A company estimates that its sales will grow continuously at a rate given by the function \[ S^{\prime}(t)=24 e^{t} \] where $S^{\prime}(t)$ is the rate at which sales are increasing, in dollars per day, on day $t$. a) Find the accumulated sales for the first 4 days. b) Find the sales from the 2 nd day through the 5 th day. (This is the integral from 1 to 5 .) a) The accumulated sales for the first 4 days is $\$ \square$. (Round to the nearest cent as needed.)
Solution
Step 1 ：The problem is asking for the accumulated sales for the first 4 days. This is a calculus problem and can be solved by integrating the given function from 0 to 4. The integral of a function gives the area under the curve, which in this case represents the total sales over a period of time.
Step 2 ：First, we set up the integral: \(\int_{0}^{4} S^\prime(t) dt = \int_{0}^{4} 24e^t dt\).
Step 3 ：Next, we calculate the integral: \(-24 + 24e^4\).
Step 4 ：This represents the total sales over the first 4 days. However, this is not in a form that is easy to interpret. We need to evaluate this expression to get a numerical value.
Step 5 ：Evaluating the expression gives us approximately 1286.36.
Step 6 ：\(\boxed{\text{The accumulated sales for the first 4 days is approximately \$1286.36.}}\)
