Let represent the Taylor Polynomial of degree n about for .If is used to approximate the value of ,then which of the following expressions is NOT less than ?

Question

​Let  represent the Taylor Polynomial of degree n about  for  .If  is used to approximate the value of  ,then which of the following expressions is NOT less than  ? ​

Quizplus · Accepted Answer

Since the given Taylor polynomial is of degree n, we can use the error formula for Taylor polynomials to get an upper bound on the error. The error formula is given by:

|f(x) - Pn(x)| <= M(x-a)^(n+1)/(n+1)!

where M is an upper bound on the (n+1)th derivative of f(x) on some interval containing x, and a is the center of the Taylor polynomial (in this case, a=11eac44f_ec1a_ef27_bedf_15a7df1ef627_TB7497_11).

Using the given information, we can approximate M and then use the error formula to compare the given expressions with the actual value of 11eac44f_ec1b_163b_bedf_27f1196e53bc_TB7497_11.

For choice A, notice that the difference between the given center (11eac44f_ec1a_ef27_bedf_15a7df1ef627_TB7497_11) and the point we are approximating (11eac44f_ec1b_163a_bedf_3b0211835aad_TB7497_11) is relatively small; the only difference is in the last 4 digits. This means that the error bound given by the error formula is likely to be small as well. In fact, if we assume that f(x) is some "nice" function (e.g. a polynomial) and we use a value of n that is relatively large (e.g. n=10 or n=20), then we can expect that the error in the approximation will be negligible and that 11eac44f_ec1b_163c_bedf_d31e7916a54b_TB7497_11 is less than 11eac44f_ec1b_163b_bedf_27f1196e53bc_TB7497_11.

For choice B, notice that the difference between the given center (11eac44f_ec1a_ef27_bedf_15a7df1ef627_TB7497_11) and the point we are approximating (11eac44f_ec1b_163a_bedf_3b0211835aad_TB7497_11) is relatively large; the only common digits are the first 6. This means that the error bound given by the error formula is likely to be very large, and we cannot say for sure whether 11eac44f_ec1b_3d4d_bedf_8fbf12c13372_TB7497_11 is less than 11eac44f_ec1b_163b_bedf_27f1196e53bc_TB7497_11.

For choice C, notice that the difference between the given center (11eac44f_ec1a_ef27_bedf_15a7df1ef627_TB7497_11) and the point we are approximating (11eac44f_ec1b_163a_bedf_3b0211835aad_TB7497_11) is exactly 1 in the last digit. This means that the error bound given by the error formula is likely to be relatively small, since the derivative of f(x) should not change too much over such a small interval. Therefore, we can expect that 11eac44f_ec1b_3d4e_bedf_d777b0318f6f_TB7497_11 is less than 11eac44f_ec1b_163b_bedf_27f1196e53bc_TB7497_11.

For choice D, notice that the difference between the given center (11eac44f_ec1a_ef27_bedf_15a7df1ef627_TB7497_11) and the point we are approximating (11eac44f_ec1b_163a_bedf_3b0211835aad_TB7497_11) is relatively large; the only common digits are the first 6. This means that the error bound given by the error formula is likely to be very large, and we cannot say for sure whether 11eac44f_ec1b_645f_bedf_83a445d71e53_TB7497_11 is less than 11eac44f_ec1b_163b_bedf_27f1196e53bc_TB7497_11.

​Let Represent the Taylor Polynomial of Degree N About

Let Represent the Taylor Polynomial of Degree N About