company estimates that its sales will grow continuously at a rate given by the function \( S^{\prime}(t)=12 e^{t} \) here \( S^{\prime}(t) \) is the rate at which sales are increasing, in dollars per day, on day \( t \). 1) Find the accumulated sales for the first 7 days. D) Find the sales from the 2 nd day through the 5 th day. (This is the integral from 1 to 5 .)

To find the accumulated sales for the first 7 days, we need to integrate the rate of sales with respect to time from 0 to 7. Given: \[ S^{\prime}(t) = 12e^{t} \] To find the accumulated sales, we need to integrate \( S^{\prime}(t) \) with respect to time from 0 to 7. \[ \text{Accumulated Sales} = \int_{0}^{7} 12e^{t} dt \] Now, let's calculate the integral. Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{0}^{7} 12e^{t} dt\) - step1: Evaluate the integral: \(\int 12e^{t} dt\) - step2: Use properties of integrals: \(12\times \int e^{t} dt\) - step3: Evaluate the integral: \(12e^{t}\) - step4: Return the limits: \(\left(12e^{t}\right)\bigg |_{0}^{7}\) - step5: Calculate the value: \(12e^{7}-12\) The accumulated sales for the first 7 days is \(12e^{7} - 12\). To find the sales from the 2nd day through the 5th day, we need to integrate the rate of sales with respect to time from 1 to 5. \[ \text{Sales from 2nd day to 5th day} = \int_{1}^{5} 12e^{t} dt \] Now, let's calculate the integral. Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{1}^{5} 12e^{t} dt\) - step1: Evaluate the integral: \(\int 12e^{t} dt\) - step2: Use properties of integrals: \(12\times \int e^{t} dt\) - step3: Evaluate the integral: \(12e^{t}\) - step4: Return the limits: \(\left(12e^{t}\right)\bigg |_{1}^{5}\) - step5: Calculate the value: \(12e^{5}-12e\) The sales from the 2nd day through the 5th day is \(12e^{5} - 12e\).