Problem 54 - 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 2.71692.

Step 1: Input Expression

Open your graphing calculator. Locate the area where you can input mathematical expressions. Enter the expression \( \left(1+\frac{1}{1000}\right)^{1000} \). This means you are raising the number \( 1.001 \) to the power of \( 1000 \).

Step 2: Calculate the Expression

After inputting the expression, press the 'Enter' or 'Calculate' button on your calculator. This will prompt the calculator to evaluate the expression and provide the numerical result.

Step 3: Interpret the Output

The calculator should display the result as approximately \( 2.716923932 \). This is a numerical approximation of the expression.

Step 4: Understand the Context

Recognize that this calculation is an approximation of the expression \( \left(1+\frac{1}{n}\right)^{n} \) as \( n \) approaches infinity, which approaches the value of Euler's number \( e \). Therefore, the result is very close to the value of \( e \), which is approximately \( 2.71828 \).

Key Concepts

Numerical Approximation

Numerical approximations are crucial when we need to obtain a value for expressions that involve complex calculations or infinite series. Often, exact values may be difficult or impossible to find, so we rely on these approximations to provide a close estimate. In our exercise, entering \[ \left(1+\frac{1}{1000}\right)^{1000} \] into a graphing calculator gives us an approximate result of 2.716923932.

• This result is not exact but rather a close estimation of the expression's real value.
• Numerical approximations can vary based on the precision and the method used.
• In technology like graphing calculators, these approximations provide significant insights without manual calculations.

By understanding how numerical approximations work, students can feel more comfortable navigating complex mathematical problems, where such approximations provide valuable insights.

Euler's Number

Euler's Number, often denoted as \( e \), is a fundamental constant in mathematics, approximately equal to 2.71828. Discovered by the Swiss mathematician Leonhard Euler, this constant is crucial in various branches of mathematics and sciences. It surfaces in processes involving growth, such as

• Continuous compound interest calculations
• Calculating limits in calculus
• Describing certain natural phenomena like population growth models

When we evaluated the expression \( \left(1+\frac{1}{1000}\right)^{1000} \), our result, 2.716923932, closely approximates \( e \). As \( n \) in the expression \( \left(1+\frac{1}{n}\right)^{n} \) grows infinitely larger, it approaches Euler's Number. Understanding \( e \) helps students grasp various mathematical concepts related to growth and change over continuous time.

Exponential Expressions

Exponential expressions involve numbers raised to a power, one of which you're familiar with from our expression \( \left(1+\frac{1}{1000}\right)^{1000} \). These types of expressions can model a variety of real-world phenomana. Here's what you need to know about exponential expressions:

• They depict growth or decay processes such as population increase or radioactive decay.
• The base number tells you the growth (or decay) rate.
• The exponent indicates the number of times the base is multiplied by itself.

Using a graphing calculator, students can quickly input complex expressions without manually multiplying numerous times. By simplifying the process, students gain more time to understand the underlying concepts of exponential growth and decay, enhancing their comprehension of both mathematical and real-world applications.