• Polynomial functions can be used to approximate elementary functions such as $\sin x$, $e^x$, and $\ln x$.

In fact, the tangent line approximation (in other words: Linearization and Differentials) is a Taylor Polynomials.

<aside> 📌 However, just like linealízatenos, the yielded number from Taylor Polynomials is a result of a approximation. Therefore, only f(x)$\approx$ P_n(x)

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Definition of a Nth-Degree Taylor Polynomials

Definition of a Maclaurin Polynomials

Occurring Taylor Polynomials

• $f(x)=\sin x \approx P_n(x)=(x-c)-\frac {(x-c)^3} {3!}+\frac {(x-c)^5} {5!}...$ Odd degrees and alternating signs
• $f(x)=\cos x \approx P_n(x)=1-\frac {(x-c)^2} {2!}+\frac {(x-c)^4} {4!}…$ Even degrees and alternating signs
• $f(x)=\ln x \approx P_n(x)=(x-c)-\frac {(x-c)^2} {2} +\frac {(x-c)^3} {3}-\frac {(x-c)^4} {4}$…alternating signs and no !
• $f(x)=\frac 1 {x+1} \approx 1-(x-c)+(x-c)^2-(x-c)^3+(x-c)^4$ …alternating signs, the exponent increases
• $f(x)=\frac 1 {1-x} \approx 1+(x-c)+(x-c)^2+(x-c)^3+(x-c)^4$ … the exponent increases
• $f(x)= \frac 1 2 (e^x+e^{-x}) \approx P_n(x)=1+\frac {(x-c)^2} {2!}+\frac {(x-c)^4} {4!}+\frac {(x-c)^6} {6!}…$ Often occurs: similar to $\cos x$ but absolute.

Derivate polynomials from the parent polynomials.

• If x of the parent function are n, then replace the x in terms in the polynomials.
• If there is a consonant or variable A that is multiple to the parent function, then multiple A to every term of the polynomials
• Combine

Manipulation of Taylor Series

Replacement

Sub Concept