Proof Let Pn(x) be the n th Taylor polynomial for f at c . Prove that Pn(c)=f(c) and P^(k)(c)=f^

Proof Let Pn(x) be the n th Taylor polynomial for f at c ....

Key Concepts

Term-by-Term Differentiation

Differentiating a Taylor polynomial term-by-term is a key concept that allows us to analyze how well the polynomial approximates the derivatives of the original function. Because each term of the polynomial includes factors of (x - c), differentiating and then evaluating at x = c yields zero for terms of higher order than the derivative being taken, except for the proper matching term. This mechanism confirms that the kth derivative of the Taylor polynomial at the center is exactly the kth derivative of the original function, ensuring that the polynomial matches the function's behavior up to the nth derivative.

Evaluation at the Center

When a Taylor polynomial is evaluated at the center (x = c), all terms involving (x - c) become zero except for the constant term. This property guarantees that the polynomial exactly equals the function value at the center. It is a fundamental aspect of Taylor polynomials because it confirms that the approximation is perfectly accurate at the point of expansion.

Taylor Polynomial

A Taylor polynomial is a finite sum used to approximate a function near a specific point, known as the center. It is constructed by taking the derivatives of a function at that center and forming a polynomial whose terms are given by these derivatives divided by factorials. This construction ensures that the polynomial reflects the local behavior of the function up to a certain degree and provides a powerful way to approximate complex functions with polynomials.

Transcript

00:01 Okay, what we want to do is kind of walk through a proof of certain aspects of a tailor polynomial.

00:08 So we're going to let piece of n of x be an nth degree tailor polynomial centered at x equal to c.

00:15 And so we know then that p of n of x is equal to function evaluated at c plus the first derivative of the function evaluated c times x minus c plus the second derivative, evaluated c over 2 factorial times x minus c squared, all the way to the very end where it's plus the nth derivative of the function evaluated c over n factorial times x minus c to the nth power.

00:45 And what we want to do is we want to show that the nth degree polynomial evaluated at c is actually equal to the function.

01:03 Evaluated at c.

01:05 And so this is why we do.

01:07 We would do the nth degree polynomial evaluated at c is just f of c, plus the first derivative evaluated c times c minus c plus the second derivative evaluated c over 2 factorial times c minus c squared, plus, and then we continue doing it until i have the nth derivative evaluated c over n factorial times c minus c to the nth power.

01:39 And so you notice all of these subsequent terms, c minus c, have c minus c in there, which is zero.

01:47 And so all of those terms would be zero.

01:50 So the nth degree polynomial evaluated at c is actually equal to your function evaluated at c.

02:01 So now the second thing we want to show is, that we want to show that the kth derivative of your nth degree polynomial evaluated c is actually equal to your kth derivative of your function evaluated at c for k values between one and in.

02:34 Okay, so if i actually think about this, if i kind of look at that let's go back and do the first derivative...

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