Mathematics for Management -- Supplementary Electronic Materials

Quiz: The Product & Quotient Rule

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1. Determine the derivative \(f'(x)\) of \(f(x) = (x^2 - 4x + 2) (2x^3 - x^2 + 4)\).

\( f'(x) = 10x^4 - 36x^3 + 24x^2 +4x - 16 \) \( f'(x) = 2x^4 - 32x^3 + 24x^2 + 4x - 16 \) \( f'(x) = 2x^4 - 36x^3 + 24x^2 + 4x - 16 \) \( f'(x) = 10x^4 - 32x^3 +24x^2 + 4x - 16 \)

2. Determine the value of \(f''(0)\) for \(f(x) = {\rm{e}}^x \cdot (x-1)\).

\( -2 \) \( -1 \) \( 0 \) \( 1 \)

3. Determine the derivative \(f'(x)\) of \(f(x) = \frac{5x}{x^3 - 4}\).

\( f'(x) = \frac{5}{3 x^2} \) \( f'(x) = \frac{-10 x^3 - 20}{(x^3 - 4)^2} \) \( f'(x) = \frac{x^4 - 15 x^3 - 4x}{x^3 - 4} \) \( f'(x) = \frac{20 x^2 - 20}{(x^3 - 4)^2} \)

4. Determine the derivative \(f'(x)\) of \(f(x) = \frac{x^2 + 4x + 3}{\sqrt{x}}\).

\( f'(x) = \frac{3}{2} x^{1/2} - 2 x^{-1/2} + \frac{3}{2} x^{-3/2} \) \( f'(x) = -\frac{3}{2} x^{1/2} + 2 x^{-1/2} - \frac{3}{2} x^{-3/2} \) \( f'(x) = -\frac{3}{2} x^{1/2} - 2 x^{-1/2} + \frac{3}{2} x^{-3/2} \) \( f'(x) = \frac{3}{2} x^{1/2} + 2 x^{-1/2} - \frac{3}{2} x^{-3/2} \)

5. Give an equation of the tangent line to the graph of \(f(x)\) at \(x=0\), where \(f(x)\) is given by \[ f(x) \, \, = \, \, \frac{-10x^2 - 3}{4x + 1} \, . \]

\( y = 12x - 3 \) \( y = -12x - 3 \) \( y = 12x + 3 \) \( y = -12x + 3 \)

6. Let \(g(x) : \mathbb{R} \to \mathbb{R}\) be a continuously differentiable function. The line tangent to the graph of \(g(x)\) at \(x = -1\) passes through the two points \((-1,3)\) and \((0,-3)\). Let \(f(x) = {\rm{e}}^x \cdot g(x)\). Determine the value of \(f'(-1)\).

\( \frac{9}{{\rm{e}}} \) \( -\frac{3}{{\rm{e}}} \) \( -\frac{6}{{\rm{e}}} \) \( -\frac{6}{{\rm{e}}} - \frac{3}{{\rm{e}}^2} \)

7. For the remaining four questions assume that the continuously differentiable functions \(f(x)\) and \(g(x)\) be given and they have the values shown in the table: \[ \begin{array}{c || c |c | c | c} x & f & f' & g & g' \\ \hline 0 & 2 & 1 & 5 & -4 \\ 1 & 3 & 2 & 3 & -3 \\ 2 & 5 & 3 & 1 & -2 \\ 3 & 10 & 4 & 0 & -1 \\ \end{array} \]
With \(f(x)\) and \(g(x)\) given as above, let \(F(x) = f(x) + 2 g(x)\). Determine the value \(F'(3)\).

\( -2 \) \( 2 \) \( 7 \) \( 8 \)

8. With \(f(x)\) and \(g(x)\) given as above, let \(F(x) = f(x) \cdot g(x)\). Determine the value \(F'(2)\).

\( -20 \) \( -7 \) \( -6 \) \( 13 \)

9. With \(f(x)\) and \(g(x)\) given as above, let \(F(x) = \frac{1}{g(x)}\). Determine the value \(F'(1)\).

\( \frac{1}{3} \) \( -\frac{1}{3} \) \( -\frac{1}{9} \) \( \frac{1}{9} \)

10. With \(f(x)\) and \(g(x)\) given as above, let \(F(x) = \frac{f(x)}{g(x)}\). Determine the value \(F'(0)\).

\( -\frac{13}{25} \) \( -\frac{1}{4} \) \( \frac{13}{25} \) \( \frac{13}{16} \)

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

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