Provide an appropriate response.
-How large should n (n an integer) be chosen for Ln and Rn for the approximation of (ln x + 1) dx to be within 0.05 of the true value?

Question

Provide an appropriate response.
-How large should n (n an integer) be chosen for Ln and Rn for the approximation of  (ln x + 1) dx to be within 0.05 of the true value?

Quizplus · Accepted Answer

According to the error estimation formula for Trapezoidal Rule, the error is proportional to (b-a)^3/n^2. To get the error within 0.05, we need (b-a)^3/n^2 <= 0.05. Since a=0 and b=e-1, we have (e-1)^3/n^2 <= 0.05, which gives n >= 34,729.05. Rounding up to the nearest integer, we get n = 34,730, which corresponds to choice C. We can use the error formula for the Trapezoidal Rule: Error <= (b-a)^3/(12n^2) * max|f''(x)| For ln(x+1), we have f''(x) = -1/(1+x)^2, and we want the error to be no more than 0.05, so we have: (b-a)^3/(12n^2) * 1/(1+x)^2 <= 0.05 Since ln(11ebb6d7_48aa_0dea_b437_db52df8551d1_TB8593_11) is approximately 2.435, we can take a = 0 and b = 2.435. Solving for n gives: n >= sqrt((11ebb6d7_48aa_0dea_b437_db52df8551d1_TB8593_11)^3/(12*(2.435)^2*0.05)) n >= 128.38 Since n has to be an integer, we should round up to the next integer, giving us a minimum value of n = 129. Therefore, the best choice is option C.

Provide an Appropriate Response