Expand f (x,y) = e^-x logy upto second degree

To expand the function \( f(x, y) = e^{-x} \log(y) \) up to the second degree, we need to use a **Taylor series expansion** around a point, typically \((x_0, y_0)\). Let’s assume we expand around the point \((x_0 = 0, y_0 = 1)\), which is a common choice for logarithmic functions.

The general form of the second-degree Taylor expansion for a function of two variables \( f(x, y) \) around \( (x_0, y_0) \) is:

\[
f(x, y) \approx f(x_0, y_0) + (x - x_0) \frac{\partial f}{\partial x} \bigg|_{(x_0, y_0)} + (y - y_0) \frac{\partial f}{\partial y} \bigg|_{(x_0, y_0)} + \frac{1}{2} \left( (x - x_0)^2 \frac{\partial^2 f}{\partial x^2} \bigg|_{(x_0, y_0)} + 2(x - x_0)(y - y_0) \frac{\partial^2 f}{\partial x \partial y} \bigg|_{(x_0, y_0)} + (y - y_0)^2 \frac{\partial^2 f}{\partial y^2} \bigg|_{(x_0, y_0)} \right)
\]

### Step 1: Calculate \( f(x_0, y_0) \)

\[
f(0, 1) = e^0 \log(1) = 1 \cdot 0 = 0
\]

###

Expand f(x, y) = e^-x logy up to second degree

This solution expands the multivariable function f(x, y) = e^-x log(y) up to the second degree using Taylor series. It covers key concepts in multivariable calculus, including partial derivatives and the Taylor series expansion around the point (x_0 = 0, y_0 = 1). Ideal for students studying undergraduate calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation