Question

Find the two-variable Maclaurin series for the following functions. \(\cos x \sinh y\)

Short answer

\[ \cos x \sinh y \approx y + \frac{y^3}{3!} - \frac{y x^2}{2!} \]

Step-by-step solution

- Recall Maclaurin series definitions

Maclaurin series is a special case of the Taylor series centered at 0. For a function of two variables, it's given by: data: Coulomb Potentiala.\[f(x,y) = f(0,0) + \left( \frac{\partial f}{\partial x} \right)_{(0,0)} x + \left( \frac{\partial f}{\partial y} \right)_{(0,0)} y + \frac{1}{2!} \left( \frac{\partial^2 f}{\partial x^2} \right)_{(0,0)} x^2 + \cdots\] degree: }Solve the requant

- Find the Maclaurin series for \( \cos x \)

The Maclaurin series for \( \cos x \) is: \[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\]

- Find the Maclaurin series for \( \sinh y \)

The Maclaurin series for \( \sinh y \) is: \[ \sinh y = y + \frac{y^3}{3!} + \frac{y^5}{5!} + \cdots\]

- Form the product of the series

To find the Maclaurin series for \( \cos x \sinh y \), multiply the two series: \[ \cos x \sinh y = \left( 1 - \frac{x^2}{2!} + \cdots \right)\left( y + \frac{y^3}{3!} + \cdots \right) \]

- Combine like terms

Combine terms up to a reasonable degree to get a simplified expansion: \[ \cos x \sinh y = y + \frac{y^3}{3!} + \frac{y x^2}{2!} + \cdots \]

Key concepts

Two-variable functions

In mathematics, a two-variable function is a function that takes two inputs. For example, in the exercise, we worked with the function \( \cos x \sinh y \.\) This means the function takes an input \( x \) and an input \( y \), and performs operations on both. These functions are very useful in various fields. Here are a few key points about two-variable functions:

• They help in representing situations with two changing variables.
• The notation \( f(x,y) = z \) shows that \( z \) is the output given inputs \( x \) and \( y \).
• These functions can be visualized as surfaces in three-dimensional space.

Understanding how operations work on these functions, such as differentiation and series expansion, is crucial for solving problems like finding Maclaurin series.

Cosine function

The cosine function, denoted as \( \cos x \,\), is a fundamental trigonometric function that appears frequently in calculus and engineering. It describes the x-coordinate of a point on the unit circle as it travels around the circle.

• Cosine function is even, which means \( \cos(-x) = \cos(x) \).
• The Taylor series expansion of \( \cos x \) around 0 (Maclaurin series) is \( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots \).
• This series is useful for approximating the value of the cosine function.

When working with Maclaurin series, you can perform multiplications and combinations of these simplified series expressions to arrive at the final result.

Hyperbolic sine function

The hyperbolic sine function, denoted as \( \sinh y \,\), is similar to the more familiar sine function, but it applies to hyperbolas instead of circles. Its series expansion, also called the Maclaurin series, is very useful for calculations.

• It is defined as \( \sinh y = \frac{e^y - e^{-y}}{2} \).
• The Maclaurin series for \( \sinh y \) is \( y + \frac{y^3}{3!} + \frac{y^5}{5!} + \cdots \).
• This series only includes odd powers of y, which makes it simpler to handle in some contexts.

Using this series, combined with the series for cosine, allows us to find the Maclaurin series of more complex expressions like \( \cos x \sinh y \). This technique is powerful in solving many calculus problems.

Taylor series expansion

A Taylor series is a way to represent a function as an infinite sum of terms calculated from the values of its derivatives at a single point. When the expansion is done around 0, it is called a Maclaurin series.

• Taylor series provide an approximation of functions that are otherwise difficult to compute.
• The general formula is \( f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots \).
• For Maclaurin series, it simplifies to \( f(0) + f'(0)x + \frac{f''(0)}{2!}x^2 + \cdots \).

By breaking down complex functions into polynomial terms, Taylor and Maclaurin series make it easier to analyze and compute various properties of these functions.