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 $p_1$.

= -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.