Polynomial Approximations Use the polynomial approximations...

Polynomial Approximations Use the polynomial approximations of the sine and cosine functions in Exercise 100 to approximate the following function values. Compare the results with those given by a calculator. Is the error in the approximation the same in each case? Explain. $$ \begin{array}{lll}{\text { (a) } \sin \frac{1}{2}} & {\text { (b) } \sin 1} & {\text { (c) } \sin \frac{\pi}{6}}\end{array} $$ $$ \begin{array}{lll}{\text { (d) } \cos (-0.5)} & {\text { (e) } \cos 1} & {\text { (f) } \cos \frac{\pi}{4}}\end{array} $$

Polynomial Approximations Use the polynomial
approximations of the sine and cosine functions in
Exercise 100 to approximate the following function
values. Compare the results with those given by a
calculator. Is the error in the approximation the same
in each case? Explain.
$$
\begin{array}{lll}{\text { (a) } \sin \frac{1}{2}} & {\text { (b) } \sin 1} & {\text { (c) } \sin \frac{\pi}{6}}\end{array}
$$
$$
\begin{array}{lll}{\text { (d) } \cos (-0.5)} & {\text { (e) } \cos 1} & {\text { (f) } \cos \frac{\pi}{4}}\end{array}
$$

Key Concepts

Taylor Series Expansion

The Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. When this series is centered at zero, it is called the Maclaurin series. For trigonometric functions like sine and cosine, the series provides a powerful method for approximating the functions using polynomials, which is especially useful when dealing with small angles or when using a limited number of terms in practical calculations.

Polynomial Approximation

Polynomial approximation involves using a finite number of terms from a function's Taylor or Maclaurin series to approximate the function's value. For functions such as sine and cosine, these polynomial forms provide an efficient means to compute approximate values, particularly when high precision is not required or when the argument lies within a specific range where the approximation is accurate.

Error Analysis

Error analysis in the context of polynomial approximations focuses on estimating the difference between the true function value and its polynomial approximation. This is typically done by examining the remainder term in the Taylor series, which gives insight into how the approximation error behaves as more terms are included, and helps in understanding why certain approximations may yield different errors based on the function input.

Convergence and Remainder Term

The convergence of a Taylor series and the behavior of its remainder term are crucial in determining the accuracy of a polynomial approximation. The remainder term quantifies the error introduced by truncating the series, and its magnitude depends on both the number of terms used and the specific value at which the function is evaluated. This concept explains why the error may differ for various inputs even when the same polynomial degree is used.

Transcript

00:01 Okay, for this problem, we will need to use these estimations to estimate values of cosine and sine, and with the functions that we are shown here.

00:08 Just a fun fact before we continue, if in calculus bc for those you who take it in high school, and calculus 2 for those who take it in college, this series can be extended indefinitely.

00:19 Like for instance, cosine of x, you can also add minus x to the 6 power over 6, x factorial, plus x the 8th power over 8 factorial, and so on and so forth, indefinitely.

00:29 Indefinitely.

00:30 The more further you go, the more accurate you get.

00:34 Now, let's start with sign first, because those values are a, b, and c.

00:38 Okay, so first thing we need to do, who is? well, start with one half.

00:49 We're using the given value.

00:52 That seems right.

00:54 Now let's try this.

01:00 Hmm, make sense.

01:05 Those values aren't exact, but they're pretty close.

01:08 Now, let's try the same thing, but with one.

01:16 That one is a little strange.

01:18 Let's try...