Chapter 11 — Quiz

Q: Write the linearization of at and use it to estimate .

and gives . So and . (True value ) (§11.2–11.3)

Q: The linearization of at gives the famous rule "?" for small . Fill in the blank and justify in one line.

. Anchoring at : , , so ; substituting for the argument gives . (§11.3)

Q: A cube has side cm, and the side is increased by cm. Use a differential to estimate the change in volume .

. (Exact — the differential is within a percent.) (§11.4)

Q: The radius of a circle is measured to within . Using the power-law error rule, what is the approximate percentage error in the area ?

Area scales as , so the relative error doubles: . The exponent is the amplification factor. (§11.5)

Q: Why does measuring a sphere's radius to precision yield a volume known only to ? State the governing rule.

Because , and for a power law the relative errors obey . With , the radius error triples to a volume error. (§11.5)

Q: Write the general Newton's-method iteration, and explain in one sentence the geometric meaning of a single step.

. Geometrically, you draw the tangent line to the curve at and slide down it to where it crosses the -axis; that intercept is the next guess. (§11.6)

Q: Set up Newton's method to compute and carry out one step from .

Use , ; the iteration simplifies to . From : . (True .) (§11.7)

Q: The error of the linear approximation at satisfies . What two qualitative facts does this bound encode, and which chapter generalizes to higher order?

(1) The error is quadratic in the distance — halving your distance from the anchor cuts the error by a factor of four. (2) The error is proportional to the curvature , so flatter functions are approximated better. Keeping more terms produces the Taylor series, developed in Chapter 23. (§11.11)

DataField.Dev

Chapter 11 — Quiz

Ten questions covering linear approximation, differentials, error propagation, and Newton's method. Try each before opening the answer. Section references point back to index.md.

Q1. Write the linearization of $f(x) = \sqrt{x}$ at $a = 16$ and use it to estimate $\sqrt{16.4}$.

Answer

$f(16) = 4$ and $f'(x) = \tfrac{1}{2\sqrt{x}}$ gives $f'(16) = \tfrac18$. So $L(x) = 4 + \tfrac18(x - 16)$ and $L(16.4) = 4 + \tfrac18(0.4) = 4.05$. (True value $\sqrt{16.4} = 4.04969\ldots$) **(§11.2–11.3)**

Q2. The linearization of $\ln x$ at $a = 1$ gives the famous rule "$\ln(1 + x) \approx \,$?" for small $x$. Fill in the blank and justify in one line.

Answer

$\ln(1+x) \approx x$. Anchoring $f(x) = \ln x$ at $1$: $f(1) = 0$, $f'(1) = 1$, so $L(x) = x - 1$; substituting $1 + x$ for the argument gives $\ln(1+x) \approx x$. **(§11.3)**

Q3. A cube has side $s = 8$ cm, and the side is increased by $ds = 0.05$ cm. Use a differential to estimate the change in volume $V = s^3$.

Answer

$dV = 3s^2\,ds = 3(64)(0.05) = 9.6\ \mathrm{cm^3}$. (Exact $\Delta V = 8.05^3 - 8^3 = 9.66\ \mathrm{cm^3}$ — the differential is within a percent.) **(§11.4)**

Q4. The radius of a circle is measured to within $2\%$. Using the power-law error rule, what is the approximate percentage error in the area $A = \pi r^2$?

Answer

Area scales as $r^2$, so the relative error doubles: $\Delta A/A \approx 2(2\%) = 4\%$. The exponent $2$ is the amplification factor. **(§11.5)**

Q5. Why does measuring a sphere's radius to $1\%$ precision yield a volume known only to $3\%$? State the governing rule.

Answer

Because $V \propto r^3$, and for a power law $y = x^n$ the relative errors obey $\Delta y/y \approx n\,\Delta x/x$. With $n = 3$, the $1\%$ radius error triples to a $3\%$ volume error. **(§11.5)**

Q6. Write the general Newton's-method iteration, and explain in one sentence the geometric meaning of a single step.

Answer

$x_{n+1} = x_n - \dfrac{f(x_n)}{f'(x_n)}$. Geometrically, you draw the tangent line to the curve at $(x_n, f(x_n))$ and slide down it to where it crosses the $x$-axis; that intercept is the next guess. **(§11.6)**

Q7. Set up Newton's method to compute $\sqrt{7}$ and carry out one step from $x_0 = 3$.

Answer

Use $f(x) = x^2 - 7$, $f'(x) = 2x$; the iteration simplifies to $x_{n+1} = \tfrac12\!\left(x_n + \tfrac{7}{x_n}\right)$. From $x_0 = 3$: $x_1 = \tfrac12(3 + 7/3) = \tfrac12(3 + 2.3333) = 2.6667$. (True $\sqrt 7 = 2.6458$.) **(§11.7)**

Q8. Newton's method exhibits quadratic convergence near a simple root. What does that mean numerically, and what is the error-recurrence that produces it?

Answer

The number of correct digits roughly doubles each step because the new error is proportional to the square of the old one: $e_{n+1} \approx \dfrac{f''(r)}{2f'(r)}\,e_n^2$. So an error of $10^{-3}$ becomes $\sim 10^{-6}$, then $\sim 10^{-12}$. **(§11.8)**

Q9. Name two distinct ways Newton's method can fail, and for each say briefly why.

Answer

Any two of: (1) a **bad starting guess** sends the tangent line away from the root; (2) a **near-zero derivative** $f'(x_n) \approx 0$ makes the step enormous and flings the guess far away; (3) **cycling**, where iterates fall into a repeating loop (e.g. $f(x) = x^3 - 2x + 2$ from $x_0 = 0$ oscillates $0 \to 1 \to 0$); (4) a **multiple root** ($f' = 0$ at the root) degrades convergence from quadratic to merely linear. **(§11.9)**

Q10. The error of the linear approximation at $a$ satisfies $|f(x) - L(x)| \le \tfrac{M}{2}(x-a)^2$. What two qualitative facts does this bound encode, and which chapter generalizes $L$ to higher order?

Answer

(1) The error is *quadratic* in the distance $x - a$ — halving your distance from the anchor cuts the error by a factor of four. (2) The error is proportional to the curvature $M = \max|f''|$, so flatter functions are approximated better. Keeping more terms produces the **Taylor series**, developed in **Chapter 23**. **(§11.11)**

Scoring Guide

Score	Interpretation
9–10	Excellent. You can linearize, propagate error, and run and diagnose Newton's method fluently. Move on to Chapter 12.
7–8	Solid. Re-read the one or two sections behind your misses (each answer cites its section).
5–6	Partial. Re-work the §11.3 and §11.7 worked examples by hand, then retry.
0–4	Revisit the chapter from §11.2, focusing on the tangent-line picture that unifies every result.