To expand $e^x \log(1 + y)$ up to the second-order terms around $(0, 0)$ using a Taylor series, we follow these steps:

1. Function Definition

Let $f(x, y) = e^x \log(1 + y)$.

We want to expand this function around $(x, y) = (0, 0)$, including terms up to second order.

2. Taylor Expansion Formula

The Taylor expansion of a function of two variables $f(x, y)$ around $(0, 0)$ is given by:

$f(x, y) \approx f(0, 0) + \frac{\partial f}{\partial x}(0, 0) x + \frac{\partial f}{\partial y}(0, 0) y + \frac{1}{2} \left( \frac{\partial^2 f}{\partial x^2}(0, 0) x^2 + 2 \frac{\partial^2 f}{\partial x \partial y}(0, 0) xy + \frac{\partial^2 f}{\partial y^2}(0, 0) y^2 \right)$

3. Compute Function and Derivatives at $(0, 0)$

$f(0, 0)$:

$f(0, 0) = e^0 \log(1 + 0) = 1 \cdot 0 = 0$

First-order partial derivatives:

• $\frac{\partial f}{\partial x} = e^x \log(1 + y)$, so at $(0, 0)$: $\frac{\partial f}{\partial x}(0, 0) = e^0 \log(1 + 0) = 1 \cdot 0 = 0$

• $\frac{\partial f}{\partial y} = \frac{e^x}{1 + y}$, so at $(0, 0)$: $\frac{\partial f}{\partial y}(0, 0) = \frac{e^0}{1 + 0} = 1$

Second-order partial derivatives:

• $\frac{\partial^2 f}{\partial x^2} = e^x \log(1 + y)$, so at $(0, 0)$: $\frac{\partial^2 f}{\partial x^2}(0, 0) = e^0 \log(1 + 0) = 1 \cdot 0 = 0$

• $\frac{\partial^2 f}{\partial y^2} = -\frac{e^x}{(1 + y)^2}$, so at $(0, 0)$: $\frac{\partial^2 f}{\partial y^2}(0, 0) = -\frac{e^0}{(1 + 0)^2} = -1$

• $\frac{\partial^2 f}{\partial x \partial y} = \frac{e^x}{1 + y}$, so at $(0, 0)$: $\frac{\partial^2 f}{\partial x \partial y}(0, 0) = \frac{e^0}{1 + 0} = 1$

4. Combine the Terms

Now, using the Taylor expansion formula, we plug in the values we've computed:

$f(x, y) \approx 0 + 0 \cdot x + 1 \cdot y + \frac{1}{2} \left( 0 \cdot x^2 + 2 \cdot 1 \cdot xy + (-1) \cdot y^2 \right)$

Simplifying:

$f(x, y) \approx y + xy - \frac{1}{2} y^2$

Final Answer:

The second-order Taylor expansion of $e^x \log(1 + y)$ around $(0, 0)$ is:

$f(x, y) \approx y + xy - \frac{1}{2} y^2$

Would you like to explore the details further or ask any questions about the steps?

Here are 5 questions to expand on this topic:

1. How do we generalize Taylor series for higher dimensions?
2. What is the significance of second-order terms in a Taylor expansion?
3. Can you expand $e^x \log(1 + y)$ to third order?
4. How does the mixed partial derivative $\frac{\partial^2 f}{\partial x \partial y}$ contribute to the expansion?
5. Why is the expansion centered around $(0, 0)$ instead of another point?

Tip: When expanding around a point, ensure you calculate all partial derivatives correctly up to the desired order.