Problem 75 - Brief Applied Calculus

Question

Find, rounding to five decimal places: a. \(\left(1+\frac{1}{100}\right)^{100}\) b. \(\left(1+\frac{1}{10,000}\right)^{10,000}\) c. \(\left(1+\frac{1}{1,000,000}\right)^{1,000,000}\) d. Do the resulting numbers seem to be approaching a limiting value? Estimate the limiting value to five decimal places. The number that you have approximated is denoted \(e\), and will be used extensively in Chapter 4 .

Step-by-Step Solution

Answer

The limiting value is approximately 2.71828.

Step 1: Calculate Part a

First, we need to calculate \( \left(1+\frac{1}{100}\right)^{100} \). Simplifying inside the parentheses:\[ 1 + \frac{1}{100} = 1.01 \]Now, raise this result to the 100th power:\[ 1.01^{100} \approx 2.70481 \] (rounded to five decimal places).

Step 2: Calculate Part b

Next, calculate \( \left(1+\frac{1}{10,000}\right)^{10,000} \). Simplifying inside the parentheses:\[ 1 + \frac{1}{10,000} = 1.0001 \]Now, raise this result to the 10,000th power:\[ 1.0001^{10,000} \approx 2.71815 \] (rounded to five decimal places).

Step 3: Calculate Part c

Finally, calculate \( \left(1+\frac{1}{1,000,000}\right)^{1,000,000} \). Simplifying inside the parentheses:\[ 1 + \frac{1}{1,000,000} = 1.000001 \]Now, raise this result to the 1,000,000th power:\[ 1.000001^{1,000,000} \approx 2.71828 \] (rounded to five decimal places).

Step 4: Evaluate the Limiting Value

Parts a, b, and c show successively increasing computations that approach a particular value. The results from parts b and c, in particular, are notably close to a known mathematical constant \( e \), which is approximately 2.71828. Therefore, the limiting value approached by these calculations is 2.71828.

Key Concepts

Exponential Growth

Exponential growth occurs when the growth rate of a mathematical quantity is proportional to its current value. This results in the quantity increasing at an accelerating speed. Exponential growth can be observed in many natural phenomena and processes, such as population growth, radioactive decay, and interest calculations. Mathematically, it can be represented as a function of the form:

• \( N(t) = N_0e^{rt} \)

Here, \( N(t) \) is the quantity at time \( t \), \( N_0 \) is the initial quantity, \( r \) is the rate of growth, and \( e \) is Euler’s Number.
Understanding exponential growth helps in modeling real-world scenarios where this type of increase applies. The exercise we're discussing introduces us to this concept by progressively approaching a value through calculations that exhibit an exponential trend.
As the iterations increase, the growth in our calculations becomes less pronounced but follows a predictable pattern, similar to natural exponential growth phenomena.

Mathematical Constant

A mathematical constant is a special number that is inherently significant in mathematics due to its unique properties or frequent appearance in mathematical expressions. Euler's Number, denoted as \( e \), is one such constant.
This constant is approximately equal to 2.71828 and appears often in calculus, particularly in scenarios involving growth based on constant rates of comparison.
Euler's Number is the base of the natural logarithm and has implications in various mathematical calculations, from defining complex logarithms to determining compound interest.
Characteristics of \( e \):

• Irrational: It cannot be expressed as a simple fraction.
• Transcendental: It is not the root of any non-zero polynomial equation with rational coefficients.

The sequence of operations in the initial exercise illustrates the gradual approximation of Euler’s Number, introducing learners to its foundational role in mathematics.

Limits in Calculus

Limits are a fundamental concept in calculus that informally describe the value that a function or sequence "approaches" as the index (or input) approaches some value.
They are crucial for defining concepts like continuity, derivatives, and integrals.
In the context of our exercise, limits help us understand how repeating a process—such as repeatedly compounding an interest factor—gradually settles at a particular value. This can be seen in how the expressions we evaluated (involving increasingly smaller fractions and larger powers) all eventually approach \( e \).
Key points about limits:

• Helps in defining the behavior of functions as they move toward a particular point or infinity.
• Foundational for calculating derivatives and integrals, which form the core of calculus.

By exploring limits through the iterative calculations, students begin to see how mathematical processes can yield constants like \( e \) once those processes are carried to infinity, or a practical approximation thereof.