(b)	Give the demand function;	(10)
q_1=f(p_1,p_2 )=36,000-0.1p_1^2-0.5p_2^2
	Determine the partial derivatives f_(p_1 )and f_(p_2 )
	If p_1=20 and p_2=10, evaluate f_(p_1 ) and f_(p_2 ) and interpret their meaning.




The demand function is given as:

\[
q_1 = f(p_1, p_2) = 36,000 - 0.1p_1^2 - 0.5p_2^2
\]

We are tasked with determining the **partial derivatives** of \( f(p_1, p_2) \) with respect to \( p_1 \) and \( p_2 \), and then evaluating them at \( p_1 = 20 \) and \( p_2 = 10 \).

### Step 1: Compute the Partial Derivatives

#### 1. Partial Derivative with respect to \( p_1 \):

To find \( \frac{\partial f}{\partial p_1} \), we treat \( p_2 \) as a constant and differentiate \( f(p_1, p_2) \) with respect to \( p_1 \).

\[
\frac{\partial f}{\partial p_1} = \frac{d}{dp_1} \left( 36,000 - 0.1p_1^2 - 0.5p_2^2 \right)
= -0.2p_1
\]

#### 2. Partial Derivative with respect to \( p_2 \):

To find \( \frac{\partial f}{\partial p_2} \), we treat \( p_1 \) as a constant and differentiate \( f(p_1, p_2) \) with respect to \( p_2 \).

\[
\frac{\partial f}{\partial p_2} = \frac{d}{dp_2} \left( 36,000 - 0.1p_1^2 - 0.5p_2^2 \right)
= -p_2
\]

### Step 2: Evaluate the Partial Derivatives at \( p_1 = 20 \) and \( p_2 = 10 \)

#### 1. Evaluating \( \frac{\partial f}{\partial p_1} \) at \( p_1 = 20 \):

\[
\frac{\partial f}{\partial p_1} \Bigg|_{p_1=20} = -0.2(20) = -4
\]

#### 2. Evaluating \( \frac{\partial f}{\partial p_2} \) at \( p_2 = 10 \):

\[
\frac{\partial f}{\partial p_2} \Bigg|_{p_2=10} = -(10) = -10
\]

### Interpretation of the Results:

1. **\( \frac{\partial f}{\partial p_1} = -4 \)** means that when \( p_1 = 20 \), for each unit increase in \( p_1 \), the demand \( q_1 \) decreases by approximately 4 units, holding \( p_2 \) constant. This reflects the **rate of change of demand with respect to the price \( p_1 \)**.

2. **\( \frac{\partial f}{\partial p_2} = -10 \)** means that when \( p_2 = 10 \), for each unit increase in \( p_2 \), the demand \( q_1 \) decreases by approximately 10 units, holding \( p_1 \) constant. This reflects the **rate of change of demand with respect to the price \( p_2 \)**.

These partial derivatives indicate how sensitive the demand \( q_1 \) is to changes in \( p_1 \) and \( p_2 \), showing that demand decreases with increases in either price.

---

Would you like further details on these calculations, or do you have any other questions?

### Related Questions:
1. What does a positive partial derivative signify in a demand function?
2. How can second-order partial derivatives help in understanding concavity or convexity of demand functions?
3. What is the interpretation of cross-partial derivatives in multivariable functions?
4. How does elasticity of demand relate to the partial derivatives of the demand function?
5. How would changes in prices \( p_1 \) and \( p_2 \) affect total revenue based on these partial derivatives?

### Tip:
When interpreting partial derivatives in economics, think of them as marginal effects—how a small change in one variable impacts the outcome, holding other variables constant.

Give the demand function: q1 = f(p1, p2) = 36,000 - 0.1p1^2 - 0.5p2^2. Determine the partial derivatives f_p1 and f_p2. If p1 = 20 and p2 = 10, evaluate f_p1 and f_p2 and interpret their meaning.

Explore how to calculate the partial derivatives of a demand function q1 = f(p1, p2) = 36,000 - 0.1p1^2 - 0.5p2^2. Evaluate these derivatives at p1 = 20 and p2 = 10 to understand the marginal effects of price changes on demand.

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