The video discusses the complexity of real-world phenomena, such as the forces on an aircraft, the stresses in a building subjected to an earthquake, or a car crash. These situations often involve differential or partial differential equations, which are often impossible to solve analytically. Instead, engineers and scientists use numerical methods and approximation techniques such as Taylor series to find solutions.
The video demonstrates how Taylor series can be used to approximate solutions to differential equations, using the Maclaurin series for e^X as an example. It shows how the approximation improves as more terms are included in the series. The video also explains that the Maclaurin series can be used to approximate other functions, such as sine(X) for small input angles, and the speed of sound in air for typical everyday temperatures.
The video concludes by emphasizing the importance of approximation in real-world scenarios. It mentions that there are entire courses dedicated to numerical approximation methods, including those offered by Brilliant.org, which the video is sponsored by.
1. The video is sponsored by Brilliant. [Source: Document(page_content="00:00:00.03: this video was sponsored by brilliant\n00:00:01.88: the world is mathematically complicated\n00:00:05.22: when you analyze the forces on an\n00:00:07.11: aircraft as it flies through the air or\n00:00:09.27: the stresses throughout a building that\n00:00:10.83: subject to an earthquake or the force\n00:00:13.17: through a car in a crash and how it will\n00:00:15.00: deform when dealing with these things it\n00:00:17.58: turns out the equations are not only\n00:00:19.38: very difficult to solve but they're\n00:00:21.42: often impossible to solve often there's\n00:00:24.15: no analytical method to find that force\n00:00:26.61: as a function of position or whatever it\n00:00:28.53: may be but engineers and scientists\n00:00:31.47: still need these solutions I mean we\n00:00:33.57: definitely want to know what the stress\n00:00:35.13: will be throughout that bridge and how\n00:00:36.66: to account for all the forces that will\n00:00:37.95: be subject to so even if the equations\n00:00:40.29: are impossible we have to get around\n00:00:41.91: that now those equations that describe\n00:00:45.18: real-world things are often differential\n00:00:47.79: or partial differential equations fluid\n00:00:51.00: flow heat flow electromagnetism\n00:00:53.45: vibrations how beams will Bend again the\n00:00:56.91: stress throughout a bridge due to\n00:00:58.11: traffic it all comes down to frequently\n00:01:00.39: impossible to solve differential\n00:01:02.10: equations so what do we do with these\n00:01:05.13: really hard equations we approximate and\n00:01:07.77: when I say that no I don't mean we round\n00:01:10.11: I mean we apply numerical methods to get\n00:01:12.63: an approximate answer that's close\n00:01:14.25: enough to the exact one and Taylor\n00:01:16.59: series are one of several ways we can\n00:01:18.72: accomplish this when you see computers\n00:01:21.87: simulating car crashes and air flow that\n00:01:25.11: are approximating the solutions to often\n00:01:26.85: differential equations in the background\n00:01:28.98: while staying within a certain margin of\n00:01:30.84: error and you know we don't need to know\n00:01:33.81: exactly how a car will deform in a crash\n00:01:35.91: for example we need accurate solutions\n00:01:38.22: that allow us to maximize safety\n00:01:39.60: minimize any prototypes that might need\n00:01:41.52: to be made and so on now as a real quick\n00:01:45.0