Chapter 5 — Quiz

10 questions, ~20 minutes. Answer each, then expand the solution to check yourself and find the section to review.

1. The average rate of change of $f$ on $[a, b]$ equals: - A) $f(b) - f(a)$ - B) $\dfrac{f(b) - f(a)}{b - a}$ - C) $\dfrac{f(a) + f(b)}{2}$ - D) the derivative at the midpoint

Answer**B)** Average rate of change is the *change in output* divided by the *change in input* — equivalently, the slope of the secant line through $(a, f(a))$ and $(b, f(b))$. Option C is the average of the two *values*, a different quantity. *Reference: Section 5.1.*

2. The derivative $f'(a)$ is defined as: - A) $f(a + 1) - f(a)$ - B) $\displaystyle\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$ - C) the maximum value of $f$ near $a$ - D) the average of $f$ over a small interval around $a$

Answer**B)** The derivative is the limit of the difference quotient — the limit of average rates of change over shrinking intervals. *Reference: Section 5.2.*

3. Using the limit definition, the slope of the tangent to $f(x) = x^2$ at $(3, 9)$ is: - A) $6$ - B) $9$ - C) $3$ - D) $0$

Answer**A) $6$.** $f'(3) = \lim_{h\to 0}\frac{(3+h)^2 - 9}{h} = \lim_{h\to 0}(6 + h) = 6$. *Reference: Section 5.3 (worked example).*

4. A particle has position $s(t) = t^2$ (meters, seconds). Its instantaneous velocity at $t = 4$ is: - A) $8$ m/s - B) $16$ m/s - C) $32$ m/s - D) $4$ m/s

Answer**A) $8$ m/s.** Velocity is the derivative of position: $s'(t) = 2t$ (from the definition), so $s'(4) = 8$. *Reference: Section 5.4.*

5. "Differentiable at $a$" implies which of the following at $a$? - A) continuity - B) boundedness only - C) monotonicity - D) none of the above

Answer**A) Continuity.** Differentiability is *stronger* than continuity: every differentiable function is continuous, but not conversely. *Reference: Section 5.7 (theorem: differentiability implies continuity).*

6. At a corner of $f$ — for example, $|x|$ at $x = 0$ — the function is: - A) differentiable - B) not continuous - C) continuous but not differentiable - D) equipped with a vertical tangent

Answer**C)** At a corner the two one-sided difference-quotient limits exist but disagree (for $|x|$ they are $-1$ and $+1$), so $f'(0)$ does not exist — yet $f$ is perfectly continuous. *Reference: Section 5.7 (case 1: corner).*

7. The equation of the tangent line to $y = f(x)$ at $x = a$ is: - A) $y = f(a) + f'(a)\cdot x$ - B) $y - f(a) = f'(a)\,(x - a)$ - C) $y = f'(a)\cdot x + f(a)$ - D) $y = f(x) + f'(a)$

Answer**B)** Point-slope form using the point $(a, f(a))$ and slope $f'(a)$. *Reference: Section 5.3.*

8. If $f(x) = c$ (a constant), then $f'(x) =$ - A) the constant $c$ itself - B) $0$ - C) undefined - D) $x$

Answer**B) $0$.** $f(a+h) - f(a) = c - c = 0$, so the difference quotient is $0$ for all $h$ and its limit is $0$. A constant function is a horizontal line — zero slope everywhere. *Reference: Section 5.9.*

9. Which of these is not a notation for the derivative of $y = f(x)$? - A) $f'(x)$ - B) $\dfrac{dy}{dx}$ - C) $\displaystyle\int f(x)\,dx$ - D) $Df$

Answer**C)** $\int f(x)\,dx$ is the (indefinite) *integral*, not the derivative. The others — Lagrange $f'(x)$, Leibniz $\frac{dy}{dx}$, and operator $Df$ — all name the derivative. *Reference: Section 5.6.*

10. Using the limit definition, the derivative of $f(x) = 1/x$ is: - A) $1/x^2$ - B) $-1/x^2$ - C) $-1/x$ - D) $\ln|x|$

Answer**B) $-1/x^2$.** $f'(x) = \lim_{h\to 0}\dfrac{\frac{1}{x+h} - \frac{1}{x}}{h} = \lim_{h\to 0}\dfrac{x - (x+h)}{h\,x(x+h)} = \lim_{h\to 0}\dfrac{-1}{x(x+h)} = -\dfrac{1}{x^2}$. *Reference: Section 5.2 (definition); see also Exercise 5.11.*

Scoring

  • 9–10: Excellent — you own the conceptual core of Part I. Move on to Part II (Chapter 6).
  • 7–8: Solid. If you missed Q5 or Q6, re-read the differentiability-implies-continuity theorem in Section 5.7.
  • 5–6: Re-read Sections 5.2 (the definition) and 5.7 (when derivatives fail), then redo Part C of the exercises.
  • Below 5: Slow down and rework the definition by hand. Redo Parts B and C of the exercises before continuing.