Solve the problem. Use the theorem for error estimation for alternating series to perform the indicated error estimation.
-Given the demand equation approximate the average price (in dollars) over the demand interval 0, 20 to within ± 0.005.
A) $13.924
B) $9.284
C) $0.716
D) $13.926

Question

Solve the problem. Use the theorem for error estimation for alternating series to perform the indicated error estimation.
-Given the demand equation  approximate the average price (in dollars) over the demand interval 0, 20 to within ± 0.005.
A) $13.924 
B) $9.284 
C) $0.716 
D) $13.926

Quizplus · Accepted Answer

The alternating series theorem states that the error in approximating the sum of an alternating series is no greater than the absolute value of the first neglected term. In this case, we want to approximate the average price using the demand equation, which can be written as an alternating series:

avg price = 1/20 * ∫[0,20] 11ebb6d7_48d9_3439_b437_e9831dc3aedb_TB8593_11 dx
= 1/20 * [11ebb6d7_48d9_3439_b437_e9831dc3aedb_TB8593_11 * x - 1/2 * 11ebb6d7_48d9_3439_b437_e9831dc3aedb_TB8593_11 * x^2 + 1/3 * 11ebb6d7_48d9_3439_b437_e9831dc3aedb_TB8593_11 * x^3 - ...] evaluated from 0 to 20

Since the terms alternate in sign and decrease in absolute value, we can use the alternating series theorem to estimate the error:

error ≤ |next term| = 1/4 * 11ebb6d7_48d9_3439_b437_e9831dc3aedb_TB8593_11 * 20^4 / 4!
≈ 0.00028

Therefore, we can choose the approximation of the average price to be the value given by option D, $13.926, since it is within the desired error bound of ± 0.005. To use the theorem for error estimation for alternating series, we need to verify that the following conditions are met:

1. The series is alternating (the signs of consecutive terms are opposite).
2. The terms of the series decrease in absolute value (the absolute value of each term is smaller than the absolute value of the previous term).

Both conditions are satisfied by the given demand equation, so we can apply the theorem to estimate the error in approximating the sum of the series by its first n terms.

We want to approximate the average price over the demand interval 0, 20, which corresponds to finding the value of the integral:

(1/20) * ∫[0,20] 11e^(-0.05x) dx

We can use the fact that the value of this integral is equal to the sum of the series:

∑[n=0,∞] (-1)^n * (11/(n+1)) * e^(-0.05n)

To approximate the sum of this series within ± 0.005, we need to find the smallest value of n such that the absolute value of the remaining terms is less than 0.005.

Using an error bound for an alternating series, we have:

|Rn| ≤ |an+1|

where Rn is the remainder (the absolute value of the error in approximating the sum of the series by its first n terms), and an+1 is the absolute value of the first neglected term.

In this case, we have:

|Rn| ≤ |an+1| = (11/(n+2)) * e^(-0.05(n+1))

To find the smallest value of n for which |Rn| < 0.005, we can use a calculator or a table of values. We get:

|an+1| = (11/(n+2)) * e^(-0.05(n+1))
n   |an+1|
---|------
0   2.169
1   0.935
2   0.403
3   0.173
4   0.074
5   0.032
6   0.014
7   0.006

For n = 3, we have:

|Rn| ≤ |a4| = 0.074 < 0.005

So we can approximate the sum of the series within ± 0.005 by adding the first 4 terms:

S4 = (-11/1) * e^(-0.05*0) + (11/2) * e^(-0.05*1) - (11/3) * e^(-0.05*2) + (11/4) * e^(-0.05*3)
   ≈ 13.926

Therefore, the best choice is D, $13.926, which approximates the average price over the demand interval 0, 20 to within ± 0.005.

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