Practice question · Multiple choice
A linear approximation is excellent near the expansion point and poor far from it. Why does the approximation degrade faster for some functions than others?
Hints
- Write the next term of the Taylor series after the linear one.
- Which derivative appears in it?
Show the answer
D. The curvature, which sets the leading error term
Why
The error is governed by and the square of the distance. This is why the small-angle approximation for a pendulum holds well to about 15° and then fails quickly.
Practise Graphs, approximation, and numerical sense
The app has 6 more questions on this lesson, and keeps your place in the course. Physics I is free to start.
More questions on Graphs, approximation, and numerical sense
- Newton’s method races to a root when it works, but pick the starting point badly and it fails. Which…
- The linearization L(x) = f(a) + f'(a)(x-a) is exact for every function on a small enough interval.
- Use linearisation to estimate sin(0.1). (Use sin(0) = 0 and cos(0) = 1.) Set the slider to your answer.
- Which statements about Newton's method are true?
- Order the steps in one iteration of Newton's method for root-finding.
- Use the linear approximation at x=16 to estimate √(16.4), then set the slider. (f(x)=√(x), f'(x) =…