{Use of Tech} Maximum error Use the remainder term to find a bound on the error in the following approximations on the given interval. Error bounds are not unique.


tan x ≈ x on [−π/6, π/6]

Verified step by step guidance

1. Identify the function and the approximation: here, the function is \(f(x) = \tan x\) and the approximation is \(f(x) \approx x\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\).
2. Recognize that the approximation \(\tan x \approx x\) is the first-degree Taylor polynomial of \(\tan x\) centered at 0, so the error can be estimated using the Lagrange remainder term for the Taylor series expansion.
3. Recall the Lagrange remainder formula for the Taylor polynomial of degree 1 centered at 0: the error \(R_1(x)\) satisfies \[R_1(x) = \frac{f^{(2)}(c)}{2!} x^2\] for some \(c\) between 0 and \(x\).
4. Calculate the second derivative of \(f(x) = \tan x\). First, \(f'(x) = \sec^2 x\), then \[f^{(2)}(x) = \frac{d}{dx} (\sec^2 x) = 2 \sec^2 x \tan x.\]
5. To find the error bound, determine the maximum value of \(|f^{(2)}(c)|\) on the interval \(\left[-\frac{\pi}{6}, \frac{\pi}{6}\right]\), then use it in the remainder formula: \[|R_1(x)| \leq \frac{\max_{c \in [-\frac{\pi}{6}, \frac{\pi}{6}]} |f^{(2)}(c)|}{2} x^2.\]

Key Concepts

Taylor Polynomial Approximation

A Taylor polynomial approximates a function near a point using derivatives at that point. For tan x near 0, the linear approximation is tan x ≈ x, which is the first-degree Taylor polynomial. Understanding this helps in estimating how close the approximation is to the actual function.

Remainder Term (Lagrange Form)

The remainder term quantifies the error between the function and its Taylor polynomial approximation. The Lagrange form expresses this error using a higher-order derivative evaluated at some point in the interval, providing a bound on the maximum possible error.

Error Bound on an Interval

To find an error bound on [−π/6, π/6], we evaluate the maximum value of the relevant derivative in that interval. This maximum derivative value, combined with the remainder formula, gives a guaranteed upper limit on the approximation error over the entire interval.