Problem 14 - Calculus with Concepts in Calculus

Question

Use a calculator to find the approximate value. $$ e^{-1.24 \times 10^{-4}} $$

Step-by-Step Solution

Answer

The approximate value of \( e^{-1.24 \times 10^{-4}} \) is 0.999876.

Step 1: Understand the Function

We want to find the approximate value of the function \( e^{-1.24 \times 10^{-4}} \), where \( e \) is the base of natural logarithms, approximately equal to 2.718.

Step 2: Simplify the Exponent

The exponent \( -1.24 \times 10^{-4} \) is a small number. This signifies a very slight attenuation of \( e \), since the exponent is negative but near zero.

Step 3: Use a Calculator to Compute

Enter \( e^{-1.24 \times 10^{-4}} \) into your calculator. Make sure to use the exponential function, often denoted as \( \exp(x) \) on calculators. Input the value \(-1.24 \times 10^{-4}\) to get the result.

Step 4: Interpret the Result

The calculator will give an approximate value. Due to the small exponent, the outcome should be very close to 1, as \( e^0 = 1 \) and the exponent is nearly zero.

Key Concepts

Natural Logarithms

Natural logarithms are a fundamental part of mathematics and they often appear in various scientific and engineering fields. The symbol \( e \) is used to represent the base of natural logarithms, which is approximately 2.718. This number is known as Euler's number. It has unique properties that make it ideal for continuous growth or decay models. For example, natural logarithms help model population growth, radioactive decay, and even interest compounding in finance. The expression \( e^x \) defines an exponential function, where \( x \) is the exponent. When you have smaller values of \( x \), especially negative values close to zero, the function approaches 1. This is because \( e^0 = 1 \), and any small deviations from zero in the exponent influence the function by a small amount. Understanding this concept helps in approximating functions with very small exponents without heavy computation.

Calculator Use

Using a calculator efficiently can greatly simplify calculations, especially those involving exponential functions. Most scientific calculators have a dedicated button for exponential functions, typically labeled \( \operatorname{exp}(x)\). This is designed to quickly compute values of \( e^x \). To solve problems like finding the approximate value of \( e^{-1.24 \times 10^{-4}} \), you simply need to:

• Turn on your calculator.
• Locate the \( \operatorname{exp}\) button.
• Input the exponent value, in this case, \(-1.24 \times 10^{-4}\).
• Press the button to calculate and view the result.

Remember to always double-check your input to avoid common mistakes such as incorrect placement of decimal points or entering the wrong exponent. These small details can significantly impact the calculation outcome.

Approximations

Approximations are useful for understanding and working with functions when exact values are unnecessary, overly complex, or impossible to compute precisely. In the context of exponential functions, approximations become very useful when dealing with very small or very large exponents.For the exercise given, \( e^{-1.24 \times 10^{-4}} \) was approximated using a calculator. However, knowing the nature of exponential functions allows us to estimate that the outcome is very close to 1. This is because the exponent \(-1.24 \times 10^{-4}\) is extremely close to zero, and thus the change from 1 is negligible. Approximations allow:

• Simplifying complex calculations.
• Gaining insights into the behavior of functions.
• Executing faster computations without sacrificing much accuracy when exactness isn't critical.

In situations where speed is of essence or the numbers are so minuscule that precision isn’t crucial, approximations offer a practical, efficient solution.