Problem 56 - Applied Calculus

Question

Use a graphing calculator to evaluate each expression. $$ \left(1+10^{-6}\right)^{10^{6}} $$

Step-by-Step Solution

Answer

The expression approxiamtely evaluates to 2.718 using a calculator.

Step 1: Understand the Expression

We need to evaluate the expression \( (1+10^{-6})^{10^{6}} \). This is a binomial expression in the form of \( (1+x)^n \).

Step 2: Approximation Insight

For expressions of the form \( (1+x)^n \) with very small \( x \) and large \( n \), it approximates to \( e^{nx} \) due to the limit \( \lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x \).

Step 3: Substitute Values into Approximation Formula

Plug \( x = 10^{-6} \) and \( n = 10^6 \) into \( e^{nx} \). We find that \( nx = 10^6 \cdot 10^{-6} = 1 \), so the approximation becomes \( e^1 \).

Step 4: Evaluate the Approximation

Calculate \( e^1 \), which is approximately 2.718.

Step 5: Use a Graphing Calculator

Input the original expression \( (1+10^{-6})^{10^{6}} \) into a graphing calculator to get a more precise result. The result will be very close to \( e \), which verifies our approximation.

Key Concepts

Understanding Binomial Expressions

A binomial expression is a mathematical expression involving two terms. When we refer to the expression \((1+x)^n\), it comprises two components: a constant and a variable part. In our exercise, the expression \((1+10^{-6})^{10^{6}}\) pertains to this form, where 1 is the constant term and \(10^{-6}\) is our variable form of \(x\). The number \(n\) represents the exponent, which is a large number in this context.

• The term binomial is derived from "bi-" meaning two and "-nomial" meaning terms; hence, it involves two terms.
• Expansion of a binomial expression could involve expanding by multiplication, however, it's not always practical with very large exponents.
• In an exercise like this, it's important to seek approximation methods such as exponential limits that simplify calculations.

Understanding this structure allows us to see how the expression can simplify in special cases, especially when using approximation techniques.

Graphing Calculator Usage

A graphing calculator is a powerful tool that helps us tackle complex calculations that would be cumbersome and lengthy by hand. In evaluating the expression \((1+10^{-6})^{10^{6}}\), a calculator can produce a precise numerical result quickly.Here's how you can use a graphing calculator effectively for such expressions:

• Input the expression exactly as it appears: \((1+10^{-6})^{10^{6}}\).
• Ensure to use parentheses to maintain correct order of operations.
• The calculator computes the result almost instantaneously, offering a more precise solution compared to manual approximations.

By using a graphing calculator, students can verify their manual approximation by comparing it to the calculated result. This approach combines both technological power and mathematical insight for thorough understanding.

Exploring Limit Approximation

Limit approximation is a method that helps simplify complex expressions by using limiting behavior. For exponential forms such as \((1+x)^n\) where \(x\) is very small and \(n\) is large, it becomes closely approximated by the base of natural logs, \(e\), through the formula \(e^{nx}\).

• The fundamental concept revolves around the limit \(\lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x\), which bridges the gap between binomial and exponential expressions.
• This means for our specific problem, turning \((1+10^{-6})^{10^{6}}\) into \(e^{(10^{6} \cdot 10^{-6})}\) simplifies the evaluation.
• This approximation shows the power of limits, using them instead of the full expansion wastes less time and avoids errors inherent in manual calculations.

By mastering limit approximation, students gain access to a key tool in calculus and advanced mathematics, enabling them to tackle otherwise intractable problems with ease.