S'(t)= -147.78e^(-0.2t)
S=7389 units when t=0
find the function that describes the number of daily sales t days after the end of the campaign
Really stuck can someone help me to just find the function so I can head in the right track for the rest of the question
To find the function that describes the number of daily sales t days after the end of the campaign, we need to integrate the rate of change of sales (S'(t)) with respect to time.
Given that S'(t) = -147.78e^(-0.2t), we can integrate this expression to find S(t).
∫S'(t) dt = ∫[-147.78e^(-0.2t)] dt
To integrate the expression, we can use the power rule of integration in reverse, where the integral of e^x is e^x divided by the derivative of x.
Using this rule, we can rewrite the integral as:
∫S'(t) dt = -147.78 ∫[e^(-0.2t)] dt
Now, let's integrate e^(-0.2t).
We can do this by letting u = -0.2t, then du = -0.2dt.
The integral becomes:
-147.78 ∫[e^u] du
To integrate e^u, we can simply use the power rule of integration, where the integral of e^x is e^x.
So, the integral becomes:
-147.78 [e^u] + C
Now, let's substitute back the original variable t:
-147.78 [e^(-0.2t)] + C
Since we are looking for the number of daily sales, we can set the constant C as the initial number of sales, S(0) = 7389 units.
So, the function that describes the number of daily sales t days after the end of the campaign is:
S(t) = -147.78 [e^(-0.2t)] + 7389 units