Example 2: Approximating sin(x) with Taylor Polynomials and Calculating its Error

mes (67)in DTube • 2 days ago

In this video I go over a very in-depth video exploring the trigonometric sine function, its Taylor polynomials approximation, and its associated error. The Maclaurin polynomial (Taylor series centered at a = 0) approximation has alternating signs, is decreasing, and the terms approach zero, thus we can use the Alternating Series Estimation Theorem to calculate the error. Later, I show that we can get the same result using Taylor's Inequality as well as via graphing the error directly with Desmos graphing calculator. When we approximate functions, it is important to center it near the value of x we want to approximate, because the series converges more rapidly. Lastly, I show how graphing multiple Maclaurin Polynomials for sin(x) get more and more accurate as we increase the number of terms, hence increasing the degree of polynomials.

Timestamps:

Example 2: Error approximation for sin(x) Taylor polynomials: 0:00
Solution to (a): Maclaurin series for sin(x): 0:41
Maclaurin sine series is alternating when x is not zero: 2:03
Can use Alternating Series Estimation Theorem to calculate the error: 5:02
- Absolute value of error is less than 4.3x10^(-8): 7:56
Approximating sin(12 degrees) by converting to radians first: 9:41
- sin(12°) = 0.207912 correct to 6 decimal places: 13:09
Solution to (b): Values of x to have accuracy less than 0.0005: 15:28
- Accuracy is within 0.0005 for when the absolute value of x is less than 0.82: 18:55
Example 2 using Taylor's Inequality: 20:27
- Solving for 7th derivative of sin(x): 22:10
- Taylor's Inequality gives same result as the Alternating Estimation Theorem: 23:52
Graphing the Remainder Function with Desmos: https://www.desmos.com/calculator/yphi6wpizp: 27:24
- Error is less than 4.3x10^(-8) just as in Part (a): 29:49
Graphing Remainder function and y = 0.00005 in Desmos: https://www.desmos.com/calculator/5uuyhdlp1y: 30:58
- Interval of error is between absolute of x less than 0.82 just like in part (b): 31:48
For different angles we can use Taylor Polynomials centered at different values of x: 32:56
Graphing various Maclaurin polynomials for sin(x) using Desmos: https://www.desmos.com/calculator/8pl3nc2zha: 33:33
- The Maclaurin polynomial with the most terms is the most accurate: 35:22
Taylor Polynomials are used in Calculators and Computers to solve functions like sine and e^x: 36:58