Problem 6 - Essential Calculus Early Transcendentals

Question

Find the exact value of each expression. (a) \(\sinh 1\)

Step-by-Step Solution

Answer

\(\sinh 1\) is approximately 1.175.

Step 1: Understanding the Hyperbolic Sine Function

The hyperbolic sine function is defined as \(\sinh x = \frac{e^x - e^{-x}}{2}\). To evaluate \(\sinh 1\), substitute \(x = 1\) into the formula.

Step 2: Substitute and Simplify

Substitute \(x = 1\) into the hyperbolic sine formula to get: \(\sinh 1 = \frac{e^1 - e^{-1}}{2}\). Simplify to \(\sinh 1 = \frac{e - \frac{1}{e}}{2}\).

Step 3: Calculate Using Approximate Values

To find an approximate value, calculate \(e \approx 2.718\) and \(e^{-1} \approx 0.368\). Substitute these into the simplified expression to get \(\sinh 1 \approx \frac{2.718 - 0.368}{2}\).

Step 4: Final Calculation

Perform the subtraction and final division: \(\sinh 1 \approx \frac{2.350}{2} = 1.175\). Hence, \(\sinh 1\) is approximately 1.175.

Key Concepts

Hyperbolic Sine Function

The hyperbolic sine function, denoted as \( \sinh x \), is a mathematical function that forms part of the hyperbolic functions family, much like its familiar counterparts, sine and cosine, in the circular functions. The formula for the hyperbolic sine function is \( \sinh x = \frac{e^x - e^{-x}}{2} \). It is built using exponentials rather than trigonometric circular measurements. This gives hyperbolic functions their unique properties compared to trigonometric functions.
A key characteristic of \( \sinh x \) is that it maps real numbers to real numbers and is defined for all real \( x \). The function is also odd, meaning that \( \sinh(-x) = -\sinh x \).
The use of exponential functions gives \( \sinh x \) unique properties in calculations and graphs, making it especially useful in many areas of engineering and physics, particularly in calculus, differential equations, and hyperbolic geometry.

Euler's Number

Euler's number, represented as \( e \), is approximately equal to 2.71828. It is an irrational number, meaning it cannot be written as a simple fraction. \( e \) is the base of natural logarithms and exhibits many interesting properties in calculus and complex analysis.
One of the most significant aspects of \( e \) is how it arises naturally in many different areas of mathematics, especially in scenarios involving growth and decay, compounding interests, and calculus' derivatives.

• Exponentially growing functions such as \( e^x \) often appear in natural processes and are used extensively in modeling.
• \( e^{-x} \) describes exponential decay, valuable in contexts like radioactive decay and cooling processes.

In the hyperbolic sine function, both \( e^x \) and \( e^{-x} \) play pivotal roles, helping to define the particular shape and values returned by this function.

Approximate Calculation

Approximating numbers, such as those involving Euler's number \( e \), is vital in many mathematical calculations, especially when an exact number is either not possible or not required. In our exercise's solution, we saw an approximate calculation for \( \sinh 1 \) using \( e \approx 2.718 \) and \( e^{-1} \approx 0.368 \).
Here is a general process to follow for similar calculations:

• First, identify the key numbers requiring approximation, typically the irrational ones like \( e \).
• Use these approximations in the original mathematical expression. In our case, it was substituting \( e \) and its reciprocal into the hyperbolic function formula.
• Finally, perform the arithmetic operations, carefully simplifying and combining terms to achieve an approximate value.

This approach balances accuracy and simplicity, often yielding useful results for practical purposes, especially when high precision is not critical.