Chapter 1 — Self-Assessment Quiz

10 multiple-choice questions, ~20 minutes. Answers and explanations in collapsible blocks below each question. Aim for 70% (7/10) or above on first attempt. If you score below 70%, re-read the relevant sections before moving on.

1. The slope of a curve at a point is best understood as:

  • A) The slope of the line connecting two arbitrary points on the curve
  • B) The limit of secant slopes as the second point approaches the point of interest
  • C) The slope of the line connecting the point to the origin
  • D) The average rate of change over a small interval
Answer **B) The limit of secant slopes as the second point approaches the point of interest.** This is the central idea of differentiation. The slope at a single point is defined by approaching it with secant slopes that have well-defined values. *Why not A:* A secant slope is an approximation, not the answer. *Why not C:* The line to the origin has nothing to do with the slope at the point. *Why not D:* An average rate is a secant slope; we need the *instantaneous* rate, which requires a limit. *Reference: Section 1.1.*

2. The Fundamental Theorem of Calculus states that:

  • A) Every continuous function has a derivative
  • B) The integral of a function equals the function evaluated at the endpoints
  • C) Differentiation and integration are inverse operations
  • D) The area under a curve is always finite
Answer **C) Differentiation and integration are inverse operations.** This is the central insight — that finding slopes and finding areas, which seem unrelated, undo each other. *Why not A:* Continuous functions can fail to be differentiable (e.g., $|x|$ at $0$). *Why not B:* This is one *consequence* of FTC (FTC Part 2 specifically), not the theorem itself; you also need the connection to derivatives. *Why not D:* Not all areas are finite — improper integrals can diverge. *Reference: Section 1.4.*

3. Why did it take two thousand years (from Archimedes to Newton/Leibniz) to develop calculus?

  • A) The Greeks didn't know about variables
  • B) The required notion of "limit" was not yet rigorously formalized
  • C) No one tried until the 17th century
  • D) Books were too expensive
Answer **B) The required notion of "limit" was not yet rigorously formalized.** Both the tangent problem (limit of secant slopes) and the area problem (limit of rectangle sums) depend on a formal theory of limits, which did not exist before the late 17th century. *Why not A:* The Greeks had variables, just in geometric form. *Why not C:* Many mathematicians (Madhava in India in 1400, Cavalieri in Italy in 1635, Fermat in France in 1638) worked on parts of the problem. *Why not D:* Not the bottleneck; manuscripts circulated widely. *Reference: Section 1.3.*

4. Using the technique from Section 1.7, the slope of $y = x^2$ at $x = a$ is:

  • A) $a$
  • B) $a^2$
  • C) $2a$
  • D) $2a^2$
Answer **C) $2a$.** The general derivation: $\dfrac{(a+h)^2 - a^2}{h} = \dfrac{2ah + h^2}{h} = 2a + h$. As $h \to 0$, this approaches $2a$. *Why not A or B:* These don't come from any correct derivation. *Why not D:* Incorrect arithmetic. *Reference: Section 1.7 and exercise 1.11.*

5. Using right-endpoint Riemann sums on $[0, 1]$ with $n$ rectangles, the area under $y = x^2$ equals:

  • A) $\dfrac{n(n+1)(2n+1)}{6}$
  • B) $\dfrac{(n+1)(2n+1)}{6n^2}$
  • C) $\dfrac{1}{n}$
  • D) $\dfrac{1}{3}$ for all $n$
Answer **B) $\dfrac{(n+1)(2n+1)}{6n^2}$.** This is the algebraic expression for the right-endpoint Riemann sum *before* taking the limit. *Why not A:* This is $\sum_{i=1}^n i^2$, not the rectangle-area sum (which divides by $n^3$). *Why not C:* This is just the rectangle width. *Why not D:* The true area *is* $1/3$, but only in the limit as $n \to \infty$ — not for finite $n$. *Reference: Section 1.2.*

6. Calculus is the mathematics of:

  • A) Geometry
  • B) Change
  • C) Number theory
  • D) Logic
Answer **B) Change.** Derivatives measure rates of change; integrals accumulate change. The unifying theme of the subject is *change* — anything that varies with time, position, or another variable. *Reference: chapter introduction and Section 1.6; theme 1 in `_continuity.md`.*

7. Which of the following is not a typical application of calculus?

  • A) Training a neural network (gradient descent)
  • B) Modeling the spread of an epidemic (SIR equations)
  • C) Listing all prime numbers below one million
  • D) Computing the area under a probability density function
Answer **C) Listing all prime numbers below one million.** That's a problem in number theory and is solved by direct enumeration (e.g., the Sieve of Eratosthenes), not calculus. The other three are textbook calculus applications. *Reference: Sections 1.5 and 1.6.*

8. In the secant-slope formula $\dfrac{f(a+h) - f(a)}{h}$, what role does $h$ play?

  • A) The slope itself
  • B) The horizontal distance between the two points used
  • C) The vertical distance between the two points used
  • D) An arbitrary constant unrelated to the problem
Answer **B) The horizontal distance between the two points used.** $h$ is the change in $x$. The two points used are $(a, f(a))$ and $(a + h, f(a + h))$. The numerator $f(a + h) - f(a)$ is the corresponding change in $y$. The ratio is rise over run. *Reference: Section 1.1.*

9. Suppose you compute secant slopes for $y = x^2$ at $x = 1$ with $h = 0.1$, $0.01$, $0.001$, and $0.0001$. The values you should observe are approximately:

  • A) $2.1$, $2.01$, $2.001$, $2.0001$
  • B) $1.1$, $1.01$, $1.001$, $1.0001$
  • C) $2.0$, $2.0$, $2.0$, $2.0$
  • D) $0.21$, $0.0201$, $0.002001$, $0.00020001$
Answer **A) $2.1$, $2.01$, $2.001$, $2.0001$.** The general formula is $\text{secant slope} = 2 + h$ (Section 1.7), so as $h$ shrinks the slope approaches $2$ from above. *Why not C:* The secant slope is never *exactly* $2$ for $h \neq 0$. *Why not D:* These are the values of $f(1+h) - f(1)$, not the slopes (which divide by $h$). *Reference: Section 1.1 numerical experiment.*

10. Which statement most accurately describes the relationship between hand computation and Python in this book?

  • A) Python replaces hand computation entirely — calculators do the work now
  • B) Hand computation is required; Python is for visualization and verification
  • C) Python is optional and can be skipped without consequence
  • D) Both are equivalent; pick whichever you prefer
Answer **B) Hand computation is required; Python is for visualization and verification.** The book takes the explicit position that you must learn to differentiate and integrate by hand to understand what is happening. Python (a) visualizes what the math means, (b) verifies hand answers, (c) handles problems too complex for hand computation, and (d) prepares you for how calculus is used in practice. *Reference: Section 1.8 ("Hand computation and machine computation work together"); `_continuity.md` Section 4.*

Scoring

  • 9–10 correct: You have a strong grasp of Chapter 1. Move on to Chapter 2.
  • 7–8 correct: Solid understanding. Re-read sections you missed before continuing.
  • 5–6 correct: Re-read the whole chapter, paying particular attention to Sections 1.1 (tangent), 1.2 (area), and 1.4 (FTC). Retake before moving on.
  • Below 5: Slow down. Read more carefully. Work through the Part A and Part B exercises. Then retake. There is no shame in revisiting — these concepts are foundational.