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Lesson 57 — Linear Approximation and Differentials

Tangent line as local approximation. Differential dy. Error estimation via second derivative. Newton-Raphson as iterated linearization.

Used in: 2nd Year of Program (Calculus I) · Equivalent Japanese Math III §4 · Equivalent German Leistungskurs Differentialrechnung · Singapore H2 Math §4.3

L(x) = f(a) + f'(a)(x - a)

Linearization of f at point a: the tangent line to the graph at $(a,\,f(a))$ is the best linear approximation of f near a. The error is $O((x-a)^2)$ — it shrinks much faster than the distance to the base point.

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Rigorous notation, full derivation, hypotheses

Rigorous Definition and Error Theory

Linearization

"If $f$ is differentiable at $a$ , then $L(x) = f(a) + f'(a)(x - a)$ is called the linearization of $f$ at $a$ . The approximation $f(x) \approx L(x)$ is called the linear approximation or tangent line approximation of $f$ at $a$ ." — OpenStax Calculus Vol.1 §4.2

Differential

"We define $dx$ and $dy$ as real variables so that the following equation holds: $dy = f'(x)\,dx$ . The differential $dy$ is a linear approximation of the actual change $\Delta y$ ." — OpenStax Calculus Vol.1 §4.2

Error Estimation via Taylor

Theorem· Linearization Error (Taylor with Remainder)

If $f$ is twice differentiable on an interval containing $a$ and $x$ , then there exists $\xi$ between $a$ and $x$ such that

$f(x) = f(a) + f'(a)(x - a) + \frac{f''(\xi)}{2}(x - a)^2.$

The error of the linearization is therefore $f(x) - L(x) = \dfrac{f''(\xi)}{2}(x-a)^2$ . If $|f''(t)| \leq M_2$ for all $t$ between $a$ and $x$ , then

$|f(x) - L(x)| \leq \frac{M_2}{2}(x - a)^2.$

The error is quadratic in the distance to the base point — it shrinks much faster than the distance.

Figure: Tangent Line as Local Approximation

The tangent line $L(x)$ touches the graph of $f$ at $(a, f(a))$ . The error (orange segment) between the curve and the line grows with the square of the distance $|x-a|$ .

Classic Approximations at $a = 0$ (Linear Maclaurin Series)

$f(x)$	$L(x)$ at $a = 0$	Valid for
$e^x$	$1 + x$	small $x$
$\sin x$	$x$	$x$ in radians, small
$\cos x$	$1$	small $x$
$\ln(1 + x)$	$x$	small $x$
$(1 + x)^n$	$1 + nx$	small $x$
$\sqrt{1 + x}$	$1 + x/2$	small $x$
$\tan x$	$x$	small $x$
$\arctan x$	$x$	small $x$

Error Propagation

For $y = f(x)$ with uncertainty $\sigma_x$ in $x$ :

$\sigma_y \approx |f'(x)|\,\sigma_x.$

For functions of several variables $y = f(x_1, \ldots, x_n)$ with independent errors $\sigma_i$ :

$\sigma_y^2 \approx \sum_{i=1}^{n} \left(\frac{\partial f}{\partial x_i}\right)^2 \sigma_i^2.$

Solved Examples

Example— 1· Linear Approximation of a Square Root (Direct Application)

Problem: Using linearization, approximate $\sqrt{9.2}$ without a calculator. Estimate the maximum error.

Strategy: We identify $f(x) = \sqrt{x}$ and choose $a = 9$ because $\sqrt{9} = 3$ is exact and $9.2$ is close to $9$ .

Solution:

$f(x) = \sqrt{x}$ , $f(9) = 3$ .
$f'(x) = \dfrac{1}{2\sqrt{x}}$ , $f'(9) = \dfrac{1}{6}$ .
Linearization: $L(x) = 3 + \dfrac{1}{6}(x - 9)$ .
At $x = 9.2$ : $L(9.2) = 3 + \dfrac{0.2}{6} = 3 + 0.0\overline{3} \approx 3.0333$ .

Error Estimation:

$f''(x) = -\dfrac{1}{4x^{3/2}}$ . On $[9, 9.2]$ : $|f''(t)| \leq \dfrac{1}{4 \cdot 9^{3/2}} = \dfrac{1}{108}$ .

$|\text{error}| \leq \frac{1/108}{2}(0.2)^2 = \frac{0.04}{216} \approx 0.000185.$

Verification: $\sqrt{9.2} \approx 3.03315$ . Actual error: $|3.0333 - 3.03315| \approx 0.00015 < 0.000185$ . Estimate confirmed.

Source. Active Calculus §1.8 — Boelkins · CC-BY-NC-SA.

Example— 2· Differential and Error Propagation in Sphere Volume (Intermediate)

Problem: The radius of a sphere is measured as $r = 6.0$ cm with a maximum error of $0.05$ cm. Estimate the maximum error in the volume $V = \frac{4}{3}\pi r^3$ .

Strategy: We use the differential $dV = V'(r)\,dr$ to linearize the error. The differential is the linear approximation of the change in $V$ .

Solution:

$V(r) = \dfrac{4}{3}\pi r^3$ , $V'(r) = 4\pi r^2$ .
With $r = 6$ and $dr = 0.05$ :

$dV = 4\pi(6)^2 (0.05) = 4\pi \cdot 36 \cdot 0.05 = 7.2\pi \approx 22.62 \text{ cm}^3.$

Nominal volume: $V(6) = \dfrac{4}{3}\pi(216) = 288\pi \approx 904.8$ cm³.
Relative error: $\dfrac{dV}{V} = \dfrac{7.2\pi}{288\pi} = \dfrac{7.2}{288} = 0.025 = 2.5\%$ .

Verification: An uncertainty of $0.8\%$ in the radius propagates to $2.5\%$ in the volume — a factor of 3, because $V \propto r^3$ (exponent 3). General rule: for $f(x) = x^n$ , the relative error in $f$ is $n$ times the relative error in $x$ .

Source. OpenStax Calculus Vol.1 §4.2 — CC-BY-NC-SA.

Example— 3· Trigonometric Function Approximation and Error Estimation (Intermediate)

Problem: Use linearization to approximate $\sin(32°)$ in terms of $\sin(30°)$ and $\cos(30°)$ . Compare with the exact value.

Strategy: We convert to radians (required in differential calculus) and linearize $\sin$ at $a = \pi/6$ .

Solution:

$f(x) = \sin x$ , $a = \pi/6$ (corresponding to $30°$ ).
$f(a) = \sin(\pi/6) = 1/2$ , $f'(a) = \cos(\pi/6) = \sqrt{3}/2$ .
$x = 32° = 32\pi/180$ radians. $x - a = 2\pi/180 = \pi/90$ .

$L(x) = \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\pi}{90} = 0.5 + 0.866 \cdot 0.03491 \approx 0.5 + 0.03023 = 0.53023.$

Comparison: $\sin(32°) = 0.52992$ . Error: $|0.53023 - 0.52992| = 0.00031$ .

Verification: $|f''(t)| = |\sin(t)| \leq \sin(32°) < 0.530$ . Bound: $\dfrac{0.530}{2}\left(\dfrac{\pi}{90}\right)^2 \approx 0.265 \cdot 0.001218 \approx 0.000323 > 0.000310$ . The bound is met.

Source. Active Calculus §1.8 — Boelkins · CC-BY-NC-SA.

Example— 4· Newton-Raphson as Iterated Linearization (Advanced)

Problem: Use Newton-Raphson to find $\sqrt[3]{2}$ to three decimal places, starting with $x_0 = 1.5$ .

Strategy: We reformulate as $f(x) = x^3 - 2 = 0$ and apply $x_{n+1} = x_n - f(x_n)/f'(x_n)$ , which comes from setting the tangent line to $f$ at $x_n$ to zero.

Solution:

$f(x) = x^3 - 2$ , $f'(x) = 3x^2$ .

Iteration	$x_n$	$f(x_n)$	$f'(x_n)$	$x_{n+1}$
0	$1.5$	$1.375$	$6.75$	$1.2963$
1	$1.2963$	$0.1786$	$5.040$	$1.2609$
2	$1.2609$	$0.00595$	$4.771$	$1.2599$
3	$1.2599$	$0.0000064$	$4.763$	$1.25992$

Verification: After 3 iterations: $x_3 = 1.25992$ , matching $\sqrt[3]{2} = 1.25992105\ldots$ to six decimal places. Convergence is quadratic: the number of correct digits roughly doubles with each iteration.

Source. Active Calculus §1.8 — Boelkins · CC-BY-NC-SA.

Example— 5· Multivariable Linearization — Ohm's Law with Two Uncertainties (Challenge)

Problem: Electrical resistance is calculated by $R = V/I$ . Measurements yield $V = 12.0 \pm 0.3$ V and $I = 2.00 \pm 0.05$ A. Estimate the uncertainty in $R$ using linear error propagation.

Strategy: We use the error propagation formula based on the total differential for functions of two variables: $\sigma_R^2 \approx (\partial R/\partial V)^2 \sigma_V^2 + (\partial R/\partial I)^2 \sigma_I^2$ .

Solution:

$R(V, I) = V/I$ . Partial derivatives:

$\frac{\partial R}{\partial V} = \frac{1}{I} = \frac{1}{2.00} = 0.5 \text{ Ω/V}, \quad \frac{\partial R}{\partial I} = -\frac{V}{I^2} = -\frac{12.0}{4.00} = -3.0 \text{ Ω/A}.$

Propagated variance:

$\sigma_R^2 \approx (0.5)^2(0.3)^2 + (-3.0)^2(0.05)^2 = 0.0225 + 0.0225 = 0.045 \text{ Ω}^2.$

$\sigma_R \approx \sqrt{0.045} \approx 0.212 \text{ Ω}.$

Result: $R = 6.00 \pm 0.21$ Ω.

Verification: The uncertainties in $V$ and $I$ contribute equally to the total uncertainty in $R$ — a characteristic of this specific combination of values.

Source. APEX Calculus §4.4 — Hartman et al. · CC-BY-NC.

Exercise list

40 exercises · 10 with worked solution (25%)

22 4 8 4 2

Sources

Active Calculus — Boelkins · 2024 · §1.8 "The tangent line approximation" · CC-BY-NC-SA. Primary source. Exercises 57.1–57.2, 57.5, 57.7, 57.11–57.12, 57.15, 57.20–57.21, 57.27, 57.29, 57.31, 57.35, 57.37, 57.39–57.40.
Calculus Volume 1 — OpenStax · 2016 · §4.2 "Linear Approximations and Differentials" · CC-BY-NC-SA. Exercises 57.3–57.4, 57.6, 57.13–57.14, 57.17–57.19, 57.23, 57.25, 57.28, 57.30, 57.33.
APEX Calculus — Hartman et al. · 2024 · v5 · §4.4 "Differentials" · CC-BY-NC. Exercises 57.8–57.10, 57.16, 57.22, 57.24, 57.26, 57.31, 57.34, 57.36, 57.38.