Table 1 shows the values of a twice–differentiable function together with two…
MATHEMATICS · Multi-part · AP · Calculus BC · Infinite Sequences and Series · Infinite Sequences and Series
Problem
Table 1 shows the values of a twice–differentiable function g together with two different polynomial approximations, P2 and P4, for several values of x near 0.
Table 1 (all values rounded to 6 decimal places)
x | g(x) | P2(x)=x−2x2 | P4(x)=x−2x2+3x3−4x4
─────┼─────────────────┼──────────────────────────────┼──────────────────────────────────────────
0.1 | 0.095310 | 0.095000 | 0.095308
0.3 | 0.262364 | 0.255000 | 0.261554
0.5 | 0.405465 | 0.375000 | 0.401042
(The actual function is g(x)=ln(1+x), but this need not be used explicitly.)
(1) Part 1: (a) Discuss how the numerical data in Table 1 illustrate the relative accuracy of P2 and P4 for estimating g(x) when x is close to 0.
(2) Part 2: (b) Provide a mathematical explanation, based on the way P2 and P4 are constructed, for why P4 generally gives a smaller error than P2 when ∣x∣ is small.
(3) Part 3: (c) Without performing any additional computation, predict which polynomial—P2 or P4—will give the more accurate estimate of g(0.8). Justify your answer in terms of the structure of the two polynomials and their agreement with g at x=0.