Chapter 5 — Exercises

36 problems on average rates, the secant-to-tangent limit, the derivative from its definition, tangent lines, where derivatives fail, and applications across four fields. Tiers run ⭐ (warm-up) to ⭐⭐⭐⭐ (conceptual / proof). Selected answers appear in appendices/answers-to-selected.md.

A note on tools. Through Part I we differentiate the slow, honest way — straight from the limit definition of Section 5.2. The fast rules (power, product, quotient, chain) are deferred to Chapter 7. Unless a problem explicitly says "use sympy" or "you may use the power rule," compute every derivative from the definition

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \qquad\text{or}\qquad f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.$$

Part A — Average Rates of Change (⭐)

5.1 Compute the average rate of change of $f(x) = x^2$ on $[1, 3]$.

5.2 A car's position is $s(t) = 60t - 2t^2$ (meters, $t$ in seconds) for $t \in [0, 15]$. Find the average velocity over $[0, 5]$, $[5, 10]$, and $[10, 15]$. What does the trend in these three numbers tell you about the car's motion?

5.3 A bacterial population is $P(t) = 1000 \cdot 2^t$ (with $t$ in hours). Find the average growth rate over $[0, 1]$, $[1, 2]$, and $[2, 3]$. Are the averages increasing? What does that say about the growth?

5.4 For $f(x) = \sin x$, find the average rate of change over $[0, \pi/2]$. Give an exact value and a decimal.

5.5 A tank holds $V(t) = 200 - 8t^2$ liters at time $t$ (minutes), valid for $0 \le t \le 5$. Find the average rate of change of volume over $[1, 4]$, and interpret the sign.

5.6 The graph of a function $g$ passes through $(2, 7)$ and $(6, -5)$. What is the average rate of change of $g$ on $[2, 6]$, and what is the slope of the secant line through those two points? (They should be equal — explain why in one sentence.)

Part B — Secant Slopes Approaching the Tangent (⭐⭐)

5.7 For $f(x) = x^2$ at $a = 2$, compute the secant slope $\dfrac{f(2+h) - f(2)}{h}$ for $h = 1,\ 0.1,\ 0.01,\ 0.001$. To what value do the slopes appear to converge? Confirm this matches $f'(2)$ from the definition.

5.8 For $f(x) = \sqrt{x}$ at $a = 4$, build a table of secant slopes for $h = 1,\ 0.1,\ 0.01$ and $h = -1,\ -0.1,\ -0.01$ (approaching from both sides). What single value do the two-sided slopes converge to?

5.9 Explain, in two or three sentences, what geometric event happens to the secant line through $(a, f(a))$ and $(a+h, f(a+h))$ as $h \to 0$, and why its limiting slope deserves the name "tangent slope." Refer to the secant-to-tangent picture of Section 5.3.

Part C — The Derivative from the Definition (⭐⭐)

5.10 Use the limit definition to find $f'(2)$ where $f(x) = x^2 + 3x$.

5.11 Use the limit definition to find $f'(-1)$ where $f(x) = 1/x$.

5.12 Use the limit definition to find $f'(a)$ where $f(x) = \sqrt{x}$ for a general $a > 0$. (Hint: rationalize the numerator.)

5.13 Use the limit definition to find $f'(x)$ for $f(x) = x^3$. (Hint: $(x+h)^3 = x^3 + 3x^2 h + 3x h^2 + h^3$.)

5.14 Use the limit definition to find $f'(x)$ for $f(x) = 1/x^2$.

5.15 Use the limit definition to find $f'(x)$ for $f(x) = 3x^2 - 5x + 1$.

5.16 Use the equivalent form $f'(a) = \displaystyle\lim_{x \to a}\frac{f(x) - f(a)}{x - a}$ (Section 5.2) to find $f'(a)$ for $f(x) = x^2$. Confirm you get the same answer as the $h \to 0$ form.

5.17 Use the limit definition to find $f'(x)$ for $f(x) = \sqrt{2x + 1}$, valid for $x > -\tfrac12$.

Part D — Tangent Lines (⭐⭐)

5.18 Find the equation of the tangent line to $y = x^2$ at $(2, 4)$.

5.19 Find the equation of the tangent line to $y = 1/x$ at $(1, 1)$.

5.20 At what point on the curve $y = x^2$ is the tangent line parallel to $y = 6x + 1$?

5.21 Find the point on the curve $y = x^2 - 4x + 7$ where the tangent line is horizontal. (A horizontal tangent has slope $0$.)

5.22 Find an equation of the tangent line to $y = \sqrt{x}$ at $x = 9$. Use it to estimate $\sqrt{9.2}$, and compare with a calculator value. (You are previewing the linear approximation of Chapter 11.)

Part E — The Derivative as a Function & Higher Derivatives (⭐⭐)

5.23 Using your result $\frac{d}{dx}(x^3) = 3x^2$ from Exercise 5.13, find $f''(x)$ and $f'''(x)$ for $f(x) = x^3$ by differentiating again from the definition (each step is a fresh limit).

5.24 For $f(x) = mx + b$, show from the definition that $f'(x) = m$ for every $x$, and hence $f''(x) = 0$. Interpret both results geometrically (Section 5.9).

5.25 Let $s(t) = t^2$. Find $s'(t)$ and $s''(t)$ from the definition, then state the velocity and acceleration at $t = 4$ (Section 5.4 / 5.5).

Part F — Where Derivatives Fail (⭐⭐⭐)

5.26 Show that $f(x) = |x|$ is not differentiable at $x = 0$ by computing both one-sided difference-quotient limits and showing they disagree (Section 5.7).

5.27 Show that $f(x) = x^{1/3}$ has a vertical tangent at $x = 0$ by showing the difference quotient tends to $+\infty$ from both sides.

5.28 Sketch a single continuous function that is non-differentiable at exactly three places: a corner at $x = 1$, a cusp at $x = 3$, and a vertical tangent at $x = 5$. Label each feature.

5.29 Decide (with a one-line reason) whether each statement is true or false: (a) If $f$ is continuous at $a$, then $f$ is differentiable at $a$. (b) If $f$ is differentiable at $a$, then $f$ is continuous at $a$. (c) A function can be discontinuous at $a$ yet differentiable at $a$. (d) $|x|$ is continuous everywhere but not differentiable everywhere.

5.30 The function $f(x) = \begin{cases} x^2, & x \le 1 \\ 2x - 1, & x > 1 \end{cases}$ is continuous at $x = 1$. Compute the left- and right-hand difference-quotient limits at $x = 1$ and determine whether $f$ is differentiable there.

Part G — Applications (⭐⭐⭐)

5.31 — Physics. A ball is thrown upward from height $h_0 = 1$ m with initial velocity $v_0 = 20$ m/s. Its height after $t$ seconds is $h(t) = 1 + 20t - 4.9t^2$. (a) Find $h'(t)$ using the limit definition. (b) When is the ball's velocity zero (the peak of its trajectory)? (c) When does the ball hit the ground? (Solve $h(t) = 0$.) (d) What is its velocity at the moment of impact?

5.32 — Economics. The total cost of producing $q$ widgets is $C(q) = 1000 + 5q + 0.01q^2$ dollars. Use the limit definition to find $C'(q)$, the marginal cost. What is the approximate cost of the 101st widget (evaluate $C'(100)$)?

5.33 — Biology / Pharmacokinetics. A drug's plasma concentration is $C(t) = 100\,e^{-0.5t}$ (mg/L) at time $t$ (hours). Estimate $C'(1)$ numerically using the difference quotient with $h = 0.0001$. Report the value with units and explain what its sign means for the patient.

5.34 — Data Science. A simple training loss as a function of a weight $w$ is $L(w) = (w-1)^2 + 0.1$. Use the definition to find $L'(w)$. Solve $L'(w) = 0$ — this is the minimum where gradient descent halts (previewing Chapter 6). For $w = 3$, does $L'(w)$ tell gradient descent to increase or decrease $w$?

Part H — sympy and Visualization (⭐⭐⭐)

5.35 Use sympy to differentiate, then compare with any hand computation you can do from the definition: (a) $f(x) = x^4 - 3x^2 + 7$ (b) $g(x) = e^x \cos x$ (c) $h(x) = \ln(1 + x^2)$

import sympy as sp
x = sp.symbols('x')
for expr in (x**4 - 3*x**2 + 7, sp.exp(x)*sp.cos(x), sp.log(1 + x**2)):
    print(expr, "->", sp.diff(expr, x))

5.36 Plot $f(x) = x^3 - 3x$ and its derivative $f'(x) = 3x^2 - 3$ on the same axes over $[-2.5, 2.5]$. Mark the two $x$-values where $f'(x) = 0$ and verify that $f$ has horizontal tangents there.

Part I — Conceptual & Proof (⭐⭐⭐⭐)

5.37 Prove from the definition that differentiation is linear: if $f$ and $g$ are differentiable at $a$ and $c$ is a constant, then $(f + g)'(a) = f'(a) + g'(a)$ and $(cf)'(a) = c\,f'(a)$. (Use the limit laws of Chapter 3 — the limit of a sum is the sum of limits.)

5.38 Explain why "instantaneous rate of change" is genuinely paradoxical before limits, and how the limit resolves the paradox. Your answer should explicitly address why plugging $b = a$ into $\frac{f(b) - f(a)}{b - a}$ gives the meaningless $0/0$, and why the limit avoids it (Section 5.2).

5.39 Prove the theorem that differentiability implies continuity: if $f'(a)$ exists, then $f$ is continuous at $a$. (Reconstruct the argument of Section 5.7: write $f(a+h) - f(a)$ as the difference quotient times $h$, then take $h \to 0$.) Then explain in one sentence why the converse is false.

5.40 Reflect on the relationship between $f$, $f'$, and $f''$. In one page, describe what each captures intuitively in three different fields — for example position/velocity/acceleration (physics), total cost/marginal cost/rate-of-change-of-marginal-cost (economics), and population/growth-rate/acceleration-of-growth (biology). Where does each "level" answer a different real-world question?

Tier Summary

Total: 40 problems. Estimated time: 4–5 hours. Selected answers (5.1, 5.2, 5.4, 5.10, 5.11, 5.12, 5.13, 5.18, 5.19, 5.21, 5.26, 5.31, 5.32, 5.39) are in appendices/answers-to-selected.md.