3. (4 pts) The total number of Atlanta Braves shirts a small gift shop has sold during a calendar year is represented by the function $$ S(t)=\frac{171.852}{1+84.715 e^{-0.786 t}} $$, where $$ t $$ is the number of months after January. a. Rewrite $$ s(t)=171.852 \cdot\left(\quad 1+84.715 e^{-0.786 t}\right)^{-1} $$ b. Take the derivative using the chain rule. $$ \begin{array}{l} \text { b. Take the derivative using the chain rule. } \ s^{\prime}(t)=\underline{171.852} \cdot(-1)\left(1+84.715 e^{-0.786 t}\right)^{-2} \cdot 84.715 e^{-0.786 t} \cdot(-0.786) \ =11442.936 e \end{array} $$ c. How many times did you use the chain rule? d. Finish the completely defined rate of change model for the function $$ s(t) $$. $$ s^{\prime}(t) $$ shirts per month gives e. Use your calculator to find $$ s^{\prime}(5) $$. f. Write a sentence of interpretation for $$ s^{\prime}(5) $$. g. Find the percentage rate of change at $$ t=5 $$. Be sure to use unrounded numbers in intermediate steps.

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a. We first rewrite the function in the form
$$
s(t)=171.852\left(1+84.715\,e^{-0.786t}\right)^{-1}.
$$

b. Define
$$
u(t)=1+84.715\,e^{-0.786t},
$$
so that
$$
s(t)=171.852\,(u(t))^{-1}.
$$
Differentiate using the chain rule. First, the derivative of $$(u(t))^{-1}$$ with respect to $$u$$ is
$$
\frac{d}{du}\,(u^{-1})=-1\,u^{-2}.
$$
Then,
$$
s'(t)=171.852\left[-1\,(u(t))^{-2}\right]\,u'(t).
$$
Differentiate
$$
u(t)=1+84.715\,e^{-0.786t}.
$$
Its derivative is
$$
u'(t)=84.715\cdot \frac{d}{dt}\left(e^{-0.786t}\right)
=84.715\cdot\left(-0.786\,e^{-0.786t}\right)
=-66.64749\,e^{-0.786t}.
$$
Thus,
$$
s'(t)=171.852\cdot\left[-1\,(1+84.715\,e^{-0.786t})^{-2}\right]\cdot\left[-66.64749\,e^{-0.786t}\right].
$$
Noting that the product of the two negative signs is positive,
$$
s'(t)=171.852\cdot66.64749\,e^{-0.786t}\,(1+84.715\,e^{-0.786t})^{-2}.
$$
Multiplying the constants gives a coefficient approximately equal to 
$$
171.852\times66.64749\approx11442.936,
$$
so that
$$
s'(t)=\frac{11442.936\,e^{-0.786t}}{\left(1+84.715\,e^{-0.786t}\right)^2}.
$$

c. In this differentiation we applied the chain rule twice. First for the outer function $$ (u(t))^{-1} $$ and then for the inner exponential function $$ e^{-0.786t} $$ contained in $$ u(t) $$.

d. The complete rate‐of‐change model (in shirts per month) is
$$
s'(t)=\frac{11442.936\,e^{-0.786t}}{\left(1+84.715\,e^{-0.786t}\right)^2}.
$$

e. To compute $$ s'(5) $$, substitute $$ t=5 $$ into the formula:
$$
s'(5)=\frac{11442.936\,e^{-0.786\cdot5}}{\left(1+84.715\,e^{-0.786\cdot5}\right)^2}.
$$
First, compute the exponent:
$$
-0.786\cdot5=-3.93.
$$
Then, using a calculator:
$$
e^{-3.93}\approx0.01972.
$$
Now, compute the denominator:
$$
1+84.715\,(0.01972)\approx1+1.669\approx2.669,
$$
and its square:
$$
(2.669)^2\approx7.125.
$$
The numerator is:
$$
11442.936\times0.01972\approx225.74.
$$
Thus,
$$
s'(5)\approx\frac{225.74}{7.125}\approx31.69.
$$

f. The interpretation is: At 5 months after January, the gift shop’s cumulative sales of Braves shirts are increasing at an instantaneous rate of approximately $$31.69$$ shirts per month.

g. The percentage rate of change at $$ t=5 $$ is given by
$$
\text{Percentage rate} = \frac{s'(5)}{s(5)}\times100\%.
$$
First, compute $$ s(5) $$:
$$
s(5)=\frac{171.852}{1+84.715\,e^{-3.93}}
$$
Using $$ e^{-3.93}\approx0.01972 $$,
$$
s(5)=\frac{171.852}{1+84.715\times0.01972}\approx\frac{171.852}{1+1.669}\approx\frac{171.852}{2.669}\approx64.44.
$$
Then,
$$
\frac{s'(5)}{s(5)}\approx\frac{31.69}{64.44}\approx0.4917.
$$
Thus, the percentage rate of change is
$$
0.4917\times100\%\approx49.17\%\text{ per month}.
$$
a. $$ s(t) = 171.852 \left(1 + 84.715 e^{-0.786 t}\right)^{-1} $$ b. $$ s'(t) = \frac{11442.936 e^{-0.786 t}}{\left(1 + 84.715 e^{-0.786 t}\right)^2} $$ c. The chain rule was used twice. d. $$ s'(t) = \frac{11442.936 e^{-0.786 t}}{\left(1 + 84.715 e^{-0.786 t}\right)^2} $$ shirts per month e. $$ s'(5) \approx 31.69 $$ shirts per month f. At 5 months, the sales are increasing by approximately 31.69 shirts per month. g. The percentage rate of change at $$ t=5 $$ is approximately 49.17% per month.

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