Problem 12 - Precalculus

Question

Approximate with a calculator. Round your answer to four decimal places. $$e^{-\sqrt{2}}$$

Step-by-Step Solution

Answer

The value of \( e^{-\sqrt{2}} \) rounded to four decimal places is 0.2431.

Step 1: Understand the Expression

The expression we need to evaluate is \( e^{-\sqrt{2}} \). This involves the natural exponential function \( e^x \) with a power of \(-\sqrt{2}\).

Step 2: Evaluate the Square Root

Calculate \( \sqrt{2} \) using a calculator. The approximate value is 1.4142 when rounded to four decimal places.

Step 3: Apply the Negative Sign

Now apply the negative sign from the exponent. This gives us \(-\sqrt{2} \approx -1.4142\).

Step 4: Calculate the Exponential Value

Use a calculator to find \( e^{-1.4142} \). Inputting this into a calculator gives an approximate value of 0.2431.

Step 5: Round the Result

The value from the calculator is already rounded to four decimal places as 0.2431.

Key Concepts

Natural Exponential Function

The natural exponential function, commonly denoted as \( e^x \), is a special exponential function where the base is the constant \( e \). This constant \( e \) is approximately equal to 2.71828, and it is a fundamental constant in mathematics, similar to \( \pi \). One important property of the natural exponential function is that it describes exponential growth or decay, depending on whether the exponent \( x \) is positive or negative.

• When \( x > 0 \), \( e^x \) represents exponential growth.
• When \( x < 0 \), \( e^x \) models exponential decay.

In our exercise, we are dealing with \( e^{-\sqrt{2}} \), which indicates an exponential decay due to the negative exponent. When computing values like this, you often end up with a very small number, as evidenced by our approximation of 0.2431.

Square Root

The square root symbol \( \sqrt{} \) essentially asks the question: 'What number, when multiplied by itself, gives me the number inside the square root?' For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \). Similarly, the square root of 2, \( \sqrt{2} \), is a number that when multiplied by itself equals 2.

• \( \sqrt{2} \) is an irrational number; it cannot be expressed as a simple fraction.
• We often use a calculator to approximate its value. Here, it approximates to 1.4142 when limited to four decimal places.

In exercises requiring precision, the exact value isn't used for practical reasons; instead, its approximation helps in calculating further expressions like \( e^{-\sqrt{2}} \).

Calculator Approximation

Approximating values with a calculator is a practical skill, especially when working with irrational numbers or complex expressions. Calculators are designed to handle these computations easily and quickly. In our exercise, several approximations occur:

• First, \( \sqrt{2} \) is approximated to 1.4142.
• Then, using the exponential function, we find \( e^{-1.4142} \).
• Finally, the approximation of the exponential function is concluded at 0.2431, rounded to four decimal places.

Rounding answers is crucial in ensuring the simplicity and readability of results, especially when precision beyond four decimal places offers negligible improvements for most practical purposes. It's important to know how to use your calculator effectively to achieve these approximations, and it often avoids unnecessary errors in lengthy calculations.