Chapter 6 — Quiz

10 questions, about 20 minutes. Each answer cites the section it comes from. Try every question before opening the answer.

1. The derivative function $f'$ is defined by: - A) $f'(x) = f(x) / x$ - B) $f'(x) = \displaystyle\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$ - C) $f'(x) = f(x + 1) - f(x)$ - D) $f'(x) = $ slope of the secant line through the origin

Answer**B.** Letting the point $x$ become a variable turns the pointwise limit into a new function $f'$, defined wherever the limit exists. *Reference: §6.2.*

2. Using the limit definition, the derivative of $f(x) = x^3$ is: - A) $x^2$ - B) $3x^2$ - C) $3x$ - D) $x^4/4$

Answer**B.** Expanding $(x+h)^3$ and cancelling $h$ gives $f'(x) = 3x^2$ (Worked Example 1). *Reference: §6.2.*

3. On an interval where $f'(x) > 0$, the function $f$ is: - A) at a local maximum - B) decreasing - C) increasing - D) constant

Answer**C.** A positive slope everywhere on the interval means the graph rises; this is the first entry in the §6.4 translation dictionary. *Reference: §6.4.*

4. A critical point of $f$ is a point where: - A) $f$ is undefined - B) $f'(x) = 0$ or $f'(x)$ does not exist - C) $f''$ changes sign - D) the graph crosses the $x$-axis

Answer**B.** Critical points are the zeros of $f'$ together with the points where $f'$ fails to exist; they are the candidates for maxima and minima. *Reference: §6.4.*

5. If $s(t)$ is position, then the second derivative $s''(t)$ is the: - A) speed - B) acceleration - C) distance travelled - D) velocity

Answer**B.** $s'$ is velocity and $s'' $ is acceleration; the third derivative $s'''$ is jerk. *Reference: §6.6.*

6. The gradient-descent update rule is $x_{n+1} = $: - A) $x_n + \alpha\, f'(x_n)$ - B) $x_n - \alpha\, f'(x_n)$ - C) $x_n - \alpha\, f(x_n)$ - D) $x_n / f'(x_n)$

Answer**B.** Step *against* the slope: the derivative is a compass pointing uphill, so the minus sign sends you downhill. *Reference: §6.8.*

7. Differentiability implies continuity. Is the converse true (does continuity imply differentiability)? - A) Yes, always - B) No - C) Only if $f$ is a polynomial - D) Only on closed intervals

Answer**B. No.** A continuous function can still have a corner (such as $|x|$ at $0$), a cusp, or a vertical tangent. Differentiability is strictly stronger. *Reference: §6.3.*

8. If $f''(x) > 0$ on an interval, then on that interval $f$ is: - A) increasing - B) decreasing - C) concave up - D) concave down

Answer**C. Concave up.** A positive second derivative means the slope is increasing, so the graph bends upward (cup-shaped). *Reference: §6.6.*

9. For $f(x) = x^4$, the fourth derivative $f^{(4)}(x)$ equals: - A) $0$ - B) $4$ - C) $24$ - D) $x^4$

Answer**C. $24$.** $f' = 4x^3$, $f'' = 12x^2$, $f''' = 24x$, $f^{(4)} = 24 = 4!$, and $f^{(5)} = 0$. *Reference: §6.6.*

10. A function $f$ is continuous at $a$ but has a cusp there. What can you conclude about $f'(a)$? - A) $f'(a) = 0$ - B) $f'(a)$ exists and is positive - C) $f'(a)$ does not exist - D) $f'(a)$ equals the value of $f(a)$

Answer**C.** At a cusp the one-sided slopes run off to $+\infty$ and $-\infty$, so no finite slope exists — one of the four failure modes. Continuity alone never guarantees a derivative. *Reference: §6.7.*

Scoring Guide

  • 9–10 correct — Strong command of the derivative as a function. Move on to Chapter 7's differentiation rules.
  • 7–8 correct — Solid. Skim the sections flagged on any misses before proceeding.
  • 5–6 correct — Re-read §6.4 (graphing $f'$) and §6.8 (gradient descent); redo Parts C and G of the exercises.
  • Below 5 — Revisit §6.2–§6.3 carefully and review the limit material of Chapter 5, then retake.