Problem 55 - Brief Applied Calculus

Question

Use a graphing calculator to evaluate each expression. $$ \left(1-\frac{1}{1000}\right)^{-1000} $$

Step-by-Step Solution

Answer

The expression evaluates to approximately 2.71815 using a graphing calculator.

Step 1: Understand the expression

The given expression is \( \left(1 - \frac{1}{1000}\right)^{-1000} \). It involves an exponentiation of a fraction base.

Step 2: Recognize the similarity to a known limit

Notice that the expression \( \left(1 - \frac{1}{1000}\right)^{-1000} \) is similar to the limit definition of the exponential constant \( e \), where \( \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{-n} = e \).

Step 3: Enter the expression into the graphing calculator

On your graphing calculator, enter the expression exactly as it appears: \( \left(1 - \frac{1}{1000}\right)^{-1000} \).

Step 4: Calculate and observe the result

After entering the expression into the calculator, evaluate it to get a result. You should find that the calculated value is approximately equal to 2.71815.

Key Concepts

Exponential Constant

The exponential constant, commonly denoted as \( e \), is a fundamental mathematical constant approximately equal to 2.71828. It is crucial in fields like calculus and complex analysis.

• The origin of \( e \) is linked to compound interest, where it resolves to continuous compounding.
• \( e \) is transcendental, meaning it is not a solution to any polynomial equation with rational coefficients.

In mathematics, \( e \) is defined via a limit involving exponentiation, showing up in expressions like \( \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{-n} = e \).
This formula indicates how \( e \) emerges naturally when growth processes are studied.
In many applications, an accurate approximation of \( e \) is essential for calculations, as it is inherently irrational and non-repeating.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It is written as \( b^n \), where \( b \) is the base and \( n \) is the exponent.

• When the exponent is a positive integer, it indicates how many times the base is multiplied by itself.
• Fractional exponents correspond to roots and are essential in expressing operations like the square root or cube root.

Negative exponents represent the reciprocal of the base raised to the opposite positive exponent.
For example, \( 3^{-2} \) is equivalent to \( \frac{1}{3^2} = \frac{1}{9} \).
In our example expression \( \left(1 - \frac{1}{1000}\right)^{-1000} \), the use of a negative exponent signifies taking the reciprocal of the expression inside the parenthesis.

Limit Definition

The concept of a limit in calculus is used to describe the behavior of a function as its input approaches a particular point. The limit is essential for defining derivatives, integrals, and continuity.

• A limit examines values that a function approaches as the input becomes arbitrarily close to a specific point.
• Limits can be found at finite points, infinity, or nonexistent if values fluctuate too wildly.

In our example, the expression \( \left(1 - \frac{1}{1000}\right)^{-1000} \) closely resembles the limit definition of \( e \).
The limit \( \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{-n} = e \) shows how \( e \) arises as \( n \) becomes very large.
Understanding limits is foundational for tackling advanced calculus topics, providing insight into the fundamental nature of changes in a system.