Mathematics for Management -- Supplementary Electronic Materials

Quiz: The Derivative

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1. By the definition, the derivative of \(f(x) = \sqrt{2x - 5}\) is

\( f'(x) = \lim_{h \to 0} \frac{\sqrt{2x + 2h - 5} - \sqrt{2x - 5}}{h} \) \( f'(x) = \lim_{h \to 0} \frac{\sqrt{2x + 2h - 5} + \sqrt{2x - 5}}{h} \) \( f'(x) = \lim_{h \to 0} \frac{\sqrt{2x + h - 5} - \sqrt{2x - 5}}{h} \) \( f'(x) = \lim_{h \to 0} \frac{\sqrt{2x + h - 5} + \sqrt{2x - 5}}{h} \)

2. For the function \(f(x) = \frac{4}{x}\) simplify the difference quotient \(\frac{f(x+h) - f(x)}{h}\).

\( -\frac{4}{x^2} \) \( \frac{-4}{x (x+h)} \) \( \frac{4h}{x(x+h)} \) \( \frac{4h-4}{x(x+h} \)

3. Let \(f(x) = x^2 - 8x + 18\). Determine the value of \(f'(5)\).

\( 5 \) \( 3 \) \( 2 \) \( 1 \)

4. Let \(f(x) = x^2 - 8x + 18\) (as in question 3). Determine the equation of the tangent to the graph of \(f(x)\) at the point \((3, f(5))\).

\( y - 3 = -5 (x - 5) \) \( y - 3 = 3 (x - 5) \) \( y - 3 = -2 (x - 5) \) \( y - 3 = 1 (x - 5) \)

5. Let \(f(x) = x^3 - 6x^2 + 8x - 2\). What is the instantaneous rate of change of \(f\) at \(x=3\)?

\( -5 \) \( -\frac{15}{4} \) \( -1 \) \( 6 \)

6. What is the inclination nature of the function \(f(x) = x^3 - 3x^2 + 4x\) on \(\mathbb{R}\)?

always increasing always decreasing always constant both, increasing and decreasing

7. Determine the interval on which the function \(f(x) = x^2 - 4x + 5\) is increasing.

\( (2, \infty) \) \( (-\infty, 2) \) \( (3, \infty) \) \( (-\infty, \infty) \)

8. Let \(f : \mathbb{R} \to \mathbb{R}\) be a differentiable function such that \(f(0) = -5\) and \(f'(x) \leq 3\) for all \(x \in \mathbb{R}\).
Of the following, which is not a possible value for \(f(2)\)?

\( -10 \) \( -5 \) \( 1 \) \( 2 \)

9. Let \(f(x) : \mathbb{R} \to \mathbb{R}\) be the piecewise given function \[ f(x) \, \, = \, \left\{ \begin{array}{l c l} x^2 + 5 \, , & & \text{if \(x < 2\)}\\ 7x - 5 \, , & & \text{if \(x \geq 2\)} \end{array} \right. \] Which of the following must be true:
I) \(f(x)\) is continuous everywhere.
II) \(f(x)\) is differentiable everywhere.
III) \(f(x)\) is not differentiable at \(x = 2\).

I) only I) and II) only I) and III) only III) only

10. Which of the following is true for the function \(f(x) = |x-1|\)?

\(f(x)\) is differentiable at \(x = 1\) and \(x = 2\). \(f(x)\) is differentiable at \(x = 1\), but not at \(x = 2\). \(f(x)\) is differentiable at \(x = 2\), but not at \(x = 1\). \(f(x)\) is not differentiable at either \(x = 1\) or \(x = 2\).

Your grade is: __ of 10

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

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