( \frac{{\rm{e}}^x}{\sqrt{1 - {\rm{e}}^{2x}}} \right)^{-4/5} \)

3 The limit \(lim_{x \to 3} \left( \ln( x-3) \right)^{x^2 -9}\) is an indeterminate form of type.

\( \infty^0 \) \( 0^0 \) \( 1^{\infty} \) \( \frac{0}{0} \)

4 What is the value of \(lim_{x \to 1^+} x^{1/\ln(x)}\) (if it exists).

\( 1 \) \( 0 \) \({\rm{e}}\) does not exist

5 The slope of the tangent line to the graph of \(y = \ln(x^2)\) at \(x = {\rm{e}}^2\) is.

\( {\rm{e}}^{-2} \) \( 2 {\rm{e}}^{-2} \) \( {\rm{e}}^{-4} \) \( 2 {\rm{e}}^{-4} \)

6 The following table gives values of continuously differentiable functions \( f(x) \) and \( g(x) \) as well as their first derivatives for various \( x \) values. Choose the correct statement. \[ \]\begin{array}{c || c c c c c c c c c} x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline \hline f(x) & 3 & -5 & 2 & 10 & 4 & -6 & 5 & -4 & 6 \\ g(x) & 1 & 0 & -1 & 1 & -2 & 3 & 8 & 9 & -3 \\ \hline f'(x) & 5 & 2 & 4 & -3 & 7 & 1 & -2 & 6 & -5 \\ g'(x) & 3 & 7 & 9 & 2 & 5 & -3 & 1 & 4 & 6 \end{array} \[ \]

\( (f \circ g)'(4) = 1\) and \((f^{-1} \circ g) (6) = 2 \) \( (f \circ g)'(4) = 5\) and \((f^{-1} \circ g) (6) = 2 \) \( (f \circ g)'(4) = 6\) and \((f^{-1} \circ g) (6) = 1 \) \( (f \circ g)'(4) = 10\) and \((f^{-1} \circ g) (6) = 1 \)

7 Determine the derivative \( f'(x) \) of the function \( f(x) \, \, = \, \, \ln\left( \frac{{\rm{e}}^x}{{\rm{e}}^x - 10} \right) \, . \)

\( f'(x) = x - \frac{{\rm{e}}^x}{{\rm{e}}^x - 10} \) \( f'(x) = -\frac{1}{{\rm{e}}^x} \) \( f'(x) = \frac{10}{10 - {\rm{e}}^x} \) \( f'(x) = \frac{{\rm{e}}^x - 20}{{\rm{e}}^x - 10} \)

8 The table below gives values of the differentiable functions \( f(x) \) and \( g(x) \) as well a the derivative \( f'(x) \) of \( f(x) \) at selected values of \( x \). \[ \] \begin{array}{c | c c c c c} x & -4 & -2 & 0 & 2 & 4 \\ \hline f(x) & 0 & 4 & 6 & 7 & 10 \\ g(x) & -9 & -7 & -4 & -3 & -2 \\ f'(x) & 5 & 4 & 2 & 1 & 3 \end{array} \[ \] Let \( g(x) = f^{-1}(x) \), then what is the value of \( g'(4) \)?

\( -\frac{1}{3} \) \( -\frac{1}{4} \) \( \frac{1}{3} \) \( \frac{1}{4} \)

9 Let \( f(x) = g^{-1}(x) \) be the inverse function of \( g(x) = \frac{1}{x} \), then what is the value of \( f'(3) \)?

\( -9 \) \( -\frac{1}{9} \) \( \frac{1}{9} \) \( 9 \)

10 Let \( g(x) = 2x^3 - 3x \). If \( f(x) = g^{-1}(x) \) is the inverse function of \( g(x) \), then what is the value of \( f'(-1) \)?

\( -1 \) \( \frac{1}{5} \) \( \frac{1}{3} \) \( 3 \)

Your grade is: __ of 10

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Solution: 1a; 2b; 3a; 4c; 5b; 6d; 7c; 8d; 9a; 10c

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.