Question

Find the two-variable Maclaurin series for the following functions. $$e^{x y}$$

Short answer

The Maclaurin series for \(e^{xy}\) is \(1 + xy\).

Step-by-step solution

Understand the Maclaurin Series

The Maclaurin series is a special case of the Taylor series, centered at 0. For a function of two variables, the Maclaurin series expansion is given by: \[ f(x,y) = f(0,0) + \frac{1}{1!}(f_x(0,0)x + f_y(0,0)y) + \frac{1}{2!}(f_{xx}(0,0)x^2 + 2f_{xy}(0,0)xy + f_{yy}(0,0)y^2) + \ \text{higher order terms} \] where \( f_x, f_y, f_{xx}, f_{xy}, f_{yy} \) are the partial derivatives of \( f \).

Calculate the Function Value at (0,0)

Evaluate \( f(0,0) \) for \( f(x,y) = e^{xy} \): \[ f(0,0) = e^{0 \times 0} = e^0 = 1 \]

Calculate the First Order Partial Derivatives

Find the first order partial derivatives \( f_x \) and \( f_y \): \[ f_x = \frac{\partial}{\partial x} e^{xy} = y e^{xy} \ f_y = \frac{\partial}{\partial y} e^{xy} = x e^{xy} \] Evaluate at the origin (0,0): \( f_x(0,0) = y e^{0} = 0 \) \( f_y(0,0) = x e^{0} = 0 \)

Calculate the Second Order Partial Derivatives

Find the second order partial derivatives \( f_{xx}, f_{xy} \) and \( f_{yy} \): \[ f_{xx} = \frac{\partial^2}{\partial x^2} e^{xy} = y^2 e^{xy} \ f_{xy} = \frac{\partial^2}{\partial x \partial y} e^{xy} = (1 + xy) e^{xy} \ f_{yy} = \frac{\partial^2}{\partial y^2} e^{xy} = x^2 e^{xy} \] Evaluate at the origin (0,0): \( f_{xx}(0,0) = y^2 e^0 = 0 \) \( f_{xy}(0,0) = 1 \times e^0 = 1 \) \( f_{yy}(0,0) = x^2 e^0 = 0 \)

Assemble the Maclaurin Series

Using the partial derivatives, construct the Maclaurin series: \[ f(x,y) = 1 + 0 \times x + 0 \times y + \frac{1}{2!}(0 \times x^2 + 2 \times 1 \times xy + 0 \times y^2) \ = 1 + xy \]

Key concepts

two-variable functions

A two-variable function involves two inputs, usually represented as \(x\) and \(y\), which produce a single output. Unlike single-variable functions, which map one input to one output, two-variable functions map a pair of inputs to one output. This type of function can be represented as \(f(x, y)\). For example, the function \(f(x, y) = e^{xy}\) takes two variables, \(x\) and \(y\), and calculates the output by raising the natural exponential constant \(e\) to the power of \(xy\).
This nature of having two variables allows us to create 3D plots where the surface height represents the output of the function for every pair of \(x\) and \(y\) values.
The more complex structure of two-variable functions makes techniques like partial derivatives and expansions even more essential in analyzing and understanding them.

partial derivatives

Partial derivatives measure how a function changes as one of its input variables changes while keeping the other constant. For a two-variable function like \(f(x, y)\), partial derivatives are denoted as \(f_x\) (the derivative with respect to \(x\)) and \(f_y\) (the derivative with respect to \(y\)).
To compute \(f_x\), treat \(y\) as a constant and differentiate with respect to \(x\). Conversely, to get \(f_y\), treat \(x\) as a constant and differentiate with respect to \(y\).
For \(f(x, y) = e^{xy}\), the partial derivatives are:

• \(f_x = \frac{\partial}{\backslash\text {partial x}} e^{xy} = ye^{xy}\)
• \(f_y = \frac{\partial}{\partial y} e^{xy} = xe^{xy}\)

Partial derivatives serve as the building blocks for more complex calculations including higher-order derivatives and Taylor series expansions.

Taylor series expansion

The Taylor series expansion approximates functions using infinite sums of their derivatives at a specific point. For a function of a single variable, it's written as:
\[f(x) = f(a) + f'(a)(x – a) + \frac{f''(a)}{2!}(x – a)^2 + \frac{f'''(a)}{3!}(x – a)^3 + \text{...}\]
For a two-variable function, a similar expansion can be done. The Maclaurin series is a special case of the Taylor series centered at \(x = 0\) and \(y = 0\). The Maclaurin series for a function \(f(x, y)\) is given by:
\[f(x,y) = f(0,0) + \frac{1}{1!}(f_x(0,0)x + f_y(0,0)y) + \frac{1}{2!}(f_{xx}(0,0)x^2 + 2f_{xy}(0,0)xy + f_{yy}(0,0)y^2) + \text{higher order terms}\]
This formula uses partial derivatives evaluated at the origin to build the series. In our example, for \(f(x, y) = e^{xy}\), the series expansion simplifies to \(1 + xy\). This shows how useful Taylor series expansions can be in approximating function behaviors using polynomials.

second order partial derivatives

Second order partial derivatives are the derivatives of the first order partial derivatives. They help to understand the curvature of the function. For a two-variable function \(f(x, y)\), these include:

• \(f_{xx}\) – The second partial derivative with respect to \(x\)
• \(f_{yy}\) – The second partial derivative with respect to \(y\)
• \(f_{xy}\) and \(f_{yx}\) – Mixed partial derivatives, representing the change first with respect to one variable and then the other

For \(f(x, y) = e^{xy}\), these second order partial derivatives are:

• \(f_{xx} = \frac{\partial^2}{\partial x^2} e^{xy} = y^2 e^{xy}\)
• \(f_{xy} = \frac{\partial^2}{\partial x \partial y} e^{xy} = (1 + xy)e^{xy}\)
• \(f_{yy} = \frac{\partial^2}{\partial y^2} e^{xy} = x^2 e^{xy}\)

Evaluating at \((0, 0)\), we get:

• \(f_{xx}(0,0) = 0\)
• \(f_{xy}(0, 0) = 1\)
• \(f_{yy}(0,0) = 0\)

These second order partial derivatives are used to construct the higher order terms in the Maclaurin series.