The black curve in each panel is $y=\cos x$. The dashed blue curve is a Maclaurin polynomial. Matc

Question

The black curve in each panel is $y=\cos x$. The dashed blue curve is a Maclaurin polynomial. Matching more derivatives at $x=0$ generally extends the close agreement around the origin, although it does not guarantee better accuracy for every $x$. $P_2(x)=1-\dfrac{x^2}{2!}$ $P_4(x)=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}$ $P_6(x)=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}$ $P_8(x)=1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\dfrac{x^8}{8!}$

Accepted Answer

$P_2(x)=1-\frac{x^2}{2}$, $P_4(x)=1-\frac{x^2}{2}+\frac{x^4}{24}$, and $P_6(x)=1-\frac{x^2}{2}+\frac{x^4}{24}-\frac{x^6}{720}$. Each added term makes the Maclaurin polynomial approximate $\cos x$ over a wider interval around $x=0$.

Visualising Successive Approximations

Visualising Successive Approximations

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