Problem 37 - Calculus

Question

Use the series for \(e^{x}\) to find the Taylor series for \(\sinh 2 x\) and \(\cosh 2 x\)

Step-by-Step Solution

Answer

The Taylor series for \(\sinh 2x\) is \(2x + \frac{(2x)^3}{3!} + \cdots\), and for \(\cosh 2x\) it is \(1 + \frac{(2x)^2}{2!} + \cdots\).

Step 1: Recall series for exponential function

The Taylor series for the exponential function is given by:\[e^{x} = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \]

Step 2: Define sinh and cosh in terms of exponential functions

The hyperbolic sine and cosine functions are defined as:\[\sinh x = \frac{e^{x} - e^{-x}}{2}\]\[\cosh x = \frac{e^{x} + e^{-x}}{2}\]

Step 3: Substitute and simplify for sinh 2x

Substitute \(x\) with \(2x\) in the definition of \(\sinh x\):\[\sinh 2x = \frac{e^{2x} - e^{-2x}}{2}\]Using the series for \(e^{x}\),\[e^{2x} = 1 + \frac{2x}{1!} + \frac{(2x)^2}{2!} + \frac{(2x)^3}{3!} + \cdots \]\[e^{-2x} = 1 - \frac{2x}{1!} + \frac{(2x)^2}{2!} - \frac{(2x)^3}{3!} + \cdots \]Subtracting these and dividing by 2,\[\sinh 2x = x + \frac{(2x)^3}{3!} + \cdots\]

Step 4: Substitute and simplify for cosh 2x

Substitute \(x\) with \(2x\) in the definition of \(\cosh x\):\[\cosh 2x = \frac{e^{2x} + e^{-2x}}{2}\]Using the series for \(e^{x}\), substitute as before:\[\cosh 2x = 1 + \frac{(2x)^2}{2!} + \frac{(2x)^4}{4!} + \cdots \]

Step 5: Final Taylor series expressions

The Taylor series for \(\sinh 2x\) is:\[\sinh 2x = 2x + \frac{(2x)^3}{3!} + \cdots\]The Taylor series for \(\cosh 2x\) is:\[\cosh 2x = 1 + \frac{(2x)^2}{2!} + \cdots\]

Key Concepts

Understanding the Exponential Function

The exponential function, often expressed as \(e^x\), is a fundamental mathematical function widely used in various fields. It is unique due to its property of being its own derivative. This means when we differentiate \(e^x\), we end up with \(e^x\) again.
The function itself can be expressed as an infinite sum through its Taylor series:

• \(e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots \)

This series expansion allows us to compute \(e^x\) for any real number \(x\) by adding up successive terms, where each term is a fraction with a power of \(x\) on top and a factorial in the denominator. This concise representation is crucial when dealing with complex mathematical problems, especially within calculus and analysis.

Exploring Hyperbolic Functions

Hyperbolic functions, similar to trigonometric functions, are vital in various mathematical applications. The hyperbolic sine and cosine functions, \(\sinh x\) and \(\cosh x\), are defined using exponential functions:

• \(\sinh x = \frac{e^x - e^{-x}}{2}\)
• \(\cosh x = \frac{e^x + e^{-x}}{2}\)

These functions are called hyperbolic because they describe the shapes of hyperbolas, similar to how trigonometric functions describe circles. Insightfully, \(\sinh\) and \(\cosh\) have properties akin to sine and cosine functions. For example, they feature identities such as \(\cosh^2 x - \sinh^2 x = 1\). When working with problems involving hyperbolic functions, we can leverage their definitions and connections to the exponential function to find series expansions, allowing us to approximate their values for various inputs.

The Art of Series Expansion

Series expansions like the Taylor series serve as powerful tools in mathematics for approximating functions that might otherwise be difficult to handle. The concept involves expressing a function as an infinite sum of terms calculated from the function's derivatives at a single point.
For example, the Taylor series for a function \(f(x)\) about 0 (Maclaurin series) is:

• \(f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \cdots \)

When using a series expansion, we transform a complex function into a sum of simpler terms, which can make analyzing and computing the function more manageable. In real-world applications, we often approximate functions by truncating their series expansions to a finite number of terms. This is essential in numerical computation, where exact results are less feasible, and we rely on these approximations to give us accurate enough results for practical use.