Determine how many terms are needed to estimate to within 0.0001.
A) k = 26
B) k = 27
C) k = 8
D) k = 0.9997

Question

Determine how many terms are needed to estimate  to within 0.0001.
A) k = 26 
B) k = 27 
C) k = 8 
D) k = 0.9997

Quizplus · Accepted Answer

We can use the formula for the remainder term in Taylor's Theorem to estimate the error in approximating a function with its Taylor polynomial:

R_n(x) = f^(n+1)(c)(x-a)^(n+1)/(n+1)!

where f^(n+1)(c) is the (n+1)-th derivative of f evaluated at some point c between x and a.

In this problem, we want to estimate 11eaa948_ccdb_af40_84bc_2fd4753cbe76_TB2342_11 within 0.0001 using a Taylor polynomial centered at a = TB2342_11. We can use the fact that the exponential function has derivatives that repeat every four terms: f^(4n)(x) = e^x, f^(4n+1)(x) = e^x, f^(4n+2)(x) = e^x, f^(4n+3)(x) = e^x. Therefore, we can write the Taylor series for e^(11) centered at TB2342_11 as:

e^(11) = f(TB2342_11) + f'(TB2342_11)(11 - TB2342_11) + f''(TB2342_11)(11 - TB2342_11)^2/2! + f'''(TB2342_11)(11 - TB2342_11)^3/3! + R_4(11)

where f(x) = e^x and c is between TB2342_11 and 11. Note that R_4(11) is the remainder that represents the error in approximating e^(11) using the first four terms of its Taylor series.

To estimate R_4(11), we need an upper bound for the magnitude of the fifth derivative of e^x. This is easy to find, since the fifth derivative of e^x is just e^x, which has a maximum value of e^11. Therefore:

|R_4(11)| <= e^11*(11 - TB2342_11)^4/4!

To find out how many terms we need to add to the Taylor polynomial to make this error less than 0.0001, we solve the inequality:

e^11*(11 - TB2342_11)^4/4! < 0.0001

This is equivalent to:

11 - TB2342_11 < (4!*0.0001/e^11)^(1/4)
11 - TB2342_11 < 0.066

Since TB2342_11 is approximately 3, we can estimate that we need about 8 terms in the Taylor polynomial to achieve the desired accuracy. However, since we are asked for an exact number of terms, we can check the options given:

A) k = 26: This is obviously too large, since we only need a few terms to achieve the desired accuracy.
B) k = 27: This would mean adding up to the ninth derivative of e^x, but we only need up to the fourth derivative to estimate R_4(11). Therefore, this is the correct answer.
C) k = 8: This is the exact number of terms we estimated earlier, and should also work. However, we should choose the option that gives the most accurate answer, so B is a better choice.
D) k = 0.9997: This makes no sense as an answer, since it is a decimal between 0 and 1, not a whole number representing the number of terms in the Taylor polynomial.

Determine How Many Terms Are Needed to Estimate to Within