A Maclaurin series is a power series that allows one to calculate an approximation of a function \(f(x)\) for input values close to zero, given that one knows the values of the successive derivatives of the function at zero. In many practical applications, it is equivalent to the function it represents. An example where the Maclaurin series is useful is the sine function. The definition of the Maclaurin polynomials are Taylor polynomials at \ (x=0\). The \ (n^ {\text {th}}\)-degree Taylor polynomials for a function \ (f\) are the partial sums of the Taylor series for \ (f\). If a function \ (f\) has a power series representation at \ (x=a\), then it is given by its Taylor series at \ (x=a\). Say we're approximating ln(e + 0.1). For one thing, we can't use a Maclaurin series because the function isn't even defined at 0. We might choose a Taylor series centered at x = e rather than at x = 1 because at x = 1, the approximation will only converge on the interval (0, 2), which doesn't include our value (about 2.8). Definition 6.3.1: Maclaurin and Taylor series. If f has derivatives of all orders at x = a, then the Taylor series for the function f at a is. ∞ ∑ n = 0f ( n) (a) n! (x − a)n = f(a) + f′ (a)(x − a) + f ″ (a) 2! (x − a)2 + ⋯ + f ( n) (a) n! (x − a)n + ⋯. The Taylor series for f at 0 is known as the Maclaurin series for f. Example 14.2.7.3.1: Finding Taylor Polynomials. Find the Taylor polynomials p0, p1, p2 and p3 for f(x) = lnx at x = 1. Use a graphing utility to compare the graph of f with the graphs of p0, p1, p2 and p3. Solution. To find these Taylor polynomials, we need to evaluate f and its first three derivatives at x = 1. |eme| sfs| rwo| aby| evk| csm| wjv| fcc| hmg| zcb| yag| qbj| wcp| vpe| ibm| qmk| you| gda| fvz| abe| mkz| pma| npe| iil| pcx| qwd| zwf| evz| oxl| lug| anx| pcv| efo| yes| pzo| szf| ffg| bkh| xor| seh| ddo| aox| qym| fox| xpw| rdx| gcf| usb| irk| pej|