Question

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

Short answer

The Maclaurin series for \(\text{cos } x \text{ sinh } y\) starts with \( y - \frac{x^2 y}{2} + \frac{y^3}{6} - \frac{x^2 y^3}{12} + \frac{y^5}{120}\).

Step-by-step solution

Express \(\text{cos } x\) and \( \text{sinh } y\) as Maclaurin Series

First, write the Maclaurin series expansions for both \( \text{cos } x \) and \( \text{sinh } y \). \[ \text{cos } x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \] \[ \text{sinh } y = y + \frac{y^3}{3!} + \frac{y^5}{5!} + \ldots \]

Multiply the Series

To find the Maclaurin series for \(\text{cos } x \text{ sinh } y\), multiply the two series obtained in Step 1 term by term. \[ \text{cos } x \text{ sinh } y = \left( 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \ldots \right) \left( y + \frac{y^3}{3!} + \frac{y^5}{5!} + \ldots \right) \]

Distribute and Combine Like Terms

Distribute the terms from each series and combine like terms. The first few terms of the product are: \[ y - \frac{x^2 y}{2!} + \frac{y^3}{3!} + \frac{x^2 y^3}{2! 3!} + \frac{y^5}{5!} - \frac{x^2 y^5}{2! 5!} + \frac{x^4 y}{4!} + \ldots \]

Simplify the Series

Combine and simplify the terms of the series. The simplified form up to the first few non-zero terms is: \[ \text{cos } x \text{ sinh } y = y - \frac{x^2 y}{2} + \frac{y^3}{6} - \frac{x^2 y^3}{12} + \frac{y^5}{120} + \ldots \]

Key concepts

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Understanding two-variable functions can broaden your knowledge of multivariable calculus. They are functions that depend on two independent variables, like \( f(x, y) \). In this problem, we deal with the function \( \text{cos } x \text{ sinh } y\).

We need to find its Maclaurin series, which involves expanding both variables around 0. Learning this concept allows you to approximate functions with more than one variable using power series.

When you encounter two-variable functions, remember:

• The function value changes based on both variables.
• We often expand these functions using series to make calculations easier.
• Understanding each variable’s effect helps in visualizing complex surfaces or curves.

By mastering two-variable functions, you can apply these techniques to solve more complex math and engineering problems.

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Series multiplication involves combining two series term-by-term. For our function \( \text{cos } x \text{ sinh } y\), we need to multiply their individual Maclaurin series together.

This step requires you to:

• Write each series clearly with separate terms.
• Multiply every term in the expansion of \( \text{cos } x \) by each term in the expansion of \( \text{sinh } y \).
• Distribute the multiplication across all terms to ensure no term is left out.

Following these steps can be tedious but ensures your final result is accurate. Be careful with coefficients and factorials, as mistakes here can quickly compound. Series multiplication is a fundamental tool for translating complex function interactions into manageable series.

Once you’ve multiplied, check your work by comparing against a simpler, manually-calculated example.

headline of the respective core concept

Term-by-term expansion is the final step to get the Maclaurin series for our function. After multiplying series, you combine like terms to simplify the result.

Key points to remember:

• Distribute each term carefully and expand correctly.
• Combine like terms, ensuring you add coefficients of the same degree terms.
• Simplify the expressions by calculating the coefficients’ values.

For \( \text{cos } x \text{ sinh } y \), distributing and combining gives: \[ \text{cos } x \text{ sinh } y = y - \frac{x^2 y}{2} + \frac{y^3}{6} - \frac{x^2 y^3}{12} + \frac{y^5}{120} + \text{ higher order terms} \]
Taking time to practice term-by-term expansion solidifies your understanding of series and expansions. This method simplifies otherwise complex interactions between variables, making it easier to visualize and solve mathematical problems.