Lesson 57 — Linear Approximation and Differentials

Approximate $\sqrt{4.1}$ using linearization at $a = 4$.

Approximate $\sqrt{9.06}$ using linearization at $a = 9$. Compare with the actual value.

Approximate $\sin(0.05)$ using linearization at $a = 0$.

Approximate $\cos(0.1)$ using linearization at $a = 0$. Explain why the result is surprisingly inaccurate.

Approximate $e^{0.03}$ using linearization at $a = 0$.

Approximate $\ln(1.05)$ using linearization at $a = 1$.

Approximate $(1.02)^{10}$ using the linearization of $(1+x)^{10}$ at $x = 0$.

Approximate $\sqrt[3]{27.5}$ using linearization at $a = 27$.

Write the linearization of $f(x) = \tan x$ at $a = 0$. This linearization is identical to that of $\sin x$ at $a = 0$ — why?

Write the linearization of $f(x) = \arctan x$ at $a = 1$.

Write the linearization of $f(x) = e^x \sin x$ at $a = 0$.

Approximate $\sqrt{50}$ using linearization at $a = 49$.

Calculate the differential $dy$ for $y = x^3$ at $x = 2$, $dx = 0.01$. Compare with the actual change $\Delta y$.

Calculate the differential $dy$ for $y = e^x$ at $x = 0$, $dx = 0.1$.

Approximate $\sin(31°)$ using the linearization of $\sin$ at $a = 30°$. Use $\sin(30°) = 0.5$ and $\cos(30°) = \sqrt{3}/2$.

Approximate $\cos(59°)$ using the linearization of $\cos$ at $a = 60°$.

What is the linearization of $f(x) = \sqrt{1 + x}$ at $a = 0$?

Approximate $1/\sqrt{4.1}$ using linearization at $a = 4$.

Write the linearization of $f(x) = \ln x$ at $a = e$.

Calculate the absolute error of the linearization of $\sqrt{4.1}$ at $a = 4$ by comparing with $\sqrt{4.1} \approx 2.024846$. Verify that the error is within the bound $M_2 h^2/2$.

Perform one iteration of Newton-Raphson to find $\sqrt{5}$, starting from $x_0 = 2$.

Perform two iterations of Newton-Raphson to solve $x^3 - 2 = 0$ with $x_0 = 1$.

A sphere has radius $r = 5.0 \pm 0.1$ cm. Estimate the maximum error in the volume $V = \frac{4}{3}\pi r^3$ using the differential.

The period of a pendulum is $T = 2\pi\sqrt{L/g}$. If the length $L$ has a relative error of $1\%$, what is the relative error in $T$?

For $R = V/I$ with independent errors $\sigma_V$ and $\sigma_I$, write the error propagation formula for $\sigma_R$ using partial derivatives.

The area of a circle is $A = \pi r^2$ with $r = 3.0 \pm 0.064$ cm. Estimate the maximum error in $A$ using the differential.

Why is the linearization error $|f(x) - L(x)|$ said to be $O((x-a)^2)$? What theorem supports this?

Under what circumstance is the linearization of $f$ at $a$ particularly inaccurate, even for $x$ close to $a$? What should be done in such cases?

Show that $\sin x \approx x$ is the linearization of $\sin$ at $a = 0$. For what values of $x$ (in radians) is the error less than 1%?

What is the relationship between $\Delta y$ (actual change) and $dy$ (differential)?

Explain where the Newton-Raphson formula $x_{n+1} = x_n - f(x_n)/f'(x_n)$ comes from in terms of linearization.

Use linearization to approximate $(1.002)^{30}$.

The side of a cube is measured with a relative error of $1\%$. What is the relative error in the volume $V = s^3$?

Write the linearization of $g(x) = 1/\sqrt{1+x}$ at $a = 0$.

Calculate the absolute and relative error of the linearization of $e^x$ at $a = 0$ when approximating $e^1 = e$. Is the result surprising? Explain.

Newton-Raphson fails when $f'(x_n) = 0$. Explain geometrically and give an example function where this occurs.

Derive the error bound $|f(x) - L(x)| \leq \frac{M_2}{2}(x-a)^2$ from Taylor's theorem with the Lagrange remainder.

Volume of a cylinder: $V = \pi r^2 h$. With $r = 5.0 \pm 0.1$ cm and $h = 10.0 \pm 0.2$ cm, estimate the maximum error in $V$ using the total differential.

Prove that $L(x) = f(a) + f'(a)(x-a)$ is the degree 1 Taylor polynomial of $f$ at $a$, and that the error is $O((x-a)^2)$.

Prove that Newton-Raphson has quadratic convergence: if $e_n = x_n - x^*$ is the error at the $n$-th iteration and $f'(x^*) \neq 0$, then $|e_{n+1}| \leq C\,|e_n|^2$ for some constant $C > 0$.