Mathematics for Management -- Supplementary Electronic Materials

Quiz: Techniques of Differentiation

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1. Let \(f(x) = 6 x^{-2} + 8 x^3 + 11x\). What is \(f'(x)\)?

\( f'(x) = -12 x^{-1} + 24 x^2 + 11 \) \( f'(x) = -12 x^{-1} + 24 x^2 \) \( f'(x) = -12 x^{-3} + 24 x^2 + 11 \) \( f'(x) = -12 x^{-3} + 24 x^2 \)

2. Find the derivative \(f'(x)\) of \(f(x) = 5 x^2 (x + 47)\) .

\( f'(x) = 15 x^2 + 470 x \) \( f'(x) = 5 x^2 + 470 x \) \( f'(x) = 10 x \) \( f'(x) = 15 x^2 - 470 x \)

3. Find the derivative \(f'(x)\) of \(f(x) = \frac{3}{5 x^3} - 4x^2 + 2x - \frac{1}{2}\).

\( f'(x) = -\frac{9}{5} x^{-4} - 8x + 2 \) \( f'(x) = \frac{9}{5} x^{-4} - 8x + 2 \) \( f'(x) = -\frac{9}{5} x^{-4} - 8x + 2 - \frac{1}{2} \) \( f'(x) = -\frac{9}{5} x^{-4} + 8x + 2 \)

4. Find the slope of the line tangent to the graph of the function \(f(x) = 9 x^{5/2} - 7 x^{3/2}\) at \(x = 4\).

\( 159 \) \( 8 \) \( 6 \) \( 96 \)

5. Find the equation of the line tangent to the graph of the function \(f(x) = x^2 - x\) at \(x = -3\).

\( y = -7x + 6 \) \( y = -7x - 9 \) \( y = -7x - 6 \) \( y = -7x + 9 \)

6. Let \(f(x) = 4 x^3 - 5x + 3\). Which of the following is an equation of the line tangent to the graph of \(f(x)\) at the point where \(x = -1\)?

\( y = 7x - 3 \) \( y = 7x + 11 \) \( y = -5x - 5 \) \( y = 7x + 7 \)

7. Which of the following is an equation of the line tangent to the graph of \(x^2 - 3y = 10\) at the point \((1 , -3)\)?

\( y + 3 = -11(x - 1) \) \( y + 3 = -\frac{7}{3}(x - 1) \) \( y + 3 = \frac{2}{3}(x - 1) \) \( y + 3 = \frac{11}{3}(x - 1) \)

8. Find all points of the graph of \(f(x) = 3x^2 + 9x\) whose tangent lines are parallel to the line \(y - 33x = 0\).

\( (4, 84) \) \( (6, 162) \) \( (7, 210) \) \( (5, 120) \)

9. At what point on the graph of \(y = \frac{1}{2} x^2\) is the tangent line parallel to the line \(2x - 4y = 3\)?

\( \left( \frac{1}{2}, -\frac{1}{2} \right) \) \( \left( \frac{1}{2}, \frac{1}{8} \right) \) \( \left( 1, -\frac{1}{4} \right) \) \( \left( 1, \frac{1}{2} \right) \)

10. The equation of the tangent line to the graph of \(f(x) = x^2 - 4x\) at the \(y\)-intercept of \(f(x)\) is

\( y = 8x - 4 \) \( y = 4x \) \( y = -4x \) \( y = 4x - 8 \)

Your grade is: __ of 10

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

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