The rate of change in the number of subscribers $S$ to a newly introduced magazine is modeled by $\frac { d S } { d t } = 1000 t ^ { 2 } e ^ { - 1 } , 0 \leq t \leq 6$ where $t$ is the time in years. Use Simpson's Rule $n = 12$ with to estimate the total increase in the number of subscribers during the first 6 years.
A) $\approx$ 1870 subscribers
B) $\approx$ 1780 subscribers
C) $\approx$ 1800 subscribers
D) $\approx$ 1878 subscribers
E) $\approx$ 1987 subscribers

Question

Quizplus · Accepted Answer

Simpson's Rule formula for $n$ intervals is given by $\frac{\Delta x}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 4f(x_{n-1}) + f(x_n)]$, where $\Delta x = \frac{b-a}{n}$ and $a$ and $b$ are the interval limits. Here, $a = 0$, $b = 6$, and $n = 12$, so $\Delta x = \frac{6-0}{12} = 0.5$. The rate of change function is $\frac{dS}{dt} = 1000t^2e^{-t}$. Applying Simpson's Rule and calculating the values at the respective points, the total increase in subscribers is approximately 1878.

The Rate of Change in the Number of Subscribers SSS To a Newly Introduced Magazine Is Modeled By

The Rate of Change in the Number of Subscribers $S$ To a Newly Introduced Magazine Is Modeled By