Remainders Find the remainder Rₙ for the nth−order Taylor poly...

Question

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = e⁻ˣ, a = 0

Accepted Answer

Welcome back, everyone. Consider the function F X equals e negative 3X. Let PNX be the Taylor polynomial of degree N centered at X equals 0 for this function. Write an expression for the remainder term RF X when approximating F of X by PN X for a general integer N. Express RNF X in a form that involves a value C between 0 and X. Now what do we know about our Taylor polynomial and our remainder term? Well, recall that the Taylor polynomial of degree N centered at X equals 0 for the function F of X equals E-3X is given by PNX equals the sum. Between K equals 0 and N of the K derivative of F of 0 divided by K factorial multiplied by X to the K. OK. And again, let's just make a note here for degree N. Centered At X equals 0, OK. What else do we know? And of course we're this is the case derivative of F of X for X equals 0. What else do we know? Well, we also know that the remainder term RN of X for the Taylor polynomial approximation is given by the Lagrange form of the remainder, which says that it is the n + 1 derivative of F of C divided by n + 1 factorial. Multiplied by X to the N + 1 for some C OK, between 0 and X. So C is between 0 and X. So now we need to find the N +1 derivative of FFX for us to write the expression for RNF X explicitly. Now, remember, F of X E equals E to the negative 3X. And since we cannot differentiate with respect to a symbolic N+1 directly, let's instead reason about the derivative pattern for FF X. So now, given F of X equals E to the negative 3 X, right? Then if we start thinking about our derivatives, the first derivative of FFX would be -3 E to the -3X. The second derivative would be equal to -32, OK, multiplied by E to the -3 X. The 3rd derivative. Would be equal to -3 cubed multiplied by E to the -3 X. And do you notice a pattern here? Well, based on this pattern. OK, if we look at our numbers, then this means that if the N + 1 derivative of F of X then would be equal to -3 multiplied or raised to the power of N + 1 multiplied by E to the -3 X because notice here, it's really the power that keeps changing, right? So now in this case, Since we have an expression for the N +1 derivative of F of X, then we can express the remainder term RF X. As -3 to the power of N + 1, OK, multiplied by E to the -3C. OK, that's F of C, divided by N + 1 factorial multiplied by X to the N + 1, OK? And that is for some C. Between 0 and X, OK, between 0 and X, OK? In other words. Where sorry X. OK. Uh sorry, C rather, C is in the interval from 0 to X. So this means that this would be our remainder, OK? This is our remainder in a form that involves a value C. Thanks a lot for watching everyone. I hope this video helped.

Remainders Find the remainder Rₙ for the nth−order Taylor polynomial centered at a for the given functions. Express the result for a general value of n.

f(x) = e⁻ˣ, a = 0

Key Concepts

Taylor Polynomial

Remainder Term (Lagrange Form)

Derivatives of the Exponential Function

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