Using the values of TU=2Q, graph TU and MU. identify the stages of utility MU should have + zero and - values

To graph the Total Utility (TU) and Marginal Utility (MU) based on the given equation \( TU = 2Q \), and to identify the stages of utility, let's proceed step by step.

### 1. **Understanding the Functions:**

- **Total Utility (TU):**  
  The Total Utility function is given as:
  \[
  TU = 2Q
  \]
  This means that the total utility increases linearly as the quantity \( Q \) increases.

- **Marginal Utility (MU):**  
  Marginal Utility is the derivative of the Total Utility with respect to the quantity \( Q \).  
  \[
  MU = \frac{d(TU)}{dQ} = \frac{d(2Q)}{dQ} = 2
  \]
  This means the Marginal Utility is constant and equal to 2 for all levels of \( Q \).

### 2. **Graphing TU and MU:**

- **Graph of TU:**
  The TU graph will be a straight line with a slope of 2, passing through the origin (0,0).

- **Graph of MU:**
  The MU graph will be a horizontal line at \( MU = 2 \) because the marginal utility is constant.

### 3. **Stages of Utility:**

In general, the stages of utility are described as follows:

1. **Increasing Utility (MU positive):**  
   When additional consumption increases total utility, \( MU > 0 \). In this case, since \( MU = 2 \) is always positive, this stage is always present.

2. **Zero Utility (MU = 0):**  
   This is the point where the additional consumption does not change the total utility, i.e., \( MU = 0 \). In this case, this never happens because \( MU \) is constant and not equal to 0.

3. **Negative Utility (MU < 0):**  
   When additional consumption decreases total utility, \( MU < 0 \). This stage also never happens here because \( MU \) is always positive.

Given that \( MU \) is constant and positive, only the **increasing utility** stage is present, with no points where MU is zero or negative.

### 4. **Graphing and Analysis:**

- **Graph of TU**:  
  A straight line starting from the origin (0, 0) and increasing linearly with slope 2.

- **Graph of MU**:  
  A horizontal line at \( MU = 2 \), showing that the marginal utility is constant for all values of \( Q \).

### 5. **Conclusion:**

- **Stages of Utility:**
  - **MU Positive:** \( MU = 2 \) for all \( Q \).
  - **MU Zero and Negative:** Not present.

Would you like a detailed explanation on how to sketch the graph or any further analysis?

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### Relative Questions:
1. What does the linearity of the TU function imply about consumer satisfaction?
2. How would the graph change if the TU function was quadratic, e.g., \( TU = Q^2 \)?
3. What real-world scenarios can be modeled by a constant marginal utility?
4. How would decreasing marginal utility be represented graphically?
5. How does the concept of diminishing marginal returns relate to the MU graph?

### Tip:
When analyzing utility functions, always derive the marginal utility function first. It gives immediate insight into consumer behavior as they increase consumption.

Learn how to graph Total Utility (TU) and Marginal Utility (MU) using the equation TU = 2Q. Understand the stages of utility and how MU remains constant at 2 for all quantities Q. Suitable for undergraduate students studying utility theory and graphing functions.

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