Chapter 1 — Exercises

35 exercises across five difficulty tiers. Aim to complete all ⭐ and ⭐⭐ problems; attempt several ⭐⭐⭐ and at least one ⭐⭐⭐⭐. Because this is the motivational opening chapter, the emphasis is conceptual and exploratory — there is deliberately little heavy computation yet. Selected answers are in appendices/answers-to-selected.md.

Part A — Conceptual (⭐ Foundational, ~5 min each)

1.1 In your own words, state the tangent problem. Why is it harder than the analogous question for straight lines?

1.2 In your own words, state the area problem. Why is it harder than the analogous question for rectangles and triangles?

1.3 Define secant line and tangent line. Draw a picture of each on a sample curve.

1.4 What does it mean for two operations to be "inverse" to each other? Give an example from arithmetic (e.g., addition and subtraction).

1.5 State the Fundamental Theorem of Calculus in plain English. (You don't need the formal statement; just the idea.)

1.6 Name three real-world domains, other than physics, where calculus is essential. For each, briefly describe what calculus is used to compute.

1.7 Why did it take two thousand years to develop calculus? What was the missing mathematical concept?

1.8 A classmate computes the secant-slope formula for $y = x^2$ at $x = 1$, simplifies it to $2 + h$, and then says "now set $h = 0$ to get the slope." A second classmate objects that you "can't divide by zero." In two or three sentences, explain who is right and why the original division-by-zero worry does not apply to the simplified expression $2 + h$. (See §1.7.)

1.9 Section 1.3 calls the tangent problem a local question and the area problem a global question. In your own words, explain what "local" and "global" mean here, and why that contrast makes the Fundamental Theorem (§1.4) so surprising.

1.10 The chapter names four anchor examples in §1.5 (gradient descent, the SIR model, the area under the normal curve, and Euler's formula). Pick the one that most interests you and write two sentences on why you would want to understand it by the end of the book.

Part B — Secant Slope Calculations (⭐⭐ Intermediate, ~10 min each)

1.11 Estimate the slope of $y = x^2$ at the point $(2, 4)$ by computing secant slopes with $h = 0.1$, $h = 0.01$, and $h = 0.001$. What value do they approach?

1.12 Estimate the slope of $y = x^3$ at the point $(1, 1)$ using $h = 0.1$, $h = 0.01$, $h = 0.001$. What value does the slope appear to approach?

1.13 Estimate the slope of $y = \sqrt{x}$ at the point $(4, 2)$ using $h = 0.1$, $h = 0.01$, $h = 0.001$. What value does the slope appear to approach? (Hint: evaluate $\sqrt{4 + h}$ for small $h$.)

1.14 For $y = x^2$, compute the secant slope formula $\dfrac{(a+h)^2 - a^2}{h}$, simplify, then let $h \to 0$. Show that the slope at $x = a$ is $2a$. (This is the general formula behind the pattern you've been observing.)

1.15 For $y = x^3$, compute the secant slope formula $\dfrac{(a+h)^3 - a^3}{h}$, simplify, then let $h \to 0$. Show that the slope at $x = a$ is $3a^2$. (Hint: expand $(a+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3$.)

1.16 Use the §1.7 method on the line $y = 5x - 2$: form $\dfrac{[5(a+h) - 2] - [5a - 2]}{h}$, simplify, and let $h \to 0$. Why is the answer the same for every $a$, and why is that exactly what you expect for a straight line?

Part C — Riemann Sum Area Estimates (⭐⭐ Intermediate, ~15 min each)

1.17 Estimate the area under $y = x$ on $[0, 1]$ using right-endpoint rectangles with $n = 10$, $n = 100$, $n = 1000$. The true answer is $1/2$ (geometrically, it's a triangle). How close do your estimates get?

1.18 Estimate the area under $y = x^2$ on $[0, 2]$ using right-endpoint rectangles with $n = 100$. (Hint: the rectangle widths are now $2/n$, not $1/n$.) The true answer is $8/3 \approx 2.667$. How close is your estimate?

1.19 Estimate the area under $y = e^x$ on $[0, 1]$ using right-endpoint rectangles with $n = 1000$. The true answer is $e - 1 \approx 1.71828$. How close is your estimate?

1.20 Estimate the area under $y = \sin x$ on $[0, \pi]$ using right-endpoint rectangles with $n = 1000$. The true answer is $2$. How close is your estimate?

1.21 For the area under $y = x$ on $[0,1]$, will right-endpoint rectangles overshoot or undershoot the true area? Will left-endpoint rectangles overshoot or undershoot? Justify with a picture, then confirm numerically with a small $n$ (say $n=4$).

Part D — Applied (⭐⭐⭐ Challenging, ~25 min each — at least two distinct fields)

1.22 — Falling object (Physics). An object falls from rest. Its height (in metres) above the ground after $t$ seconds is $h(t) = 100 - 4.9t^2$. Estimate the object's downward velocity at $t = 2$ s by computing $-[h(2.01) - h(2)] / 0.01$. (The minus sign makes downward velocity positive, since height is decreasing.) Then find the exact velocity using the §1.7 method, the time the object hits the ground, and its impact speed.

1.23 — Bacterial growth (Biology). A bacterial population doubles every hour. With $1{,}000$ cells at $t = 0$, the population is $P(t) = 1000 \cdot 2^t$ (with $t$ in hours). Estimate the growth rate (cells per hour) at $t = 3$ by computing $[P(3.01) - P(3)] / 0.01$. What pattern do you notice in the growth rate as $t$ increases, and what does it say about exponential growth?

1.24 — Marginal revenue (Economics). A company's revenue from selling $q$ widgets is $R(q) = 100q - 0.01q^2$ dollars. Estimate the marginal revenue at $q = 1000$ by computing $[R(1001) - R(1000)] / 1$, then by shrinking the step to $0.01$. Compare with the calculus answer $R'(q) = 100 - 0.02q$ at $q = 1000$. Why is the small-step estimate closer?

1.25 — Drug concentration (Pharmacology). A drug's blood concentration $t$ hours after a dose is $C(t) = 10 e^{-0.3t}$ mg/L. Estimate the rate at which the concentration is decreasing at $t = 5$ h by computing $-[C(5.01) - C(5)] / 0.01$. What does this rate mean physiologically, and why is it negative before the sign flip?

1.26 — Distance from speed (Physics, area version). A car's speed is $v(t) = 9.2\,t$ m/s for $0 \le t \le 3$. Estimate the distance travelled by summing right-endpoint rectangles under $v$ with $n = 3$, $n = 100$, and $n = 1000$. The exact distance is $41.4$ m. Notice that this is an area problem, the mirror image of the slope problems above.

Part E — Computational (Python, ⭐⭐⭐ Challenging, ~30 min each)

1.27 Write a Python function secant_slope(f, a, h) that computes the secant slope of f at x = a with step h. Use it to estimate the slope of $y = \sin x$ at $x = 0$, $x = \pi/2$, and $x = \pi$ (use small h). What pattern do you notice? (You are estimating the derivative of $\sin$; you'll meet it for real in Chapter 7.)

def secant_slope(f, a, h):
    return (f(a + h) - f(a)) / h

1.28 Write a Python function riemann_sum(f, a, b, n) that estimates the area under f on $[a, b]$ using $n$ right-endpoint rectangles. Use it to estimate the area under $y = 1/x$ on $[1, 2]$ with $n = 10{,}000$. The true answer is $\ln 2 \approx 0.6931$. How close do you get?

def riemann_sum(f, a, b, n):
    width = (b - a) / n
    total = 0.0
    for i in range(1, n + 1):
        x = a + i * width
        total += f(x) * width
    return total

1.29 Modify riemann_sum to use midpoint rectangles (the height at the midpoint of each subinterval). Repeat exercise 1.28 with midpoints, $n = 10{,}000$. Is the midpoint estimate more accurate, less accurate, or the same? Make a conjecture, then try $n = 100$ to see whether the gap is more dramatic on coarser grids.

1.30 Using secant_slope, build a table of slope estimates for $y = x^2$ at $x = 1$ as h runs through $10^{-1}, 10^{-3}, 10^{-6}, 10^{-10}, 10^{-14}$. The estimates approach $2$, then worsen for the tiniest h. Explain what goes wrong numerically. (This is catastrophic cancellation in floating-point arithmetic — a preview of Chapter 16's care with numerical methods. You need not fix it, only observe and describe it.)

Part F — Reflective, Exploratory, and Proof (⭐⭐⭐⭐ Advanced, open-ended)

1.31 The function $f(x) = |x|$ has a sharp corner at $x = 0$. Estimate the slope at $x = 0$ using secant slopes with $h = 0.1$, then $h = -0.1$. What happens? Does the curve have a well-defined slope at the corner? Why or why not? (We formalize this in Chapters 5–6; for now give your best intuitive explanation.)

1.32 — A genuine derivation (proof-flavoured). Prove that the area under $y = x$ on $[0, 1]$ equals $1/2$ using a Riemann sum, not geometry. Set up the right-endpoint sum $\sum_{i=1}^n \frac{i}{n}\cdot\frac{1}{n}$, apply the identity $\sum_{i=1}^n i = \frac{n(n+1)}{2}$, simplify, and take $n \to \infty$. Show every step, mirroring the $y = x^2$ derivation in §1.2.

1.33 Archimedes (c. 250 BC) computed the area under a parabolic arc with a clever geometric argument. Look up his "quadrature of the parabola." In one paragraph, describe his method, and explain how it relates to the Riemann-sum idea of §1.2.

1.34 Reflect on the §1.6 claim that "almost every modern quantitative field uses calculus." Name one quantitative field where you would expect calculus not to appear (number theory? combinatorics? graph theory?). Spend ten minutes checking whether it actually shows up there, and report what you find.

1.35 Write a one-page response: Why did calculus appear independently in two places (Newton in England, Leibniz in Germany) at almost the same time? What about the intellectual context of 17th-century Europe made calculus, in some sense, inevitable? Use the further-reading.md references for background.

Solutions to Selected Exercises

Selected solutions appear in appendices/answers-to-selected.md. For full solutions to all exercises, instructors may request the solutions manual; see instructor-guide/overview.md.

Difficulty Distribution

Total: 35 exercises, ~4 hours of focused work.