In structural dynamics and mechanical engineering, the motion of a simple pendulum is governed by the differential equation $\frac{d^2\theta}{dt^2} + \frac{g}{L}\sin\theta = 0$. Because the sine function makes this equation non-linear and difficult to solve explicitly, engineers use the Maclaurin series approximation for sine: $\sin(\theta) \approx \theta - \frac{\theta^3}{3!} + \dots$. For very small angles (typically less than $15^\circ$), all terms after the first become negligible, allowing engineers to use the Small Angle Approximation: $\sin(\theta) \approx \theta$. This simplifies the equation to a linear harmonic oscillator, $\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$, which is trivial to solve and design around.

A surveying team is measuring the height of a distant building using a transit. They measure the horizontal distance $x$ to the building and the angle of elevation $\theta$. The height is calculated as $h = x \tan\theta$. Because no instrument is perfectly precise, both the distance measurement and the angle measurement have inherent tolerances (errors), denoted by $dx$ and $d\theta$. Using Differentials, the total propagated error in the calculated height is approximated by the total differential: $dh = \frac{\partial h}{\partial x} dx + \frac{\partial h}{\partial \theta} d\theta$. This allows the head surveyor to calculate the maximum possible error in the building's calculated height and determine if the equipment used meets the project's strict precision requirements.

Use linear approximation to estimate $\sqrt{26}$.

Use linear approximation to estimate the value of $\sin(0.1)$ radians.

Use a linear approximation to estimate $(1.99)^4$.

The radius of a sphere is measured as 10 cm with a possible error of $\pm 0.05$ cm. Estimate the maximum error in the calculated volume.

A surveyor measures the side of a square plot of land as 200 m with a possible error of 0.2 m. What is the estimated percentage error in calculating the area?

If a measured quantity $x$ has a relative error of $2\%$, estimate the relative error in calculating $y = \sqrt{x}$.

Find the 3rd-degree Maclaurin polynomial for $f(x) = e^x$.

Find the 2nd-degree Taylor polynomial for $f(x) = \ln(x)$ centered at $a = 1$.

Find the 3rd-degree Maclaurin polynomial for $f(x) = \sin(x)$.

Use Newton's Method to approximate the positive root of $f(x) = x^2 - 2$. Start with an initial guess of $x_0 = 1$ and compute $x_2$.

Use one iteration of Newton's Method to estimate a root of $f(x) = x^3 - x - 1$, starting with $x_0 = 1$.

What happens if you use an initial guess of $x_0 = 0$ for $f(x) = x^3 - 3x + 1$?