Suppose that the marginal revenue for a product is given by the following, where x is the number of units and revenue is in dollars.
MR =
−28
(2x + 2)2
+ 28
Find the total revenue.
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To find the total revenue, we need to integrate the marginal revenue function with respect to x.
Since the marginal revenue function is given as MR = -28 / ((2x + 2)^2) + 28, we need to find the antiderivative of this function and evaluate it from the starting level of x to the ending level of x.
To integrate the function, we first write it as a sum of two terms:
MR = (-28 / (2x + 2)^2) + 28
The first term, (-28 / (2x + 2)^2), can be integrated using a u-substitution. Let u = 2x + 2, then du = 2 dx. Rearranging, we have dx = du / 2. The integral becomes:
∫(-28 / (2x + 2)^2) dx = ∫(-28 / u^2) (du / 2) = (-14/u) + C1.
The second term, 28, is a constant, so its integral is simply 28x.
Therefore, the total revenue is given by:
TR = ∫MR dx = ∫((-14/u) + 28)dx = (-14ln|u| + 28x) + C,
where C is the constant of integration.
Finally, we can replace the variable u with its original expression in terms of x:
TR = (-14ln|2x + 2| + 28x) + C.
So, the total revenue function is given by TR = -14ln|2x + 2| + 28x + C.