Suppose that the marginal revenue for a product is given by
MR = 
−34
(2x + 1)2
 + 34
where x is the number of units and revenue is in dollars. Find the total revenue, 
R(x).
R(x) =

The marginal revenue (MR) represents the rate of change of the total revenue function \(R(x)\), meaning \(MR = \frac{dR(x)}{dx}\). To find the total revenue function \(R(x)\), we integrate the given marginal revenue function with respect to \(x\).

Given:
\[
MR = -\frac{34}{(2x + 1)^2} + 34
\]

Now, we integrate to find \(R(x)\):

\[
R(x) = \int \left(-\frac{34}{(2x + 1)^2} + 34\right) dx
\]

### Step 1: Split the integral
We can split the integral into two parts:

\[
R(x) = \int -\frac{34}{(2x + 1)^2} dx + \int 34 dx
\]

### Step 2: Solve each integral
#### First integral:
\[
\int -\frac{34}{(2x + 1)^2} dx
\]
Let \( u = 2x + 1 \), hence \( du = 2dx \), or \( dx = \frac{du}{2} \).

Substitute into the integral:
\[
= -34 \int \frac{1}{u^2} \cdot \frac{du}{2}
\]
\[
= -\frac{34}{2} \int u^{-2} du
\]
\[
= -17 \left(-\frac{1}{u}\right)
\]
\[
= \frac{17}{2x + 1}
\]

#### Second integral:
\[
\int 34 dx = 34x
\]

### Step 3: Combine the results
Now, combine the two parts:
\[
R(x) = \frac{17}{2x + 1} + 34x + C
\]
where \(C\) is the constant of integration.

Thus, the total revenue function is:
\[
R(x) = \frac{17}{2x + 1} + 34x + C
\]

Would you like to find the constant \(C\) using an initial condition, or do you have any questions on this?

---

Here are 5 questions to expand your understanding:
1. What is the significance of the constant of integration in this problem?
2. How would you determine the constant \(C\) if given an initial condition, like \(R(0)\)?
3. How does the shape of the marginal revenue function affect the behavior of \(R(x)\)?
4. What is the long-term behavior of the total revenue function as \(x\) grows large?
5. How can we interpret the units of the terms in the total revenue function?

**Tip:** Always check whether an initial condition is provided to determine the specific constant of integration for practical applications of the function.

Suppose that the marginal revenue for a product is given by MR = -34 / (2x + 1)^2 + 34 where x is the number of units and revenue is in dollars. Find the total revenue, R(x).

Discover how to calculate the total revenue function R(x) from the marginal revenue (MR) function given by MR = -34 / (2x + 1)^2 + 34. This problem involves calculus concepts such as integration and marginal revenue analysis, making it suitable for high school seniors or college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation