Chapter 3: Applications of Differentiation
Section 3.3: Taylor Polynomials
Example 3.3.3
For fx=sinx with a=0 in Taylor's Formula, obtain the general form of Pnx.
Solution
Define the function f
Type fx=… Context Panel: Assign Function
fx=sinx→assign as functionf
Obtain fk0 for k=0,…,8
Notation for kth derivative evaluated at x=0: Context Panel: Sequence≻k Choose k=0 to k=8
fk0→sequence w.r.t. k0,1,0,−1,0,1,0,−1,0
Infer Pnx
The cyclic nature of the values of the derivatives at x=0 suggest only odd-powers of x have nonzero coefficients, and these coefficients are either ±1 divided by the factorial of an odd integer. These observations suggest
Pnx=∑k=0n−1kx2 x+12 k+1!
Corroborate with a specific Taylor polynomial
Type sinx
Context Panel: Series≻Series≻x
sinx→series in xx−16x3+1120x5−15040x7
Verify
Context Panel: Series≻Formal Power Series See Figure 3.3.3(a) for details.
sinx→formal series∑k=0∞−1kx2k+12k+1!
A Formal Power Series is an infinite sum, the truncation of which is then just a polynomial.
The general term in the formal power series is the general term in Pnx, the Taylor polynomial of degree n.
Figure 3.3.3(a) Formal Power Series dialog
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